s_euclidean_plane | axiom | Euclidean plane ℝ²
s_euclidean_3_space | axiom | Euclidean 3-space (sphere-packing context) ;; s_euclidean_solid_geometry
s_real_numbers | axiom | Real numbers ℝ ;; s_real_line
s_real_line_or_circle | axiom | ℝ or 𝕋¹
s_complex_numbers | axiom | Complex numbers ℂ
s_integers | axiom | Integers ℤ
s_naturals_with_multiplication | axiom | Naturals ℕ (with multiplication)
s_divisibility_definition | axiom | Divisibility relation
s_euclid_lemma | axiom | Euclid's Lemma
s_zfc_axioms | axiom | ZFC axioms
s_first_order_peano_arithmetic | axiom | Peano arithmetic (first-order)
s_turing_machine_model | axiom | Turing machine model
s_probability_axioms | axiom | Probability axioms (Kolmogorov)
s_newtonian_inverse_square_force | axiom | Newtonian inverse-square force law
s_conic_sections | axiom | Conic sections
s_graph_definition | axiom | Graph (V, E)
s_projective_plane | axiom | Projective plane ℙ²
s_projective_space_axioms | axiom | Projective space axioms
s_area_additivity | axiom | Area additivity
s_circle_definition | axiom | Circle (as locus)
s_polygon_area_formula | axiom | Polygon area formula
s_isosceles_triangle_base_angles_equal | axiom | Pons asinorum
s_similar_triangle_criterion | axiom | AA similarity
s_coprime_pair | axiom | Coprime pair
s_prime_p | axiom | Prime p
s_prime_p_equiv_1_mod_4 | axiom | Prime p ≡ 1 (mod 4)
s_prime_power_divisor_p_n | axiom | Prime-power divisor pⁿ
s_pell_equation_x2_minus_N_y2 | axiom | Pell equation x² − Ny² = 1
s_radical_extension_tower | axiom | Radical extension tower
s_field_extension_L_over_K | axiom | Field extension L/K
s_group_action | axiom | Group action
s_differential_form | axiom | Differential form
s_divisor_on_curve | axiom | Divisor on curve
s_sublinear_functional_p | axiom | Sublinear functional p
s_linear_functional_on_subspace | axiom | Linear functional on subspace
s_family_of_compact_spaces | axiom | Family of compact spaces
s_product_topology | axiom | Product topology
s_iid_sequence_finite_variance | axiom | IID sequence (finite variance)
s_smooth_function | axiom | Smooth function
s_analytic_exponential_series | axiom | Exponential series Σ zⁿ/n!
s_unit_circle_in_C | axiom | Unit circle in ℂ
s_convex_polyhedron | axiom | Convex polyhedron
s_topological_sphere_S2 | axiom | Topological sphere S²
s_sine_function | axiom | Sine function sin(x)
s_real_analysis | axiom | Real analysis framework
s_continuous_function_on_interval | axiom | Continuous function on [a,b]
s_mean_value_theorem | axiom | Mean value theorem (lemma)
s_euler_four_square_identity | axiom | Euler four-square identity
s_prime_pair_p_q | axiom | Prime pair (p,q)
s_smooth_surface_in_R3 | axiom | Smooth surface in ℝ³
s_first_fundamental_form | axiom | First fundamental form
s_holomorphic_function_on_domain | axiom | Holomorphic function on domain
s_polynomial_ring_over_Q | axiom | Polynomial ring ℚ[x]
s_simply_connected_proper_domain_in_C | axiom | Simply connected proper domain in ℂ
s_compact_riemann_surface | axiom | Compact Riemann surface
s_primes_in_naturals | axiom | Primes in ℕ ;; s_prime_numbers
s_riemann_zeta_function | axiom | Riemann zeta function ζ(s)
s_finite_group | axiom | Finite group G
s_closed_ball_D_n | axiom | Closed ball Dⁿ
s_continuous_self_map | axiom | Continuous self-map
s_lagrangian_action_integral | axiom | Lagrangian action integral
s_lie_group | axiom | Lie group
s_complete_metric_space | axiom | Complete metric space
s_strict_contraction | axiom | Strict contraction
s_real_vector_space | axiom | Real vector space
s_noetherian_ring_R | axiom | Noetherian ring R
s_polynomial_ring | axiom | Polynomial ring R[x]
s_algebraically_closed_field_k | axiom | Algebraically closed field k
s_polynomial_ring_in_n_vars | axiom | Polynomial ring k[x₁,…,xₙ]
s_infinite_set | axiom | Infinite set
s_k_coloring_of_pairs | axiom | k-coloring of pairs
s_measure_preserving_transformation | axiom | Measure-preserving transformation
s_L2_function_space | axiom | L² function space
s_compact_smooth_manifold | axiom | Compact smooth manifold
s_elliptic_operator_D | axiom | Elliptic operator D
s_finite_simple_group | axiom | Finite simple group
s_planar_graph | axiom | Planar graph
s_proper_vertex_coloring | axiom | Proper vertex coloring
s_smooth_projective_curve_over_Q | axiom | Smooth projective curve over ℚ
s_jacobian_variety | axiom | Jacobian variety
s_elliptic_curve_over_Q | axiom | Elliptic curve over ℚ
s_modular_form | axiom | Modular form
s_galois_representation | axiom | Galois representation
s_closed_3_manifold | axiom | Closed 3-manifold
s_riemannian_metric | axiom | Riemannian metric
s_unit_balls_sphere_packing | axiom | Unit ball sphere packing
s_szemeredi_theorem | axiom | Szemerédi's theorem (input to Green–Tao)
s_admissible_k_tuple | axiom | Admissible k-tuple
s_minor_ordering | axiom | Minor ordering on graphs
s_positive_density_subset | axiom | Positive-density subset (of ℤ)
s_smooth_manifold_with_boundary | axiom | Smooth manifold with boundary
s_right_triangle_in_plane | state | Right triangle in plane
s_two_similar_subtriangles | state | Two similar subtriangles
s_segment_length_identity_on_hypotenuse | state | Segment-length identity on hypotenuse
s_triangle_inscribed_in_semicircle | state | Triangle inscribed in semicircle
s_two_isosceles_subtriangles | state | Two isosceles subtriangles
s_finite_list_of_primes | state | Finite list of primes {p₁,…,pₖ}
s_new_number_N_coprime_to_all_primes_in_list | state | N = p₁⋯pₖ + 1
s_uniqueness_of_prime_factorization | state | Uniqueness of prime factorization
s_circle | state | Circle (as 2D object)
s_inscribed_circumscribed_96_gons | state | Inscribed/circumscribed 96-gons
s_sphere | state | Sphere (as 3D object)
s_sphere_as_solid_of_revolution | state | Sphere as solid of revolution
s_cyclic_quadrilateral | state | Cyclic quadrilateral
s_pair_of_similar_triangles_on_diagonal | state | Pair of similar triangles on diagonal
s_pair_of_coprime_moduli | state | Pair of coprime moduli (m,n)
s_idempotent_pair_mod_mn | state | Idempotent pair mod mn
s_near_solution_triple_a_b_k | state | Near-solution triple (a,b,k)
s_composed_triple_mod_scaling | state | Composed triple (after bhāvanā)
s_general_cubic_ax3_plus_bx2_plus_cx_plus_d | state | General cubic ax³+bx²+cx+d
s_depressed_cubic_t3_plus_pt_plus_q | state | Depressed cubic t³+pt+q
s_system_sum_and_product_of_cubes | state | System u+v=?, uv=?
s_general_quartic | state | General quartic
s_depressed_quartic | state | Depressed quartic
s_resolvent_cubic | state | Resolvent cubic
s_two_triangles_in_perspective_in_plane | state | Two triangles in perspective (plane)
s_two_triangles_in_perspective_in_space | state | Two triangles in perspective (space)
s_axis_of_perspective | state | Axis of perspective
s_multiplicative_group_mod_p | state | (ℤ/p)*
s_orbit_of_a_mod_p | state | Orbit of a mod p
s_existence_of_x_with_x2_equiv_minus1 | state | x with x² ≡ -1 (mod p)
s_small_a_b_with_a2_plus_b2_equiv_0_mod_p | state | (a,b) with a²+b² ≡ 0 (mod p)
s_integral_as_limit_of_sums | state | Integral as limit of Riemann sums
s_tycho_brahe_observation_table | state | Tycho Brahe observation table
s_T_squared_prop_a_cubed_conjecture | state | T² ∝ a³ conjecture
s_third_law | state | Third law (verified)
s_polynomial_approximation_of_order_n | state | Order-n polynomial approximation
s_complex_number_cos_theta_plus_i_sin_theta | state | cos θ + i sin θ
s_multiplicative_law_on_unit_circle | state | Multiplicative law on S¹
s_series_for_e_i_theta | state | Series for e^{iθ}
s_triangulated_sphere_S2 | state | Triangulated S²
s_sin_x_as_infinite_product | state | sin(x) = x ∏(1 - x²/n²π²)
s_konigsberg_bridge_configuration | state | Königsberg bridge configuration
s_abstract_multigraph | state | Abstract multigraph
s_complex_polynomial_p_z | state | Complex polynomial p(z)
s_two_real_algebraic_curves_in_plane | state | Two real algebraic curves in ℝ²
s_intersection_exists | state | Intersection exists (lemma)
s_auxiliary_mp_equals_sum_of_four_squares | state | mp = a²+b²+c²+d² (with small m)
s_prime_as_sum_of_four_squares | state | p = sum of four squares
s_characteristic_function_of_sum | state | Characteristic function of sum
s_limit_characteristic_function_equals_gaussian | state | Limit characteristic function = e^{-t²/2}
s_legendre_symbol_table | state | Legendre symbol table
s_reciprocity_conjecture | state | Reciprocity conjecture
s_refined_reciprocity_conjecture | state | Refined reciprocity conjecture
s_gauss_curvature_K | state | Gaussian curvature K
s_compact_oriented_surface_without_boundary | state | Compact oriented surface ∂=∅
s_geodesic_triangulation | state | Geodesic triangulation
s_local_angle_defect_identity | state | Local angle-defect identity
s_cauchy_integral_theorem | state | Cauchy integral theorem (lemma)
s_quintic_polynomial | state | Generic quintic polynomial
s_galois_group_S5 | state | Galois group S₅
s_A5_is_simple_non_abelian | state | A₅ is simple non-abelian
s_finite_normal_separable_extension_L_over_K | state | Finite normal separable extension L/K
s_galois_group | state | Galois group Gal(L/K)
s_intermediate_fields_of_L | state | Lattice of intermediate fields
s_galois_correspondence | state | Galois correspondence
s_heat_conduction_on_rod | state | Heat conduction on rod (physical)
s_heat_equation_PDE | state | Heat equation PDE ∂u/∂t = α∂²u/∂x²
s_mode_by_mode_ODE_system | state | Mode-by-mode ODE system
s_differential_form_omega_on_manifold | state | Differential form ω on M
s_d_omega_in_local_coords | state | dω in local coordinates
s_family_F_of_injective_holomorphic_maps_to_unit_disk | state | Family F of injective hol maps to 𝔻
s_extremal_map_f_star | state | Extremal map f*
s_sheaf_O_D_on_curve | state | Sheaf O(D) on curve
s_euler_characteristic_chi_O_D | state | Euler char χ(O(D))
s_divisor_D | state | Divisor D (specific)
s_euler_product_zeta | state | Euler product ζ(s) = ∏(1-p⁻ˢ)⁻¹
s_meromorphic_zeta_on_plane | state | Meromorphic ζ on ℂ \ {1}
s_zeta_nonvanishing_on_line_Re_1 | state | ζ(1+it) ≠ 0
s_finite_group_G_with_order_divisible_by_p_n | state | Finite G with pⁿ | |G|
s_set_of_p_subgroups_with_G_action | state | p-subgroups with G-conjugation action
s_alleged_enumeration_of_reals_in_0_1 | state | Alleged enumeration of ℝ ∩ [0,1]
s_real_not_in_list | state | Real not in list
s_continuous_function_on_closed_interval | state | Continuous f on closed interval
s_smooth_approximant_f_epsilon | state | Smooth approximant f_ε
s_ideal_in_R_x | state | Ideal in R[x]
s_finite_generating_set_of_ideal | state | Finite generating set of ideal
s_maximal_ideal_m_in_k_x_1_x_n | state | Maximal ideal m ⊂ k[x₁,…,xₙ]
s_point_in_affine_n_space | state | Point in affine n-space
s_ideal_I | state | Ideal I
s_vanishing_variety_V_I | state | Vanishing variety V(I)
s_continuous_self_map_of_disk_without_fixed_point | state | Continuous self-map of Dⁿ (no fixed pt)
s_retraction_to_boundary_sphere | state | Retraction Dⁿ → ∂Dⁿ
s_contradiction_no_retraction | state | No retraction Dⁿ → ∂Dⁿ (contradiction)
s_infinitesimal_action_variation | state | Infinitesimal action variation
s_syntactic_predicates_as_arithmetic_predicates | state | Syntactic predicates arithmetized
s_self_referential_godel_sentence_G | state | Gödel sentence G
s_alleged_decider_H_for_halting | state | Alleged halting decider H
s_encoding_of_machines_as_data | state | Encoding of TMs as data
s_self_contradictory_machine_D | state | Self-contradictory machine D
s_model_L_of_ZFC_plus_GCH | state | Model L (constructible universe)
s_godel_L_model | state | Gödel L model (alias) ;; s_model_L_of_ZFC_plus_GCH
s_nested_monochromatic_sequence | state | Nested monochromatic sequence
s_ultrafilter_limit_in_product | state | Ultrafilter limit in ∏Xᵢ
s_invariant_subspace_decomposition | state | Invariant subspace decomposition
s_elliptic_operator_D_on_manifold | state | Elliptic operator D on M
s_principal_symbol_in_K_theory_of_TM | state | Principal symbol in K(T*M)
s_topological_index_class_in_K_of_point | state | Topological index in K(pt)=ℤ
s_component_type_and_characteristic_p_type_cases | state | Component/char-p type cases
s_case_exhaustion_complete | state | Case exhaustion complete
s_model_of_ZFC_plus_not_CH | state | Model of ZFC + ¬CH
s_finite_list_of_1500_configurations | state | Unavoidable set of ~1500 configs
s_four_color_theorem_machine_certified | state | Four Color Theorem (machine-certified)
s_curve_of_genus_at_least_2 | state | Curve C, genus ≥ 2
s_curve_inside_abelian_variety | state | C ↪ Jacobian J_C
s_finiteness_of_isogeny_class | state | Finiteness of isogeny class
s_hypothetical_FLT_solution_a_n_plus_b_n_equals_c_n | state | Hypothetical FLT solution aⁿ+bⁿ=cⁿ
s_frey_elliptic_curve | state | Frey elliptic curve y²=x(x-aⁿ)(x+bⁿ)
s_non_modular_galois_representation_required | state | Non-modular mod-ℓ rep (required)
s_semistable_modularity_theorem | state | Semistable modularity theorem
s_ricci_flow_equation | state | Ricci flow eq. ∂g/∂t = -2Ric(g)
s_long_time_decomposition_into_geometric_pieces | state | Long-time decomposition into geometric pieces
s_thurston_eight_geometries_classification | state | Thurston 8 geometries
s_sphere_packing_density_functional | state | Sphere-packing density functional
s_finite_family_of_local_star_configurations | state | Finite family of local star configs
s_kepler_conjecture_machine_certified | state | Kepler conjecture (Flyspeck-verified)
s_primes_with_density_zero | state | Primes with density 0
s_relative_szemeredi_for_pseudorandom_majorants | state | Relative Szemerédi (pseudorandom majorants)
s_aps_in_pseudorandom_dense_subset | state | APs in pseudorandom dense subset
s_goldston_yildirim_sieve_majorant | state | Goldston–Yıldırım sieve majorant
s_positive_lower_bound_on_prime_pairs_in_tuple | state | Positive lower bound on prime pairs in tuple
s_gap_bound_246 | state | Gap bound 246 (Polymath8)
s_integer_N_odd_and_larger_than_5 | state | Odd integer N > 5
s_circle_integral_for_r3_N | state | Circle integral for r₃(N)
s_major_arc_asymptotic_plus_minor_arc_error | state | Major/minor arc decomposition
s_infinite_sequence_of_graphs | state | Infinite sequence of graphs
s_tree_width_decomposition | state | Tree-width decomposition
s_positive_density_subset_of_integers | state | Positive-density subset of ℤ
s_furstenberg_system_with_positive_measure_A | state | Furstenberg system with μ(A)>0
s_multiple_recurrence_for_A | state | Multiple recurrence for A
s_one_dim_extension_step | state | One-dim extension step (Hahn–Banach)
s_pythagorean_theorem | theorem | Pythagorean theorem
s_thales_theorem | theorem | Thales's theorem
s_infinitude_of_primes | theorem | Infinitude of primes
s_fundamental_theorem_of_arithmetic | theorem | Fundamental theorem of arithmetic
s_area_of_circle | theorem | Area of circle = πr²
s_volume_of_sphere | theorem | Volume of sphere = 4/3·πr³
s_ptolemys_theorem | theorem | Ptolemy's theorem
s_chinese_remainder_theorem | theorem | Chinese remainder theorem
s_solvability_of_pell_equation | theorem | Solvability of Pell's equation (Chakravāla)
s_cardano_cubic_formula | theorem | Cardano's cubic formula
s_ferrari_quartic_formula | theorem | Ferrari's quartic formula
s_desargues_theorem | theorem | Desargues's theorem
s_fermat_little_theorem | theorem | Fermat's little theorem
s_fermat_two_squares | theorem | Fermat's sum of two squares
s_fundamental_theorem_of_calculus | theorem | Fundamental theorem of calculus
s_kepler_three_laws | theorem | Kepler's three laws
s_taylor_theorem | theorem | Taylor's theorem
s_de_moivre_formula | theorem | De Moivre's formula
s_euler_formula | theorem | Euler's formula
s_euler_polyhedron_formula | theorem | Euler polyhedron formula V-E+F=2
s_basel_identity | theorem | Basel identity Σ1/n² = π²/6
s_eulerian_path_criterion | theorem | Eulerian path criterion
s_fundamental_theorem_of_algebra | theorem | Fundamental theorem of algebra
s_lagrange_four_squares | theorem | Lagrange four-square theorem
s_central_limit_theorem | theorem | Central limit theorem
s_quadratic_reciprocity | theorem | Quadratic reciprocity
s_theorema_egregium | theorem | Theorema Egregium
s_gauss_bonnet_theorem | theorem | Gauss–Bonnet theorem
s_cauchy_integral_formula | theorem | Cauchy integral formula
s_abel_ruffini | theorem | Abel–Ruffini theorem
s_fundamental_theorem_of_galois_theory | theorem | Fundamental theorem of Galois theory
s_fourier_theorem_heat | theorem | Fourier's theorem on the heat equation
s_stokes_theorem | theorem | Stokes's theorem
s_riemann_mapping_theorem | theorem | Riemann mapping theorem
s_riemann_roch_theorem | theorem | Riemann–Roch theorem
s_prime_number_theorem | theorem | Prime number theorem
s_sylow_theorems | theorem | Sylow theorems
s_uncountability_of_reals | theorem | Uncountability of ℝ
s_weierstrass_approximation | theorem | Weierstrass approximation
s_hilbert_basis_theorem | theorem | Hilbert basis theorem
s_nullstellensatz | theorem | Hilbert's Nullstellensatz
s_brouwer_fpt | theorem | Brouwer fixed-point theorem
s_noether_theorem | theorem | Noether's theorem
s_godel_incompleteness | theorem | Gödel's incompleteness theorems
s_banach_fpt | theorem | Banach fixed-point theorem
s_hahn_banach | theorem | Hahn–Banach theorem
s_tychonoff_theorem | theorem | Tychonoff's theorem
s_undecidability_of_halting | theorem | Undecidability of halting
s_con_zfc_gch | theorem | Con(ZFC) ⇒ Con(ZFC+GCH) (Gödel)
s_ramsey_theorem_infinite | theorem | Infinite Ramsey theorem
s_birkhoff_ergodic_theorem | theorem | Birkhoff ergodic theorem
s_atiyah_singer_index_theorem | theorem | Atiyah–Singer index theorem
s_cfsg | theorem | Classification of finite simple groups
s_ch_independent_of_zfc | theorem | CH independent of ZFC
s_four_color_theorem | theorem | Four color theorem
s_mordell_faltings | theorem | Mordell–Faltings theorem
s_flt | theorem | Fermat's Last Theorem
s_poincare_conjecture | theorem | Poincaré conjecture
s_geometrization_theorem | theorem | Geometrization theorem
s_kepler_conjecture | theorem | Kepler conjecture
s_green_tao | theorem | Green–Tao theorem
s_bounded_gaps_between_primes | theorem | Bounded gaps between primes
s_ternary_goldbach | theorem | Ternary Goldbach (Helfgott)
s_graph_minor_theorem | theorem | Robertson–Seymour graph minor theorem
s_szemeredi_theorem_terminal | theorem | Szemerédi's theorem (as terminal)
t_spot_pattern_in_table | technique | Spot pattern in table
t_verify_on_special_cases | technique | Verify on special cases
t_complete_the_square | technique | Complete the square
t_reduce_to_canonical_form | technique | Reduce to canonical form
t_compose_with_identity | technique | Compose with identity
t_symmetry_reduction | technique | Symmetry reduction
t_conserved_quantity | technique | Conserved quantity
t_duality | technique | Duality
t_character_decomposition_count | technique | Character decomposition count
t_exhaustion_squeeze | technique | Exhaustion / squeeze
t_interpolate_and_continue | technique | Interpolate and continue
t_frequency_decomposition | technique | Frequency decomposition
t_axiomatize_from_instances | technique | Axiomatize from instances
t_structural_isomorphism | technique | Structural isomorphism
t_ultraproduct_transfer | technique | Ultraproduct transfer
t_raise_dimension | technique | Raise dimension
t_obstruction_class | technique | Obstruction class
t_compactness_argument | technique | Compactness argument
t_deformation_cohomology | technique | Deformation cohomology
t_rescale_for_asymptotic_geometry | technique | Rescale for asymptotic geometry
t_diagonalize | technique | Diagonalize
t_arithmetize_syntax | technique | Arithmetize syntax
t_force_independence | technique | Force independence
t_contraction_fixed_point | technique | Contraction fixed point
t_infinite_descent | technique | Infinite descent
t_flow_with_surgery | technique | Flow with surgery
t_physics_to_pde | technique | Physics to PDE
t_complex_analysis_to_integers | technique | Complex analysis to integers
t_analysis_algebra_topology_bridge | technique | Analysis–algebra–topology bridge
t_major_minor_arc_decomposition | technique | Major/minor arc decomposition
t_ergodic_correspondence | technique | Ergodic correspondence
t_finite_case_check | technique | Finite case check
t_formal_verify | technique | Formal verify
t_distributed_collaboration | technique | Distributed collaboration
t_probabilistic_existence | technique | Probabilistic existence
t_pigeonhole_collision | technique | Pigeonhole collision
t_sieve_by_optimized_quadratic | technique | Sieve by optimized quadratic
t_group_complete_exact_category | technique | Group-complete exact category (K-theory)
t_sheafify_on_grothendieck_topology | technique | Sheafify on Grothendieck topology
t_representable_functor_trick | technique | Representable functor trick
t_polynomial_method | technique | Polynomial method
t_double_centralizer_decompose | technique | Double centralizer decompose
t_fourier_transform | technique | Fourier transform (umbrella)
t_svd_and_spectral_decomposition | technique | SVD / spectral decomposition
t_galois_correspondence | technique | Galois correspondence (composite)
t_ricci_flow_with_surgery | technique | Ricci flow with surgery (composite)
t_wiles_modularity | technique | Wiles modularity (composite)
t_godel_numbering | technique | Gödel numbering (composite)
t_atiyah_singer_index_machinery | technique | Atiyah–Singer index machinery (composite)
t_selberg_sieve_method | technique | Selberg sieve method (composite)
t_circle_method | technique | Circle method (composite)
t_furstenberg_correspondence_principle | technique | Furstenberg correspondence (composite)
t_category_theoretic_colimits_and_adjoints | technique | Category-theoretic colimits/adjoints (composite)
t_auxiliary_construction | technique | Auxiliary construction
t_conjecture_refinement | technique | Conjecture refinement
t_reductio_ad_absurdum | technique | Reductio ad absurdum
t_projection_to_subspace | technique | Projection to subspace
t_sheaf_cohomology_bridge | technique | Sheaf cohomology bridge
t_k_theoretic_index_bridge | technique | K-theoretic index bridge
t_heights_and_galois_rep_bridge | technique | Heights / Galois representation bridge
t_level_lowering_bridge | technique | Level-lowering bridge (Ribet)
t_transference_bridge | technique | Transference bridge
s_heron_formula | theorem | Heron's Formula
s_incircle_tangent_decomposition | state | incircle tangent decomposition
s_area_squared_equals_s_times_product | state | area squared equals s times product
s_menelaus_theorem | theorem | Menelaus's Theorem
s_perpendiculars_from_vertices_to_transversal | state | perpendiculars from vertices to transversal
s_ratio_product_cancels_heights | state | ratio product cancels heights
s_brahmagupta_formula | theorem | Brahmagupta's Formula
s_two_triangles_with_supplementary_angles | state | two triangles with supplementary angles
s_area_squared_symmetric_in_four_sides | state | area squared symmetric in four sides
s_brahmagupta_midpoint_theorem | theorem | Brahmagupta's Theorem
s_orthodiagonal_cyclic_quadrilateral | state | orthodiagonal cyclic quadrilateral
s_perpendicular_from_diagonal_intersection_to_side | state | perpendicular from diagonal intersection to side
s_aryabhata_sine_pi_kuttaka | theorem | Aryabhata's Theorems
s_sine_second_difference_recurrence | state | sine second difference recurrence
s_aryabhata_sine_table | state | aryabhata sine table
s_aryabhata_pi_3_1416 | state | aryabhata pi 3 1416
s_kuttaka_extended_euclid | state | kuttaka extended euclid
s_alkhwarizmi_six_quadratic_types | theorem | Al-Khwārizmī's Quadratic Framework
s_six_canonical_quadratic_forms | state | six canonical quadratic forms
s_geometric_completion_of_square | state | geometric completion of square
s_khayyam_cubic_geometric_solution | theorem | Omar Khayyām's Geometric Solution of Cubics
s_fourteen_irreducible_cubic_types | state | fourteen irreducible cubic types
s_cubic_as_intersection_of_two_conics | state | cubic as intersection of two conics
s_viete_formulas | theorem | Viète's Formulas
s_polynomial_as_product_of_root_factors | state | polynomial as product of root factors
s_elementary_symmetric_polynomials | state | elementary symmetric polynomials
s_flt_n_equals_4 | theorem | Fermat's Last Theorem, Origin — n = 4 descent
s_flt_general_conjecture | state | flt general conjecture
s_flt_n_4_strengthened_claim | state | flt n 4 strengthened claim
s_descartes_rule_of_signs | theorem | Descartes' Rule of Signs
s_sign_change_parity_conjecture | state | sign change parity conjecture
s_sign_change_increment_under_positive_factor | state | sign change increment under positive factor
s_descartes_angular_defect | theorem | Descartes' Theorem on Total Angular Defect
s_vertex_angular_defect | state | vertex angular defect
s_total_defect_equals_2pi_chi | state | total defect equals 2pi chi
s_pascal_mystic_hexagram | theorem | Pascal's Theorem on Hexagons in a Conic / Mystic Hexagram
s_inscribed_hexagon_in_circle | state | inscribed hexagon in circle
s_three_opposite_side_intersection_points | state | three opposite side intersection points
s_pascal_line | state | pascal line
s_pascal_triangle_identity | theorem | Pascal's Triangle Identity
s_binomial_recurrence_conjecture | state | binomial recurrence conjecture
s_distinguished_element_bijection | state | distinguished element bijection
s_newton_binomial_theorem | theorem | Newton's Generalized Binomial Theorem
s_generalized_binomial_coefficient | state | generalized binomial coefficient
s_vandermonde_convolution_extended | state | vandermonde convolution extended
s_newton_identities | theorem | Newton's Identities
s_log_derivative_generating_function | state | log derivative generating function
s_power_sum_series | state | power sum series
s_rolle_theorem | theorem | Rolle's Theorem
s_attained_extrema_on_closed_interval | state | attained extrema on closed interval
s_interior_extremum_or_constant | state | interior extremum or constant
s_torricelli_law | theorem | Torricelli's Law
s_galilean_free_fall_speed | state | galilean free fall speed
s_fluid_column_energy_balance | state | fluid column energy balance
s_wallis_product | theorem | Wallis Product for π
s_wallis_integer_quadrature_table | state | wallis integer quadrature table
s_wallis_half_integer_interpolation | state | wallis half integer interpolation
s_bernoulli_lln | theorem | Bernoulli's Law of Large Numbers
s_binomial_tail_ratios | state | binomial tail ratios
s_binomial_tail_bound | state | binomial tail bound
s_de_moivre_laplace | theorem | De Moivre–Laplace Theorem
s_stirling_central_term_asymptotic | state | stirling central term asymptotic
s_gaussian_density_as_limit | state | gaussian density as limit
s_euler_totient_theorem | theorem | Euler's Totient Theorem
s_unit_group_mod_n | state | unit group mod n
s_orbit_of_a_mod_n | state | orbit of a mod n
s_euler_homogeneous_functions | theorem | Euler's Theorem on Homogeneous Functions
s_homogeneous_function_definition | state | homogeneous function definition
s_scaling_vector_field_identity | state | scaling vector field identity
s_lagrange_mvt | theorem | Lagrange's Mean Value Theorem
s_auxiliary_function_with_equal_endpoints | state | auxiliary function with equal endpoints
s_rolle_interior_critical_point | state | rolle interior critical point
s_wilson_theorem | theorem | Wilson's Theorem
s_polynomial_x_p_minus_1_factored_mod_p | state | polynomial x p minus 1 factored mod p
s_factored_form_of_cyclic_unit_polynomial | state | factored form of cyclic unit polynomial
s_bayes_theorem | theorem | Bayes's Theorem
s_conditional_probability_definition | state | conditional probability definition
s_symmetric_chain_rule_for_intersection | state | symmetric chain rule for intersection
s_cauchy_schwarz | theorem | Cauchy–Schwarz Inequality
s_inner_product_space | state | inner product space
s_non_negative_quadratic_in_t | state | non negative quadratic in t
s_bolzano_ivt | theorem | Bolzano's Intermediate Value Theorem
s_supremum_set_S | state | supremum set S
s_candidate_root_c_as_sup | state | candidate root c as sup
s_bolzano_weierstrass | theorem | Bolzano–Weierstrass Theorem
s_bounded_sequence_in_Rn | state | bounded sequence in Rn
s_bounded_sequence_in_closed_box | state | bounded sequence in closed box
s_nested_intervals_with_infinitely_many_terms | state | nested intervals with infinitely many terms
s_greens_theorem | theorem | Green's Theorem
s_simple_region_decomposition_in_R2 | state | simple region decomposition in R2
s_per_slice_ftoc_identity | state | per slice ftoc identity
s_divergence_theorem | theorem | Divergence Theorem / Gauss–Ostrogradsky
s_z_simple_region_decomposition_in_R3 | state | z simple region decomposition in R3
s_per_axis_flux_identity | state | per axis flux identity
s_liouville_bounded_entire | theorem | Liouville's Theorem (complex analysis)
s_cauchy_derivative_bound | state | cauchy derivative bound
s_vanishing_derivative_on_all_of_C | state | vanishing derivative on all of C
s_cayley_hamilton | theorem | Cayley–Hamilton Theorem
s_square_matrix_with_char_poly | state | square matrix with char poly
s_adjugate_matrix_polynomial_identity | state | adjugate matrix polynomial identity
s_cantor_bernstein_schroeder | theorem | Cantor–Bernstein–Schroeder Theorem
s_pair_of_injections_with_orphans | state | pair of injections with orphans
s_trajectory_partition_of_A | state | trajectory partition of A
s_poincare_recurrence | theorem | Poincaré Recurrence Theorem
s_measurable_set_of_positive_measure | state | measurable set of positive measure
s_non_returning_subset_B | state | non returning subset B
s_disjoint_preimages_of_B_sum_bounded | state | disjoint preimages of B sum bounded
s_zermelo_well_ordering | theorem | Zermelo's Well-Ordering Theorem
s_axiom_of_choice_function_phi | state | axiom of choice function phi
s_maximal_gamma_set_on_M | state | maximal gamma set on M
s_noether_isomorphism_theorems | theorem | Noether's Isomorphism Theorems
s_normal_subgroup_N_in_G | state | normal subgroup N in G
s_group_homomorphism_f | state | group homomorphism f
s_abstract_subobject_quotient_morphism_pattern | state | abstract subobject quotient morphism pattern
s_canonical_quotient_by_kernel_map | state | canonical quotient by kernel map
s_godel_completeness_theorem | theorem | Gödel's Completeness Theorem
s_first_order_logic_language | axiom | first order logic language
s_consistent_first_order_theory_T | state | consistent first order theory T
s_consistent_theory_with_witnesses | state | consistent theory with witnesses
s_maximal_consistent_henkin_theory_T_star | state | maximal consistent henkin theory T star
s_banach_tarski_paradox | theorem | Banach–Tarski Paradox
s_free_group_F2_on_two_generators | state | free group F2 on two generators
s_paradoxical_decomposition_of_F2 | state | paradoxical decomposition of F2
s_F2_as_subgroup_of_SO3 | state | F2 as subgroup of SO3
s_paradoxical_sphere_decomposition | state | paradoxical sphere decomposition
s_stone_weierstrass_theorem | theorem | Stone–Weierstrass Theorem
s_compact_hausdorff_space | axiom | compact hausdorff space
s_point_separating_unital_subalgebra_A_of_C_X_R | state | point separating unital subalgebra A of C X R
s_closure_of_A_is_a_lattice | state | closure of A is a lattice
s_pointwise_matching_family_f_xy | state | pointwise matching family f xy
s_stone_representation_theorem | theorem | Stone Representation Theorem
s_boolean_algebra_B | state | boolean algebra B
s_ultrafilter_spectrum_Spec_B | state | ultrafilter spectrum Spec B
s_compact_totally_disconnected_stone_space | state | compact totally disconnected stone space
s_full_modularity_theorem_BCDT | theorem | Modularity Theorem — full, Breuil–Conrad–Diamond–Taylor 2001
s_ramification_stratification_of_elliptic_curves | state | ramification stratification of elliptic curves
s_non_semistable_modularity_cases_handled | state | non semistable modularity cases handled
s_abc_conjecture_mochizuki_claimed | theorem | abc Conjecture — Mochizuki's disputed proof
s_coprime_triple_a_b_c_with_a_plus_b_equals_c | state | coprime triple a b c with a plus b equals c
s_mochizuki_IUT_framework | state | mochizuki IUT framework
s_anabelian_reconstruction_of_E | state | anabelian reconstruction of E
s_theta_link_between_hodge_theaters | state | theta link between hodge theaters
s_log_volume_inequality | state | log volume inequality
s_compactness_theorem_fol | theorem | Compactness theorem
s_lowenheim_skolem | theorem | Löwenheim–Skolem theorem
s_church_rosser | theorem | Church–Rosser theorem
s_deduction_theorem | theorem | Deduction theorem
s_los_theorem | theorem | Łoś's theorem
s_craig_interpolation | theorem | Craig's interpolation theorem
s_beth_definability | theorem | Beth's definability theorem
s_lindstrom_theorem | theorem | Lindström's theorem
s_morley_categoricity | theorem | Morley's categoricity theorem
s_martin_borel_determinacy | theorem | Martin's Borel determinacy theorem
s_cohen_forcing | theorem | Cohen's forcing theorems
s_mrdp_theorem | theorem | Matiyasevich's theorem (MRDP)
s_paris_harrington | theorem | Paris–Harrington theorem
s_gentzen_consistency | theorem | Gentzen's consistency proof
s_ac_equivalents | theorem | Axiom of choice equivalents
s_schroder_bernstein | theorem | Schröder–Bernstein theorem
s_hartogs_theorem | theorem | Hartogs's theorem
s_konig_cardinal_theorem | theorem | König's theorem (cardinal)
s_reflection_principle | theorem | Reflection principle
s_silver_theorem | theorem | Silver's theorem
s_hall_marriage | theorem | Hall's marriage theorem
s_konig_lemma | theorem | König's lemma
s_konig_bipartite | theorem | Kőnig's theorem (bipartite)
s_dilworth_theorem | theorem | Dilworth's theorem
s_mirsky_theorem | theorem | Mirsky's theorem
s_turan_theorem | theorem | Turán's theorem
s_erdos_ko_rado | theorem | Erdős–Ko–Rado theorem
s_menger_theorem | theorem | Menger's theorem
s_maxflow_mincut | theorem | Max-flow min-cut theorem
s_kuratowski_planarity | theorem | Kuratowski's theorem
s_wagner_planarity | theorem | Wagner's theorem
s_schur_ramsey | theorem | Schur's theorem (Ramsey)
s_van_der_waerden | theorem | Van der Waerden's theorem
s_hales_jewett | theorem | Hales–Jewett theorem
s_hindman_theorem | theorem | Hindman's theorem
s_cayley_formula | theorem | Cayley's formula
s_matrix_tree | theorem | Matrix-tree theorem
s_best_theorem | theorem | BEST theorem
s_burnside_lemma | theorem | Burnside's lemma
s_polya_enumeration | theorem | Pólya enumeration theorem
s_pigeonhole_principle | theorem | Pigeonhole principle
s_fermat_two_square_theorem | theorem | Fermat's two-square theorem
s_lagrange_four_square_theorem | theorem | Lagrange's four-square theorem
s_legendre_three_square_theorem | theorem | Legendre's three-square theorem
s_bertrand_postulate | theorem | Bertrand's postulate
s_dirichlet_primes_in_ap | theorem | Dirichlet's theorem on primes in arithmetic progressions
s_dirichlet_unit_theorem | theorem | Dirichlet's unit theorem
s_minkowski_convex_body_theorem | theorem | Minkowski's theorem on convex bodies
s_hilbert_waring_theorem | theorem | Hilbert–Waring theorem
s_thue_theorem | theorem | Thue's theorem
s_roth_approximation_theorem | theorem | Roth's theorem
s_siegel_integral_points_theorem | theorem | Siegel's theorem on integral points
s_mordell_weil_theorem | theorem | Mordell–Weil theorem
s_hasse_minkowski_theorem | theorem | Hasse–Minkowski theorem
s_hasse_elliptic_bound | theorem | Hasse's theorem on elliptic curves
s_weil_conjectures_deligne | theorem | Weil conjectures (Deligne's theorem)
s_chebotarev_density_theorem | theorem | Chebotarev density theorem
s_class_field_theory_main | theorem | Main theorem of class field theory
s_artin_reciprocity_law | theorem | Artin reciprocity law
s_erdos_ginzburg_ziv_theorem | theorem | Erdős–Ginzburg–Ziv theorem
s_chen_theorem | theorem | Chen's theorem
s_vinogradov_three_primes_theorem | theorem | Vinogradov's theorem
s_eisenstein_criterion | theorem | Eisenstein's criterion
s_gauss_lemma_polynomials | theorem | Gauss's lemma (polynomials)
s_primitive_element_theorem | theorem | Primitive element theorem
s_artin_schreier_theorem | theorem | Artin–Schreier theorem
s_hilbert_theorem_90 | theorem | Hilbert's theorem 90
s_luroth_theorem | theorem | Lüroth's theorem
s_nakayama_lemma | theorem | Nakayama's lemma
s_krull_hauptidealsatz | theorem | Krull's principal ideal theorem (Hauptidealsatz)
s_krull_intersection_theorem | theorem | Krull intersection theorem
s_krull_akizuki_theorem | theorem | Krull–Akizuki theorem
s_cohen_structure_theorem | theorem | Cohen structure theorem
s_hilbert_syzygy_theorem | theorem | Hilbert's syzygy theorem
s_auslander_buchsbaum_formula | theorem | Auslander–Buchsbaum formula
s_going_up_going_down_theorem | theorem | Going-up and going-down theorems
s_bezout_theorem | theorem | Bézout's theorem
s_chow_theorem | theorem | Chow's theorem
s_hironaka_resolution | theorem | Hironaka's resolution of singularities
s_serre_duality | theorem | Serre duality
s_grothendieck_riemann_roch | theorem | Grothendieck–Riemann–Roch
s_zariski_main_theorem | theorem | Zariski's main theorem
s_kodaira_vanishing_theorem | theorem | Kodaira vanishing theorem
s_lefschetz_hyperplane_theorem | theorem | Lefschetz hyperplane theorem
s_rank_nullity_theorem | theorem | Rank–nullity theorem
s_spectral_theorem_finite_dim | theorem | Spectral theorem (finite dim)
s_schur_decomposition | theorem | Schur decomposition
s_jordan_normal_form | theorem | Jordan normal form
s_singular_value_decomposition | theorem | Singular value decomposition
s_perron_frobenius_theorem | theorem | Perron–Frobenius theorem
s_gershgorin_circle_theorem | theorem | Gershgorin circle theorem
s_sylvester_law_of_inertia | theorem | Sylvester's law of inertia
s_gram_schmidt_process | theorem | Gram–Schmidt process
s_cramer_rule | theorem | Cramer's rule
s_cauchy_group_theorem | theorem | Cauchy's theorem (group theory)
s_jordan_holder_theorem | theorem | Jordan–Hölder theorem
s_krull_schmidt_theorem | theorem | Krull–Schmidt theorem
s_feit_thompson_theorem | theorem | Feit–Thompson theorem
s_schur_zassenhaus_theorem | theorem | Schur–Zassenhaus theorem
s_fg_abelian_structure_theorem | theorem | Structure theorem for finitely generated abelian groups
s_frattini_argument | theorem | Frattini's argument
s_jordan_finite_linear_theorem | theorem | Jordan's theorem on finite linear groups
s_burnside_paqb_theorem | theorem | Burnside's p^a q^b theorem
s_hnn_theorem | theorem | Higman–Neumann–Neumann theorem
s_maschke_theorem | theorem | Maschke's theorem
s_schur_lemma | theorem | Schur's lemma
s_wedderburn_finite_division_ring | theorem | Wedderburn's theorem (finite division rings)
s_artin_wedderburn_theorem | theorem | Artin–Wedderburn theorem
s_peter_weyl_theorem | theorem | Peter–Weyl theorem
s_frobenius_reciprocity | theorem | Frobenius reciprocity
s_heine_borel | theorem | Heine–Borel theorem
s_urysohn_lemma | theorem | Urysohn's lemma
s_normal_hausdorff_space | axiom | normal hausdorff space
s_tietze_extension | theorem | Tietze extension theorem
s_urysohn_metrization | theorem | Urysohn metrization theorem
s_second_countable_regular_hausdorff_space | axiom | second countable regular hausdorff space
s_alexandroff_compactification | theorem | Alexandroff compactification (one-point)
s_locally_compact_hausdorff_space | axiom | locally compact hausdorff space
s_borsuk_ulam | theorem | Borsuk–Ulam theorem
s_continuous_map_Sn_to_Rn | axiom | continuous map Sn to Rn
s_ham_sandwich | theorem | Ham sandwich theorem
s_n_measurable_bodies_in_Rn | axiom | n measurable bodies in Rn
s_jordan_curve_theorem | theorem | Jordan curve theorem
s_simple_closed_curve_in_plane | axiom | simple closed curve in plane
s_hairy_ball | theorem | Hairy ball theorem
s_continuous_tangent_vector_field | axiom | continuous tangent vector field
s_poincare_hopf | theorem | Poincaré–Hopf theorem
s_invariance_of_domain | theorem | Brouwer's invariance of domain
s_continuous_injection_between_Rn_opens | axiom | continuous injection between Rn opens
s_de_rham_theorem | theorem | de Rham's theorem
s_whitney_embedding | theorem | Whitney embedding theorem
s_nash_embedding | theorem | Nash embedding theorem
s_mostow_rigidity | theorem | Mostow rigidity theorem
s_closed_hyperbolic_manifold_dim_ge_3 | axiom | closed hyperbolic manifold dim ge 3
s_h_cobordism | theorem | Smale's h-cobordism theorem
s_simply_connected_manifold_dim_ge_5 | axiom | simply connected manifold dim ge 5
s_h_cobordism_between_manifolds | axiom | h cobordism between manifolds
s_freedman_theorem | theorem | Freedman's theorem
s_topological_4_manifold | axiom | topological 4 manifold
s_donaldson_theorem | theorem | Donaldson's theorem
s_smooth_4_manifold | axiom | smooth 4 manifold
s_alexander_duality | theorem | Alexander duality
s_subspace_of_Sn | axiom | subspace of Sn
s_poincare_duality | theorem | Poincaré duality
s_closed_oriented_n_manifold | axiom | closed oriented n manifold
s_hopf_rinow | theorem | Hopf–Rinow theorem
s_connected_riemannian_manifold | axiom | connected riemannian manifold
s_cartan_hadamard | theorem | Cartan–Hadamard theorem
s_complete_simply_connected_nonpositive_curvature_manifold | axiom | complete simply connected nonpositive curvature manifold
s_myers_theorem | theorem | Myers's theorem
s_ricci_curvature_positive_lower_bound | axiom | ricci curvature positive lower bound
s_hodge_theorem | theorem | Hodge theorem
s_compact_riemannian_manifold | axiom | compact riemannian manifold
s_chern_gauss_bonnet | theorem | Chern–Gauss–Bonnet theorem
s_uniformization_theorem | theorem | Uniformization theorem
s_simply_connected_riemann_surface | axiom | simply connected riemann surface
s_monotone_convergence_theorem | theorem | Monotone convergence theorem
s_fatou_lemma | theorem | Fatou's lemma
s_dominated_convergence_theorem | theorem | Dominated convergence theorem
s_fubini_theorem | theorem | Fubini's theorem
s_tonelli_theorem | theorem | Tonelli's theorem
s_radon_nikodym_theorem | theorem | Radon–Nikodym theorem
s_lebesgue_differentiation_theorem | theorem | Lebesgue differentiation theorem
s_vitali_covering_lemma | theorem | Vitali covering lemma
s_egorov_theorem | theorem | Egorov's theorem
s_lusin_theorem | theorem | Lusin's theorem
s_rademacher_theorem | theorem | Rademacher's theorem
s_sard_theorem | theorem | Sard's theorem
s_baire_category_theorem | theorem | Baire category theorem
s_arzela_ascoli_theorem | theorem | Arzelà–Ascoli theorem
s_picard_little_theorem | theorem | Picard's little theorem
s_picard_great_theorem | theorem | Picard's great theorem
s_casorati_weierstrass_theorem | theorem | Casorati–Weierstrass theorem
s_residue_theorem | theorem | Residue theorem
s_argument_principle | theorem | Argument principle
s_rouche_theorem | theorem | Rouché's theorem
s_morera_theorem | theorem | Morera's theorem
s_schwarz_lemma | theorem | Schwarz lemma
s_phragmen_lindelof_principle | theorem | Phragmén–Lindelöf principle
s_mittag_leffler_theorem | theorem | Mittag-Leffler theorem
s_weierstrass_factorization_theorem | theorem | Weierstrass factorization theorem
s_montel_theorem | theorem | Montel's theorem
s_hurwitz_theorem | theorem | Hurwitz's theorem
s_bloch_theorem | theorem | Bloch's theorem
s_gauss_lucas_theorem | theorem | Gauss–Lucas theorem
s_banach_steinhaus_theorem | theorem | Banach–Steinhaus theorem
s_open_mapping_theorem | theorem | Open mapping theorem
s_closed_graph_theorem | theorem | Closed graph theorem
s_banach_alaoglu_theorem | theorem | Banach–Alaoglu theorem
s_krein_milman_theorem | theorem | Kreĭn–Milman theorem
s_riesz_representation_theorems | theorem | Riesz representation theorems
s_spectral_theorem_self_adjoint | theorem | Spectral theorem for self-adjoint operators
s_gelfand_naimark_theorem | theorem | Gelfand–Naimark theorem
s_lax_milgram_lemma | theorem | Lax–Milgram lemma
s_sobolev_embedding_theorem | theorem | Sobolev embedding theorem
s_rellich_kondrachov_theorem | theorem | Rellich–Kondrachov theorem
s_stone_one_parameter_unitary_theorem | theorem | Stone's theorem on one-parameter unitary groups
s_riesz_fischer_theorem | theorem | Riesz–Fischer theorem
s_plancherel_theorem | theorem | Plancherel theorem
s_riesz_thorin_interpolation_theorem | theorem | Riesz–Thorin interpolation theorem
s_marcinkiewicz_interpolation_theorem | theorem | Marcinkiewicz interpolation theorem
s_carleson_theorem | theorem | Carleson's theorem
s_hardy_littlewood_maximal_inequality | theorem | Hardy–Littlewood maximal inequality
s_picard_lindelof_theorem | theorem | Picard–Lindelöf theorem
s_peano_existence_theorem | theorem | Peano existence theorem
s_cauchy_kovalevskaya_theorem | theorem | Cauchy–Kovalevskaya theorem
s_frobenius_theorem_distributions | theorem | Frobenius theorem (DE / distributions)
s_poincare_bendixson_theorem | theorem | Poincaré–Bendixson theorem
s_hartman_grobman_theorem | theorem | Hartman–Grobman theorem
s_kam_theorem | theorem | KAM theorem
s_kolmogorov_zero_one_law | theorem | Kolmogorov's 0–1 law
s_kolmogorov_extension_theorem | theorem | Kolmogorov's extension theorem
s_borel_cantelli_lemma | theorem | Borel–Cantelli lemma
s_doob_martingale_convergence | theorem | Doob's martingale convergence theorem
s_doob_optional_stopping | theorem | Doob's optional stopping theorem
s_ito_lemma | theorem | Itô's lemma
s_feynman_kac_formula | theorem | Feynman–Kac formula
s_girsanov_theorem | theorem | Girsanov's theorem
s_glivenko_cantelli | theorem | Glivenko–Cantelli theorem
s_donsker_theorem | theorem | Donsker's theorem
s_lindeberg_feller_clt | theorem | Lindeberg–Feller central limit theorem
s_sharkovskii_theorem | theorem | Sharkovskii's theorem
s_oseledets_multiplicative_ergodic | theorem | Oseledets multiplicative ergodic theorem
s_yoneda_lemma | theorem | Yoneda lemma
s_diagram_in_C | axiom | diagram in C
s_adjoint_functor_theorem | theorem | Adjoint functor theorem
s_rice_theorem | theorem | Rice's theorem
s_cook_levin_theorem | theorem | Cook–Levin theorem
s_time_hierarchy_theorem | theorem | Time hierarchy theorem
s_space_hierarchy_theorem | theorem | Space hierarchy theorem
s_ladner_theorem | theorem | Ladner's theorem
s_immerman_szelepcsenyi | theorem | Immerman–Szelepcsényi theorem
s_pcp_theorem | theorem | PCP theorem
s_godel_speed_up_theorem | theorem | Gödel speed-up theorem
s_savitch_theorem | theorem | Savitch's theorem
s_shannon_source_coding | theorem | Shannon's source coding theorem
s_shannon_noisy_channel | theorem | Shannon's noisy-channel coding theorem
s_shannon_hartley_theorem | theorem | Shannon–Hartley theorem
s_kraft_inequality | theorem | Kraft's inequality
s_cpt_theorem | theorem | CPT theorem
s_spin_statistics_theorem | theorem | Spin-statistics theorem
s_haag_theorem | theorem | Haag's theorem
s_noether_second_theorem | theorem | Noether's second theorem
s_reeh_schlieder_theorem | theorem | Reeh–Schlieder theorem
s_bell_theorem | theorem | Bell's theorem
s_kochen_specker_theorem | theorem | Kochen–Specker theorem
s_von_neumann_minimax | theorem | Von Neumann's minimax theorem
s_nash_existence_theorem | theorem | Nash's existence theorem
s_arrow_impossibility | theorem | Arrow's impossibility theorem
s_gibbard_satterthwaite | theorem | Gibbard–Satterthwaite theorem
s_finite_voting_profiles | axiom | Finite voting profiles
s_cauchy_theorem | theorem | Cauchy's theorem
s_cauchy_p_tuple_set | state | cauchy p tuple set
s_cyclic_action_on_p_tuples | state | cyclic action on p tuples
s_fixed_point_count_divisible_by_p | state | fixed point count divisible by p
s_class_equation | theorem | Class equation
s_conjugation_action_on_G | state | conjugation action on G
s_orbit_stabilizer_for_conjugation | state | orbit stabilizer for conjugation
s_lagrange_theorem | theorem | Lagrange's theorem
s_subgroup_H | axiom | subgroup H
s_coset_partition_of_G | state | coset partition of G
s_cosets_equinumerous_to_H | state | cosets equinumerous to H
s_orbit_stabilizer_theorem | theorem | Orbit-stabilizer theorem
s_coset_to_orbit_map | state | coset to orbit map
s_orbit_isomorphic_to_coset_space | state | orbit isomorphic to coset space
s_burnside_counting_lemma | theorem | Burnside's lemma
s_fix_incidence_set | state | fix incidence set
s_double_count_fix_stab | state | double count fix stab
s_burnside_paqb_solvable | theorem | Burnside's p^a q^b solvability theorem
s_finite_group_with_order_paqb | axiom | finite group with order paqb
s_hypothetical_simple_paqb_group | state | hypothetical simple paqb group
s_character_table_of_G | state | character table of G
s_algebraic_integer_relation | state | algebraic integer relation
s_central_element_exists | state | central element exists
s_schreier_refinement_theorem | theorem | Schreier refinement theorem
s_group_with_two_subnormal_series | axiom | group with two subnormal series
s_zassenhaus_refined_terms | state | zassenhaus refined terms
s_isomorphic_factor_pairs | state | isomorphic factor pairs
s_fundamental_theorem_fg_abelian | theorem | Fundamental theorem of finitely generated abelian groups
s_finitely_generated_abelian_group | axiom | finitely generated abelian group
s_presentation_matrix_A | state | presentation matrix A
s_smith_normal_form_of_A | state | smith normal form of A
s_invariant_factor_decomposition | state | invariant factor decomposition
s_pid_structure_theorem | theorem | Structure theorem for f.g. modules over a PID
s_principal_ideal_domain | axiom | principal ideal domain
s_finitely_generated_module_over_pid | axiom | finitely generated module over pid
s_module_presentation_over_R | state | module presentation over R
s_smith_form_module_presentation | state | smith form module presentation
s_invariant_factor_module_decomposition | state | invariant factor module decomposition
s_smith_normal_form_theorem | theorem | Smith normal form
s_matrix_over_pid | axiom | matrix over pid
s_pivot_reduction_sequence | state | pivot reduction sequence
s_descent_on_ideal_chain | state | descent on ideal chain
s_cayley_theorem | theorem | Cayley's theorem
s_symmetric_group_Sn | axiom | symmetric group Sn
s_left_regular_representation | state | left regular representation
s_nielsen_schreier_theorem | theorem | Nielsen–Schreier theorem
s_free_group_F | axiom | free group F
s_cayley_tree_of_F | state | cayley tree of F
s_quotient_graph_by_H_action | state | quotient graph by H action
s_pi1_of_quotient_is_free | state | pi1 of quotient is free
s_kurosh_subgroup_theorem | theorem | Kurosh subgroup theorem
s_free_product_of_groups | axiom | free product of groups
s_bass_serre_tree_for_free_product | state | bass serre tree for free product
s_H_action_on_bass_serre_tree | state | H action on bass serre tree
s_grushko_decomposition_theorem | theorem | Grushko decomposition theorem
s_finitely_generated_group_G | axiom | finitely generated group G
s_generating_set_of_free_product | state | generating set of free product
s_folded_graph_for_generators | state | folded graph for generators
s_stallings_ends_theorem | theorem | Stallings theorem on ends of groups
s_cayley_graph_of_G | axiom | cayley graph of G
s_ends_invariant_of_G | state | ends invariant of G
s_end_count_classification | state | end count classification
s_G_acts_on_tree_with_finite_edge_stabilizers | state | G acts on tree with finite edge stabilizers
s_gromov_polynomial_growth_theorem | theorem | Gromov's theorem on groups of polynomial growth
s_polynomial_growth_function | axiom | polynomial growth function
s_asymptotic_cone_of_G | state | asymptotic cone of G
s_lie_group_acting_on_cone | state | lie group acting on cone
s_tits_alternative_theorem | theorem | Tits alternative
s_finitely_generated_linear_group | axiom | finitely generated linear group
s_field_of_characteristic_zero | axiom | field of characteristic zero
s_zariski_closure_G | state | zariski closure G
s_semisimple_or_solvable_part | state | semisimple or solvable part
s_free_subgroup_via_ping_pong | state | free subgroup via ping pong
s_fitting_theorem | theorem | Fitting's theorem
s_nilpotent_normal_subgroup | axiom | nilpotent normal subgroup
s_product_of_two_nilpotent_normals | state | product of two nilpotent normals
s_nilpotency_class_bound | state | nilpotency class bound
s_first_isomorphism_theorem | theorem | Fundamental theorem on homomorphisms / first isomorphism theorem
s_group_homomorphism | axiom | group homomorphism
s_kernel_image_of_phi | state | kernel image of phi
s_induced_quotient_map | state | induced quotient map
s_second_isomorphism_theorem | theorem | Second isomorphism theorem (diamond)
s_normal_subgroup_N | axiom | normal subgroup N
s_HN_and_intersection | state | HN and intersection
s_diamond_iso_map | state | diamond iso map
s_third_isomorphism_theorem | theorem | Third isomorphism theorem
s_normal_subgroup_K | axiom | normal subgroup K
s_nested_normal_chain | state | nested normal chain
s_correspondence_theorem | theorem | Correspondence theorem (lattice isomorphism)
s_quotient_group_G_mod_N | axiom | quotient group G mod N
s_subgroup_correspondence_map | state | subgroup correspondence map
s_schurs_lemma | theorem | Schur's lemma
s_simple_module | axiom | simple module
s_module_homomorphism | axiom | module homomorphism
s_kernel_image_submodules | state | kernel image submodules
s_simplicity_dichotomy | state | simplicity dichotomy
s_wedderburn_artin_theorem | theorem | Wedderburn–Artin theorem
s_semisimple_artinian_ring | axiom | semisimple artinian ring
s_division_ring | axiom | division ring
s_simple_module_decomp_of_R | state | simple module decomp of R
s_endomorphism_ring_product | state | endomorphism ring product
s_matrix_rings_over_division_rings | state | matrix rings over division rings
s_jacobson_density_theorem | theorem | Jacobson density theorem
s_M_as_D_vector_space | state | M as D vector space
s_interpolation_problem_in_M | state | interpolation problem in M
s_double_centralizer_theorem | theorem | Double centralizer theorem
s_central_simple_algebra | axiom | central simple algebra
s_simple_subalgebra_B | axiom | simple subalgebra B
s_B_otimes_Aop_action | state | B otimes Aop action
s_centralizer_dimension_formula | state | centralizer dimension formula
s_skolem_noether_theorem | theorem | Skolem–Noether theorem
s_pair_of_B_embeddings | state | pair of B embeddings
s_bimodule_isomorphism | state | bimodule isomorphism
s_wedderburn_little_theorem | theorem | Wedderburn's little theorem
s_finite_division_ring | axiom | finite division ring
s_assumed_noncommutative_D | state | assumed noncommutative D
s_class_equation_for_D_units | state | class equation for D units
s_cyclotomic_divisibility_obstruction | state | cyclotomic divisibility obstruction
s_brauer_group_theorem | theorem | Brauer group periodicity
s_field_K | axiom | field K
s_brauer_equivalence_classes | state | brauer equivalence classes
s_brauer_group_structure | state | brauer group structure
s_hilbert_90_theorem | theorem | Hilbert's theorem 90
s_cyclic_galois_extension | axiom | cyclic galois extension
s_norm_one_element | state | norm one element
s_h1_vanishing | state | h1 vanishing
s_kummer_theory_theorem | theorem | Kummer theory
s_field_containing_nth_roots_of_unity | axiom | field containing nth roots of unity
s_cyclic_extension_of_degree_n | axiom | cyclic extension of degree n
s_kummer_pairing | state | kummer pairing
s_kummer_cohomology_iso | state | kummer cohomology iso
s_isomorphism_extension_theorem | theorem | Isomorphism extension theorem
s_partial_extension_family | state | partial extension family
s_maximal_extension_via_zorn | state | maximal extension via zorn
s_steinitz_algebraic_closure_theorem | theorem | Steinitz's theorem on field extensions
s_algebraic_closure | axiom | algebraic closure
s_algebraic_extensions_poset | state | algebraic extensions poset
s_maximal_algebraic_extension | state | maximal algebraic extension
s_frobenius_reciprocity_theorem | theorem | Frobenius reciprocity
s_ind_res_functor_pair | state | ind res functor pair
s_ind_res_adjunction | state | ind res adjunction
s_mackey_decomposition_theorem | theorem | Mackey decomposition / double coset formula
s_double_coset_decomposition | state | double coset decomposition
s_restriction_per_double_coset | state | restriction per double coset
s_character_orthogonality_theorem | theorem | Character orthogonality relations
s_irreducible_representation | axiom | irreducible representation
s_intertwiner_operator | state | intertwiner operator
s_g_equivariant_operator | state | g equivariant operator
s_schur_scalar_evaluation | state | schur scalar evaluation
s_burnside_character_vanish_theorem | theorem | Burnside's theorem on character vanishing (zeros in character table)
s_chi_is_sum_of_roots_of_unity | state | chi is sum of roots of unity
s_size_bound_on_chi_ratio | state | size bound on chi ratio
s_brauer_induced_character_theorem | theorem | Brauer's theorem on induced characters
s_irreducible_character | axiom | irreducible character
s_virtual_character_ring | state | virtual character ring
s_elementary_subgroup_generators | state | elementary subgroup generators
s_artin_induction_theorem | theorem | Artin's induction theorem
s_cyclic_subgroup | axiom | cyclic subgroup
s_induced_from_cyclic_subgroups | state | induced from cyclic subgroups
s_rational_combination_from_cyclics | state | rational combination from cyclics
s_weyl_complete_reducibility_theorem | theorem | Weyl's theorem on complete reducibility
s_semisimple_lie_algebra | axiom | semisimple lie algebra
s_finite_dim_representation | axiom | finite dim representation
s_casimir_central_element | state | casimir central element
s_casimir_eigenspace_decomposition | state | casimir eigenspace decomposition
s_highest_weight_theorem | theorem | Theorem of the highest weight
s_dominant_integral_weight | axiom | dominant integral weight
s_borel_subalgebra_b | state | borel subalgebra b
s_verma_module_M_lambda | state | verma module M lambda
s_unique_irreducible_quotient_L_lambda | state | unique irreducible quotient L lambda
s_borel_weil_bott_theorem | theorem | Borel–Weil–Bott theorem
s_semisimple_complex_lie_group | axiom | semisimple complex lie group
s_flag_variety_G_over_B | axiom | flag variety G over B
s_line_bundle_L_lambda | state | line bundle L lambda
s_sheaf_cohomology_of_L_lambda | state | sheaf cohomology of L lambda
s_weyl_shift_to_dominant | state | weyl shift to dominant
s_lie_kolchin_theorem | theorem | Lie–Kolchin theorem
s_connected_solvable_linear_algebraic_group | axiom | connected solvable linear algebraic group
s_g_action_on_flags | state | g action on flags
s_invariant_complete_flag | state | invariant complete flag
s_engel_theorem | theorem | Engel's theorem
s_lie_algebra_over_field | axiom | lie algebra over field
s_nilpotent_endomorphism | axiom | nilpotent endomorphism
s_all_adjoints_nilpotent | state | all adjoints nilpotent
s_common_zero_vector_constructed | state | common zero vector constructed
s_cartan_dieudonne_theorem | theorem | Cartan–Dieudonné theorem
s_quadratic_space_over_field | axiom | quadratic space over field
s_orthogonal_group_On | axiom | orthogonal group On
s_reflection_generators | state | reflection generators
s_descent_on_fixed_vectors | state | descent on fixed vectors
s_crystallographic_restriction_theorem | theorem | Crystallographic restriction theorem
s_lattice_in_euclidean_space | axiom | lattice in euclidean space
s_finite_rotation_subgroup | axiom | finite rotation subgroup
s_integer_trace_constraint | state | integer trace constraint
s_allowed_rotation_orders | state | allowed rotation orders
s_artin_rees_lemma | theorem | Artin–Rees lemma
s_finitely_generated_module | axiom | finitely generated module
s_rees_algebra_construction | state | rees algebra construction
s_rees_module_fg | state | rees module fg
s_going_up_theorem | theorem | Going-up theorem (Cohen–Seidenberg)
s_integral_ring_extension | axiom | integral ring extension
s_prime_ideal_chain | axiom | prime ideal chain
s_partial_lift_to_B | state | partial lift to B
s_integral_quotient_extension | state | integral quotient extension
s_going_down_theorem | theorem | Going-down theorem
s_integral_extension_of_integrally_closed_domains | axiom | integral extension of integrally closed domains
s_localized_extension | state | localized extension
s_galois_action_on_primes | state | galois action on primes
s_krull_pit_theorem | theorem | Krull's principal ideal theorem (Hauptidealsatz)
s_principal_ideal | axiom | principal ideal
s_minimal_prime_over_a | state | minimal prime over a
s_localized_at_p_setup | state | localized at p setup
s_chain_stabilizes | state | chain stabilizes
s_krull_height_theorem | theorem | Krull dimension theorem / height theorem
s_ideal_with_n_generators | axiom | ideal with n generators
s_induction_setup_on_generators | state | induction setup on generators
s_pit_as_base_case | state | pit as base case
s_quillen_suslin_theorem | theorem | Quillen–Suslin theorem (Serre's conjecture)
s_finitely_generated_projective_module | axiom | finitely generated projective module
s_local_freeness_of_P | state | local freeness of P
s_extended_module_certified | state | extended module certified
s_dimension_reduction_via_suslin | state | dimension reduction via suslin
s_auslander_buchsbaum_ufd_theorem | theorem | Auslander–Buchsbaum theorem (regular local ⇒ UFD)
s_regular_local_ring | axiom | regular local ring
s_unique_factorization_domain | axiom | unique factorization domain
s_height_one_prime_p | state | height one prime p
s_ext_class_vanishes | state | ext class vanishes
s_height_one_principal | state | height one principal
s_hopkins_levitzki_theorem | theorem | Hopkins–Levitzki theorem
s_artinian_ring | axiom | artinian ring
s_nilpotent_jacobson_radical | state | nilpotent jacobson radical
s_semisimple_filtration_quotients | state | semisimple filtration quotients
s_goldie_theorem | theorem | Goldie's theorem
s_semiprime_right_goldie_ring | axiom | semiprime right goldie ring
s_classical_ring_of_quotients | axiom | classical ring of quotients
s_ore_set_of_regular_elements | state | ore set of regular elements
s_classical_quotient_ring | state | classical quotient ring
s_lasker_noether_theorem | theorem | Primary decomposition (Lasker–Noether)
s_irreducible_intersection_decomp | state | irreducible intersection decomp
s_primary_intersection_decomp | state | primary intersection decomp
s_hom_tensor_adjunction_theorem | theorem | Hom-tensor adjunction
s_ring_R | axiom | ring R
s_module_category_R_Mod | axiom | module category R Mod
s_tensor_hom_functor_pair | state | tensor hom functor pair
s_flatness_tor_criterion_theorem | theorem | Flatness via Tor vanishing (Lazard's theorem flavor)
s_module_M | axiom | module M
s_tor_with_quotient_modules | state | tor with quotient modules
s_tor_vanishing_equivalence | state | tor vanishing equivalence
s_eilenberg_watts_theorem | theorem | Eilenberg–Watts theorem
s_colimit_preserving_functor | axiom | colimit preserving functor
s_colim_preserving_F | state | colim preserving F
s_bimodule_B_extracted | state | bimodule B extracted
s_mitchell_embedding_theorem | theorem | Mitchell's embedding theorem
s_abelian_category | axiom | abelian category
s_small_abelian_category | state | small abelian category
s_yoneda_embedding_setup | state | yoneda embedding setup
s_snake_lemma | theorem | Snake lemma
s_short_exact_sequence_diagram | axiom | short exact sequence diagram
s_connecting_homomorphism_delta | state | connecting homomorphism delta
s_six_term_long_sequence | state | six term long sequence
s_five_lemma | theorem | Five lemma
s_commutative_diagram_5x2 | axiom | commutative diagram 5x2
s_diagram_chase_argument | state | diagram chase argument
s_long_exact_sequence_theorem | theorem | Long exact sequence in cohomology
s_short_exact_sequence_of_chain_complexes | axiom | short exact sequence of chain complexes
s_chain_complex | axiom | chain complex
s_connecting_boundary_map | state | connecting boundary map
s_snake_lemma_applied_to_homology | state | snake lemma applied to homology
s_universal_coefficient_theorem | theorem | Universal coefficient theorem
s_chain_complex_over_pid | axiom | chain complex over pid
s_abelian_group_coefficients | axiom | abelian group coefficients
s_split_resolution_of_homology | state | split resolution of homology
s_tor_ext_terms_computed | state | tor ext terms computed
s_kunneth_formula_theorem | theorem | Künneth formula
s_tensor_product_of_complexes | axiom | tensor product of complexes
s_external_product_map | state | external product map
s_tor_correction_term | state | tor correction term
s_hilbert_serre_theorem | theorem | Hilbert–Serre theorem on Hilbert series rationality
s_graded_noetherian_ring | axiom | graded noetherian ring
s_finitely_generated_graded_module | axiom | finitely generated graded module
s_hilbert_series_of_M | state | hilbert series of M
s_induction_on_generators | state | induction on generators
s_cayley_hamilton_theorem | theorem | Cayley–Hamilton theorem
s_commutative_ring | axiom | commutative ring
s_square_matrix_over_ring | axiom | square matrix over ring
s_adjugate_identity | state | adjugate identity
s_substitution_x_eq_A | state | substitution x eq A
s_jordan_normal_form_theorem | theorem | Jordan normal form theorem
s_finite_dim_vector_space | axiom | finite dim vector space
s_v_as_kt_module | state | v as kt module
s_primary_cyclic_decomposition | state | primary cyclic decomposition
s_rational_canonical_form_theorem | theorem | Rational canonical form
s_v_as_kt_module_general | state | v as kt module general
s_invariant_factor_decomp_for_V | state | invariant factor decomp for V
s_chevalley_warning_theorem | theorem | Chevalley–Warning theorem
s_finite_field_Fq | axiom | finite field Fq
s_polynomial_system | axiom | polynomial system
s_indicator_via_fermat | state | indicator via fermat
s_indicator_sum_evaluation | state | indicator sum evaluation
s_ax_grothendieck_theorem | theorem | Ax–Grothendieck theorem
s_polynomial_map_C_n_to_C_n | axiom | polynomial map C n to C n
s_finite_field_case | state | finite field case
s_finite_field_injective_implies_surjective | state | finite field injective implies surjective
s_amitsur_levitzki_theorem | theorem | Amitsur–Levitzki theorem
s_matrix_algebra_M_n | axiom | matrix algebra M n
s_polynomial_identity | axiom | polynomial identity
s_standard_polynomial_2n | state | standard polynomial 2n
s_symmetric_trace_relations | state | symmetric trace relations
s_hochster_roberts_theorem | theorem | Hochster–Roberts theorem
s_reductive_group_action | axiom | reductive group action
s_invariant_subring_R_G | state | invariant subring R G
s_reynolds_splitting | state | reynolds splitting
s_eakin_nagata_theorem | theorem | Eakin–Nagata theorem
s_finite_ring_extension | axiom | finite ring extension
s_ascending_chain_in_S | state | ascending chain in S
s_chain_pulled_back_to_R | state | chain pulled back to R
s_schur_commutator_theorem | theorem | Schur's theorem on commutator subgroup
s_finite_index_center | axiom | finite index center
s_commutator_value_count | state | commutator value count
s_commutator_subgroup_size_bound | state | commutator subgroup size bound
s_frobenius_real_division_theorem | theorem | Frobenius's theorem on real division algebras
s_finite_dim_associative_real_division_algebra | axiom | finite dim associative real division algebra
s_pure_imaginary_subspace | state | pure imaginary subspace
s_dimension_dichotomy | state | dimension dichotomy
s_hurwitz_composition_theorem | theorem | Hurwitz's theorem on composition algebras
s_normed_division_algebra_over_R | axiom | normed division algebra over R
s_cayley_dickson_tower | state | cayley dickson tower
s_structural_property_loss | state | structural property loss
s_fundamental_theorem_symmetric_polynomials | theorem | Fundamental theorem of symmetric polynomials
s_symmetric_group_Sn_action | axiom | symmetric group Sn action
s_lex_reduction_algorithm | state | lex reduction algorithm
s_newton_identities_theorem | theorem | Newton's identities
s_generating_series_setup | state | generating series setup
s_log_derivative_relation | state | log derivative relation
s_bezout_identity_theorem | theorem | Bezout's identity in PIDs
s_two_ring_elements | axiom | two ring elements
s_ideal_a_b_in_R | state | ideal a b in R
s_ideal_principal_d | state | ideal principal d
s_euclidean_algorithm_theorem | theorem | Euclid's algorithm in Euclidean domains
s_euclidean_domain | axiom | euclidean domain
s_euclidean_division_step | state | euclidean division step
s_termination_via_norm_descent | state | termination via norm descent
s_solvability_by_radicals_theorem | theorem | Galois solvability ⇔ solvable Galois group
s_subgroup_chain_for_tower | state | subgroup chain for tower
s_abelian_quotient_chain | state | abelian quotient chain
s_cyclotomic_galois_theorem | theorem | Galois theorem on cyclotomic extensions
s_cyclotomic_field_Q_zeta_n | axiom | cyclotomic field Q zeta n
s_cyclotomic_minimal_polynomial | state | cyclotomic minimal polynomial
s_phi_n_irreducible_over_Q | state | phi n irreducible over Q
s_dedekind_splitting_theorem | theorem | Dedekind's theorem on splitting of primes
s_galois_extension_of_number_fields | axiom | galois extension of number fields
s_decomposition_inertia_groups | state | decomposition inertia groups
s_galois_transitive_on_primes | state | galois transitive on primes
s_hasse_arf_theorem | theorem | Hasse–Arf theorem
s_local_field | axiom | local field
s_abelian_extension | axiom | abelian extension
s_ramification_filtration | state | ramification filtration
s_herbrand_psi_function | state | herbrand psi function
s_local_tate_duality_theorem | theorem | Local Tate duality
s_cup_product_pairing | state | cup product pairing
s_finiteness_of_galois_cohomology | state | finiteness of galois cohomology
s_norm_residue_isomorphism_theorem | theorem | Norm residue isomorphism (Milnor / Bloch–Kato)
s_milnor_k_theory | axiom | milnor k theory
s_milnor_K_mod_ell | state | milnor K mod ell
s_norm_residue_map | state | norm residue map
s_motivic_cohomology_comparison | state | motivic cohomology comparison
s_brauer_three_main_theorems | theorem | Brauer's three main theorems (block theory)
s_p_block_of_kG | axiom | p block of kG
s_brauer_block_correspondence | state | brauer block correspondence
s_brauer_first_main | state | brauer first main
s_frobenius_determinant_theorem | theorem | Frobenius determinant theorem
s_group_determinant | axiom | group determinant
s_group_determinant_polynomial | state | group determinant polynomial
s_factorization_into_irreducible_factors | state | factorization into irreducible factors
s_hilbert_burch_theorem | theorem | Hilbert–Burch theorem
s_noetherian_local_ring | axiom | noetherian local ring
s_perfect_ideal_of_height_2 | axiom | perfect ideal of height 2
s_two_step_resolution | state | two step resolution
s_ideal_as_maximal_minors | state | ideal as maximal minors
s_jacobson_bourbaki_theorem | theorem | Jacobson–Bourbaki theorem
s_field_L | axiom | field L
s_subring_of_endomorphisms | axiom | subring of endomorphisms
s_subring_subfield_pairing | state | subring subfield pairing
s_double_centralizer_field_endring | state | double centralizer field endring
s_gabriel_quiver_theorem | theorem | Gabriel's theorem on quivers
s_quiver_Q | axiom | quiver Q
s_tits_quadratic_form | state | tits quadratic form
s_ade_dynkin_classification | state | ade dynkin classification
s_hurwitz_automorphism_theorem | theorem | Hurwitz's automorphism theorem
s_automorphism_group | axiom | automorphism group
s_quotient_riemann_surface | state | quotient riemann surface
s_riemann_hurwitz_relation | state | riemann hurwitz relation
s_burnside_basis_theorem | theorem | Burnside's basis theorem
s_finite_p_group | axiom | finite p group
s_frattini_subgroup | axiom | frattini subgroup
s_frattini_subgroup_definition | state | frattini subgroup definition
s_quotient_is_fp_vsp | state | quotient is fp vsp
s_higman_embedding_theorem | theorem | Higman's embedding theorem
s_recursively_presented_group | axiom | recursively presented group
s_finitely_presented_group | axiom | finitely presented group
s_arithmetized_relator_set | state | arithmetized relator set
s_hnn_machine_simulation | state | hnn machine simulation
s_adian_rabin_theorem | theorem | Adian–Rabin theorem on Markov properties
s_markov_property_of_groups | axiom | markov property of groups
s_word_problem_encoded | state | word problem encoded
s_diagonal_reduction_to_word_problem | state | diagonal reduction to word problem
s_cartan_brauer_hua_theorem | theorem | Cartan–Brauer–Hua theorem
s_subdivision_ring_K | axiom | subdivision ring K
s_conjugation_invariant_subring | state | conjugation invariant subring
s_commutator_argument | state | commutator argument
s_maclane_coherence_theorem | theorem | Mac Lane's coherence theorem
s_monoidal_category | axiom | monoidal category
s_associator_unitor_isomorphisms | axiom | associator unitor isomorphisms
s_pentagon_and_triangle_axioms | state | pentagon and triangle axioms
s_coherence_diagram_setup | state | coherence diagram setup
s_finite_field_classification_theorem | theorem | Fundamental theorem of finite division rings of squares (Frobenius–Stickelberger / classification of finite fields)
s_prime_power_q | axiom | prime power q
s_splitting_field_of_xq_minus_x | state | splitting field of xq minus x
s_q_roots_form_subfield | state | q roots form subfield
s_tensor_product_universal_property | theorem | Tensor product universal property
s_bilinear_map | axiom | bilinear map
s_tensor_product_construction | state | tensor product construction
s_universal_property_via_hom | state | universal property via hom
s_cohn_irreducibility_criterion | theorem | Cohn's irreducibility criterion
s_base_b_digits_of_prime | state | base b digits of prime
s_root_location_bound | state | root location bound
s_hua_identity_theorem | theorem | Hua's identity
s_hua_candidate_identity | state | hua candidate identity
s_inverse_manipulation | state | inverse manipulation
s_ito_character_degree_theorem | theorem | Itô's theorem on character degrees
s_abelian_normal_subgroup | axiom | abelian normal subgroup
s_clifford_induced_characters | state | clifford induced characters
s_orbit_sum_on_A | state | orbit sum on A
s_localization_universal_property | theorem | Universal property of localization
s_multiplicative_subset | axiom | multiplicative subset
s_construction_of_localization | state | construction of localization
s_splitting_field_theorem | theorem | Splitting field existence and uniqueness
s_iterated_root_adjunction | state | iterated root adjunction
s_terminating_construction | state | terminating construction
s_normal_basis_theorem | theorem | Normal basis theorem
s_L_as_kG_module | state | L as kG module
s_dedekind_linear_independence | state | dedekind linear independence
s_stickelberger_theorem | theorem | Stickelberger's theorem on Gauss sums
s_gauss_sum_definition | state | gauss sum definition
s_gauss_sum_prime_factorization | state | gauss sum prime factorization
s_tate_cohomology_cyclic_theorem | theorem | Mac Lane's theorem on cohomology of cyclic groups
s_finite_cyclic_group | axiom | finite cyclic group
s_g_module_M | axiom | g module M
s_periodic_resolution | state | periodic resolution
s_two_periodic_cohomology | state | two periodic cohomology
s_schur_multiplier_theorem | theorem | Schur multiplier and central extensions
s_central_extension | axiom | central extension
s_h2_classifies_projective_reps | state | h2 classifies projective reps
s_schur_cover_construction | state | schur cover construction
s_wigner_eckart_theorem | theorem | Wigner–Eckart theorem
s_compact_lie_group | axiom | compact lie group
s_tensor_operator | axiom | tensor operator
s_irreducible_tensor_operator | state | irreducible tensor operator
s_clebsch_gordan_decomposition | state | clebsch gordan decomposition
s_multiplicity_one_theorem | theorem | Multiplicity-one theorem
s_reductive_group_GL_n | axiom | reductive group GL n
s_irreducible_automorphic_representation | axiom | irreducible automorphic representation
s_whittaker_model_for_pi | state | whittaker model for pi
s_local_whittaker_uniqueness | state | local whittaker uniqueness
s_galois_descent_theorem | theorem | Galois descent
s_galois_extension_L_over_K | axiom | galois extension L over K
s_object_over_L_with_descent_data | axiom | object over L with descent data
s_descent_cocycle | state | descent cocycle
s_h1_classification_of_forms | state | h1 classification of forms
s_intermediate_value_theorem | theorem | Intermediate Value theorem
s_normalized_sign_change_setup | state | normalized sign change setup
s_nested_intervals_with_sign_change | state | nested intervals with sign change
s_extreme_value_theorem | theorem | Extreme Value theorem
s_image_is_bounded | state | image is bounded
s_maximizing_sequence_in_domain | state | maximizing sequence in domain
s_mean_value_theorem_terminal | theorem | Mean Value theorem
s_differentiable_on_open_interval | axiom | differentiable on open interval
s_tilted_function_with_equal_endpoints | state | tilted function with equal endpoints
s_rolle_theorem_point | state | rolle theorem point
s_cauchy_mean_value_theorem | theorem | Cauchy mean value theorem
s_paired_function_with_equal_endpoints | state | paired function with equal endpoints
s_lhopital_rule | theorem | L'Hôpital's rule
s_indeterminate_form_zero_over_zero | axiom | indeterminate form zero over zero
s_normalized_0_over_0_pair | state | normalized 0 over 0 pair
s_ratio_equals_derivative_ratio_at_xi | state | ratio equals derivative ratio at xi
s_darboux_theorem_analysis | theorem | Darboux's theorem (intermediate value for derivatives)
s_auxiliary_g_with_critical_point | state | auxiliary g with critical point
s_interior_critical_point_of_g | state | interior critical point of g
s_cantor_intersection_theorem | theorem | Cantor's intersection theorem
s_nested_nonempty_closed_sets_shrinking_diameter | axiom | nested nonempty closed sets shrinking diameter
s_cauchy_sequence_from_nested_sets | state | cauchy sequence from nested sets
s_limit_point_in_each_Fn | state | limit point in each Fn
s_banach_steinhaus | theorem | Banach–Steinhaus uniform boundedness principle
s_banach_space | axiom | banach space
s_family_of_pointwise_bounded_operators | axiom | family of pointwise bounded operators
s_closed_covering_of_banach_space | state | closed covering of banach space
s_E_n_with_interior_ball | state | E n with interior ball
s_arzela_ascoli | theorem | Arzelà–Ascoli theorem
s_compact_metric_space | axiom | compact metric space
s_uniformly_bounded_equicontinuous_family | axiom | uniformly bounded equicontinuous family
s_family_indexed_by_dense_subset_D | state | family indexed by dense subset D
s_pointwise_convergent_on_D_subsequence | state | pointwise convergent on D subsequence
s_stone_weierstrass | theorem | Stone–Weierstrass theorem
s_closure_is_lattice | state | closure is lattice
s_two_point_interpolation_property | state | two point interpolation property
s_tietze_extension_theorem | theorem | Tietze extension theorem
s_normal_topological_space | axiom | normal topological space
s_bounded_continuous_function_on_closed_subset | axiom | bounded continuous function on closed subset
s_urysohn_function_for_each_level | state | urysohn function for each level
s_uniformly_convergent_extension_series | state | uniformly convergent extension series
s_dini_theorem | theorem | Dini's theorem
s_monotone_pointwise_convergent_continuous_sequence | axiom | monotone pointwise convergent continuous sequence
s_open_cover_from_pointwise_bound | state | open cover from pointwise bound
s_uniform_bound_on_X | state | uniform bound on X
s_caratheodory_extension | theorem | Carathéodory extension theorem
s_premeasure_on_algebra_of_sets | axiom | premeasure on algebra of sets
s_sigma_algebra_generated | axiom | sigma algebra generated
s_outer_measure_on_powerset | state | outer measure on powerset
s_sigma_algebra_of_measurable_sets | state | sigma algebra of measurable sets
s_riesz_markov_representation | theorem | Riesz–Markov representation theorem
s_positive_linear_functional_on_Cc_X | axiom | positive linear functional on Cc X
s_outer_premeasure_on_opens | state | outer premeasure on opens
s_borel_measure_candidate | state | borel measure candidate
s_vitali_hahn_saks | theorem | Vitali–Hahn–Saks theorem
s_sequence_of_uniformly_absolutely_continuous_measures | axiom | sequence of uniformly absolutely continuous measures
s_finite_measure_space | axiom | finite measure space
s_complete_metric_space_of_classes | state | complete metric space of classes
s_equicontinuous_at_some_set | state | equicontinuous at some set
s_lebesgue_monotone_differentiation | theorem | Lebesgue's monotone differentiation theorem
s_monotone_function_on_interval | axiom | monotone function on interval
s_lebesgue_measure_on_R | axiom | lebesgue measure on R
s_dini_derivate_quadruple | state | dini derivate quadruple
s_zero_measure_disagreement_set | state | zero measure disagreement set
s_riesz_representation_Lp | theorem | Riesz representation theorem for L^p
s_sigma_finite_measure_space | axiom | sigma finite measure space
s_bounded_linear_functional_on_Lp | axiom | bounded linear functional on Lp
s_finite_signed_measure_nu_absolutely_continuous | state | finite signed measure nu absolutely continuous
s_fefferman_stein_vector_valued | theorem | Fefferman–Stein vector-valued maximal inequality
s_lebesgue_measure_on_Rn | axiom | lebesgue measure on Rn
s_lp_sequence_of_l1_functions | axiom | lp sequence of l1 functions
s_randomized_linear_combination | state | randomized linear combination
s_dual_pairing_to_scalar_maximal | state | dual pairing to scalar maximal
s_calderon_zygmund_decomposition | theorem | Calderón–Zygmund decomposition
s_l1_function_on_Rn | axiom | l1 function on Rn
s_stopping_cubes_with_average_exceeding_lambda | state | stopping cubes with average exceeding lambda
s_good_bad_decomposition | state | good bad decomposition
s_calderon_zygmund_Lp_bound | theorem | Calderón–Zygmund singular integral L^p bound
s_calderon_zygmund_kernel_on_Rn | axiom | calderon zygmund kernel on Rn
s_lp_function_on_Rn | axiom | lp function on Rn
s_L2_bound_for_singular_integral | state | L2 bound for singular integral
s_weak_11_bound_via_CZ_decomp | state | weak 11 bound via CZ decomp
s_ramsey_theorem_finite | theorem | Ramsey's theorem (finite version)
s_finite_ramsey_conjecture | state | finite ramsey conjecture
s_recursive_majority_color_neighbourhood | state | recursive majority color neighbourhood
s_ramsey_number_recurrence_R_r_s | state | ramsey number recurrence R r s
s_schur_theorem | theorem | Schur's theorem
s_edge_colored_complete_graph_from_schur_coloring | state | edge colored complete graph from schur coloring
s_monochromatic_triangle_to_schur_triple | state | monochromatic triangle to schur triple
s_van_der_waerden_theorem | theorem | Van der Waerden's theorem
s_vdw_conjecture_W_r_k | state | vdw conjecture W r k
s_color_focused_progressions | state | color focused progressions
s_vdw_finite_intersection_lemma | state | vdw finite intersection lemma
s_hales_jewett_theorem | theorem | Hales–Jewett theorem
s_finite_combinatorial_cube_word_space | axiom | finite combinatorial cube word space
s_combinatorial_line_target | state | combinatorial line target
s_shelah_cube_lemma | state | shelah cube lemma
s_graham_rothschild_theorem | theorem | Graham–Rothschild theorem
s_parameter_word_partition_setup | state | parameter word partition setup
s_iterated_hj_combinatorial_subspace | state | iterated hj combinatorial subspace
s_folkman_theorem | theorem | Folkman's theorem
s_folkman_finite_target | state | folkman finite target
s_folkman_via_hj_reduction | state | folkman via hj reduction
s_gallai_theorem_multidim_vdw | theorem | Gallai's theorem (multidimensional VdW)
s_homothetic_copy_target | state | homothetic copy target
s_gallai_via_hj_reduction | state | gallai via hj reduction
s_erdos_rado_theorem | theorem | Erdős–Rado theorem
s_uncountable_partition_target | state | uncountable partition target
s_transfinite_color_canonization | state | transfinite color canonization
s_carlson_simpson_theorem | theorem | Carlson–Simpson theorem
s_infinitary_hj_target | state | infinitary hj target
s_ultrafilter_on_variable_words | state | ultrafilter on variable words
s_erdos_ko_rado_theorem | theorem | Erdős–Ko–Rado theorem
s_finite_set_family | axiom | finite set family
s_intersecting_family_property | axiom | intersecting family property
s_max_intersecting_k_uniform_problem | state | max intersecting k uniform problem
s_cyclic_arc_counting_bound | state | cyclic arc counting bound
s_double_counting_arc_intersection | state | double counting arc intersection
s_kruskal_katona_theorem | theorem | Kruskal–Katona theorem
s_shadow_operator_on_set_family | axiom | shadow operator on set family
s_shadow_minimisation_problem | state | shadow minimisation problem
s_colex_initial_family_extremum | state | colex initial family extremum
s_compression_preserves_size_reduces_shadow | state | compression preserves size reduces shadow
s_lym_inequality | theorem | LYM inequality
s_antichain_in_boolean_lattice | axiom | antichain in boolean lattice
s_random_maximal_chain_on_boolean_lattice | state | random maximal chain on boolean lattice
s_chain_meets_antichain_at_most_once | state | chain meets antichain at most once
s_sperner_theorem | theorem | Sperner's theorem
s_middle_layer_bound_for_antichain | state | middle layer bound for antichain
s_bollobas_set_pair_inequality | theorem | Bollobás set-pair inequality
s_cross_intersecting_pair_system | axiom | cross intersecting pair system
s_random_linear_order_on_ground_set | state | random linear order on ground set
s_probability_event_disjoint_pairs | state | probability event disjoint pairs
s_sauer_shelah_lemma | theorem | Sauer–Shelah lemma (VC dimension)
s_vc_dimension_definition | axiom | vc dimension definition
s_vc_count_bound_target | state | vc count bound target
s_down_compressed_shattering_preserved | state | down compressed shattering preserved
s_shattering_bound_sum_binomials | state | shattering bound sum binomials
s_frankl_wilson_theorem | theorem | Frankl–Wilson theorem
s_mod_p_intersection_pattern_target | state | mod p intersection pattern target
s_incidence_polynomial_basis_in_F_p | state | incidence polynomial basis in F p
s_ahlswede_khachatrian_theorem | theorem | Ahlswede–Khachatrian theorem
s_t_intersecting_family_property | axiom | t intersecting family property
s_complete_intersection_problem | state | complete intersection problem
s_frankl_family_F_r_candidate | state | frankl family F r candidate
s_johnson_scheme_eigenvalue_optimum | state | johnson scheme eigenvalue optimum
s_hall_marriage_theorem | theorem | Hall's marriage theorem
s_bipartite_graph | axiom | bipartite graph
s_perfect_matching_definition | axiom | perfect matching definition
s_hall_condition_statement | state | hall condition statement
s_minimum_vertex_cover_smaller_than_A | state | minimum vertex cover smaller than A
s_augmenting_path_yields_violator_S | state | augmenting path yields violator S
s_konig_theorem | theorem | König's theorem
s_min_vertex_cover_definition | axiom | min vertex cover definition
s_lp_relaxation_matching_vs_cover | state | lp relaxation matching vs cover
s_konig_cover_constructed_from_max_matching | state | konig cover constructed from max matching
s_max_flow_min_cut_theorem | theorem | Max-flow min-cut theorem
s_directed_graph_with_capacities | axiom | directed graph with capacities
s_flow_conservation_axiom | axiom | flow conservation axiom
s_max_flow_lp_dual | state | max flow lp dual
s_residual_graph_no_augmenting_path_yields_cut | state | residual graph no augmenting path yields cut
s_berge_augmenting_theorem | theorem | Berge's theorem (augmenting path)
s_matching_in_graph | axiom | matching in graph
s_berge_target | state | berge target
s_symmetric_difference_M_Mprime_decomposes_into_paths | state | symmetric difference M Mprime decomposes into paths
s_petersen_theorem | theorem | Petersen's theorem (3-regular bridgeless)
s_three_regular_bridgeless_graph | axiom | three regular bridgeless graph
s_petersen_target | state | petersen target
s_tutte_odd_component_count_under_3regular_bridgeless | state | tutte odd component count under 3regular bridgeless
s_tutte_perfect_matching_theorem | theorem | Tutte's theorem (perfect matching)
s_tutte_condition_target | state | tutte condition target
s_maximal_counterexample_extremal_S | state | maximal counterexample extremal S
s_block_structure_contradicts_tutte_count | state | block structure contradicts tutte count
s_gale_ryser_theorem | theorem | Gale–Ryser theorem (degree sequence)
s_degree_sequence | axiom | degree sequence
s_gale_ryser_target | state | gale ryser target
s_transportation_polytope_integrality_certificate | state | transportation polytope integrality certificate
s_erdos_gallai_theorem | theorem | Erdős–Gallai theorem (graphic sequence)
s_graphic_sequence_target | state | graphic sequence target
s_sorted_degree_partial_sums_inequality | state | sorted degree partial sums inequality
s_erdos_stone_theorem | theorem | Erdős–Stone theorem
s_chromatic_number_definition | axiom | chromatic number definition
s_erdos_stone_target | state | erdos stone target
s_blow_up_complete_multipartite_found | state | blow up complete multipartite found
s_kovari_sos_turan_theorem | theorem | Kővári–Sós–Turán theorem
s_forbidden_subgraph_K_s_t | axiom | forbidden subgraph K s t
s_kst_extremal_target | state | kst extremal target
s_jensen_lower_bound_on_K_s_copies | state | jensen lower bound on K s copies
s_bondy_simonovits_theorem | theorem | Bondy–Simonovits theorem (even cycles)
s_forbidden_even_cycle_C_2k | axiom | forbidden even cycle C 2k
s_even_cycle_extremal_target | state | even cycle extremal target
s_bfs_layer_count_doubling | state | bfs layer count doubling
s_szemeredi_regularity_lemma | theorem | Szemerédi regularity lemma
s_density_between_vertex_sets | axiom | density between vertex sets
s_regularity_target | state | regularity target
s_mean_square_density_index_q | state | mean square density index q
s_graph_removal_lemma | theorem | Graph removal lemma
s_cleaned_reduced_graph_H_free_or_dense | state | cleaned reduced graph H free or dense
s_counting_lemma_dense_H_implication | state | counting lemma dense H implication
s_corners_theorem | theorem | Corners theorem (Ajtai–Szemerédi)
s_two_dimensional_lattice | axiom | two dimensional lattice
s_corner_target | state | corner target
s_triangle_removal_yields_corner | state | triangle removal yields corner
s_szemeredi_trotter_theorem | theorem | Szemerédi–Trotter theorem (incidence)
s_finite_point_set_in_plane | axiom | finite point set in plane
s_finite_line_set_in_plane | axiom | finite line set in plane
s_szemeredi_trotter_target | state | szemeredi trotter target
s_cutting_into_cells_count_incidences_per_cell | state | cutting into cells count incidences per cell
s_erdos_posa_theorem | theorem | Erdős–Pósa theorem
s_disjoint_cycle_packing_vs_cover | axiom | disjoint cycle packing vs cover
s_erdos_posa_dual_target | state | erdos posa dual target
s_dfs_back_edges_yield_disjoint_cycles_or_small_FVS | state | dfs back edges yield disjoint cycles or small FVS
s_q_analog_ekr_theorem | theorem | Erdős–Ko–Rado for vector spaces (q-analog)
s_finite_dim_vector_space_over_F_q | axiom | finite dim vector space over F q
s_q_ekr_target | state | q ekr target
s_grassmann_scheme_eigen_optimum | state | grassmann scheme eigen optimum
s_matrix_tree_theorem | theorem | Kirchhoff's matrix-tree theorem
s_laplacian_matrix_of_graph | axiom | laplacian matrix of graph
s_incidence_matrix_factorisation_L_eq_B_B_T | state | incidence matrix factorisation L eq B B T
s_cauchy_binet_expansion_of_principal_minor | state | cauchy binet expansion of principal minor
s_lgv_lemma | theorem | Lindström–Gessel–Viennot lemma
s_acyclic_directed_graph_with_weights | axiom | acyclic directed graph with weights
s_non_intersecting_lattice_paths_problem | axiom | non intersecting lattice paths problem
s_path_matrix_M_setup | state | path matrix M setup
s_sign_reversing_involution_on_crossing_path_systems | state | sign reversing involution on crossing path systems
s_rsk_correspondence | theorem | RSK correspondence
s_finite_matrix_nonneg_integer_entries | axiom | finite matrix nonneg integer entries
s_pair_of_young_tableaux | axiom | pair of young tableaux
s_rsk_target | state | rsk target
s_row_insertion_algorithm_well_defined | state | row insertion algorithm well defined
s_hook_length_formula | theorem | Hook-length formula
s_partition_lambda | axiom | partition lambda
s_standard_young_tableau | axiom | standard young tableau
s_hook_formula_conjecture | state | hook formula conjecture
s_hook_walk_bijection_setup | state | hook walk bijection setup
s_uniform_random_tableau_distribution_argument | state | uniform random tableau distribution argument
s_polya_enumeration_theorem | theorem | Pólya enumeration theorem
s_group_action_on_set_of_colorings | axiom | group action on set of colorings
s_cycle_index_polynomial | state | cycle index polynomial
s_burnside_orbit_counting | theorem | Burnside's lemma (orbit counting)
s_incidence_relation_R | state | incidence relation R
s_double_count_fix_stabilizer_identity | state | double count fix stabilizer identity
s_lagrange_inversion_theorem | theorem | Lagrange inversion theorem
s_formal_power_series_ring | axiom | formal power series ring
s_compositional_inverse_problem | axiom | compositional inverse problem
s_lagrange_inversion_target | state | lagrange inversion target
s_residue_calculation_of_coefficient | state | residue calculation of coefficient
s_macmahon_master_theorem | theorem | MacMahon's master theorem
s_macmahon_target | state | macmahon target
s_residue_extraction_diagonal_coefficient | state | residue extraction diagonal coefficient
s_stanley_reciprocity_theorem | theorem | Stanley's reciprocity theorem
s_rational_cone_in_R_n | axiom | rational cone in R n
s_lattice_point_enumerator | axiom | lattice point enumerator
s_reciprocity_target | state | reciprocity target
s_polar_duality_on_cones | state | polar duality on cones
s_five_color_theorem | theorem | Five color theorem
s_5_color_target | state | 5 color target
s_min_degree_at_most_five_lemma | state | min degree at most five lemma
s_kempe_chain_recoloring_argument | state | kempe chain recoloring argument
s_brooks_theorem | theorem | Brooks' theorem
s_brooks_target | state | brooks target
s_good_vertex_ordering_with_two_nonadjacent_priors | state | good vertex ordering with two nonadjacent priors
s_vizing_theorem | theorem | Vizing's theorem
s_edge_chromatic_number | axiom | edge chromatic number
s_vizing_target | state | vizing target
s_vizing_fan_recoloring_setup | state | vizing fan recoloring setup
s_grotzsch_theorem | theorem | Grötzsch's theorem
s_triangle_free_graph_property | axiom | triangle free graph property
s_grotzsch_target | state | grotzsch target
s_reducible_configurations_for_3_coloring | state | reducible configurations for 3 coloring
s_kuratowski_theorem | theorem | Kuratowski's theorem (planarity)
s_planar_embedding_definition | axiom | planar embedding definition
s_kuratowski_target | state | kuratowski target
s_minimal_non_planar_obstructions_K5_K33 | state | minimal non planar obstructions K5 K33
s_wagner_theorem | theorem | Wagner's theorem
s_wagner_target | state | wagner target
s_minor_vs_topological_minor_equivalence | state | minor vs topological minor equivalence
s_fary_theorem | theorem | Fáry's theorem
s_straight_line_embedding_definition | axiom | straight line embedding definition
s_fary_target | state | fary target
s_inductive_straightening_of_edges | state | inductive straightening of edges
s_steinitz_theorem | theorem | Steinitz's theorem (polytopes ↔ planar 3-connected)
s_steinitz_target | state | steinitz target
s_tutte_rubber_band_realisation | state | tutte rubber band realisation
s_lovasz_perfect_graph_theorem | theorem | Perfect graph theorem (Lovász)
s_perfect_graph_definition | axiom | perfect graph definition
s_perfect_complement_target | state | perfect complement target
s_complement_duality_on_perfection | state | complement duality on perfection
s_replication_lemma | state | replication lemma
s_strong_perfect_graph_theorem | theorem | Strong perfect graph theorem
s_spgt_target | state | spgt target
s_berge_graph_decomposition_skeleton | state | berge graph decomposition skeleton
s_lovasz_local_lemma | theorem | Lovász local lemma
s_dependency_graph_of_events | axiom | dependency graph of events
s_lll_target | state | lll target
s_weight_certificate_for_lll | state | weight certificate for lll
s_erdos_ramsey_lower_bound | theorem | Erdős probabilistic lower bound on R(k,k)
s_uniform_random_two_coloring_of_K_n | state | uniform random two coloring of K n
s_expected_count_less_than_one | state | expected count less than one
s_spencer_six_sigma_theorem | theorem | Spencer's six standard deviations
s_set_system_on_n_points | axiom | set system on n points
s_signed_coloring_problem | axiom | signed coloring problem
s_spencer_target | state | spencer target
s_partial_coloring_lemma_entropy_bucket | state | partial coloring lemma entropy bucket
s_beck_fiala_theorem | theorem | Beck–Fiala theorem
s_beck_fiala_target | state | beck fiala target
s_floating_lp_iteration | state | floating lp iteration
s_beck_three_coloring_theorem | theorem | Beck's three-coloring theorem
s_beck_three_color_target | state | beck three color target
s_random_two_coloring_and_local_correction | state | random two coloring and local correction
s_erdos_renyi_connectivity_threshold | theorem | Erdős–Rényi threshold for connectivity
s_random_graph_G_n_p_model | axiom | random graph G n p model
s_threshold_target | state | threshold target
s_first_moment_isolated_vertices | state | first moment isolated vertices
s_second_moment_estimate | state | second moment estimate
s_erdos_szekeres_theorem | theorem | Erdős–Szekeres monotone subsequence theorem
s_finite_sequence_of_reals | axiom | finite sequence of reals
s_monotone_subsequence_problem | axiom | monotone subsequence problem
s_es_target | state | es target
s_pair_labels_distinct | state | pair labels distinct
s_friendship_theorem | theorem | Friendship theorem (Erdős–Rényi–Sós)
s_friendship_condition_every_pair_unique_common_neighbor | axiom | friendship condition every pair unique common neighbor
s_friendship_target | state | friendship target
s_adjacency_spectral_argument | state | adjacency spectral argument
s_ore_theorem | theorem | Ore's theorem
s_hamilton_cycle_problem | axiom | hamilton cycle problem
s_ore_target | state | ore target
s_longest_path_endpoint_degree_sum_bound | state | longest path endpoint degree sum bound
s_bondy_chvatal_closure_theorem | theorem | Bondy–Chvátal closure theorem
s_closure_target | state | closure target
s_closure_operator_well_defined | state | closure operator well defined
s_chvatal_erdos_theorem | theorem | Chvátal–Erdős theorem
s_chvatal_erdos_target | state | chvatal erdos target
s_disjoint_paths_argument | state | disjoint paths argument
s_posa_rotation_extension_theorem | theorem | Pósa's rotation-extension lemma
s_posa_target | state | posa target
s_endpoint_set_grows_under_rotation | state | endpoint set grows under rotation
s_tutte_hamilton_4connected_planar | theorem | Tutte's theorem on Hamilton cycles in 4-connected planar
s_tutte_4conn_target | state | tutte 4conn target
s_tutte_path_construction | state | tutte path construction
s_fleischner_theorem | theorem | Fleischner's theorem
s_fleischner_target | state | fleischner target
s_dfs_ear_traversal_yields_hamilton_cycle_in_square | state | dfs ear traversal yields hamilton cycle in square
s_cauchy_davenport_theorem | theorem | Cauchy–Davenport theorem
s_subsets_of_Z_p | axiom | subsets of Z p
s_cauchy_davenport_target | state | cauchy davenport target
s_nullstellensatz_certificate_for_sumset | state | nullstellensatz certificate for sumset
s_davenport_constant_theorem_for_p_groups | theorem | Davenport constant for abelian groups
s_finite_abelian_group | axiom | finite abelian group
s_zero_sum_sequence_problem | axiom | zero sum sequence problem
s_davenport_target | state | davenport target
s_chevalley_warning_for_abelian_p_group | state | chevalley warning for abelian p group
s_kneser_addition_theorem | theorem | Kneser's addition theorem
s_sumset_definition | axiom | sumset definition
s_kneser_target | state | kneser target
s_quotient_by_stabilizer_gives_strict_sumset_growth | state | quotient by stabilizer gives strict sumset growth
s_plunnecke_ruzsa_inequality | theorem | Plünnecke–Ruzsa inequality
s_plunnecke_target | state | plunnecke target
s_petridis_magnification_argument | state | petridis magnification argument
s_freiman_theorem | theorem | Freiman's theorem
s_freiman_target | state | freiman target
s_ruzsa_covering_applied | state | ruzsa covering applied
s_bogolyubov_bohr_set_inside_2A_minus_2A | state | bogolyubov bohr set inside 2A minus 2A
s_behrend_construction_theorem | theorem | Behrend's construction (large AP-free set)
s_three_term_AP_free_property | axiom | three term AP free property
s_behrend_target | state | behrend target
s_sphere_in_Z_m_d_is_AP_free | state | sphere in Z m d is AP free
s_roth_theorem_3AP | theorem | Roth's theorem on 3-APs
s_roth_target | state | roth target
s_large_fourier_coefficient_dichotomy | state | large fourier coefficient dichotomy
s_density_increment_iteration | state | density increment iteration
s_kleene_recursion_theorem | theorem | Recursion theorem (Kleene's second)
s_godel_numbering_of_programs | state | godel numbering of programs
s_diagonal_program_construction | state | diagonal program construction
s_smn_theorem | theorem | Smn theorem (parameter theorem)
s_uniform_program_index | state | uniform program index
s_re_not_recursive | theorem | Existence of RE but non-recursive sets
s_halting_set_K | state | halting set K
s_K_is_re | state | K is re
s_karp_lipton_theorem | theorem | Karp–Lipton theorem (NP ⊆ P/poly ⇒ PH = Σ₂)
s_hypothetical_sat_circuit_family | state | hypothetical sat circuit family
s_self_reducible_circuit_finder | state | self reducible circuit finder
s_toda_theorem | theorem | Toda's theorem (PH ⊆ P^#P)
s_isolation_lemma_witness | state | isolation lemma witness
s_ph_inside_bpp_parity_p | state | ph inside bpp parity p
s_parity_p_in_p_sharp_p | state | parity p in p sharp p
s_sipser_gacs_lautemann | theorem | Sipser–Gács–Lautemann theorem (BPP ⊆ Σ₂)
s_amplified_bpp_acceptance_set | state | amplified bpp acceptance set
s_translation_cover_witness | state | translation cover witness
s_adleman_theorem | theorem | Adleman's theorem (BPP ⊆ P/poly)
s_amplified_bpp_machine | state | amplified bpp machine
s_universal_advice_string | state | universal advice string
s_yao_minimax_principle | theorem | Yao's minimax principle
s_randomized_alg_as_mixed_strategy | state | randomized alg as mixed strategy
s_razborov_smolensky | theorem | Razborov–Smolensky lower bound on AC⁰[p]
s_acc0_p_circuit_model | state | acc0 p circuit model
s_low_degree_polynomial_approximation | state | low degree polynomial approximation
s_parity_not_in_ac0 | theorem | Furst–Saxe–Sipser (parity ∉ AC⁰)
s_ac0_circuit_model | state | ac0 circuit model
s_random_restriction_argument | state | random restriction argument
s_depth_reduction_to_constant | state | depth reduction to constant
s_hastad_switching_lemma | theorem | Håstad's switching lemma
s_random_restriction_on_dnf | state | random restriction on dnf
s_encoding_of_bad_paths | state | encoding of bad paths
s_isolation_lemma_mvv | theorem | Isolation lemma (Mulmuley–Vazirani–Vazirani)
s_random_weighted_set_system | state | random weighted set system
s_schwartz_zippel_lemma | theorem | Schwartz–Zippel lemma
s_polynomial_and_sample_set | state | polynomial and sample set
s_inductive_root_bound | state | inductive root bound
s_valiant_vazirani | theorem | Valiant–Vazirani theorem (UNIQUE-SAT randomized hardness)
s_sat_instance_with_unknown_count | state | sat instance with unknown count
s_hashed_sat_instance | state | hashed sat instance
s_ip_equals_pspace | theorem | IP = PSPACE (Shamir)
s_tqbf_canonical_form | state | tqbf canonical form
s_arithmetized_tqbf | state | arithmetized tqbf
s_sum_check_protocol | state | sum check protocol
s_mip_equals_nexp | theorem | MIP = NEXP (Babai–Fortnow–Lund)
s_multi_prover_model | state | multi prover model
s_scaled_arithmetization_nexp | state | scaled arithmetization nexp
s_two_prover_oracle_protocol | state | two prover oracle protocol
s_gmw_zk_for_np | theorem | GMW zero-knowledge for NP
s_three_coloring_canonical_target | state | three coloring canonical target
s_commit_and_challenge_protocol | state | commit and challenge protocol
s_shannon_channel_coding | theorem | Shannon noisy channel coding theorem
s_dmc_capacity_definition | state | dmc capacity definition
s_random_codebook_achievability | state | random codebook achievability
s_kraft_mcmillan_inequality | theorem | Kraft–McMillan inequality
s_uniquely_decodable_code_model | state | uniquely decodable code model
s_extension_counting_bound | state | extension counting bound
s_slepian_wolf_theorem | theorem | Slepian–Wolf theorem
s_correlated_source_model | state | correlated source model
s_random_binning_scheme | state | random binning scheme
s_wyner_ziv_theorem | theorem | Wyner–Ziv theorem
s_side_info_source_model | state | side info source model
s_auxiliary_u_construction | state | auxiliary u construction
s_singleton_bound | theorem | Singleton bound
s_block_code_model | state | block code model
s_punctured_code_injection | state | punctured code injection
s_plotkin_bound | theorem | Plotkin bound
s_high_distance_binary_code | state | high distance binary code
s_double_count_pairwise_distances | state | double count pairwise distances
s_hamming_bound | theorem | Hamming (sphere-packing) bound
s_t_error_correcting_code | state | t error correcting code
s_disjoint_hamming_balls | state | disjoint hamming balls
s_gilbert_varshamov_bound | theorem | Gilbert–Varshamov bound
s_greedy_code_construction | state | greedy code construction
s_mrrw_lp_bound | theorem | MRRW (linear programming) bound
s_distance_distribution_polynomials | state | distance distribution polynomials
s_delsarte_linear_program | state | delsarte linear program
s_mrrw_dual_polynomial | state | mrrw dual polynomial
s_reed_solomon_distance | theorem | Reed–Solomon distance / BCH bound
s_rs_code_definition | state | rs code definition
s_root_count_argument | state | root count argument
s_helly_theorem | theorem | Helly's theorem
s_helly_local_hypothesis | state | helly local hypothesis
s_radon_partition_step | state | radon partition step
s_caratheodory_convex | theorem | Carathéodory's theorem (convex hull)
s_convex_combination_with_k_points | state | convex combination with k points
s_affine_dependence_relation | state | affine dependence relation
s_radon_theorem | theorem | Radon's theorem
s_d_plus_two_points | state | d plus two points
s_linear_dependence_in_d_plus_one | state | linear dependence in d plus one
s_tverberg_theorem | theorem | Tverberg's theorem
s_tverberg_input_count | state | tverberg input count
s_deleted_join_configuration_space | state | deleted join configuration space
s_colored_tverberg | theorem | Colored Tverberg (Bárány–Larman / Blagojević–Matschke–Ziegler)
s_rainbow_partition_problem | state | rainbow partition problem
s_chessboard_complex | state | chessboard complex
s_barany_point_selection | theorem | Bárány's point-selection theorem
s_simplex_family_on_n_points | state | simplex family on n points
s_lp_strong_duality | theorem | LP strong duality
s_primal_dual_lp_pair | state | primal dual lp pair
s_lp_weak_duality | state | lp weak duality
s_farkas_lemma | theorem | Farkas's lemma
s_farkas_alternatives | state | farkas alternatives
s_separating_hyperplane_for_cone | state | separating hyperplane for cone
s_gordan_theorem | theorem | Gordan's theorem of alternatives
s_gordan_alternatives | state | gordan alternatives
s_stiemke_theorem | theorem | Stiemke's theorem
s_stiemke_alternatives | state | stiemke alternatives
s_khachiyan_ellipsoid | theorem | Ellipsoid method polynomial-time (Khachiyan)
s_separation_oracle_lp | state | separation oracle lp
s_ellipsoid_volume_shrinkage_iteration | state | ellipsoid volume shrinkage iteration
s_karmarkar_polynomial_lp | theorem | Karmarkar interior-point polynomial-time
s_projective_lp_form | state | projective lp form
s_karmarkar_potential_decrease | state | karmarkar potential decrease
s_max_flow_min_cut | theorem | Max-flow min-cut theorem (Ford–Fulkerson)
s_network_flow_model | state | network flow model
s_residual_graph_construction | state | residual graph construction
s_no_augmenting_path_implies_min_cut | state | no augmenting path implies min cut
s_edmonds_matroid_intersection | theorem | Edmonds' matroid intersection theorem
s_two_matroids_on_e | state | two matroids on e
s_exchange_digraph_construction | state | exchange digraph construction
s_aks_primality_theorem | theorem | AKS primality test (deterministic polynomial-time)
s_aks_polynomial_identity | state | aks polynomial identity
s_aks_reduced_identity_modulo_x_r_minus_1 | state | aks reduced identity modulo x r minus 1
s_introspective_set_bound | state | introspective set bound
s_cantor_zassenhaus | theorem | Cantor–Zassenhaus factorization
s_equal_degree_factor_setting | state | equal degree factor setting
s_random_h_power_construction | state | random h power construction
s_berlekamp_factorization | theorem | Berlekamp factorization
s_quotient_ring_for_f | state | quotient ring for f
s_berlekamp_subalgebra | state | berlekamp subalgebra
s_subalgebra_crt_decomposition | state | subalgebra crt decomposition
s_lll_algorithm_theorem | theorem | LLL lattice basis reduction
s_lattice_basis_with_gso | state | lattice basis with gso
s_lll_swap_and_reduce_step | state | lll swap and reduce step
s_lll_potential_decrease | state | lll potential decrease
s_master_theorem | theorem | Master theorem for divide-and-conquer recurrences
s_divide_and_conquer_recurrence | state | divide and conquer recurrence
s_recursion_tree_decomposition | state | recursion tree decomposition
s_pumping_lemma_regular | theorem | Pumping lemma for regular languages
s_dfa_with_p_states | state | dfa with p states
s_revisited_state_loop | state | revisited state loop
s_myhill_nerode_theorem | theorem | Myhill–Nerode theorem
s_nerode_equivalence_relation | state | nerode equivalence relation
s_minimal_dfa_isomorphism | state | minimal dfa isomorphism
s_kleene_regex_dfa | theorem | Kleene's theorem (regex = DFA)
s_regex_syntax | state | regex syntax
s_thompson_nfa_construction | state | thompson nfa construction
s_chomsky_schutzenberger | theorem | Chomsky–Schützenberger representation theorem
s_cfl_definition | state | cfl definition
s_dyck_language_universal | state | dyck language universal
s_parikh_theorem | theorem | Parikh's theorem
s_parikh_map | state | parikh map
s_cf_parse_tree_pumping_step | state | cf parse tree pumping step
s_ogden_lemma | theorem | Ogden's lemma
s_marked_string_in_cfl | state | marked string in cfl
s_marked_parse_path_pumping | state | marked parse path pumping
s_rabin_scott_subset_construction | theorem | Rabin–Scott subset construction (NFA = DFA)
s_nfa_model | state | nfa model
s_subset_dfa_construction | state | subset dfa construction
s_trakhtenbrot_theorem | theorem | Trakhtenbrot's theorem (finite-model satisfiability undecidable)
s_fo_encoding_of_tm_computation | state | fo encoding of tm computation
s_finite_model_iff_halting | state | finite model iff halting
s_khinchin_recurrence | theorem | Khinchin recurrence theorem
s_set_of_positive_measure_A | axiom | set of positive measure A
s_ergodic_average_of_indicator | state | ergodic average of indicator
s_orthogonal_projection_onto_invariants | state | orthogonal projection onto invariants
s_density_lower_bound_on_return_times | state | density lower bound on return times
s_furstenberg_multiple_recurrence | theorem | Furstenberg multiple recurrence theorem
s_furstenberg_structure_tower | state | furstenberg structure tower
s_distal_factor_with_AP_returns | state | distal factor with AP returns
s_von_neumann_mean_ergodic | theorem | von Neumann mean ergodic theorem
s_unitary_operator_on_hilbert_space | axiom | unitary operator on hilbert space
s_spectral_measure_on_unit_circle | state | spectral measure on unit circle
s_orthogonal_decomposition_fixed_plus_coboundary | state | orthogonal decomposition fixed plus coboundary
s_wiener_ergodic_l1 | theorem | Wiener ergodic theorem (L¹)
s_L1_function | axiom | L1 function
s_hardy_littlewood_maximal_for_T | state | hardy littlewood maximal for T
s_weak_type_inequality_for_averages | state | weak type inequality for averages
s_kingman_subadditive_ergodic | theorem | Kingman subadditive ergodic theorem
s_subadditive_cocycle | axiom | subadditive cocycle
s_invariant_limsup_liminf_pair | state | invariant limsup liminf pair
s_block_decomposition_with_controlled_error | state | block decomposition with controlled error
s_oseledec_multiplicative_ergodic | theorem | Oseledec multiplicative ergodic theorem
s_matrix_cocycle_over_T | axiom | matrix cocycle over T
s_lyapunov_growth_function | state | lyapunov growth function
s_singular_value_filtration | state | singular value filtration
s_lyapunov_spectrum_a_e | state | lyapunov spectrum a e
s_pesin_entropy_formula | theorem | Pesin entropy formula
s_smooth_volume_preserving_diffeomorphism | axiom | smooth volume preserving diffeomorphism
s_oseledec_splitting_for_f | state | oseledec splitting for f
s_local_unstable_manifold_family | state | local unstable manifold family
s_conditional_entropy_along_unstable | state | conditional entropy along unstable
s_margulis_ruelle_inequality | theorem | Margulis–Ruelle inequality
s_C1_diffeomorphism_of_compact_manifold | axiom | C1 diffeomorphism of compact manifold
s_invariant_borel_probability | axiom | invariant borel probability
s_lyapunov_exponents_ae | state | lyapunov exponents ae
s_partition_subordinate_to_expanding | state | partition subordinate to expanding
s_variational_principle_entropy | theorem | Variational principle for topological entropy
s_continuous_self_map_of_compact_space | axiom | continuous self map of compact space
s_topological_entropy_h_top | state | topological entropy h top
s_measure_entropy_bounded_by_h_top | state | measure entropy bounded by h top
s_supremum_attained_by_invariant_measure | state | supremum attained by invariant measure
s_krylov_bogolyubov_invariant_measure | theorem | Krylov–Bogolyubov existence of invariant measures
s_cesaro_average_of_dirac_orbits | state | cesaro average of dirac orbits
s_weak_star_limit_point_of_averages | state | weak star limit point of averages
s_hopf_decomposition | theorem | Hopf decomposition theorem
s_conservative_dissipative_split | state | conservative dissipative split
s_invariant_partition_C_D | state | invariant partition C D
s_ergodic_decomposition_theorem | theorem | Ergodic decomposition theorem
s_invariant_probability_measure | axiom | invariant probability measure
s_conditional_measure_family | state | conditional measure family
s_ergodic_components_disintegration | state | ergodic components disintegration
s_bowen_specification_unique_equilibrium | theorem | Bowen's specification property implies unique equilibrium
s_expansive_homeomorphism_with_specification | axiom | expansive homeomorphism with specification
s_holder_potential_phi | axiom | holder potential phi
s_bowen_partition_function_Z_n | state | bowen partition function Z n
s_topological_pressure_P_phi | state | topological pressure P phi
s_eigenmeasure_for_L_phi | state | eigenmeasure for L phi
s_bowen_dimension_formula | theorem | Bowen formula for Hausdorff dimension of repellers
s_conformal_expanding_repeller_J | axiom | conformal expanding repeller J
s_geometric_pressure_function_P_s | state | geometric pressure function P s
s_bowen_root_s_star | state | bowen root s star
s_curtis_hedlund_lyndon | theorem | Curtis–Hedlund–Lyndon theorem
s_shift_space_A_Z | axiom | shift space A Z
s_continuous_shift_commuting_map | axiom | continuous shift commuting map
s_uniform_continuity_via_compactness | state | uniform continuity via compactness
s_local_rule_of_radius_r | state | local rule of radius r
s_williams_shift_equivalence | theorem | Williams classification of SFTs up to shift equivalence
s_subshift_of_finite_type | axiom | subshift of finite type
s_transition_matrix_presentation_A | state | transition matrix presentation A
s_strong_shift_equivalence_chain | state | strong shift equivalence chain
s_singer_theorem | theorem | Singer's theorem (negative Schwarzian implies bounded attractors)
s_interval_map_with_negative_schwarzian | axiom | interval map with negative schwarzian
s_negative_schwarzian_preserved_under_iteration | state | negative schwarzian preserved under iteration
s_basin_with_no_interior_critical_point_or_boundary | state | basin with no interior critical point or boundary
s_thurston_pcf_rigidity | theorem | Misiurewicz–Thurston rigidity for postcritically finite maps
s_postcritically_finite_branched_cover_of_S2 | axiom | postcritically finite branched cover of S2
s_teichmuller_pullback_map_sigma_f | state | teichmuller pullback map sigma f
s_fixed_point_or_obstruction_dichotomy | state | fixed point or obstruction dichotomy
s_hadamard_perron_stable_manifold | theorem | Hadamard–Perron stable/unstable manifold theorem
s_hyperbolic_fixed_point_of_diffeomorphism | axiom | hyperbolic fixed point of diffeomorphism
s_graph_transform_operator | state | graph transform operator
s_invariant_lipschitz_graph_W_s | state | invariant lipschitz graph W s
s_anosov_closing_lemma | theorem | Anosov closing lemma
s_anosov_diffeomorphism | axiom | anosov diffeomorphism
s_epsilon_pseudo_periodic_orbit | axiom | epsilon pseudo periodic orbit
s_local_product_box | state | local product box
s_unique_periodic_orbit_near_pseudo | state | unique periodic orbit near pseudo
s_anosov_shadowing_lemma | theorem | Anosov shadowing lemma
s_delta_pseudo_orbit | axiom | delta pseudo orbit
s_correction_sequence_space | state | correction sequence space
s_unique_shadowing_orbit | state | unique shadowing orbit
s_smale_horseshoe | theorem | Smale horseshoe
s_diffeomorphism_with_transverse_homoclinic_point | axiom | diffeomorphism with transverse homoclinic point
s_markov_rectangle_R | state | markov rectangle R
s_smale_birkhoff_homoclinic | theorem | Smale–Birkhoff homoclinic theorem
s_iterated_neighborhood_of_homoclinic | state | iterated neighborhood of homoclinic
s_horseshoe_in_f_N | state | horseshoe in f N
s_smale_spectral_decomposition | theorem | Smale spectral decomposition theorem
s_axiom_A_diffeomorphism | axiom | axiom A diffeomorphism
s_non_wandering_set_Omega | state | non wandering set Omega
s_basic_set_partition | state | basic set partition
s_srb_measure_existence | theorem | Sinai–Ruelle–Bowen (SRB) measure existence for Axiom A
s_axiom_A_attractor | axiom | axiom A attractor
s_markov_partition_of_attractor | state | markov partition of attractor
s_sft_conjugate_to_attractor | state | sft conjugate to attractor
s_gibbs_state_for_unstable_potential | state | gibbs state for unstable potential
s_manning_volume_entropy | theorem | Manning's theorem (volume entropy = topological entropy)
s_compact_negatively_curved_riemannian_manifold | axiom | compact negatively curved riemannian manifold
s_volume_growth_function | state | volume growth function
s_volume_entropy_h_vol | state | volume entropy h vol
s_topological_entropy_of_geodesic_flow | state | topological entropy of geodesic flow
s_pesin_set_theorem | theorem | Pesin set / Pesin block existence
s_smooth_diffeomorphism_with_nonzero_lyapunov_spectrum | axiom | smooth diffeomorphism with nonzero lyapunov spectrum
s_lyapunov_charts | state | lyapunov charts
s_pesin_block_Lambda_ell | state | pesin block Lambda ell
s_mane_bowen_formula | theorem | Mañé–Bowen formula for repellers
s_conformal_repeller_with_holder_jacobian | axiom | conformal repeller with holder jacobian
s_pressure_function_of_s | state | pressure function of s
s_bowen_root_for_repeller | state | bowen root for repeller
s_bowen_periodic_equidistribution | theorem | Bowen's equidistribution of periodic orbits
s_axiom_A_basic_set | axiom | axiom A basic set
s_weighted_periodic_orbit_measure | state | weighted periodic orbit measure
s_convergence_to_equilibrium_state | state | convergence to equilibrium state
s_julia_fatou_dichotomy | theorem | Julia–Fatou dichotomy
s_rational_map_on_riemann_sphere | axiom | rational map on riemann sphere
s_normality_locus_F_f | state | normality locus F f
s_montel_normal_family_criterion | state | montel normal family criterion
s_sullivan_no_wandering | theorem | Sullivan no-wandering-domains theorem
s_teichmuller_space_of_lamination | state | teichmuller space of lamination
s_finite_dimensional_deformation_space | state | finite dimensional deformation space
s_mane_sad_sullivan_stability | theorem | Mañé–Sad–Sullivan structural stability
s_lambda_lemma_extension | state | lambda lemma extension
s_douady_hubbard_straightening | theorem | Douady–Hubbard straightening theorem
s_polynomial_like_map_of_degree_d | axiom | polynomial like map of degree d
s_invariant_beltrami_coefficient | state | invariant beltrami coefficient
s_quasiconformal_conjugacy_to_polynomial | state | quasiconformal conjugacy to polynomial
s_yoccoz_local_connectivity | theorem | Yoccoz's theorem on local connectivity of Mandelbrot at finitely renormalizable points
s_finitely_renormalizable_quadratic | axiom | finitely renormalizable quadratic
s_yoccoz_puzzle_at_level_n | state | yoccoz puzzle at level n
s_a_priori_moduli_bound | state | a priori moduli bound
s_mane_hyperbolicity_theorem | theorem | Misiurewicz–Mañé theorem on hyperbolicity from non-recurrent critical orbits
s_rational_map_with_nonrecurrent_critical_orbits | axiom | rational map with nonrecurrent critical orbits
s_local_expansion_estimate | state | local expansion estimate
s_uniform_expansion_on_J | state | uniform expansion on J
s_brolin_lyubich_measure | theorem | Brolin–Lyubich measure of maximal entropy for rational maps
s_rational_map_of_degree_d_at_least_2 | axiom | rational map of degree d at least 2
s_pullback_point_measure_sequence | state | pullback point measure sequence
s_potential_theoretic_limit | state | potential theoretic limit
s_ratner_measure_rigidity | theorem | Ratner measure rigidity theorem
s_unipotent_one_parameter_subgroup_u_t | axiom | unipotent one parameter subgroup u t
s_lattice_subgroup_Gamma | axiom | lattice subgroup Gamma
s_unipotent_q_shift_invariance | state | unipotent q shift invariance
s_extra_invariance_under_subgroup_H | state | extra invariance under subgroup H
s_invariant_measure_is_homogeneous | state | invariant measure is homogeneous
s_ratner_orbit_closure | theorem | Ratner orbit closure theorem
s_orbit_closure_candidate | state | orbit closure candidate
s_homogeneous_subset_closure | state | homogeneous subset closure
s_ratner_equidistribution | theorem | Ratner equidistribution theorem
s_unipotent_time_average | state | unipotent time average
s_unique_limit_is_homogeneous_haar | state | unique limit is homogeneous haar
s_margulis_arithmeticity | theorem | Margulis arithmeticity theorem
s_semisimple_lie_group_of_higher_rank | axiom | semisimple lie group of higher rank
s_irreducible_lattice_subgroup | axiom | irreducible lattice subgroup
s_finite_dim_linear_rep_of_lattice | state | finite dim linear rep of lattice
s_extension_to_algebraic_rep_of_G | state | extension to algebraic rep of G
s_q_structure_yielding_arithmetic_lattice | state | q structure yielding arithmetic lattice
s_margulis_super_rigidity | theorem | Margulis super-rigidity
s_lattice_representation_rho | state | lattice representation rho
s_boundary_map_phi | state | boundary map phi
s_equivariant_boundary_value | state | equivariant boundary value
s_margulis_normal_subgroup | theorem | Margulis normal subgroup theorem
s_amenable_property_T_clash | state | amenable property T clash
s_rudolph_measure_rigidity | theorem | Furstenberg ×2 ×3 ergodic measure rigidity (Rudolph)
s_circle_R_mod_Z | axiom | circle R mod Z
s_jointly_invariant_ergodic_measure_for_x2_x3 | axiom | jointly invariant ergodic measure for x2 x3
s_conditional_entropy_for_x2_given_x3 | state | conditional entropy for x2 given x3
s_positive_entropy_implies_full_support_lebesgue | state | positive entropy implies full support lebesgue
s_host_normal_numbers | theorem | Host's theorem on normal numbers in base 2 and 3
s_x2_invariant_ergodic_measure | axiom | x2 invariant ergodic measure
s_x3_orbit_fourier_correlation | state | x3 orbit fourier correlation
s_decay_of_dual_iterates | state | decay of dual iterates
s_mahler_compactness_criterion | theorem | Mahler compactness criterion
s_space_of_unimodular_lattices_SL_n_R_over_SL_n_Z | axiom | space of unimodular lattices SL n R over SL n Z
s_shortest_vector_function | state | shortest vector function
s_compact_subset_via_lambda1_bound | state | compact subset via lambda1 bound
s_dani_correspondence | theorem | Dani correspondence for Diophantine approximation
s_irrational_vector_alpha_in_R_n | axiom | irrational vector alpha in R n
s_lattice_associated_to_alpha | state | lattice associated to alpha
s_diagonal_flow_orbit | state | diagonal flow orbit
s_oppenheim_conjecture | theorem | Oppenheim conjecture (Margulis)
s_indefinite_irrational_quadratic_form_Q_in_n_at_least_3 | axiom | indefinite irrational quadratic form Q in n at least 3
s_SO_Q_orbit_in_homogeneous_space | state | SO Q orbit in homogeneous space
s_orbit_dense_in_full_space | state | orbit dense in full space
s_linnik_equidistribution_spheres | theorem | Linnik's equidistribution of integer points on spheres
s_integer_points_on_sphere_x2_plus_y2_plus_z2_equals_N | axiom | integer points on sphere x2 plus y2 plus z2 equals N
s_hecke_orbit_on_homogeneous_space | state | hecke orbit on homogeneous space
s_torus_orbit_equidistribution | state | torus orbit equidistribution
s_selberg_trace_formula | theorem | Selberg trace formula
s_compact_hyperbolic_surface_Gamma_minus_H2 | axiom | compact hyperbolic surface Gamma minus H2
s_convolution_kernel_operator | state | convolution kernel operator
s_spectral_side_of_trace | state | spectral side of trace
s_geometric_side_of_trace | state | geometric side of trace
s_lindenstrauss_quantum_unique_ergodicity | theorem | Lindenstrauss arithmetic quantum unique ergodicity
s_hecke_eigenfunctions_on_arithmetic_surface | axiom | hecke eigenfunctions on arithmetic surface
s_microlocal_lift_measure_mu_inf | state | microlocal lift measure mu inf
s_diagonal_plus_hecke_invariant_measure | state | diagonal plus hecke invariant measure
s_only_invariant_measure_is_haar | state | only invariant measure is haar
s_ekl_littlewood_partial | theorem | Einsiedler–Katok–Lindenstrauss theorem on Littlewood (partial)
s_diagonal_action_on_SL_3_R_over_SL_3_Z | axiom | diagonal action on SL 3 R over SL 3 Z
s_A_invariant_ergodic_measure | state | A invariant ergodic measure
s_positive_entropy_under_some_one_parameter_flow | state | positive entropy under some one parameter flow
s_measure_classification | state | measure classification
s_poincare_linearization_theorem | theorem | Lyapunov–Poincaré linearization theorem
s_analytic_vector_field_with_hyperbolic_singular_point | axiom | analytic vector field with hyperbolic singular point
s_formal_conjugacy_series_h_z | state | formal conjugacy series h z
s_non_resonance_condition_on_eigenvalues | state | non resonance condition on eigenvalues
s_siegel_linearization | theorem | Siegel linearization theorem for irrational rotation numbers
s_holomorphic_germ_with_diophantine_rotation_number | axiom | holomorphic germ with diophantine rotation number
s_newton_iteration_of_conjugacies | state | newton iteration of conjugacies
s_small_divisor_bound | state | small divisor bound
s_floquet_theorem | theorem | Floquet theorem for periodic linear ODE
s_linear_ode_with_t_periodic_coefficient_A_t | axiom | linear ode with t periodic coefficient A t
s_monodromy_matrix_M | state | monodromy matrix M
s_floquet_exponent_matrix_B | state | floquet exponent matrix B
s_lyapunov_stability_theorem | theorem | Lyapunov stability theorem
s_ode_with_equilibrium_x_star | axiom | ode with equilibrium x star
s_lyapunov_function_V | axiom | lyapunov function V
s_lyapunov_function_decreasing | state | lyapunov function decreasing
s_trapped_in_level_set | state | trapped in level set
s_sternberg_linearization | theorem | Hartman's theorem on smoothness of conjugacies
s_smooth_vector_field_with_nonresonant_hyperbolic_point | axiom | smooth vector field with nonresonant hyperbolic point
s_normal_form_at_jet_k | state | normal form at jet k
s_sternberg_normal_form | state | sternberg normal form
s_center_manifold_theorem | theorem | Center manifold theorem
s_smooth_vector_field_with_partially_hyperbolic_fixed_point | axiom | smooth vector field with partially hyperbolic fixed point
s_graph_transform_on_center | state | graph transform on center
s_invariant_center_graph | state | invariant center graph
s_peixoto_structural_stability | theorem | Peixoto's theorem on structural stability in dimension 2
s_smooth_vector_field_on_compact_2_manifold | axiom | smooth vector field on compact 2 manifold
s_morse_smale_condition | state | morse smale condition
s_structural_stability_for_morse_smale | state | structural stability for morse smale
s_aubry_mather_theorem | theorem | Aubry–Mather theory
s_monotone_twist_map_of_annulus | axiom | monotone twist map of annulus
s_action_minimizing_configurations | state | action minimizing configurations
s_minimizing_set_M_omega | state | minimizing set M omega
s_mather_connecting_orbits | theorem | Mather's variational principle for connecting orbits
s_tonelli_lagrangian_on_T_M | axiom | tonelli lagrangian on T M
s_alpha_beta_function_pair | state | alpha beta function pair
s_dual_pair_alpha_beta | state | dual pair alpha beta
s_denjoy_theorem | theorem | Denjoy's theorem on circle diffeomorphisms
s_C2_circle_diffeomorphism_with_irrational_rotation_number | axiom | C2 circle diffeomorphism with irrational rotation number
s_distortion_control_via_BV_log_derivative | state | distortion control via BV log derivative
s_no_wandering_intervals | state | no wandering intervals
s_herman_smooth_conjugacy | theorem | Herman's theorem on smooth conjugacy of circle diffeomorphisms
s_smooth_circle_diffeomorphism_with_diophantine_rotation_number | axiom | smooth circle diffeomorphism with diophantine rotation number
s_herman_renormalization_scheme | state | herman renormalization scheme
s_small_divisor_estimate_circle_case | state | small divisor estimate circle case
s_sinai_billiard_ergodicity | theorem | Sinai dispersing billiards ergodicity
s_sinai_billiard_table_with_dispersing_walls | axiom | sinai billiard table with dispersing walls
s_invariant_cone_fields_for_billiard_map | state | invariant cone fields for billiard map
s_pesin_theory_for_billiards | state | pesin theory for billiards
s_bunimovich_stadium_ergodicity | theorem | Bunimovich stadium ergodicity
s_bunimovich_stadium_table | axiom | bunimovich stadium table
s_defocusing_property_on_arcs | state | defocusing property on arcs
s_hyperbolicity_for_stadium | state | hyperbolicity for stadium
s_boltzmann_grad_limit_chaos | theorem | Bunimovich–Spohn–Boldrighini Boltzmann-Grad limit
s_finitely_many_hard_disks_in_box | axiom | finitely many hard disks in box
s_bbgky_hierarchy | state | bbgky hierarchy
s_boltzmann_grad_scaling_limit | state | boltzmann grad scaling limit
s_veech_dichotomy | theorem | Veech dichotomy for translation surfaces
s_veech_translation_surface | axiom | veech translation surface
s_teichmuller_geodesic | state | teichmuller geodesic
s_closed_sl2_orbit_in_strata | state | closed sl2 orbit in strata
s_masur_veech_ergodicity | theorem | Masur–Veech ergodicity of Teichmüller flow on moduli of abelian differentials
s_stratum_of_abelian_differentials | axiom | stratum of abelian differentials
s_masur_veech_measure | state | masur veech measure
s_finite_volume_normalization | state | finite volume normalization
s_conley_index_theorem | theorem | Morse–Conley index theorem
s_continuous_dynamical_system_on_locally_compact_space | axiom | continuous dynamical system on locally compact space
s_isolated_invariant_set_S | axiom | isolated invariant set S
s_conley_index_pair | state | conley index pair
s_conley_index_class_h_S | state | conley index class h S
s_morse_inequalities | theorem | Morse inequalities (Smale gradient version)
s_morse_smale_gradient_flow_on_compact_manifold | axiom | morse smale gradient flow on compact manifold
s_morse_chain_complex | state | morse chain complex
s_morse_homology_equals_h_star | state | morse homology equals h star
s_hopf_bifurcation_theorem | theorem | Andronov–Hopf bifurcation theorem
s_one_parameter_family_of_vector_fields_with_pair_of_eigenvalues_crossing_imaginary_axis | axiom | one parameter family of vector fields with pair of eigenvalues crossing imaginary axis
s_normal_form_for_hopf | state | normal form for hopf
s_first_lyapunov_coefficient | state | first lyapunov coefficient
s_saddle_node_bifurcation | theorem | Saddle-node (fold) bifurcation theorem
s_one_parameter_family_with_zero_eigenvalue_at_mu_zero | axiom | one parameter family with zero eigenvalue at mu zero
s_saddle_node_normal_form | state | saddle node normal form
s_equilibrium_curve_in_mu_x_plane | state | equilibrium curve in mu x plane
s_feigenbaum_universality | theorem | Period-doubling cascade (Feigenbaum universality)
s_unimodal_family_of_interval_maps | axiom | unimodal family of interval maps
s_renormalization_operator_R | state | renormalization operator R
s_feigenbaum_fixed_point_g_star | state | feigenbaum fixed point g star
s_feigenbaum_constant_delta | state | feigenbaum constant delta
s_avila_reducibility | theorem | Avila's reducibility theorem for quasi-periodic cocycles
s_quasi_periodic_SL_2_R_cocycle_with_diophantine_frequency | axiom | quasi periodic SL 2 R cocycle with diophantine frequency
s_cocycle_renormalization_orbit | state | cocycle renormalization orbit
s_small_divisor_estimate_for_cocycle | state | small divisor estimate for cocycle
s_ten_martini_theorem | theorem | Avila–Damanik almost Mathieu spectrum (Ten Martini Problem)
s_almost_mathieu_operator_with_irrational_frequency | axiom | almost mathieu operator with irrational frequency
s_lyapunov_exponent_along_E | state | lyapunov exponent along E
s_aubry_dual_operator | state | aubry dual operator
s_global_phase_picture_of_amo | state | global phase picture of amo
s_avila_krikorian_theorem | theorem | Avila–Krikorian renormalization theorem
s_schrodinger_cocycle_with_irrational_frequency | axiom | schrodinger cocycle with irrational frequency
s_renormalization_fixed_point_for_cocycle | state | renormalization fixed point for cocycle
s_positive_measure_of_regular_energies | state | positive measure of regular energies
s_avila_global_theory | theorem | Avila global theory of one-frequency Schrödinger operators
s_one_frequency_quasi_periodic_schrodinger_operator | axiom | one frequency quasi periodic schrodinger operator
s_complexified_lyapunov_function | state | complexified lyapunov function
s_piecewise_linear_lyapunov_in_strip | state | piecewise linear lyapunov in strip
s_birkhoff_minimal_recurrence | theorem | Birkhoff's theorem on transitive points (recurrence)
s_minimal_subsystem | axiom | minimal subsystem
s_minimal_invariant_subset_M | state | minimal invariant subset M
s_uniform_recurrence_in_M | state | uniform recurrence in M
s_weyl_equidistribution | theorem | Weyl equidistribution theorem
s_irrational_real_alpha | axiom | irrational real alpha
s_weyl_exponential_sum | state | weyl exponential sum
s_decay_of_exponential_sum | state | decay of exponential sum
s_furstenberg_kesten_theorem | theorem | Furstenberg–Kesten theorem (Lyapunov exponents from cocycles)
s_log_integrable_matrix_cocycle | axiom | log integrable matrix cocycle
s_subadditive_log_norm | state | subadditive log norm
s_pointwise_limit_lambda_top | state | pointwise limit lambda top
s_glasner_weiss_structure | theorem | Glasner–Weiss strong proximality / structure theorem
s_compact_metric_dynamical_system_with_group_action | axiom | compact metric dynamical system with group action
s_proximal_equivalence_relation | state | proximal equivalence relation
s_maximal_equicontinuous_factor | state | maximal equicontinuous factor
s_hindman_finite_sums | theorem | Hindman's finite-sums theorem
s_finite_coloring_of_naturals | axiom | finite coloring of naturals
s_beta_n_compactification | state | beta n compactification
s_minimal_idempotent_p | state | minimal idempotent p
s_furstenberg_distal_structure | theorem | Furstenberg's structure theorem for distal systems
s_minimal_distal_topological_dynamical_system | axiom | minimal distal topological dynamical system
s_equicontinuous_tower | state | equicontinuous tower
s_transfinite_isometric_tower | state | transfinite isometric tower
s_ornstein_isomorphism | theorem | Bernoulli isomorphism theorem (Ornstein)
s_two_bernoulli_shifts_of_equal_entropy | axiom | two bernoulli shifts of equal entropy
s_vwb_tower_decomposition | state | vwb tower decomposition
s_finitely_determined_invariant | state | finitely determined invariant
s_hopf_geodesic_flow_ergodicity | theorem | Hopf ergodicity of geodesic flow on negative-curvature surfaces
s_horocycle_foliations | state | horocycle foliations
s_invariant_function_is_constant | state | invariant function is constant
s_pesin_stable_manifold_theorem | theorem | Smale's stable manifold theorem for non-uniformly hyperbolic systems (Pesin)
s_smooth_measure_preserving_diffeomorphism_with_nonzero_lyapunov_exponents | axiom | smooth measure preserving diffeomorphism with nonzero lyapunov exponents
s_anisotropic_lyapunov_chart | state | anisotropic lyapunov chart
s_pesin_local_stable_graph | state | pesin local stable graph
s_hopf_ratio_ergodic | theorem | Hopf ratio ergodic theorem
s_two_L1_functions_f_g | axiom | two L1 functions f g
s_birkhoff_ratio_sequence | state | birkhoff ratio sequence
s_hopf_maximal_inequality | state | hopf maximal inequality
s_furstenberg_zimmer_structure | theorem | Furstenberg–Zimmer structure theorem
s_ergodic_measure_preserving_system | axiom | ergodic measure preserving system
s_hilbert_module_decomposition | state | hilbert module decomposition
s_tower_of_compact_extensions | state | tower of compact extensions
s_host_kra_nilfactor_structure | theorem | Host–Kra structure theorem (nilfactors)
s_uniformity_seminorms_U_k | state | uniformity seminorms U k
s_nilfactor_Z_k | state | nilfactor Z k
s_tao_ziegler_nilequidistribution | theorem | Tao–Ziegler equidistribution of nilsequences
s_polynomial_nilorbit_in_nilmanifold_G_over_Gamma | axiom | polynomial nilorbit in nilmanifold G over Gamma
s_green_tao_factorization | state | green tao factorization
s_subnilmanifold_descent_step | state | subnilmanifold descent step
s_pugh_shub_stable_ergodicity | theorem | Stable ergodicity of partially hyperbolic systems (Pugh–Shub)
s_partially_hyperbolic_diffeomorphism_with_accessible_center | axiom | partially hyperbolic diffeomorphism with accessible center
s_accessibility_class_decomposition | state | accessibility class decomposition
s_invariant_functions_constant_on_classes | state | invariant functions constant on classes
s_brin_katok_local_entropy | theorem | Brin–Katok local entropy formula
s_measure_preserving_homeomorphism_of_compact_metric_space | axiom | measure preserving homeomorphism of compact metric space
s_bowen_ball_measure_decay | state | bowen ball measure decay
s_pointwise_entropy_function | state | pointwise entropy function
s_bowen_limit_set_dimension | theorem | Bowen 1979 dimension formula (Hausdorff dim of limit sets of Kleinian groups)
s_convex_cocompact_kleinian_group_Gamma | axiom | convex cocompact kleinian group Gamma
s_poincare_series_critical_exponent_delta | state | poincare series critical exponent delta
s_transfer_operator_fixed_point | state | transfer operator fixed point
s_sarnak_mobius_horocycle | theorem | Sarnak's Möbius randomness in horocycle flows
s_unipotent_horocycle_flow_on_modular_surface | axiom | unipotent horocycle flow on modular surface
s_mobius_weighted_average | state | mobius weighted average
s_disjointness_with_mobius | state | disjointness with mobius
s_sarig_symbolic_extension | theorem | Buzzi–Pollicott–Sarig classification of measures for surface diffeomorphisms
s_C_inf_surface_diffeomorphism_with_positive_topological_entropy | axiom | C inf surface diffeomorphism with positive topological entropy
s_countable_markov_partition | state | countable markov partition
s_countable_sft_extension | state | countable sft extension
s_eskin_mirzakhani_mohammadi | theorem | Eskin–Mirzakhani–Mohammadi rigidity for SL(2,ℝ) action on moduli space
s_stratum_of_translation_surfaces_with_SL2R_action | axiom | stratum of translation surfaces with SL2R action
s_P_invariant_ergodic_measure | state | P invariant ergodic measure
s_full_SL2R_invariance | state | full SL2R invariance
s_affine_invariant_submanifold_closure | state | affine invariant submanifold closure
s_mcmullen_renormalization | theorem | McMullen's renormalization theorem for complex dynamics
s_infinitely_renormalizable_quadratic_with_bounded_combinatorics | axiom | infinitely renormalizable quadratic with bounded combinatorics
s_renormalization_operator_on_germs | state | renormalization operator on germs
s_renormalization_fixed_point_or_attractor | state | renormalization fixed point or attractor
s_sullivan_conformal_expanding_classification | theorem | Sullivan's classification of conformal expanding repellers
s_conformal_expanding_repeller_in_complex_plane | axiom | conformal expanding repeller in complex plane
s_sullivan_conformal_measure | state | sullivan conformal measure
s_unique_eigenmeasure_for_T_delta | state | unique eigenmeasure for T delta
s_pesin_sinai_srb_existence | theorem | Pesin–Sinai SRB physical-measure existence
s_dissipative_attractor_with_one_positive_lyapunov_exponent | axiom | dissipative attractor with one positive lyapunov exponent
s_ac_conditional_on_unstable_manifold | state | ac conditional on unstable manifold
s_cesaro_limit_along_unstable | state | cesaro limit along unstable
s_newhouse_phenomenon | theorem | Newhouse phenomenon (persistence of homoclinic tangencies)
s_C2_diffeomorphism_with_homoclinic_tangency | axiom | C2 diffeomorphism with homoclinic tangency
s_thick_horseshoe_Lambda | state | thick horseshoe Lambda
s_residual_intersection_of_cantor_sets | state | residual intersection of cantor sets
s_palis_takens_density | theorem | Palis–Takens density of hyperbolicity for one-dim endomorphisms
s_C2_family_of_unimodal_interval_maps | axiom | C2 family of unimodal interval maps
s_hyperbolic_or_stochastic_parameter_set | state | hyperbolic or stochastic parameter set
s_full_measure_parameters | state | full measure parameters
s_krieger_generator_theorem | theorem | Krieger's generator theorem
s_ergodic_measure_preserving_transformation_with_finite_entropy | axiom | ergodic measure preserving transformation with finite entropy
s_candidate_generating_partition | state | candidate generating partition
s_partition_size_upper_bound | state | partition size upper bound
s_rohlin_lemma | theorem | Rohlin lemma (Rokhlin tower)
s_aperiodic_measure_preserving_transformation | axiom | aperiodic measure preserving transformation
s_candidate_tower_base_B | state | candidate tower base B
s_exhaustion_of_X_by_tower | state | exhaustion of X by tower
s_ks_entropy_invariance | theorem | Kolmogorov–Sinai entropy invariant theorem
s_partition_entropy_function | state | partition entropy function
s_ks_entropy_h_T | state | ks entropy h T
s_mautner_phenomenon | theorem | Mautner phenomenon
s_unitary_representation_of_simple_lie_group_G | axiom | unitary representation of simple lie group G
s_contracting_decomposition_in_G | state | contracting decomposition in G
s_extra_invariance_propagation | state | extra invariance propagation
s_howe_moore_mixing | theorem | Howe–Moore mixing for semisimple groups
s_unitary_representation_of_semisimple_lie_group_no_invariant_vectors | axiom | unitary representation of semisimple lie group no invariant vectors
s_matrix_coefficient_function | state | matrix coefficient function
s_decay_estimate_on_K_a_K_decomposition | state | decay estimate on K a K decomposition
s_hahn_banach_separation | theorem | Hahn–Banach separation theorem
s_disjoint_convex_sets_A_B | axiom | disjoint convex sets A B
s_sublinear_gauge_with_zero_in_interior | state | sublinear gauge with zero in interior
s_extended_functional_separating_A_from_B | state | extended functional separating A from B
s_mazur_lemma | theorem | Mazur's lemma (weak closure equals norm closure for convex sets)
s_weakly_convergent_sequence | axiom | weakly convergent sequence
s_separating_hyperplane_required | state | separating hyperplane required
s_hahn_banach_separates_limit_from_convex_hull | state | hahn banach separates limit from convex hull
s_banach_alaoglu | theorem | Banach–Alaoglu theorem
s_normed_space_X | axiom | normed space X
s_dual_space_X_star | axiom | dual space X star
s_weak_star_topology_on_X_star | state | weak star topology on X star
s_embedding_into_product_of_closed_intervals | state | embedding into product of closed intervals
s_product_compact_target_space | state | product compact target space
s_goldstine_theorem | theorem | Goldstine's theorem
s_bidual_X_double_star | axiom | bidual X double star
s_canonical_embedding_into_bidual | state | canonical embedding into bidual
s_unit_ball_of_X_in_weak_star_X_double_star | state | unit ball of X in weak star X double star
s_krein_milman | theorem | Krein–Milman theorem
s_locally_convex_topological_vector_space | axiom | locally convex topological vector space
s_compact_convex_subset_K | axiom | compact convex subset K
s_family_of_extremal_faces | state | family of extremal faces
s_minimal_face_is_singleton_extreme_point | state | minimal face is singleton extreme point
s_choquet_theorem | theorem | Choquet's theorem
s_metrizable_compact_convex_set | axiom | metrizable compact convex set
s_evaluation_functional_at_x | state | evaluation functional at x
s_representing_probability_measure_on_K | state | representing probability measure on K
s_maximal_boundary_measure | state | maximal boundary measure
s_bauer_maximum_principle | theorem | Bauer maximum principle
s_upper_semicontinuous_convex_function | axiom | upper semicontinuous convex function
s_maximum_attained_on_K | state | maximum attained on K
s_argmax_face_of_K | state | argmax face of K
s_uniform_boundedness_principle | theorem | Uniform boundedness principle (Banach–Steinhaus)
s_family_of_bounded_operators_pointwise_bounded | axiom | family of bounded operators pointwise bounded
s_closed_sublevel_sets_E_n | state | closed sublevel sets E n
s_some_E_n_has_interior | state | some E n has interior
s_hellinger_toeplitz | theorem | Hellinger–Toeplitz theorem
s_hilbert_space | axiom | hilbert space
s_everywhere_defined_symmetric_operator | axiom | everywhere defined symmetric operator
s_closed_graph_of_T | state | closed graph of T
s_hilbert_projection_theorem | theorem | Hilbert projection theorem
s_closed_convex_subset_C | axiom | closed convex subset C
s_minimizing_sequence_in_C | state | minimizing sequence in C
s_cauchy_minimizing_sequence | state | cauchy minimizing sequence
s_riesz_repr_hilbert | theorem | Riesz representation theorem (Hilbert space)
s_continuous_linear_functional | axiom | continuous linear functional
s_closed_kernel_hyperplane | state | closed kernel hyperplane
s_one_dim_orthogonal_complement | state | one dim orthogonal complement
s_riesz_markov_kakutani | theorem | Riesz–Markov–Kakutani representation theorem
s_outer_measure_from_functional | state | outer measure from functional
s_riesz_repr_Lp | theorem | Riesz representation for L^p
s_continuous_linear_functional_on_Lp | axiom | continuous linear functional on Lp
s_density_g_in_Lq | state | density g in Lq
s_lax_milgram | theorem | Lax–Milgram theorem
s_continuous_coercive_bilinear_form | axiom | continuous coercive bilinear form
s_bounded_operator_A_associated_to_form | state | bounded operator A associated to form
s_A_bounded_below_with_closed_range | state | A bounded below with closed range
s_A_bijective | state | A bijective
s_stampacchia_theorem | theorem | Stampacchia theorem (variational inequality)
s_closed_convex_subset_C_and_coercive_form | axiom | closed convex subset C and coercive form
s_projection_iteration_map_T | state | projection iteration map T
s_unique_fixed_point_u_in_C | state | unique fixed point u in C
s_babuska_lax_milgram | theorem | Babuška–Lax–Milgram theorem
s_pair_of_hilbert_spaces_U_V | axiom | pair of hilbert spaces U V
s_inf_sup_stable_bilinear_form_b | axiom | inf sup stable bilinear form b
s_operator_B_with_inf_sup_bound | state | operator B with inf sup bound
s_B_injective_closed_range | state | B injective closed range
s_B_bijective_between_U_V | state | B bijective between U V
s_spectral_theorem_compact_sa | theorem | Spectral theorem for compact self-adjoint operators
s_compact_self_adjoint_operator_T | axiom | compact self adjoint operator T
s_maximizer_x_1_yields_eigenpair | state | maximizer x 1 yields eigenpair
s_first_eigenvector_extracted | state | first eigenvector extracted
s_iterative_orthogonal_eigenbasis | state | iterative orthogonal eigenbasis
s_spectral_theorem_bounded_sa | theorem | Spectral theorem for bounded self-adjoint operators
s_bounded_self_adjoint_operator_A | axiom | bounded self adjoint operator A
s_polynomial_functional_calculus | state | polynomial functional calculus
s_spectral_measures_for_vector_pairs | state | spectral measures for vector pairs
s_spectral_theorem_unbounded_sa | theorem | Spectral theorem for unbounded self-adjoint operators
s_unbounded_self_adjoint_operator_T | axiom | unbounded self adjoint operator T
s_unitary_cayley_transform | state | unitary cayley transform
s_spectral_resolution_of_U | state | spectral resolution of U
s_spectral_measure_E_on_R_from_F | state | spectral measure E on R from F
s_spectral_theorem_normal | theorem | Spectral theorem for normal operators
s_bounded_normal_operator_N | axiom | bounded normal operator N
s_real_imag_self_adjoint_pair | state | real imag self adjoint pair
s_joint_spectral_measure_on_C | state | joint spectral measure on C
s_stone_theorem | theorem | Stone's theorem on one-parameter unitary groups
s_strongly_continuous_one_param_unitary_group | axiom | strongly continuous one param unitary group
s_skew_adjoint_generator_iA | state | skew adjoint generator iA
s_self_adjoint_generator_A | state | self adjoint generator A
s_spectral_measure_for_A | state | spectral measure for A
s_hille_yosida | theorem | Hille–Yosida theorem
s_densely_defined_closed_operator_A | axiom | densely defined closed operator A
s_yosida_bounded_approximants | state | yosida bounded approximants
s_resolvent_bounded_uniformly | state | resolvent bounded uniformly
s_lumer_phillips | theorem | Lumer–Phillips theorem
s_densely_defined_dissipative_operator_A | axiom | densely defined dissipative operator A
s_dissipativity_in_semi_inner_product | state | dissipativity in semi inner product
s_trotter_product_formula | theorem | Trotter product formula
s_pair_of_semigroup_generators_A_B | axiom | pair of semigroup generators A B
s_sum_closure_is_generator | axiom | sum closure is generator
s_split_step_approximant | state | split step approximant
s_chernoff_limit_yields_semigroup | state | chernoff limit yields semigroup
s_gelfand_mazur | theorem | Gelfand–Mazur theorem
s_complex_banach_division_algebra | axiom | complex banach division algebra
s_spectrum_nonempty | axiom | spectrum nonempty
s_entire_bounded_resolvent_for_division_algebra | state | entire bounded resolvent for division algebra
s_gelfand_representation | theorem | Gelfand representation (commutative Banach algebra)
s_commutative_unital_banach_algebra | axiom | commutative unital banach algebra
s_gelfand_transform_continuous_homomorphism | state | gelfand transform continuous homomorphism
s_gelfand_naimark_commutative | theorem | Gelfand–Naimark theorem for commutative C*-algebras
s_commutative_unital_c_star_algebra | axiom | commutative unital c star algebra
s_gelfand_transform_to_continuous_functions | state | gelfand transform to continuous functions
s_star_homomorphism_preserving_involution | state | star homomorphism preserving involution
s_isometric_star_homomorphism | state | isometric star homomorphism
s_gns_construction | theorem | GNS construction (Gelfand–Naimark–Segal)
s_unital_c_star_algebra | axiom | unital c star algebra
s_pre_inner_product_on_A | state | pre inner product on A
s_gelfand_naimark_general | theorem | Gelfand–Naimark theorem (every C*-algebra embeds in B(H))
s_general_c_star_algebra | axiom | general c star algebra
s_family_of_pure_states | state | family of pure states
s_family_of_irreducible_representations | state | family of irreducible representations
s_universal_faithful_representation | state | universal faithful representation
s_stinespring_dilation | theorem | Stinespring dilation theorem
s_pre_inner_product_on_A_otimes_H | state | pre inner product on A otimes H
s_naimark_dilation | theorem | Naimark dilation theorem
s_hilbert_space_H | axiom | hilbert space H
s_positive_operator_valued_measure_POVM | axiom | positive operator valued measure POVM
s_stinespring_for_commutative_target | state | stinespring for commutative target
s_sz_nagy_dilation | theorem | Sz.-Nagy dilation theorem
s_contraction_T_on_H | axiom | contraction T on H
s_defect_operator_and_defect_space | state | defect operator and defect space
s_enlarged_hilbert_space_K | state | enlarged hilbert space K
s_unitary_dilation_of_T | state | unitary dilation of T
s_choi_theorem | theorem | Choi's theorem on completely positive maps
s_choi_matrix | state | choi matrix
s_positive_choi_matrix_characterization | state | positive choi matrix characterization
s_kraus_operators_V_k_from_eigenvectors | state | kraus operators V k from eigenvectors
s_kadison_transitivity | theorem | Kadison transitivity theorem
s_finite_set_of_target_vectors | axiom | finite set of target vectors
s_strong_density_in_unit_ball | state | strong density in unit ball
s_kaplansky_density | theorem | Kaplansky density theorem
s_double_commutant_A_double_prime | axiom | double commutant A double prime
s_sa_part_dense_via_functional_calculus | state | sa part dense via functional calculus
s_unit_ball_approximation_by_truncation | state | unit ball approximation by truncation
s_bicommutant_theorem | theorem | Von Neumann bicommutant theorem
s_commutant_M_prime | axiom | commutant M prime
s_cyclic_subspaces_with_projections_in_M_prime | state | cyclic subspaces with projections in M prime
s_strong_approximation_of_T_by_M | state | strong approximation of T by M
s_mvn_factor_classification | theorem | Murray–von Neumann classification of factors
s_factor_M_trivial_center | axiom | factor M trivial center
s_dimension_function_d | state | dimension function d
s_three_dimension_patterns_observed | state | three dimension patterns observed
s_type_axioms_I_II_III | state | type axioms I II III
s_tomita_takesaki | theorem | Tomita–Takesaki modular theory
s_von_neumann_algebra_M | axiom | von neumann algebra M
s_antilinear_operator_S_closable | state | antilinear operator S closable
s_connes_injective_classification | theorem | Connes' classification of injective factors
s_injective_factor_M | axiom | injective factor M
s_modular_flow_on_M | state | modular flow on M
s_connes_invariants_S_T | state | connes invariants S T
s_atkinson_theorem | theorem | Atkinson's theorem (Fredholm operators)
s_bounded_operator_T_on_banach_space | axiom | bounded operator T on banach space
s_calkin_algebra_quotient | state | calkin algebra quotient
s_fredholm_iff_invertible_modulo_compacts | state | fredholm iff invertible modulo compacts
s_toeplitz_index_formula | theorem | Index of Toeplitz operators on the circle
s_hardy_space_H2 | axiom | hardy space H2
s_calkin_index_map | theorem | Calkin algebra index map
s_six_term_K_theory_sequence | state | six term K theory sequence
s_fuglede_theorem | theorem | Fuglede's theorem
s_normal_operator_N | axiom | normal operator N
s_entire_bounded_operator_valued_function | state | entire bounded operator valued function
s_putnam_fuglede | theorem | Putnam–Fuglede theorem
s_pair_of_normal_operators_M_N | axiom | pair of normal operators M N
s_block_normal_operator | state | block normal operator
s_block_intertwiner_commutes_with_adjoint | state | block intertwiner commutes with adjoint
s_banach_stone | theorem | Banach–Stone theorem
s_compact_hausdorff_pair_K_L | axiom | compact hausdorff pair K L
s_isometry_sends_point_masses_to_point_masses | state | isometry sends point masses to point masses
s_bochner_theorem | theorem | Bochner's theorem
s_continuous_positive_definite_function_on_R | axiom | continuous positive definite function on R
s_fourier_inversion_program | axiom | fourier inversion program
s_state_on_commutative_group_C_star | state | state on commutative group C star
s_peter_weyl | theorem | Peter–Weyl theorem
s_compact_topological_group_G | axiom | compact topological group G
s_haar_measure_on_G | axiom | haar measure on G
s_compact_self_adjoint_convolution_operators | state | compact self adjoint convolution operators
s_stone_von_neumann | theorem | Stone–von Neumann theorem
s_irreducibility_assumption | axiom | irreducibility assumption
s_weyl_form_of_CCR | state | weyl form of CCR
s_heisenberg_group_representation | state | heisenberg group representation
s_fixed_central_character_irreducible | state | fixed central character irreducible
s_schwartz_kernel_theorem | theorem | Schwartz kernel theorem
s_continuous_bilinear_form_on_test_functions | axiom | continuous bilinear form on test functions
s_schwartz_space_distributions_pair | axiom | schwartz space distributions pair
s_jointly_continuous_bilinear_form | state | jointly continuous bilinear form
s_nuclear_space_grothendieck | theorem | Nuclear space characterization (Grothendieck)
s_locally_convex_space_X | axiom | locally convex space X
s_canonical_maps_between_local_banach_spaces | state | canonical maps between local banach spaces
s_nuclear_canonical_maps | state | nuclear canonical maps
s_uniqueness_of_tensor_product_topology | state | uniqueness of tensor product topology
s_sobolev_embedding | theorem | Sobolev embedding theorem
s_riesz_potential_estimate | axiom | riesz potential estimate
s_bessel_potential_representation | state | bessel potential representation
s_dyadic_block_decomposition_of_u | state | dyadic block decomposition of u
s_hardy_littlewood_sobolev_inequality | state | hardy littlewood sobolev inequality
s_rellich_kondrachov | theorem | Rellich–Kondrachov compactness theorem
s_frechet_kolmogorov | theorem | Fréchet–Kolmogorov compactness theorem
s_subset_F_of_Lp | axiom | subset F of Lp
s_translation_continuity_uniform | axiom | translation continuity uniform
s_mollified_family_uniformly_close_to_F | state | mollified family uniformly close to F
s_precompact_after_mollification | state | precompact after mollification
s_trace_theorem | theorem | Trace theorem for Sobolev spaces
s_smooth_functions_dense_in_W1p | state | smooth functions dense in W1p
s_boundary_trace_estimate | state | boundary trace estimate
s_calderon_extension | theorem | Calderón extension theorem
s_local_half_space_reduction | state | local half space reduction
s_reflection_extension_in_each_chart | state | reflection extension in each chart
s_gagliardo_nirenberg | theorem | Gagliardo–Nirenberg interpolation inequality
s_smooth_compactly_supported_function | axiom | smooth compactly supported function
s_lebesgue_index_triple_p_q_r | axiom | lebesgue index triple p q r
s_scaling_dimensional_balance | state | scaling dimensional balance
s_interpolation_inequality_on_dyadic_blocks | state | interpolation inequality on dyadic blocks
s_full_interpolation_estimate | state | full interpolation estimate
s_riesz_thorin | theorem | Riesz–Thorin interpolation theorem
s_linear_operator_bounded_at_two_endpoints | axiom | linear operator bounded at two endpoints
s_lebesgue_endpoint_pairs | axiom | lebesgue endpoint pairs
s_analytic_family_in_complex_strip | state | analytic family in complex strip
s_marcinkiewicz_interpolation | theorem | Marcinkiewicz interpolation theorem
s_sublinear_operator_weak_type_at_endpoints | axiom | sublinear operator weak type at endpoints
s_lorentz_spaces_Lp_q | axiom | lorentz spaces Lp q
s_layer_cake_decomposition | state | layer cake decomposition
s_distribution_function_bounds | state | distribution function bounds
s_strong_type_estimate | state | strong type estimate
s_hormander_mikhlin | theorem | Hörmander–Mikhlin multiplier theorem
s_bounded_fourier_multiplier_m | axiom | bounded fourier multiplier m
s_mikhlin_derivative_estimates | axiom | mikhlin derivative estimates
s_convolution_operator_with_kernel_K_m | state | convolution operator with kernel K m
s_dyadic_pieces_with_bounded_derivatives | state | dyadic pieces with bounded derivatives
s_CZ_singular_integral_bounds | state | CZ singular integral bounds
s_lebesgue_differentiation | theorem | Lebesgue differentiation theorem
s_locally_integrable_function_on_Rn | axiom | locally integrable function on Rn
s_eberlein_smulian | theorem | Eberlein–Šmulian theorem
s_weakly_compact_subset_K | axiom | weakly compact subset K
s_K_in_double_dual_with_weak_star_compact_image | state | K in double dual with weak star compact image
s_metrizable_weak_star_compact_image | state | metrizable weak star compact image
s_james_theorem | theorem | James's theorem
s_banach_space_X | axiom | banach space X
s_every_continuous_linear_functional_attains_norm_on_B_X | axiom | every continuous linear functional attains norm on B X
s_non_attainment_in_some_functional_construction | state | non attainment in some functional construction
s_non_norm_attaining_functional | state | non norm attaining functional
s_milman_pettis | theorem | Milman–Pettis theorem
s_uniformly_convex_banach_space | axiom | uniformly convex banach space
s_quantitative_strict_convexity | state | quantitative strict convexity
s_canonical_embedding_surjective_on_unit_balls | state | canonical embedding surjective on unit balls
s_bishop_phelps | theorem | Bishop–Phelps theorem
s_support_cone_partial_order | state | support cone partial order
s_maximal_element_in_cone | state | maximal element in cone
s_krein_smulian | theorem | Krein–Smulian theorem
s_dual_banach_space_X_star | axiom | dual banach space X star
s_polar_set_in_X | state | polar set in X
s_weak_star_closure_recoverable_levelwise | state | weak star closure recoverable levelwise
s_krein_rutman | theorem | Krein–Rutman theorem
s_compact_positive_operator_on_cone | axiom | compact positive operator on cone
s_solid_pointed_cone_K | axiom | solid pointed cone K
s_rescaled_iteration_on_cone | state | rescaled iteration on cone
s_fixed_point_in_cone | state | fixed point in cone
s_schauder_fpt | theorem | Schauder fixed-point theorem
s_compact_convex_subset_of_banach | axiom | compact convex subset of banach
s_finite_dim_approximation_of_K | state | finite dim approximation of K
s_markov_kakutani | theorem | Markov–Kakutani fixed-point theorem
s_compact_convex_subset_in_locally_convex_space | axiom | compact convex subset in locally convex space
s_commuting_family_of_continuous_affine_maps | axiom | commuting family of continuous affine maps
s_cesaro_averages_of_maps | state | cesaro averages of maps
s_nonempty_intersection_per_map | state | nonempty intersection per map
s_ryll_nardzewski | theorem | Ryll-Nardzewski fixed-point theorem
s_weakly_compact_convex_set_K | axiom | weakly compact convex set K
s_noncontracting_semigroup_of_isometries | axiom | noncontracting semigroup of isometries
s_minimal_invariant_set_K_0 | state | minimal invariant set K 0
s_K_0_singleton_via_contradiction | state | K 0 singleton via contradiction
s_ekeland_variational_principle | theorem | Ekeland's variational principle
s_lower_semicontinuous_bounded_below_function | axiom | lower semicontinuous bounded below function
s_partial_order_on_X | state | partial order on X
s_browder_minty | theorem | Browder–Minty theorem (monotone operators)
s_reflexive_banach_space | axiom | reflexive banach space
s_monotone_coercive_hemicontinuous_operator | axiom | monotone coercive hemicontinuous operator
s_finite_dim_galerkin_approximants | state | finite dim galerkin approximants
s_finite_dim_solutions_x_n | state | finite dim solutions x n
s_mercer_theorem | theorem | Mercer's theorem
s_continuous_symmetric_positive_kernel | axiom | continuous symmetric positive kernel
s_compact_domain_X | axiom | compact domain X
s_integral_operator_compact_self_adjoint | state | integral operator compact self adjoint
s_spectral_decomposition_of_T_K | state | spectral decomposition of T K
s_uniformly_convergent_kernel_series | state | uniformly convergent kernel series
s_hilbert_schmidt_theorem | theorem | Hilbert–Schmidt theorem
s_hilbert_schmidt_integral_operator | axiom | hilbert schmidt integral operator
s_min_max_theorem | theorem | Min-max theorem (Courant–Fischer–Weyl)
s_eigenvalues_ordered_decreasing | axiom | eigenvalues ordered decreasing
s_eigenpair_sequence | state | eigenpair sequence
s_rayleigh_quotient_with_extremal_property | state | rayleigh quotient with extremal property
s_weyl_essential_spectrum | theorem | Weyl's theorem on essential spectrum
s_self_adjoint_operator_A | axiom | self adjoint operator A
s_compact_perturbation_K | axiom | compact perturbation K
s_weyl_sequence_preserved_under_compact_perturbation | state | weyl sequence preserved under compact perturbation
s_lomonosov_invariant_subspace | theorem | Lomonosov's invariant subspace theorem
s_bounded_operator_commuting_with_nonzero_compact | axiom | bounded operator commuting with nonzero compact
s_banach_space_infinite_dim | axiom | banach space infinite dim
s_orbit_density_assumption | state | orbit density assumption
s_self_map_with_compact_image | state | self map with compact image
s_compact_eigenvector_yields_invariant_subspace | state | compact eigenvector yields invariant subspace
s_closed_range_theorem | theorem | Closed range theorem
s_densely_defined_closed_operator_T | axiom | densely defined closed operator T
s_adjoint_T_star | axiom | adjoint T star
s_orthogonality_relation_range_kernel | state | orthogonality relation range kernel
s_equivalent_closedness_conditions | state | equivalent closedness conditions
s_fredholm_alternative | theorem | Fredholm alternative
s_compact_operator_K_on_banach | axiom | compact operator K on banach
s_operator_I_minus_K | axiom | operator I minus K
s_finite_dim_kernel | state | finite dim kernel
s_index_zero_fredholm_operator | state | index zero fredholm operator
s_analytic_fredholm | theorem | Analytic Fredholm theorem
s_connected_domain_D | axiom | connected domain D
s_meromorphic_resolvent | state | meromorphic resolvent
s_local_invertibility_or_finite_kernel | state | local invertibility or finite kernel
s_cotlar_stein | theorem | Cotlar–Stein lemma
s_almost_orthogonal_family_of_operators_T_j | axiom | almost orthogonal family of operators T j
s_l1_summable_correlation_estimates | axiom | l1 summable correlation estimates
s_expanded_2n_th_power | state | expanded 2n th power
s_chain_correlation_bound | state | chain correlation bound
s_t1_theorem | theorem | T(1) theorem (David–Journé)
s_singular_integral_operator_T_with_CZ_kernel | axiom | singular integral operator T with CZ kernel
s_paraproduct_corrected_operator_T_0 | state | paraproduct corrected operator T 0
s_almost_orthogonal_pieces_T_jk | state | almost orthogonal pieces T jk
s_hormander_symbol_class | theorem | Hörmander's pseudodifferential symbol class theorem
s_oscillatory_integral_definition | axiom | oscillatory integral definition
s_dyadic_oscillatory_pieces | state | dyadic oscillatory pieces
s_symbol_calculus_modulo_lower_order | state | symbol calculus modulo lower order
s_hormander_l2_dbar | theorem | Hörmander L² ∂̄ estimates
s_pseudoconvex_domain_in_Cn | axiom | pseudoconvex domain in Cn
s_bochner_kodaira_morrey_identity | state | bochner kodaira morrey identity
s_garding_inequality | theorem | Garding's inequality
s_pointwise_positivity_of_symbol | state | pointwise positivity of symbol
s_local_constant_coefficient_pieces | state | local constant coefficient pieces
s_aubin_lions | theorem | Aubin–Lions lemma
s_ehrling_interpolation_inequality | state | ehrling interpolation inequality
s_equicontinuity_in_time_in_B | state | equicontinuity in time in B
s_meyers_serrin | theorem | Meyers–Serrin theorem (H = W)
s_locally_finite_cover_with_pou | state | locally finite cover with pou
s_local_mollification_summed | state | local mollification summed
s_mazur_ulam | theorem | Mazur–Ulam theorem
s_real_normed_spaces | axiom | real normed spaces
s_midpoint_set_characterization | state | midpoint set characterization
s_midpoint_preserved_under_reflection | state | midpoint preserved under reflection
s_kirszbraun_theorem | theorem | Kirszbraun theorem (Lipschitz extension)
s_subset_S_of_hilbert_space | axiom | subset S of hilbert space
s_intersection_of_balls_at_each_x | state | intersection of balls at each x
s_nonempty_intersection_for_each_x | state | nonempty intersection for each x
s_michael_selection | theorem | Michael selection theorem
s_paracompact_space | axiom | paracompact space
s_lower_semicontinuous_correspondence_with_convex_values_in_banach | axiom | lower semicontinuous correspondence with convex values in banach
s_cauchy_sequence_of_approximate_selections | state | cauchy sequence of approximate selections
s_dvoretzky_theorem | theorem | Dvoretzky's theorem
s_infinite_dim_banach_space | axiom | infinite dim banach space
s_concentration_of_measure_on_sphere | axiom | concentration of measure on sphere
s_random_high_dim_subspace_selection | state | random high dim subspace selection
s_concentration_of_norm_on_sphere | state | concentration of norm on sphere
s_almost_euclidean_section_existence | state | almost euclidean section existence
s_grothendieck_inequality | theorem | Grothendieck inequality
s_unit_vectors_x_i_y_j_in_hilbert | axiom | unit vectors x i y j in hilbert
s_gaussian_rounding_estimator | state | gaussian rounding estimator
s_arcsin_kernel_relation | state | arcsin kernel relation
s_connes_embedding_statement | theorem | Connes embedding problem / theorem statement
s_separable_II_1_factor_M | axiom | separable II 1 factor M
s_ultrapower_of_hyperfinite_R_omega | axiom | ultrapower of hyperfinite R omega
s_tracial_ultrapower_target | state | tracial ultrapower target
s_finite_dim_matricial_microstates | state | finite dim matricial microstates
s_bessel_inequality | theorem | Bessel's inequality
s_orthonormal_sequence_e_n | axiom | orthonormal sequence e n
s_partial_orthogonal_projection | state | partial orthogonal projection
s_parseval_identity | theorem | Parseval's identity
s_complete_orthonormal_basis_e_n | axiom | complete orthonormal basis e n
s_bessel_becomes_equality_for_complete_basis | state | bessel becomes equality for complete basis
s_holder_inequality | theorem | Hölder's inequality
s_measure_space | axiom | measure space
s_young_pointwise_inequality | state | young pointwise inequality
s_propagation_of_singularities | theorem | Propagation of singularities theorem
s_pseudodifferential_operator_P_real_principal_symbol | axiom | pseudodifferential operator P real principal symbol
s_hamiltonian_flow_bicharacteristics | state | hamiltonian flow bicharacteristics
s_microlocal_propagation_estimate | state | microlocal propagation estimate
s_commutator_inequality_yields_regularity_transport | state | commutator inequality yields regularity transport
s_open_mapping_frechet | theorem | Open mapping theorem (Fréchet space version)
s_pair_of_frechet_spaces | axiom | pair of frechet spaces
s_continuous_surjective_linear_map_T | axiom | continuous surjective linear map T
s_translation_invariant_metric_setup | state | translation invariant metric setup
s_wold_decomposition | theorem | Wold decomposition
s_hilbert_space_with_isometry_V | axiom | hilbert space with isometry V
s_unitary_part_H_u | state | unitary part H u
s_shift_part_H_s_as_orthogonal_sum | state | shift part H s as orthogonal sum
s_commutant_lifting | theorem | Commutant lifting theorem
s_contraction_T_with_isometric_dilation_V | axiom | contraction T with isometric dilation V
s_intertwining_operator_X_commuting_with_T | axiom | intertwining operator X commuting with T
s_minimal_isometric_dilation_space | state | minimal isometric dilation space
s_norm_preserving_extension_problem | state | norm preserving extension problem
s_parrott_completion_yields_Y | state | parrott completion yields Y
s_russo_dye | theorem | Russo–Dye theorem
s_convex_hull_of_unitaries | axiom | convex hull of unitaries
s_two_unitary_decomposition_of_strict_contraction | state | two unitary decomposition of strict contraction
s_explicit_unitary_pair | state | explicit unitary pair
s_crossed_product_construction | theorem | Crossed product construction (von Neumann)
s_l2_G_with_M_valued_functions | axiom | l2 G with M valued functions
s_covariant_representation | state | covariant representation
s_continuous_functional_calculus | theorem | Continuous functional calculus
s_normal_element_a_in_c_star_algebra | axiom | normal element a in c star algebra
s_polynomial_star_calculus | state | polynomial star calculus
s_isometry_to_polynomials_on_spectrum | state | isometry to polynomials on spectrum
s_borel_functional_calculus | theorem | Borel functional calculus
s_normal_operator_on_hilbert | axiom | normal operator on hilbert
s_spectral_measure_per_vector_pair | state | spectral measure per vector pair
s_borel_functional_calculus_per_pair | state | borel functional calculus per pair
s_direct_integral_decomposition | theorem | Direct integral decomposition (von Neumann)
s_separable_hilbert_space | axiom | separable hilbert space
s_abelian_von_neumann_subalgebra_A | axiom | abelian von neumann subalgebra A
s_a_acts_as_multiplication_on_measure_space_X | state | a acts as multiplication on measure space X
s_measurable_hilbert_field | state | measurable hilbert field
s_direct_integral_isomorphism | state | direct integral isomorphism
s_banach_mazur_embedding | theorem | Banach–Mazur theorem (separable Banach spaces embed in C[0,1])
s_separable_banach_space_X | axiom | separable banach space X
s_unit_ball_of_X_star | axiom | unit ball of X star
s_compact_metrizable_dual_unit_ball_K | state | compact metrizable dual unit ball K
s_plancherel_nonabelian | theorem | Plancherel for non-abelian compact groups
s_compact_lie_group_G | axiom | compact lie group G
s_peter_weyl_decomposition | state | peter weyl decomposition
s_matrix_coefficient_orthogonality | state | matrix coefficient orthogonality
s_operator_valued_fourier_transform | state | operator valued fourier transform
s_pontryagin_duality | theorem | Pontryagin duality
s_locally_compact_abelian_group_G | axiom | locally compact abelian group G
s_canonical_double_dual_map | state | canonical double dual map
s_choi_effros_lifting | theorem | Aubert–Choi continuity / Choi–Effros lifting theorem
s_nuclear_separable_c_star_algebra | axiom | nuclear separable c star algebra
s_cp_approximation_diagrams | state | cp approximation diagrams
s_diagram_lifted_into_B | state | diagram lifted into B
s_coherent_lift_of_CP_diagrams | state | coherent lift of CP diagrams
s_glimm_dichotomy | theorem | Glimm's dichotomy / type I vs non-type-I
s_separable_c_star_algebra | axiom | separable c star algebra
s_irreducible_representation_class | axiom | irreducible representation class
s_pattern_T_0_dual_indicates_type_I | state | pattern T 0 dual indicates type I
s_non_type_I_dichotomy_witness | state | non type I dichotomy witness
s_connes_noncommutative_index | theorem | Connes' noncommutative index formula
s_chern_connes_pairing | axiom | chern connes pairing
s_k_homology_class_of_D | state | k homology class of D
s_pairing_with_K_theory_of_A | state | pairing with K theory of A
s_chern_connes_character_in_periodic_cyclic | state | chern connes character in periodic cyclic
s_pettis_measurability | theorem | Pettis theorem (weakly measurable = strongly measurable for separable values)
s_simple_function_approximant_pointwise | state | simple function approximant pointwise
s_borel_measurability_in_weak_topology | state | borel measurability in weak topology
s_eidelheit_separation | theorem | Eidelheit separation theorem
s_convex_set_with_nonempty_interior_in_topological_vector_space | axiom | convex set with nonempty interior in topological vector space
s_disjoint_convex_set | axiom | disjoint convex set
s_minkowski_gauge_continuous | state | minkowski gauge continuous
s_continuous_separating_functional | state | continuous separating functional
s_nash_moser | theorem | Nash–Moser implicit function theorem
s_tame_frechet_space_pair | axiom | tame frechet space pair
s_smoothing_operators_S_t | axiom | smoothing operators S t
s_smoothing_family_S_t | state | smoothing family S t
s_modified_newton_iteration | state | modified newton iteration
s_convergent_quadratic_iteration | state | convergent quadratic iteration
s_weak_formulation_elliptic | theorem | Weak formulation of elliptic BVP / Lax–Milgram applied
s_sobolev_space_H1_0 | axiom | sobolev space H1 0
s_coercive_continuous_bilinear_form | state | coercive continuous bilinear form
s_levi_civita_connection | theorem | Fundamental theorem of Riemannian geometry (Levi-Civita)
s_smooth_manifold | axiom | smooth manifold
s_connection_axioms_metric_torsion_free | state | connection axioms metric torsion free
s_koszul_formula_for_connection | state | koszul formula for connection
s_hopf_rinow_theorem | theorem | Hopf–Rinow theorem
s_completeness_notions_for_riemannian_manifold | state | completeness notions for riemannian manifold
s_closed_bounded_sets_compact | state | closed bounded sets compact
s_minimizing_geodesic_between_any_two_points | state | minimizing geodesic between any two points
s_cartan_hadamard_theorem | theorem | Cartan–Hadamard theorem
s_complete_nonpositively_curved_manifold | state | complete nonpositively curved manifold
s_no_conjugate_points | state | no conjugate points
s_exp_map_is_covering | state | exp map is covering
s_bonnet_myers_theorem | theorem | Bonnet–Myers theorem
s_complete_manifold_with_ricci_lower_bound | state | complete manifold with ricci lower bound
s_second_variation_index_form | state | second variation index form
s_diameter_bound_pi_over_sqrt_k | state | diameter bound pi over sqrt k
s_synge_theorem | theorem | Synge's theorem
s_compact_even_dim_orientable_positively_curved | state | compact even dim orientable positively curved
s_closed_geodesic_minimizer | state | closed geodesic minimizer
s_parallel_vector_field_along_geodesic | state | parallel vector field along geodesic
s_cheng_diameter_rigidity | theorem | Cheng's diameter rigidity theorem
s_extremal_diameter_case_of_bonnet_myers | state | extremal diameter case of bonnet myers
s_volume_equality_with_round_sphere | state | volume equality with round sphere
s_cheeger_gromoll_splitting | theorem | Cheeger–Gromoll splitting theorem
s_complete_nonneg_ricci_with_line | state | complete nonneg ricci with line
s_pair_of_busemann_functions | state | pair of busemann functions
s_harmonic_busemann_function_pair | state | harmonic busemann function pair
s_soul_theorem | theorem | Soul theorem
s_open_nonneg_curved_manifold | state | open nonneg curved manifold
s_nested_totally_convex_sets | state | nested totally convex sets
s_soul_submanifold | state | soul submanifold
s_rauch_comparison_theorem | theorem | Rauch comparison theorem
s_jacobi_equation_along_geodesic | state | jacobi equation along geodesic
s_sturm_comparison_for_jacobi_norm | state | sturm comparison for jacobi norm
s_toponogov_theorem | theorem | Toponogov comparison theorem
s_lower_curvature_bounded_manifold | state | lower curvature bounded manifold
s_geodesic_triangle_pair_M_and_Mk | state | geodesic triangle pair M and Mk
s_angle_inequality_M_geq_Mk | state | angle inequality M geq Mk
s_bishop_gromov_comparison | theorem | Bishop–Gromov volume comparison
s_ricci_lower_bounded_manifold | state | ricci lower bounded manifold
s_jacobian_of_exp_map_in_polar | state | jacobian of exp map in polar
s_pointwise_jacobian_bound | state | pointwise jacobian bound
s_gromov_compactness_riemannian | theorem | Gromov compactness for Riemannian manifolds
s_class_of_manifolds_with_uniform_geometry | state | class of manifolds with uniform geometry
s_gromov_hausdorff_distance_on_class | state | gromov hausdorff distance on class
s_total_boundedness_of_class | state | total boundedness of class
s_cheeger_finiteness | theorem | Cheeger finiteness theorem
s_uniformly_bounded_geometry_class | state | uniformly bounded geometry class
s_uniform_injectivity_radius | state | uniform injectivity radius
s_finite_diffeomorphism_types | state | finite diffeomorphism types
s_sphere_theorem_quarter_pinched | theorem | Sphere theorem (1/4-pinched)
s_strictly_quarter_pinched_manifold | state | strictly quarter pinched manifold
s_lower_injectivity_radius_pi | state | lower injectivity radius pi
s_two_exponential_charts_cover_M | state | two exponential charts cover M
s_killing_hopf_theorem | theorem | Killing–Hopf theorem
s_space_form_class | state | space form class
s_universal_cover_is_model_space | state | universal cover is model space
s_myers_steenrod_theorem | theorem | Myers–Steenrod theorem
s_isometry_group_topological | state | isometry group topological
s_isometry_group_locally_compact | state | isometry group locally compact
s_nash_embedding_theorem | theorem | Nash embedding theorem
s_free_embedding_existence | state | free embedding existence
s_perturbative_isometric_embedding | state | perturbative isometric embedding
s_preissmann_theorem | theorem | Preissmann's theorem
s_compact_negatively_curved_manifold | state | compact negatively curved manifold
s_pi1_action_on_Cartan_Hadamard_space | state | pi1 action on Cartan Hadamard space
s_flat_strip_contradiction | state | flat strip contradiction
s_mostow_rigidity_theorem | theorem | Mostow rigidity theorem
s_finite_volume_hyperbolic_manifold | state | finite volume hyperbolic manifold
s_quasiconformal_boundary_map | state | quasiconformal boundary map
s_boundary_map_is_conformal | state | boundary map is conformal
s_hodge_decomposition | theorem | Hodge decomposition theorem
s_hodge_laplacian | state | hodge laplacian
s_eigenform_orthogonal_decomp | state | eigenform orthogonal decomp
s_orthogonal_decomp_im_d_im_dstar_harmonic | state | orthogonal decomp im d im dstar harmonic
s_frobenius_integrability | theorem | Frobenius theorem (integrable distributions)
s_smooth_distribution | state | smooth distribution
s_involutive_distribution | state | involutive distribution
s_local_foliation_chart | state | local foliation chart
s_generalized_stokes_theorem | theorem | Generalized Stokes theorem
s_FTC_and_Greens_theorem_special_cases | state | FTC and Greens theorem special cases
s_local_to_global_integration_identity | state | local to global integration identity
s_cartan_formula | theorem | Cartan formula for Lie derivative
s_three_derivations_on_omega_star | state | three derivations on omega star
s_cartan_identity_on_low_degree | state | cartan identity on low degree
s_fundamental_theorem_of_curves | theorem | Fundamental theorem of curves
s_frenet_frame_ODE | state | frenet frame ODE
s_curve_recovered_from_kappa_tau | state | curve recovered from kappa tau
s_four_vertex_theorem | theorem | Four vertex theorem
s_smooth_convex_closed_curve | state | smooth convex closed curve
s_kappa_prime_sign_change_contradiction | state | kappa prime sign change contradiction
s_fenchel_theorem | theorem | Fenchel's theorem
s_tantrix_spherical_curve | state | tantrix spherical curve
s_total_curvature_equals_tantrix_length | state | total curvature equals tantrix length
s_hilbert_no_isometric_immersion_hyperbolic_plane | theorem | Hilbert's theorem on hyperbolic plane
s_hyperbolic_plane_as_abstract_surface | state | hyperbolic plane as abstract surface
s_chebyshev_net_with_angle_omega | state | chebyshev net with angle omega
s_sine_gordon_equation | state | sine gordon equation
s_bertrand_diguet_puiseux | theorem | Bertrand–Diguet–Puiseux theorem
s_geodesic_disk_circumference_C_r | state | geodesic disk circumference C r
s_taylor_expansion_of_circumference | state | taylor expansion of circumference
s_tennis_ball_theorem | theorem | Tennis ball theorem
s_area_bisecting_curve_on_sphere | state | area bisecting curve on sphere
s_zero_signed_geodesic_curvature_integral | state | zero signed geodesic curvature integral
s_three_geodesics_theorem | theorem | Theorem of the three geodesics
s_loop_space_energy_functional | state | loop space energy functional
s_min_max_critical_values | state | min max critical values
s_three_distinct_simple_closed_geodesics | state | three distinct simple closed geodesics
s_lyusternik_fet_theorem | theorem | Lyusternik–Fet theorem
s_free_loop_space_energy | state | free loop space energy
s_nontrivial_loop_homotopy_class | state | nontrivial loop homotopy class
s_gage_hamilton_grayson | theorem | Gage–Hamilton–Grayson theorem
s_curve_shortening_flow_eqn | state | curve shortening flow eqn
s_convexification_of_embedded_curve | state | convexification of embedded curve
s_round_point_limit | state | round point limit
s_willmore_conjecture | theorem | Willmore conjecture (Marques–Neves)
s_willmore_energy_functional | state | willmore energy functional
s_conformally_invariant_energy | state | conformally invariant energy
s_min_max_widths_of_torus_class | state | min max widths of torus class
s_bernstein_problem | theorem | Bernstein's problem
s_minimal_surface_equation | state | minimal surface equation
s_entire_minimal_graph_in_dim_n | state | entire minimal graph in dim n
s_tangent_cone_minimal_cone | state | tangent cone minimal cone
s_tennenbaum_theorem | theorem | Tennenbaum's theorem
s_countable_nonstandard_model_of_PA | state | countable nonstandard model of PA
s_re_inseparable_pair_coded_in_M | state | re inseparable pair coded in M
s_recursive_separation_contradiction | state | recursive separation contradiction
s_goodstein_theorem | theorem | Goodstein's theorem
s_ordinal_arithmetic_below_epsilon_0 | axiom | ordinal arithmetic below epsilon 0
s_natural_number_n | state | natural number n
s_goodstein_sequence_definition | state | goodstein sequence definition
s_ordinal_majorant_sequence_below_epsilon_0 | state | ordinal majorant sequence below epsilon 0
s_ordinal_sequence_must_hit_zero | state | ordinal sequence must hit zero
s_kirby_paris_independence | theorem | Goodstein independence from PA (Kirby–Paris)
s_total_function_growth_of_Goodstein | state | total function growth of Goodstein
s_hardy_hierarchy_majorant_at_epsilon_0 | state | hardy hierarchy majorant at epsilon 0
s_hydra_theorem | theorem | Kirby–Paris hydra theorem
s_finite_rooted_tree | axiom | finite rooted tree
s_hydra_game_state | state | hydra game state
s_hydra_ordinal_invariant | state | hydra ordinal invariant
s_hydra_terminates | state | hydra terminates
s_kanamori_mcaloon_theorem | theorem | Kanamori–McAloon theorem
s_regressive_finite_ramsey_statement | state | regressive finite ramsey statement
s_indicator_for_PA_models | state | indicator for PA models
s_kreisel_basis_theorem | theorem | Kreisel's basis theorem
s_pi_0_1_class_of_reals | axiom | pi 0 1 class of reals
s_recursive_binary_tree_T | state | recursive binary tree T
s_existence_of_infinite_branch | state | existence of infinite branch
s_leftmost_branch_recursive_in_0_prime | state | leftmost branch recursive in 0 prime
s_hyperarithmetic_hierarchy | theorem | Hyperarithmetic hierarchy theorem
s_turing_jump_operator | axiom | turing jump operator
s_recursive_ordinal | axiom | recursive ordinal
s_transfinite_jump_hierarchy_H_a | state | transfinite jump hierarchy H a
s_delta_1_1_equals_hyperarithmetic | state | delta 1 1 equals hyperarithmetic
s_strict_hyperarithmetic_hierarchy | state | strict hyperarithmetic hierarchy
s_friedman_finite_form_kruskal | theorem | Friedman's finite gap (Kruskal-style) theorem
s_finite_labelled_trees | axiom | finite labelled trees
s_wqo_axioms | axiom | wqo axioms
s_kruskal_wqo_statement | state | kruskal wqo statement
s_finitary_TREE_function_statement | state | finitary TREE function statement
s_mostowski_collapse | theorem | Mostowski collapse lemma
s_wellfounded_extensional_relation | axiom | wellfounded extensional relation
s_collapse_function_pi | state | collapse function pi
s_pi_total_and_injective | state | pi total and injective
s_mostowski_absoluteness | theorem | Mostowski absoluteness
s_transitive_model_of_ZF | axiom | transitive model of ZF
s_sigma_1_1_as_wellfoundedness | state | sigma 1 1 as wellfoundedness
s_wellfoundedness_absolute_for_transitive_models | state | wellfoundedness absolute for transitive models
s_shoenfield_absoluteness | theorem | Shoenfield absoluteness theorem
s_sigma_1_2_formula | state | sigma 1 2 formula
s_sigma_1_2_normal_form | state | sigma 1 2 normal form
s_shoenfield_tree_with_absolute_wellfoundedness | state | shoenfield tree with absolute wellfoundedness
s_absoluteness_to_L | state | absoluteness to L
s_levy_reflection | theorem | Levy reflection principle
s_cumulative_hierarchy_V_alpha | axiom | cumulative hierarchy V alpha
s_reflection_schema_for_finite_subtheory | state | reflection schema for finite subtheory
s_closure_under_skolem_functions | state | closure under skolem functions
s_constructible_universe_L | theorem | Constructible universe L (Gödel)
s_definable_powerset_operator | axiom | definable powerset operator
s_L_hierarchy_L_alpha | state | L hierarchy L alpha
s_L_models_ZF | state | L models ZF
s_canonical_wellorder_of_L | state | canonical wellorder of L
s_GCH_in_L | theorem | V=L implies GCH in L
s_condensation_lemma | axiom | condensation lemma
s_countable_elementary_substructure | state | countable elementary substructure
s_collapsed_substructure_is_L_alpha | state | collapsed substructure is L alpha
s_definability_in_L_alpha | axiom | definability in L alpha
s_elementary_substructure_of_L_alpha | state | elementary substructure of L alpha
s_transitive_isomorph_M_bar | state | transitive isomorph M bar
s_M_bar_equals_L_beta | state | M bar equals L beta
s_jensen_covering | theorem | Jensen's covering lemma
s_zero_sharp_does_not_exist | axiom | zero sharp does not exist
s_uncovered_set_hypothesis | state | uncovered set hypothesis
s_fine_structure_witness_in_L | state | fine structure witness in L
s_construction_of_zero_sharp | state | construction of zero sharp
s_zero_sharp_existence | theorem | Zero sharp (0#) existence
s_measurable_cardinal_axiom | axiom | measurable cardinal axiom
s_ultrapower_embedding_j | state | ultrapower embedding j
s_nontrivial_elementary_j_restricted_to_L | state | nontrivial elementary j restricted to L
s_silver_indiscernibles_in_L | state | silver indiscernibles in L
s_silver_indiscernibles_theorem | theorem | Silver indiscernibles theorem
s_EM_blueprint_for_L | state | EM blueprint for L
s_indiscernibility_of_uncountable_cardinals | state | indiscernibility of uncountable cardinals
s_solovay_model | theorem | Solovay's model (every set Lebesgue-measurable)
s_inaccessible_cardinal_axiom | axiom | inaccessible cardinal axiom
s_levy_collapse_model | state | levy collapse model
s_HOD_R_ordinals_inner_model | state | HOD R ordinals inner model
s_every_set_definable_from_real | state | every set definable from real
s_random_real_forcing | theorem | Solovay random real forcing
s_borel_measure_algebra | axiom | borel measure algebra
s_complete_BA_for_random_forcing | state | complete BA for random forcing
s_generic_random_real | state | generic random real
s_random_real_avoids_null_sets | state | random real avoids null sets
s_mansfield_solovay | theorem | Mansfield–Solovay theorem
s_sigma_1_2_set_of_reals | axiom | sigma 1 2 set of reals
s_sigma_1_2_as_tree_projection | state | sigma 1 2 as tree projection
s_perfect_subtree_case | state | perfect subtree case
s_dichotomy_perfect_or_inside_L | state | dichotomy perfect or inside L
s_vaught_two_cardinal | theorem | Vaught's two-cardinal theorem
s_first_order_theory_T | axiom | first order theory T
s_elementary_substructure_property | axiom | elementary substructure property
s_two_cardinal_model_M | state | two cardinal model M
s_elementary_chain_template | state | elementary chain template
s_vaught_never_22_status | theorem | Vaught's conjecture (status / never-22)
s_complete_countable_first_order_theory | axiom | complete countable first order theory
s_morley_analysis_of_countable_models | axiom | morley analysis of countable models
s_isomorphism_class_count_I_T | state | isomorphism class count I T
s_morley_tree_analysis | state | morley tree analysis
s_morley_dichotomy_via_scott_analysis | state | morley dichotomy via scott analysis
s_shelah_main_gap | theorem | Shelah's main gap theorem
s_complete_first_order_theory | axiom | complete first order theory
s_classification_invariants_dop_otop | axiom | classification invariants dop otop
s_classification_dichotomy_invariants | state | classification dichotomy invariants
s_tree_of_models_decomposition | state | tree of models decomposition
s_dimensional_decomposition_in_classifiable_case | state | dimensional decomposition in classifiable case
s_stability_spectrum_theorem | theorem | Shelah stability spectrum
s_type_space_S_n_T | axiom | type space S n T
s_order_property_dichotomy | state | order property dichotomy
s_stability_classification | state | stability classification
s_omitting_types_theorem | theorem | Morley's theorem on ω-stable theories (omitted-types)
s_countable_language | axiom | countable language
s_henkin_construction_with_omitting | state | henkin construction with omitting
s_at_each_step_omit_type | state | at each step omit type
s_pillay_steinhorn_o_minimal_cell_decomposition | theorem | Pillay–Steinhorn o-minimality theorem
s_ordered_field_structure | axiom | ordered field structure
s_o_minimal_definability | axiom | o minimal definability
s_monotonicity_lemma | state | monotonicity lemma
s_cell_decomposition_in_R_n | state | cell decomposition in R n
s_hrushovski_construction | theorem | Hrushovski's predimension construction
s_class_of_finite_structures_K | axiom | class of finite structures K
s_predimension_function_delta | axiom | predimension function delta
s_predimension_paired_class | state | predimension paired class
s_fraisse_amalgamation_with_predimension | state | fraisse amalgamation with predimension
s_hrushovski_generic_structure | state | hrushovski generic structure
s_tarski_RCF_quantifier_elimination | theorem | Tarski's quantifier elimination for RCF
s_real_closed_field | axiom | real closed field
s_polynomial_ring_over_RCF | axiom | polynomial ring over RCF
s_sturm_sign_change_count | state | sturm sign change count
s_one_step_quantifier_elimination | state | one step quantifier elimination
s_quantifier_free_equivalent | state | quantifier free equivalent
s_ACF_quantifier_elimination | theorem | Tarski's quantifier elimination for ACF
s_polynomial_solvability_predicate | state | polynomial solvability predicate
s_resultant_eliminates_existential_quantifier | state | resultant eliminates existential quantifier
s_ax_kochen_theorem | theorem | Ax–Kochen theorem
s_henselian_valued_field | axiom | henselian valued field
s_residue_field_and_value_group | axiom | residue field and value group
s_ax_kochen_ershov_axioms | state | ax kochen ershov axioms
s_elementary_equivalence_of_ultraproducts | state | elementary equivalence of ultraproducts
s_ax_kochen_transfer | state | ax kochen transfer
s_presburger_decidability | theorem | Decidability of Presburger arithmetic
s_naturals_with_addition | axiom | naturals with addition
s_first_order_logic_with_equality | axiom | first order logic with equality
s_presburger_with_congruences | state | presburger with congruences
s_qe_procedure_for_presburger | state | qe procedure for presburger
s_quantifier_free_form | state | quantifier free form
s_rice_shapiro_theorem | theorem | Rice–Shapiro theorem
s_re_set_of_partial_recursive_functions | axiom | re set of partial recursive functions
s_re_index_set_property | state | re index set property
s_finite_witness_property | state | finite witness property
s_kleene_second_recursion_theorem | theorem | Kleene's second recursion theorem
s_s_m_n_theorem | axiom | s m n theorem
s_diagonal_program_d_x | state | diagonal program d x
s_fixed_point_index_e | state | fixed point index e
s_universal_turing_machine | axiom | universal turing machine
s_program_with_hardcoded_input | state | program with hardcoded input
s_index_function_s_e_x | state | index function s e x
s_friedberg_muchnik_theorem | theorem | Friedberg–Muchnik theorem
s_re_sets | axiom | re sets
s_turing_reducibility | axiom | turing reducibility
s_priority_requirement_list | state | priority requirement list
s_finite_injury_construction | state | finite injury construction
s_each_requirement_satisfied | state | each requirement satisfied
s_sacks_density_theorem | theorem | Sacks density theorem
s_re_turing_degrees | axiom | re turing degrees
s_priority_method | axiom | priority method
s_intermediate_degree_requirements | state | intermediate degree requirements
s_infinite_injury_construction | state | infinite injury construction
s_true_path_argument_succeeds | state | true path argument succeeds
s_sacks_splitting_theorem | theorem | Sacks splitting theorem
s_re_set_A_nonrecursive | axiom | re set A nonrecursive
s_splitting_requirement_list | state | splitting requirement list
s_splitting_construction | state | splitting construction
s_sacks_jump_inversion | theorem | Sacks jump inversion theorem
s_re_degree_zero_prime | axiom | re degree zero prime
s_jump_inversion_requirements | state | jump inversion requirements
s_jump_inversion_construction | state | jump inversion construction
s_jump_target_achieved | state | jump target achieved
s_wadge_lemma | theorem | Wadge's lemma
s_borel_sets_in_baire_space | axiom | borel sets in baire space
s_axiom_of_determinacy_for_borel_games | axiom | axiom of determinacy for borel games
s_wadge_game | state | wadge game
s_wadge_game_determined | state | wadge game determined
s_wadge_hierarchy_theorem | theorem | Wadge hierarchy / antichain structure
s_wadge_ordering | state | wadge ordering
s_wadge_wellfounded | state | wadge wellfounded
s_martin_analytic_determinacy | theorem | Martin's theorem: Σ¹₁-determinacy from measurable
s_analytic_set_in_baire_space | axiom | analytic set in baire space
s_tree_representation_of_analytic_set | state | tree representation of analytic set
s_homogeneous_tree_for_T | state | homogeneous tree for T
s_tower_argument_winning_strategy | state | tower argument winning strategy
s_AD_in_L_R | theorem | AD in L(ℝ) from large cardinals (Woodin)
s_infinitely_many_woodin_cardinals | axiom | infinitely many woodin cardinals
s_constructible_closure_L_R | axiom | constructible closure L R
s_iteration_trees_on_V | state | iteration trees on V
s_universally_baire_pointclass | state | universally baire pointclass
s_projective_determinacy | state | projective determinacy
s_souslin_theorem | theorem | Souslin's theorem (Σ¹₁ ∩ Π¹₁ = Borel)
s_polish_space | axiom | polish space
s_lusin_separation_for_disjoint_analytics | state | lusin separation for disjoint analytics
s_separating_borel_set_for_A_and_comp_A | state | separating borel set for A and comp A
s_kondo_uniformization | theorem | Kondô's uniformization theorem
s_pi_1_1_set_in_product_space | axiom | pi 1 1 set in product space
s_wellfounded_tree_T_x | state | wellfounded tree T x
s_canonical_section_selector | state | canonical section selector
s_transfinite_recursion_theorem | theorem | Transfinite recursion theorem
s_wellordered_class_ON | axiom | wellordered class ON
s_set_of_attempts | state | set of attempts
s_uniqueness_of_attempts | state | uniqueness of attempts
s_erdos_tarski_theorem | theorem | Erdős–Tarski theorem on weakly compact cardinals
s_partition_relation_kappa_to_kappa_2_2 | axiom | partition relation kappa to kappa 2 2
s_tree_property_for_kappa | state | tree property for kappa
s_pi_1_1_indescribability | state | pi 1 1 indescribability
s_weak_compactness_characterization | state | weak compactness characterization
s_scott_theorem_V_not_L | theorem | Scott's theorem (no measurable cardinal in L)
s_kunen_inconsistency_in_L | state | kunen inconsistency in L
s_kunen_inconsistency | theorem | Kunen inconsistency theorem
s_elementary_embedding_j_V_to_V | axiom | elementary embedding j V to V
s_iterated_critical_points | state | iterated critical points
s_problematic_definable_subset | state | problematic definable subset
s_reflection_in_L | theorem | Reflection theorem in L
s_phi_reflection_schema_in_L | state | phi reflection schema in L
s_levy_reflection_in_L | state | levy reflection in L
s_mostowski_rank_function | theorem | Mostowski rank theorem
s_wellfounded_relation_R | axiom | wellfounded relation R
s_rank_recursive_definition | state | rank recursive definition
s_rank_total_and_uniquely_defined | state | rank total and uniquely defined
s_friedman_borel_diagonalization | theorem | Friedman's harmonious functor theorem
s_subsystem_RCA_0 | axiom | subsystem RCA 0
s_class_of_borel_functors | axiom | class of borel functors
s_BDT_statement | state | BDT statement
s_BDT_not_in_ATR_0 | state | BDT not in ATR 0
s_BDT_provable_in_pi_1_1_CA_0 | state | BDT provable in pi 1 1 CA 0
s_strong_ramsey_independence | theorem | Ramsey-type Paris–Harrington-style independence (general PA-unprovable)
s_relatively_large_ramsey_statement | state | relatively large ramsey statement
s_indicator_for_PA_growth | state | indicator for PA growth
s_skolem_paradox | theorem | Skolem's paradox
s_downward_lowenheim_skolem | axiom | downward lowenheim skolem
s_countable_elementary_substructure_of_V | state | countable elementary substructure of V
s_countable_transitive_model | state | countable transitive model
s_PA_proof_theoretic_ordinal_epsilon_0 | theorem | Specker–Ackermann ordinal of PA / Gentzen analysis
s_proof_ordinal_assignment | state | proof ordinal assignment
s_cut_elimination_bound | state | cut elimination bound
s_consistency_via_descent | state | consistency via descent
s_cut_elimination_theorem | theorem | Cut-elimination theorem (Gentzen Hauptsatz)
s_sequent_calculus_LK | axiom | sequent calculus LK
s_proof_tree | axiom | proof tree
s_local_cut_reduction_rules | state | local cut reduction rules
s_double_induction_termination | state | double induction termination
s_curry_howard_correspondence | theorem | Curry–Howard correspondence
s_simply_typed_lambda_calculus | axiom | simply typed lambda calculus
s_intuitionistic_propositional_logic | axiom | intuitionistic propositional logic
s_types_as_propositions_map | state | types as propositions map
s_proofs_as_terms_map | state | proofs as terms map
s_iso_STLC_IPL_proofs | state | iso STLC IPL proofs
s_strong_normalization_STLC | theorem | Strong normalization for STLC
s_type_assignment | axiom | type assignment
s_reducibility_candidates | state | reducibility candidates
s_candidate_closure_properties | state | candidate closure properties
s_lob_theorem | theorem | Löb's theorem
s_provability_predicate_box | axiom | provability predicate box
s_HB_provability_axioms | state | HB provability axioms
s_lob_diagonal_sentence | state | lob diagonal sentence
s_tarski_undefinability | theorem | Tarski's undefinability of truth
s_arithmetical_definability | axiom | arithmetical definability
s_godel_numbering_of_arithmetic | state | godel numbering of arithmetic
s_liar_via_truth_predicate | state | liar via truth predicate
s_robinson_joint_consistency | theorem | Robinson's joint consistency theorem
s_two_consistent_first_order_theories_T1_T2 | axiom | two consistent first order theories T1 T2
s_common_subtheory_T_0 | axiom | common subtheory T 0
s_common_complete_subtheory | state | common complete subtheory
s_no_disagreement_via_interpolant | state | no disagreement via interpolant
s_keisler_shelah_theorem | theorem | Keisler–Shelah ultrapower characterization of elementary equivalence
s_two_first_order_structures_M_N | axiom | two first order structures M N
s_ultrafilter_on_index_set | axiom | ultrafilter on index set
s_ultrapowers_M_U_N_U | state | ultrapowers M U N U
s_saturated_ultrapowers | state | saturated ultrapowers
s_isomorphism_of_saturated_models | state | isomorphism of saturated models
s_saturated_model_existence | theorem | Existence of saturated models
s_continuum_hypothesis_or_GCH | axiom | continuum hypothesis or GCH
s_elementary_chain_realizing_types | state | elementary chain realizing types
s_type_realization_succeeds | state | type realization succeeds
s_vaught_test | theorem | Vaught's test
s_first_order_theory_T_no_finite_models | axiom | first order theory T no finite models
s_kappa_categoricity | axiom | kappa categoricity
s_two_distinct_completions_T1_T2 | state | two distinct completions T1 T2
s_two_kappa_sized_models_distinct | state | two kappa sized models distinct
s_ryll_nardzewski_theorem | theorem | Ryll-Nardzewski theorem
s_complete_countable_theory_T | axiom | complete countable theory T
s_stone_space_of_T | state | stone space of T
s_isolated_types_dense_iff_omega_categorical | state | isolated types dense iff omega categorical
s_DCF_0_model_complete | theorem | Robinson's theorem on differentially closed fields
s_differential_field | axiom | differential field
s_existential_closure | axiom | existential closure
s_DCF_0_axioms | state | DCF 0 axioms
s_DCF_0_qe | state | DCF 0 qe
s_DCF_0_omega_stable | state | DCF 0 omega stable
s_hrushovski_lascar_group_configuration | theorem | Hrushovski–Lascar group configuration / definable groups in stable theories
s_stable_theory_T | axiom | stable theory T
s_group_configuration_in_types | axiom | group configuration in types
s_associativity_diagram_of_types | state | associativity diagram of types
s_germ_of_group_multiplication | state | germ of group multiplication
s_hrushovski_zilber_trichotomy | theorem | Trichotomy in Zariski geometries / Hrushovski–Zilber
s_strongly_minimal_set | axiom | strongly minimal set
s_geometry_of_algebraic_closure | axiom | geometry of algebraic closure
s_zariski_geometry_axioms | state | zariski geometry axioms
s_local_modularity_dichotomy | state | local modularity dichotomy
s_ACF_interpretation_in_non_modular_case | state | ACF interpretation in non modular case
s_levy_solovay_theorem | theorem | Lévy–Solovay theorem on indestructibility of large cardinals under small forcing
s_small_forcing_P_below_kappa | axiom | small forcing P below kappa
s_ground_normal_measure_U | state | ground normal measure U
s_generic_extension_V_G | state | generic extension V G
s_lifted_measure_U_star | state | lifted measure U star
s_easton_theorem | theorem | Easton's theorem on continuum function
s_regular_cardinal_arithmetic | axiom | regular cardinal arithmetic
s_easton_admissible_F | state | easton admissible F
s_easton_class_forcing | state | easton class forcing
s_generic_extension_with_F | state | generic extension with F
s_silver_singular_cardinals_theorem | theorem | Silver's theorem on singular cardinals
s_singular_cardinal_kappa_of_uncountable_cofinality | axiom | singular cardinal kappa of uncountable cofinality
s_stationary_GCH_carrier | state | stationary GCH carrier
s_generic_ultrapower | state | generic ultrapower
s_2_kappa_bounded_in_quotient | state | 2 kappa bounded in quotient
s_solovay_stationary_splitting | theorem | Solovay's stationary splitting theorem
s_regular_uncountable_cardinal_kappa | axiom | regular uncountable cardinal kappa
s_stationary_subset_of_kappa | axiom | stationary subset of kappa
s_regressive_function_on_S | state | regressive function on S
s_fodor_constancy_on_stationary_subset | state | fodor constancy on stationary subset
s_fodor_lemma | theorem | Fodor's pressing-down lemma
s_regressive_function_on_stationary_set | axiom | regressive function on stationary set
s_each_fibre_nonstationary | state | each fibre nonstationary
s_diagonal_intersection_club | state | diagonal intersection club
s_solovay_RVM_theorem | theorem | Solovay's theorem on real-valued measurable cardinals
s_real_valued_measurable_cardinal | axiom | real valued measurable cardinal
s_measure_extension_problem | axiom | measure extension problem
s_random_real_extension_over_measurable | state | random real extension over measurable
s_RVM_continuum_witnessed | state | RVM continuum witnessed
s_diamond_implies_suslin_tree | theorem | Suslin's hypothesis / ◊ implies a Suslin tree (Jensen)
s_diamond_principle_on_omega_1 | axiom | diamond principle on omega 1
s_tree_of_height_omega_1 | axiom | tree of height omega 1
s_diamond_anticipates_antichains | state | diamond anticipates antichains
s_construct_tree_killing_each_predicted_antichain | state | construct tree killing each predicted antichain
s_MA_implies_SH | theorem | Martin's axiom and Suslin's hypothesis
s_martins_axiom_MA_omega_1 | axiom | martins axiom MA omega 1
s_ccc_partial_order | axiom | ccc partial order
s_hypothetical_suslin_tree | state | hypothetical suslin tree
s_T_as_ccc_poset | state | T as ccc poset
s_omega_1_branch_kills_suslin | state | omega 1 branch kills suslin
s_PFA_consequences | theorem | Proper Forcing Axiom consequences (Todorčević)
s_proper_forcing_axiom | axiom | proper forcing axiom
s_countable_proper_forcing_class | axiom | countable proper forcing class
s_OCA_from_PFA | state | OCA from PFA
s_structural_dichotomy_on_separable_metric_spaces | state | structural dichotomy on separable metric spaces
s_halpern_lauchli_theorem | theorem | Halpern–Läuchli theorem
s_finite_product_of_finitely_branching_trees | axiom | finite product of finitely branching trees
s_finite_coloring | axiom | finite coloring
s_strong_subtree_template | state | strong subtree template
s_per_level_monochromatic_choice | state | per level monochromatic choice
s_galvin_prikry_theorem | theorem | Galvin–Prikry theorem
s_borel_subset_of_omega_omega | axiom | borel subset of omega omega
s_infinite_subsets_of_omega | axiom | infinite subsets of omega
s_open_case_reduction | state | open case reduction
s_galvin_prikry_game | state | galvin prikry game
s_winning_strategy_yields_homogeneous_A | state | winning strategy yields homogeneous A
s_novikov_boone_theorem | theorem | Word problem for groups (Novikov–Boone)
s_semigroup_encoding_of_TM | state | semigroup encoding of TM
s_group_simulating_TM | state | group simulating TM
s_liouville_phase_volume_theorem | theorem | Liouville's theorem on phase-space volume
s_hamiltonian_phase_space | axiom | hamiltonian phase space
s_hamilton_canonical_equations | axiom | hamilton canonical equations
s_liouville_equation | state | liouville equation
s_divergence_free_hamiltonian_flow | state | divergence free hamiltonian flow
s_hamilton_jacobi_theorem | theorem | Hamilton–Jacobi theorem
s_hamilton_jacobi_pde | state | hamilton jacobi pde
s_characteristics_match_hamilton_equations | state | characteristics match hamilton equations
s_complete_integral_generating_canonical_transformation | state | complete integral generating canonical transformation
s_arnold_liouville_theorem | theorem | Arnold–Liouville theorem (commuting integrals ⇒ integrability)
s_n_commuting_independent_first_integrals | axiom | n commuting independent first integrals
s_invariant_lagrangian_submanifold | state | invariant lagrangian submanifold
s_level_set_is_n_torus | state | level set is n torus
s_action_angle_coordinates_exist | state | action angle coordinates exist
s_bertrand_theorem | theorem | Bertrand's theorem (only 1/r and r² have closed orbits)
s_central_force_problem | axiom | central force problem
s_effective_radial_potential | state | effective radial potential
s_apsidal_angle_perturbation | state | apsidal angle perturbation
s_only_inverse_square_and_harmonic_close_at_first_order | state | only inverse square and harmonic close at first order
s_poincare_recurrence_theorem | theorem | Poincaré recurrence theorem
s_some_iterates_must_overlap_in_measure | state | some iterates must overlap in measure
s_almost_every_point_returns | state | almost every point returns
s_gromov_nonsqueezing_theorem | theorem | Symplectic non-squeezing (Gromov)
s_symplectic_manifold | axiom | symplectic manifold
s_symplectic_embedding | axiom | symplectic embedding
s_pseudoholomorphic_disks_in_target | state | pseudoholomorphic disks in target
s_existence_of_area_bounded_disk_through_image | state | existence of area bounded disk through image
s_marsden_weinstein_reduction_theorem | theorem | Marsden–Weinstein symplectic reduction
s_hamiltonian_lie_group_action_with_moment_map | axiom | hamiltonian lie group action with moment map
s_moment_map_fiber_mu_inverse_zero | state | moment map fiber mu inverse zero
s_quotient_manifold_M_red | state | quotient manifold M red
s_darboux_theorem | theorem | Darboux theorem (local symplectic uniqueness)
s_closed_nondegenerate_2_form | axiom | closed nondegenerate 2 form
s_moser_homotopy_of_symplectic_forms | state | moser homotopy of symplectic forms
s_diffeomorphism_pulling_back_to_standard_form | state | diffeomorphism pulling back to standard form
s_stone_von_neumann_theorem | theorem | Stone–von Neumann uniqueness theorem
s_canonical_commutation_relations_weyl_form | axiom | canonical commutation relations weyl form
s_irreducible_unitary_representation | axiom | irreducible unitary representation
s_weyl_unitary_ccr_algebra | state | weyl unitary ccr algebra
s_schrodinger_representation | state | schrodinger representation
s_wigner_symmetry_theorem | theorem | Wigner's theorem (symmetries of quantum states)
s_projective_hilbert_space | axiom | projective hilbert space
s_ray_preserving_transition_probability | axiom | ray preserving transition probability
s_inner_product_preserved_up_to_phase | state | inner product preserved up to phase
s_unique_lift_to_U_or_anti_U | state | unique lift to U or anti U
s_gleason_theorem | theorem | Gleason's theorem
s_hilbert_space_dim_geq_3 | axiom | hilbert space dim geq 3
s_frame_function_on_projections | axiom | frame function on projections
s_continuous_frame_function | state | continuous frame function
s_positive_trace_class_operator_rho | state | positive trace class operator rho
s_robertson_schrodinger_uncertainty | theorem | Heisenberg uncertainty (Robertson–Schrödinger inequality)
s_hilbert_space_with_self_adjoint_operators | axiom | hilbert space with self adjoint operators
s_centered_observables | state | centered observables
s_cauchy_schwarz_variance_bound | state | cauchy schwarz variance bound
s_ehrenfest_theorem | theorem | Ehrenfest theorem
s_schrodinger_equation | axiom | schrodinger equation
s_heisenberg_equation_of_motion | state | heisenberg equation of motion
s_d_dt_expectation_equals_i_hbar_commutator | state | d dt expectation equals i hbar commutator
s_rage_theorem | theorem | RAGE theorem (spectral decomposition of dynamics)
s_self_adjoint_hamiltonian_H | axiom | self adjoint hamiltonian H
s_unitary_schrodinger_flow | axiom | unitary schrodinger flow
s_lebesgue_spectral_decomposition_of_H | state | lebesgue spectral decomposition of H
s_decay_of_continuous_part_in_compact_set | state | decay of continuous part in compact set
s_wightman_reconstruction_theorem | theorem | Wightman reconstruction theorem
s_wightman_axioms | axiom | wightman axioms
s_tempered_distribution_n_point_functions | axiom | tempered distribution n point functions
s_wightman_functional_satisfies_GNS_positivity | state | wightman functional satisfies GNS positivity
s_borchers_algebra_GNS_representation | state | borchers algebra GNS representation
s_osterwalder_schrader_theorem | theorem | Osterwalder–Schrader reconstruction
s_euclidean_schwinger_functions | axiom | euclidean schwinger functions
s_OS_axioms_reflection_positivity | axiom | OS axioms reflection positivity
s_euclidean_schwinger_data | state | euclidean schwinger data
s_physical_hilbert_space_from_RP | state | physical hilbert space from RP
s_wightman_functions_recovered | state | wightman functions recovered
s_bisognano_wichmann_theorem | theorem | Bisognano–Wichmann theorem
s_wedge_local_algebra_R_W | axiom | wedge local algebra R W
s_modular_operator_for_wedge | state | modular operator for wedge
s_modular_flow_equals_boost | state | modular flow equals boost
s_coleman_mandula_theorem | theorem | Coleman–Mandula theorem
s_relativistic_s_matrix_axioms | axiom | relativistic s matrix axioms
s_internal_symmetry_lie_algebra_G | axiom | internal symmetry lie algebra G
s_bosonic_charge_algebra_constraints | state | bosonic charge algebra constraints
s_charges_commute_with_translation_generators | state | charges commute with translation generators
s_HLS_theorem | theorem | Haag–Łopuszański–Sohnius theorem
s_graded_lie_algebra_extension | axiom | graded lie algebra extension
s_super_charge_anticommutator_structure | state | super charge anticommutator structure
s_unique_super_poincare_extension | state | unique super poincare extension
s_glimm_jaffe_phi4_construction | theorem | Glimm–Jaffe construction of φ⁴ in 2 and 3 dimensions
s_phi_4_lagrangian_lattice_regularization | axiom | phi 4 lagrangian lattice regularization
s_lattice_phi_4_probability_measure | state | lattice phi 4 probability measure
s_renormalization_group_flow_to_continuum | state | renormalization group flow to continuum
s_limit_measure_exists_in_d2_d3 | state | limit measure exists in d2 d3
s_mermin_wagner_theorem | theorem | Mermin–Wagner theorem
s_lattice_spin_model_continuous_symmetry | axiom | lattice spin model continuous symmetry
s_short_range_interaction_in_d_le_2 | axiom | short range interaction in d le 2
s_putative_order_parameter | state | putative order parameter
s_bogoliubov_inequality_for_correlations | state | bogoliubov inequality for correlations
s_order_parameter_must_vanish | state | order parameter must vanish
s_lee_yang_theorem | theorem | Lee–Yang circle theorem
s_ferromagnetic_ising_partition_function | axiom | ferromagnetic ising partition function
s_polynomial_in_fugacity_z | axiom | polynomial in fugacity z
s_partition_function_polynomial_P_z | state | partition function polynomial P z
s_inductive_zero_locus_constraint | state | inductive zero locus constraint
s_onsager_2d_ising_solution | theorem | Onsager exact solution of 2D Ising
s_2d_ising_lattice_square | axiom | 2d ising lattice square
s_nearest_neighbor_coupling | axiom | nearest neighbor coupling
s_transfer_matrix_formulation | state | transfer matrix formulation
s_jordan_wigner_fermionization | state | jordan wigner fermionization
s_eigenvalues_in_closed_form | state | eigenvalues in closed form
s_peierls_phase_transition | theorem | Peierls argument for spontaneous magnetization
s_low_temperature_regime | axiom | low temperature regime
s_peierls_contour_ensemble | state | peierls contour ensemble
s_summable_contour_weight_bound | state | summable contour weight bound
s_lieb_robinson_bound | theorem | Lieb–Robinson bound
s_lattice_local_hamiltonian | axiom | lattice local hamiltonian
s_bounded_local_interaction_norm | axiom | bounded local interaction norm
s_heisenberg_evolution_operator | state | heisenberg evolution operator
s_commutator_norm_grows_at_most_exponentially_in_distance_minus_v_t | state | commutator norm grows at most exponentially in distance minus v t
s_lieb_thirring_inequality | theorem | Lieb–Thirring inequalities
s_schrodinger_operator_minus_laplacian_plus_V | axiom | schrodinger operator minus laplacian plus V
s_lp_potential_V | axiom | lp potential V
s_sum_over_negative_spectrum | state | sum over negative spectrum
s_birman_schwinger_kernel_bound | state | birman schwinger kernel bound
s_stability_of_matter_theorem | theorem | Lieb's stability of matter
s_many_body_coulomb_hamiltonian | axiom | many body coulomb hamiltonian
s_fermionic_antisymmetry | axiom | fermionic antisymmetry
s_thomas_fermi_lower_bound | state | thomas fermi lower bound
s_extensive_kinetic_energy_bound | state | extensive kinetic energy bound
s_dlr_theorem | theorem | Dobrushin–Lanford–Ruelle (DLR) characterization of Gibbs states
s_specifications_family_of_conditional_distributions | axiom | specifications family of conditional distributions
s_consistent_specification | state | consistent specification
s_set_of_DLR_solutions_nonempty_compact_convex | state | set of DLR solutions nonempty compact convex
s_kramers_wannier_duality_theorem | theorem | Kramers–Wannier duality
s_low_temperature_high_temperature_expansions | axiom | low temperature high temperature expansions
s_partition_function_duality_relation | state | partition function duality relation
s_self_dual_temperature_equation | state | self dual temperature equation
s_gibbs_variational_principle | theorem | Gibbs variational principle
s_relative_entropy_functional | axiom | relative entropy functional
s_free_energy_legendre_dual | state | free energy legendre dual
s_pressure_attains_supremum_at_gibbs_state | state | pressure attains supremum at gibbs state
s_birkhoff_gr_theorem | theorem | Birkhoff's theorem (GR)
s_einstein_field_equations_vacuum | axiom | einstein field equations vacuum
s_spherical_symmetry_ansatz | axiom | spherical symmetry ansatz
s_spherically_symmetric_metric_ansatz | state | spherically symmetric metric ansatz
s_radial_einstein_equations | state | radial einstein equations
s_staticity_emerges | state | staticity emerges
s_penrose_singularity_theorem | theorem | Penrose singularity theorem
s_einstein_field_equations | axiom | einstein field equations
s_null_energy_condition | axiom | null energy condition
s_trapped_surface_exists | axiom | trapped surface exists
s_raychaudhuri_focusing_inequality | state | raychaudhuri focusing inequality
s_null_geodesic_incompleteness_obstruction | state | null geodesic incompleteness obstruction
s_hawking_singularity_theorem | theorem | Hawking singularity theorem
s_strong_energy_condition | axiom | strong energy condition
s_globally_hyperbolic_spacetime | axiom | globally hyperbolic spacetime
s_timelike_focusing_inequality | state | timelike focusing inequality
s_finite_proper_time_to_focal_point | state | finite proper time to focal point
s_hawking_area_theorem | theorem | Hawking's area theorem
s_cosmic_censorship | axiom | cosmic censorship
s_horizon_generator_focusing | state | horizon generator focusing
s_nonnegative_expansion_on_horizon | state | nonnegative expansion on horizon
s_schoen_yau_positive_mass_theorem | theorem | Schoen–Yau positive mass theorem
s_asymptotically_flat_initial_data_set | axiom | asymptotically flat initial data set
s_dominant_energy_condition | axiom | dominant energy condition
s_adm_mass_definition | state | adm mass definition
s_minimal_surface_reduction | state | minimal surface reduction
s_no_zero_mass_initial_data_unless_minkowski | state | no zero mass initial data unless minkowski
s_bekenstein_hawking_entropy | theorem | Bekenstein–Hawking entropy formula
s_black_hole_thermodynamics_axioms | axiom | black hole thermodynamics axioms
s_area_acts_as_entropy_proxy | state | area acts as entropy proxy
s_hawking_temperature_kappa_over_2pi | state | hawking temperature kappa over 2pi
s_cosmic_no_hair_theorem | theorem | Cosmic no-hair theorem (de Sitter attractor)
s_einstein_field_equations_with_positive_lambda | axiom | einstein field equations with positive lambda
s_initially_inhomogeneous_data | axiom | initially inhomogeneous data
s_constraint_equations_with_cosmological_constant | state | constraint equations with cosmological constant
s_anisotropy_decays_like_e_minus_Ht | state | anisotropy decays like e minus Ht
s_geroch_spinor_obstruction_theorem | theorem | Geroch's no-go for global spinor structure
s_orientable_lorentzian_manifold | axiom | orientable lorentzian manifold
s_spin_lift_existence_question | axiom | spin lift existence question
s_w2_vanishing_required | state | w2 vanishing required
s_spin_lift_exists_iff_w2_zero | state | spin lift exists iff w2 zero
s_israel_uniqueness_theorem | theorem | Israel uniqueness of Schwarzschild
s_static_asymptotically_flat_black_hole | axiom | static asymptotically flat black hole
s_static_metric_ansatz | state | static metric ansatz
s_elliptic_system_on_riemannian_slice | state | elliptic system on riemannian slice
s_newman_janis_algorithm_theorem | theorem | Newman–Janis algorithm
s_schwarzschild_metric_in_null_coordinates | axiom | schwarzschild metric in null coordinates
s_complex_coordinate_shift_recipe | axiom | complex coordinate shift recipe
s_complexified_null_tetrad | state | complexified null tetrad
s_shifted_metric_in_real_form | state | shifted metric in real form
s_choquet_bruhat_well_posedness | theorem | Choquet–Bruhat well-posedness of Einstein vacuum
s_initial_data_satisfying_constraints | axiom | initial data satisfying constraints
s_harmonic_gauge_reduces_to_quasilinear_wave | state | harmonic gauge reduces to quasilinear wave
s_hyperbolic_cauchy_problem | state | hyperbolic cauchy problem
s_local_existence_and_uniqueness | state | local existence and uniqueness
s_raychaudhuri_equation_theorem | theorem | Raychaudhuri equation
s_lorentzian_manifold | axiom | lorentzian manifold
s_geodesic_congruence | axiom | geodesic congruence
s_kinematic_decomposition_of_congruence | state | kinematic decomposition of congruence
s_dtheta_ds_equation | state | dtheta ds equation
s_faddeev_popov_partition_function_theorem | theorem | Faddeev–Popov gauge-fixing procedure
s_yang_mills_action | axiom | yang mills action
s_path_integral_with_gauge_redundancy | axiom | path integral with gauge redundancy
s_gauge_slice_with_jacobian | state | gauge slice with jacobian
s_grassmann_representation_of_FP_determinant | state | grassmann representation of FP determinant
s_bps_bound_and_saturation_theorem | theorem | BPS bound and BPS saturation
s_supersymmetry_algebra_with_central_charge | axiom | supersymmetry algebra with central charge
s_massive_state_in_rep_of_algebra | axiom | massive state in rep of algebra
s_positivity_of_Q_dagger_Q | state | positivity of Q dagger Q
s_bps_mass_inequality | state | bps mass inequality
s_witten_index_theorem | theorem | Witten index
s_supersymmetric_quantum_mechanics | axiom | supersymmetric quantum mechanics
s_supercharge_Q_with_Q_squared_equals_H | axiom | supercharge Q with Q squared equals H
s_graded_trace_definition | state | graded trace definition
s_only_zero_modes_contribute | state | only zero modes contribute
s_wess_zumino_descent_theorem | theorem | Wess–Zumino consistency / descent equations
s_gauge_symmetry_with_anomaly_functional | axiom | gauge symmetry with anomaly functional
s_brst_cohomology | axiom | brst cohomology
s_anomaly_satisfies_cocycle_condition | state | anomaly satisfies cocycle condition
s_anomaly_class_in_local_BRST_cohomology | state | anomaly class in local BRST cohomology
s_atiyah_bott_fixed_point_theorem | theorem | Atiyah–Bott Lefschetz fixed point (equivariant)
s_smooth_self_map_with_isolated_fixed_points | axiom | smooth self map with isolated fixed points
s_equivariant_index_localized_to_fixed_points | state | equivariant index localized to fixed points
s_localization_to_sum_over_fixed_points | state | localization to sum over fixed points
s_chiral_anomaly_from_index_theorem | theorem | Atiyah–Singer applied to anomalies (Stora–Zumino)
s_chiral_dirac_operator | axiom | chiral dirac operator
s_gauge_bundle_with_connection | axiom | gauge bundle with connection
s_index_of_chiral_dirac_equals_A_roof_ch_E | state | index of chiral dirac equals A roof ch E
s_perturbative_anomaly_coefficient | state | perturbative anomaly coefficient
s_donaldson_intersection_form_theorem | theorem | Donaldson's theorem on smooth 4-manifolds
s_smooth_simply_connected_4_manifold | axiom | smooth simply connected 4 manifold
s_definite_intersection_form | axiom | definite intersection form
s_ASD_moduli_space | state | ASD moduli space
s_dimension_of_moduli_via_index | state | dimension of moduli via index
s_seiberg_witten_invariants_theorem | theorem | Seiberg–Witten invariants existence
s_smooth_4_manifold_with_spin_c_structure | axiom | smooth 4 manifold with spin c structure
s_seiberg_witten_equations | axiom | seiberg witten equations
s_monopole_moduli_space | state | monopole moduli space
s_compact_smooth_moduli_under_generic_metric | state | compact smooth moduli under generic metric
s_dimension_of_SW_moduli_via_index | state | dimension of SW moduli via index
s_verlinde_formula_theorem | theorem | Verlinde formula for SU(N) WZW
s_WZW_model_at_level_k_for_SU_N | axiom | WZW model at level k for SU N
s_modular_S_matrix_on_characters | axiom | modular S matrix on characters
s_fusion_rules_as_tensor_product_decomposition | state | fusion rules as tensor product decomposition
s_S_matrix_diagonalizes_fusion | state | S matrix diagonalizes fusion
s_cft_modular_invariance_theorem | theorem | Modular invariance of CFT partition function (S, T action)
s_2d_cft_on_torus | axiom | 2d cft on torus
s_partition_function_Z_tau | axiom | partition function Z tau
s_partition_function_depends_only_on_modulus | state | partition function depends only on modulus
s_Z_as_quadratic_form_in_characters | state | Z as quadratic form in characters
s_lax_pair_conserved_quantities_theorem | theorem | Lax pair representation yields conserved quantities
s_evolution_equation_with_lax_pair_L_M | axiom | evolution equation with lax pair L M
s_isospectral_flow | state | isospectral flow
s_spectrum_independent_of_time | state | spectrum independent of time
s_kdv_inverse_scattering_theorem | theorem | Inverse scattering transform for KdV
s_kdv_equation | axiom | kdv equation
s_schrodinger_scattering_operator_with_potential_u | axiom | schrodinger scattering operator with potential u
s_lax_pair_for_kdv | state | lax pair for kdv
s_scattering_data_evolve_linearly | state | scattering data evolve linearly
s_gelfand_levitan_marchenko_reconstruction | state | gelfand levitan marchenko reconstruction
s_yang_baxter_integrability_theorem | theorem | Yang–Baxter equation as integrability condition
s_two_particle_scattering_matrix_R | axiom | two particle scattering matrix R
s_factorized_n_particle_S_matrix | axiom | factorized n particle S matrix
s_triangle_consistency_condition | state | triangle consistency condition
s_yang_baxter_equation | state | yang baxter equation
s_bethe_ansatz_xxx_theorem | theorem | Bethe ansatz solution of the Heisenberg XXX chain
s_heisenberg_xxx_spin_chain | axiom | heisenberg xxx spin chain
s_plane_wave_ansatz_with_rapidities | axiom | plane wave ansatz with rapidities
s_magnon_sector_decomposition | state | magnon sector decomposition
s_two_particle_S_matrix_phase | state | two particle S matrix phase
s_bethe_equations | state | bethe equations
s_hirota_soliton_theorem | theorem | Hirota bilinear form ⇒ soliton solutions
s_integrable_pde_eg_kdv_or_KP | axiom | integrable pde eg kdv or KP
s_tau_function_substitution | state | tau function substitution
s_bilinear_pde_in_tau | state | bilinear pde in tau
s_riemann_hilbert_method_theorem | theorem | Riemann–Hilbert reformulation of integrable PDE
s_integrable_pde_with_lax_pair | axiom | integrable pde with lax pair
s_jump_matrix_on_contour | axiom | jump matrix on contour
s_jump_data_on_spectral_curve | state | jump data on spectral curve
s_matrix_RH_problem | state | matrix RH problem
s_calogero_moser_integrability_theorem | theorem | Calogero–Moser integrability
s_calogero_moser_hamiltonian_with_1_over_x_sq_pair | axiom | calogero moser hamiltonian with 1 over x sq pair
s_lax_pair_for_calogero_moser | state | lax pair for calogero moser
s_n_commuting_first_integrals_constructed | state | n commuting first integrals constructed
s_toda_lattice_integrability_theorem | theorem | Toda lattice integrability
s_toda_lattice_hamiltonian_exponential_potential | axiom | toda lattice hamiltonian exponential potential
s_flaschka_variable_form | state | flaschka variable form
s_jacobi_matrix_isospectral_flow | state | jacobi matrix isospectral flow
s_sklyanin_sov_theorem | theorem | Sklyanin separation of variables
s_quantum_integrable_system_with_transfer_matrix | axiom | quantum integrable system with transfer matrix
s_separation_coordinates_from_B_zeros | state | separation coordinates from B zeros
s_baxter_TQ_relation | state | baxter TQ relation
s_painleve_classification_theorem | theorem | Painlevé property and isomonodromy
s_second_order_ode_with_movable_singularities | axiom | second order ode with movable singularities
s_painleve_property_definition | state | painleve property definition
s_50_canonical_ODEs_reduced_to_6_irreducible | state | 50 canonical ODEs reduced to 6 irreducible
s_wolstenholme_theorem | theorem | Wolstenholme's theorem
s_normalized_central_binomial_mod_p4 | state | normalized central binomial mod p4
s_harmonic_congruence_mod_p3 | state | harmonic congruence mod p3
s_euclid_euler_perfect_numbers | theorem | Euclid–Euler theorem (even perfect numbers)
s_mersenne_prime_2p_minus_1 | axiom | mersenne prime 2p minus 1
s_sigma_multiplicative | state | sigma multiplicative
s_euclid_perfect_sufficient | state | euclid perfect sufficient
s_pentagonal_number_theorem | theorem | Pentagonal number theorem (Euler)
s_formal_power_series_over_z | axiom | formal power series over z
s_partition_generating_function_q_pochhammer | axiom | partition generating function q pochhammer
s_sparse_pentagonal_signature | state | sparse pentagonal signature
s_franklin_involution_cancellation | state | franklin involution cancellation
s_zeckendorf_theorem | theorem | Zeckendorf's theorem
s_fibonacci_numbers | axiom | fibonacci numbers
s_greedy_fibonacci_decomposition | state | greedy fibonacci decomposition
s_fermat_right_triangle_no_rational_square_area | theorem | Fermat's right-triangle theorem
s_pythagorean_triple_primitive_parametrization | axiom | pythagorean triple primitive parametrization
s_smallest_squared_area_triple | state | smallest squared area triple
s_descended_smaller_triple | state | descended smaller triple
s_sophie_germain_theorem | theorem | Sophie Germain's theorem
s_flt_exponent_p_first_case | axiom | flt exponent p first case
s_aux_sophie_germain_prime_q | state | aux sophie germain prime q
s_flt_residue_obstruction_mod_q | state | flt residue obstruction mod q
s_fermat_polygonal_number_theorem | theorem | Fermat polygonal number theorem
s_gauss_triangular_eureka | state | gauss triangular eureka
s_jacobi_four_square_formula | theorem | Jacobi's four-square theorem
s_r4_formula_8_times_sigma | state | r4 formula 8 times sigma
s_three_gap_theorem | theorem | Three-gap theorem (Steinhaus)
s_reals_with_topology | axiom | reals with topology
s_circle_t_1 | axiom | circle t 1
s_orbit_of_n_alpha_mod_1 | state | orbit of n alpha mod 1
s_at_most_three_gap_lengths | state | at most three gap lengths
s_hurwitz_diophantine_approximation | theorem | Hurwitz's theorem on Diophantine approximation
s_continued_fraction_expansion | axiom | continued fraction expansion
s_convergent_best_approximation | state | convergent best approximation
s_markov_spectrum_minimal_gap_sqrt5 | state | markov spectrum minimal gap sqrt5
s_dirichlet_approximation | theorem | Dirichlet's approximation theorem
s_unit_interval | axiom | unit interval
s_two_close_kalpha_points | state | two close kalpha points
s_mihailescu_catalan_theorem | theorem | Mihăilescu's theorem (Catalan)
s_cyclotomic_field_q_zeta_p | axiom | cyclotomic field q zeta p
s_catalan_equation_canonical | state | catalan equation canonical
s_class_group_divisibility_obstruction | state | class group divisibility obstruction
s_cubic_reciprocity | theorem | Cubic reciprocity (Eisenstein)
s_eisenstein_integers_z_omega | axiom | eisenstein integers z omega
s_primary_eisenstein_primes | state | primary eisenstein primes
s_cubic_residue_symbol_reciprocal_law | state | cubic residue symbol reciprocal law
s_quartic_reciprocity | theorem | Quartic / biquadratic reciprocity
s_gaussian_integers_z_i | axiom | gaussian integers z i
s_primary_gaussian_primes | state | primary gaussian primes
s_quartic_residue_symbol | state | quartic residue symbol
s_jacobi_triple_product | theorem | Jacobi triple product
s_q_pochhammer_symbol | axiom | q pochhammer symbol
s_quasiperiodic_shift_z_qz | state | quasiperiodic shift z qz
s_hasse_minkowski | theorem | Hasse–Minkowski theorem
s_rationals_q | axiom | rationals q
s_p_adic_completions_q_p | axiom | p adic completions q p
s_quadratic_form_over_q | axiom | quadratic form over q
s_diagonal_quadratic_form | state | diagonal quadratic form
s_hilbert_symbol_product_formula | state | hilbert symbol product formula
s_meyer_theorem_indefinite | theorem | Meyer's theorem (indefinite quadratic in ≥5 variables)
s_indefinite_form_rank_ge_5 | state | indefinite form rank ge 5
s_brauer_forms_theorem | theorem | Brauer's theorem on forms
s_finite_field_f_q | axiom | finite field f q
s_homogeneous_form_degree_d | axiom | homogeneous form degree d
s_chevalley_warning_zero_count | state | chevalley warning zero count
s_bhargava_290_theorem | theorem | Bhargava–Hanke 15 and 290 theorems
s_positive_definite_integral_quadratic_form | axiom | positive definite integral quadratic form
s_universality_witness_set_290 | state | universality witness set 290
s_hilbert_irreducibility | theorem | Hilbert's irreducibility theorem
s_polynomial_ring_in_two_vars_q_t_x | axiom | polynomial ring in two vars q t x
s_thin_set_in_an | axiom | thin set in an
s_generic_specialization_remains_irreducible | state | generic specialization remains irreducible
s_thin_exceptional_set | state | thin exceptional set
s_dedekind_discriminant_theorem | theorem | Dedekind discriminant theorem
s_number_field_k | axiom | number field k
s_ring_of_integers_o_k | axiom | ring of integers o k
s_different_ideal_definition | state | different ideal definition
s_dedekind_kummer_theorem | theorem | Dedekind–Kummer theorem
s_minimal_polynomial_theta | axiom | minimal polynomial theta
s_minimal_poly_factorization_mod_p | state | minimal poly factorization mod p
s_kummer_cyclotomic_factorization | theorem | Kummer's theorem on cyclotomic ideal factorization
s_cyclotomic_galois_structure | state | cyclotomic galois structure
s_kummer_congruence | theorem | Kummer's congruence (Bernoulli)
s_bernoulli_numbers_b_n | axiom | bernoulli numbers b n
s_p_adic_l_function | state | p adic l function
s_von_staudt_clausen_theorem | theorem | Von Staudt–Clausen theorem
s_bernoulli_integer_part_formula | state | bernoulli integer part formula
s_apery_zeta3_irrational | theorem | Apéry's theorem (ζ(3) irrational)
s_apery_recurrence_sequences | state | apery recurrence sequences
s_super_dirichlet_approximation_of_zeta3 | state | super dirichlet approximation of zeta3
s_chebotarev_density | theorem | Chebotarev density theorem
s_galois_extension_l_over_k | axiom | galois extension l over k
s_frobenius_conjugacy_class_distribution | state | frobenius conjugacy class distribution
s_density_via_character_sums | state | density via character sums
s_chebyshev_pi_bounds | theorem | Chebyshev's bounds on π(x)
s_chebyshev_function_theta_psi | axiom | chebyshev function theta psi
s_psi_squeeze_via_central_binomial | state | psi squeeze via central binomial
s_mertens_theorems | theorem | Mertens' theorems
s_log_p_over_p_partial_sums | state | log p over p partial sums
s_mertens_constant_M | state | mertens constant M
s_brun_twin_prime_constant_finite | theorem | Brun's theorem on twin primes
s_twin_prime_set | axiom | twin prime set
s_pi_2_x_upper_bound | state | pi 2 x upper bound
s_brun_titchmarsh_inequality | theorem | Brun–Titchmarsh inequality
s_dirichlet_character_mod_q | axiom | dirichlet character mod q
s_sifted_count_primes_in_ap | state | sifted count primes in ap
s_selberg_erdos_elementary_pnt | theorem | Selberg–Erdős elementary proof of PNT
s_selberg_symmetry_formula | state | selberg symmetry formula
s_psi_minus_x_decay | state | psi minus x decay
s_wiener_ikehara_theorem | theorem | Wiener–Ikehara Tauberian theorem
s_dirichlet_series | axiom | dirichlet series
s_simple_pole_at_s_eq_1 | state | simple pole at s eq 1
s_tauberian_fourier_input | state | tauberian fourier input
s_vinogradov_three_prime_theorem | theorem | Vinogradov's three-prime theorem
s_major_minor_arc_decomposition_for_3p | state | major minor arc decomposition for 3p
s_minor_arc_bound | state | minor arc bound
s_vinogradov_mean_value_theorem | theorem | Vinogradov mean-value theorem
s_polynomial_ring_z_x | axiom | polynomial ring z x
s_exponential_sum | axiom | exponential sum
s_vinogradov_system_count | state | vinogradov system count
s_efficient_congruencing_bound | state | efficient congruencing bound
s_bombieri_vinogradov | theorem | Bombieri–Vinogradov theorem
s_large_sieve_inequality | state | large sieve inequality
s_zero_density_combined | state | zero density combined
s_linnik_smallest_prime_in_ap | theorem | Linnik's theorem on smallest prime in AP
s_dirichlet_l_function | axiom | dirichlet l function
s_zero_density_and_repulsion | state | zero density and repulsion
s_polya_vinogradov_inequality | theorem | Pólya–Vinogradov inequality
s_character_in_additive_frequencies | state | character in additive frequencies
s_burgess_bound | theorem | Burgess bound on character sums
s_shifted_character_moment | state | shifted character moment
s_short_character_sum_bound | state | short character sum bound
s_kim_sarnak_bound | theorem | Selberg's 1/4 conjecture status / Kim–Sarnak bound
s_modular_forms_holomorphic_or_maass | axiom | modular forms holomorphic or maass
s_symmetric_power_l_functions | state | symmetric power l functions
s_hardy_ramanujan_partition_asymptotic | theorem | Hardy–Ramanujan partition asymptotic
s_integer_partitions | axiom | integer partitions
s_eta_function_dedekind | axiom | eta function dedekind
s_modular_transformation_of_eta | state | modular transformation of eta
s_farey_arc_contributions | state | farey arc contributions
s_rademacher_exact_formula_partition | theorem | Rademacher exact formula for p(n)
s_rademacher_contour_integral | state | rademacher contour integral
s_erdos_kac_theorem | theorem | Erdős–Kac theorem
s_additive_function_omega | axiom | additive function omega
s_quasi_independence_decomposition | state | quasi independence decomposition
s_clt_for_omega | state | clt for omega
s_erdos_wintner_theorem | theorem | Erdős–Wintner theorem
s_additive_function_f | axiom | additive function f
s_three_series_criterion | state | three series criterion
s_turan_kubilius_inequality | theorem | Turán–Kubilius inequality
s_second_moment_bound | state | second moment bound
s_erdos_fuchs_theorem | theorem | Erdős–Fuchs theorem
s_additive_basis | axiom | additive basis
s_proposed_smooth_error_term | state | proposed smooth error term
s_fourier_error_obstruction | state | fourier error obstruction
s_roth_theorem_3_aps | theorem | Roth's theorem on 3-APs
s_finite_subset_a_in_z | axiom | finite subset a in z
s_fourier_density_increment | state | fourier density increment
s_roth_diophantine_approximation | theorem | Roth's theorem on Diophantine approximation
s_algebraic_number_alpha | axiom | algebraic number alpha
s_aux_multivariate_polynomial | state | aux multivariate polynomial
s_index_height_obstruction | state | index height obstruction
s_schmidt_subspace_theorem | theorem | Subspace theorem (Schmidt)
s_linear_form_l_i | axiom | linear form l i
s_higher_dim_diophantine_inequality | state | higher dim diophantine inequality
s_lindemann_weierstrass | theorem | Lindemann–Weierstrass theorem
s_exponential_function_complex | axiom | exponential function complex
s_hermite_integral | state | hermite integral
s_integer_vs_small_obstruction | state | integer vs small obstruction
s_gelfond_schneider_theorem | theorem | Gelfond–Schneider theorem
s_logarithm_function_complex | axiom | logarithm function complex
s_gelfond_aux_function | state | gelfond aux function
s_extrapolation_zeroes | state | extrapolation zeroes
s_baker_theorem | theorem | Baker's theorem on linear forms in logarithms
s_baker_auxiliary_many_vars | state | baker auxiliary many vars
s_baker_nonzero_integer_too_small | state | baker nonzero integer too small
s_six_exponentials_theorem | theorem | Six exponentials theorem
s_2x3_interpolation_determinant | state | 2x3 interpolation determinant
s_vanishing_determinant_pigeonhole | state | vanishing determinant pigeonhole
s_schneider_lang_theorem | theorem | Schneider–Lang theorem
s_meromorphic_function_finite_order | axiom | meromorphic function finite order
s_meromorphic_algebraic_ring | state | meromorphic algebraic ring
s_schneider_lang_aux | state | schneider lang aux
s_wustholz_analytic_subgroup_theorem | theorem | Wüstholz analytic subgroup theorem
s_commutative_algebraic_group_g_over_q_bar | axiom | commutative algebraic group g over q bar
s_wustholz_analytic_subgroup_statement | state | wustholz analytic subgroup statement
s_weil_conjectures_complete | theorem | Weil conjectures (over finite fields)
s_smooth_projective_variety_over_fq | axiom | smooth projective variety over fq
s_etale_cohomology | axiom | etale cohomology
s_zeta_function_of_variety | state | zeta function of variety
s_etale_trace_formula | state | etale trace formula
s_hasse_weil_bound | theorem | Hasse–Weil bound for curves over F_q
s_smooth_projective_curve_over_fq | axiom | smooth projective curve over fq
s_riemann_hypothesis_for_curves | axiom | riemann hypothesis for curves
s_curve_l_function_eigenvalues | state | curve l function eigenvalues
s_honda_tate_theorem | theorem | Honda–Tate theorem
s_abelian_variety_over_fq | axiom | abelian variety over fq
s_weil_q_numbers | axiom | weil q numbers
s_frobenius_to_weil_number | state | frobenius to weil number
s_tate_isogeny_theorem | theorem | Tate's isogeny theorem
s_l_adic_tate_module | axiom | l adic tate module
s_galois_action_on_tate_module | state | galois action on tate module
s_nagell_lutz_theorem | theorem | Nagell–Lutz theorem
s_elliptic_curve_over_q | axiom | elliptic curve over q
s_torsion_subgroup | axiom | torsion subgroup
s_minimal_weierstrass_form | state | minimal weierstrass form
s_mazur_torsion_theorem | theorem | Mazur's torsion theorem
s_modular_curve_x0_n | axiom | modular curve x0 n
s_rational_points_on_x1_n | state | rational points on x1 n
s_modular_curves_no_extra_rational_points | state | modular curves no extra rational points
s_merel_uniform_boundedness | theorem | Merel's uniform boundedness
s_modular_curve_x1_n | axiom | modular curve x1 n
s_eisenstein_quotient_bound | state | eisenstein quotient bound
s_manin_drinfeld_theorem | theorem | Manin–Drinfeld theorem
s_cusp_divisors | axiom | cusp divisors
s_eisenstein_divisor_rationality | state | eisenstein divisor rationality
s_tunnell_theorem | theorem | Tunnell's theorem (congruent numbers)
s_weight_three_halves_form | state | weight three halves form
s_modularity_theorem_full | theorem | Modularity theorem (full, Breuil–Conrad–Diamond–Taylor)
s_l_adic_galois_representation_of_E | state | l adic galois representation of E
s_full_modularity_lift_machine | state | full modularity lift machine
s_ribet_level_lowering | theorem | Ribet's level-lowering theorem
s_galois_representation_residual | axiom | galois representation residual
s_residual_unramified_at_p_consequence | state | residual unramified at p consequence
s_eichler_shimura_congruence | theorem | Eichler–Shimura congruence
s_hecke_operator | axiom | hecke operator
s_hecke_mod_p_decomposition | state | hecke mod p decomposition
s_khare_wintenberger_serre_modularity | theorem | Khare–Wintenberger (Serre's modularity conjecture)
s_modularity_lifting_for_residual_rep | state | modularity lifting for residual rep
s_waldspurger_theorem | theorem | Waldspurger's theorem
s_l_function_central_value | axiom | l function central value
s_shimura_correspondence_pair | state | shimura correspondence pair
s_gross_zagier_formula | theorem | Gross–Zagier formula
s_heegner_points | axiom | heegner points
s_heegner_height | state | heegner height
s_kolyvagin_theorem | theorem | Kolyvagin's theorem
s_euler_system_kolyvagin | axiom | euler system kolyvagin
s_anti_cyclotomic_compatible_system | state | anti cyclotomic compatible system
s_iwasawa_main_conjecture | theorem | Iwasawa main conjecture (Mazur–Wiles, total)
s_cyclotomic_z_p_extension | axiom | cyclotomic z p extension
s_class_group_inverse_limit_module | state | class group inverse limit module
s_class_number_formula | theorem | Class number formula (Dirichlet)
s_dedekind_zeta_function | axiom | dedekind zeta function
s_residue_of_dedekind_zeta | state | residue of dedekind zeta
s_minkowski_lattice_theorem | theorem | Minkowski's theorem on lattices
s_lattice_in_rn | axiom | lattice in rn
s_convex_symmetric_body | axiom | convex symmetric body
s_volume_dominance_condition | state | volume dominance condition
s_minkowski_class_number_bound | theorem | Minkowski bound for class group
s_class_rep_below_minkowski_bound | state | class rep below minkowski bound
s_stark_rank_0_abelian | theorem | Stark conjectures (rank-0 abelian case, proven)
s_artin_l_function | axiom | artin l function
s_abelian_extension_of_number_field | axiom | abelian extension of number field
s_stark_unit_candidate | state | stark unit candidate
s_brauer_siegel_theorem | theorem | Brauer–Siegel theorem
s_log_hr_asymptotic | state | log hr asymptotic
s_heegner_class_number_one | theorem | Heegner's theorem (class number 1)
s_imaginary_quadratic_field | axiom | imaginary quadratic field
s_class_number | axiom | class number
s_hilbert_class_polynomial | state | hilbert class polynomial
s_goldfeld_effective_class_number | theorem | Goldfeld's effective lower bound on class number
s_gross_zagier_input | state | gross zagier input
s_faltings_product_theorem | theorem | Faltings' product theorem
s_smooth_projective_variety_over_q | axiom | smooth projective variety over q
s_arakelov_geometry | axiom | arakelov geometry
s_high_index_arakelov_section | state | high index arakelov section
s_vojta_height_inequality_statement | theorem | Lang–Vojta conjecture (status / Vojta height inequality)
s_naive_height_on_an | axiom | naive height on an
s_vojta_diophantine_dictionary | state | vojta diophantine dictionary
s_maynard_bounded_gaps_improved | theorem | Bounded gaps between primes (Zhang / Maynard) (cite: https://en.wikipedia.org/wiki/Bounded_gaps_between_primes) — extension of `s_bounded_gaps_between_primes`
s_multidim_selberg_weight | state | multidim selberg weight
s_heath_brown_three_cubes_density | theorem | Heath-Brown three-cubes representation
s_three_cubes_minor_arc_bound | state | three cubes minor arc bound
s_hooley_artin_primitive_root_grh | theorem | Hooley on Artin's primitive-root conjecture (under GRH)
s_grh_for_dedekind_zeta | axiom | grh for dedekind zeta
s_grh_chebotarev_explicit | state | grh chebotarev explicit
s_hardy_littlewood_singular_series_predictor | theorem | Hardy–Littlewood prime k-tuples conjecture (status / circle-method input) (cite: https://en.wikipedia.org/wiki/First_Hardy%E2%80%93Littlewood_conjecture) — note: a conjecture, but the circle-method singular-series predictor is a theorem
s_singular_series_major_arc_count | state | singular series major arc count
s_fundamental_lemma_of_sieve | theorem | Brun–Hooley sieve / linear sieve fundamental lemma
s_sieve_problem_axiomatic | axiom | sieve problem axiomatic
s_selberg_upper_lower_bounds | state | selberg upper lower bounds
s_hasse_principle_failure_genus_one | theorem | Hasse principle failure / Reichardt–Lind counterexample
s_smooth_projective_curve_over_q | axiom | smooth projective curve over q
s_reichardt_lind_curve_specific | state | reichardt lind curve specific
s_birch_theorem_systems_of_forms | theorem | Birch's theorem on systems of forms
s_birch_circle_method_setup | state | birch circle method setup
s_mahler_compactness_theorem | theorem | Mahler's compactness theorem
s_space_of_unimodular_lattices | axiom | space of unimodular lattices
s_mahler_pre_compact_criterion | state | mahler pre compact criterion
s_skolem_mahler_lech_theorem | theorem | Skolem–Mahler–Lech theorem
s_linear_recurrence_sequence | axiom | linear recurrence sequence
s_p_adic_analytic_extension | state | p adic analytic extension
s_mahler_continuous_p_adic | theorem | Mahler's theorem on continuous p-adic functions
s_continuous_function_qp_to_qp | axiom | continuous function qp to qp
s_mahler_expansion_basis | state | mahler expansion basis
s_e_transcendental | theorem | Mahler's theorem on transcendence of e (special case)
s_hermite_integral_for_e | state | hermite integral for e
s_pi_transcendental | theorem | Transcendence of π (Lindemann)
s_prime_local_factor_geometric | state | prime local factor geometric
s_zeta_functional_equation | theorem | Functional equation of ζ(s) (Riemann)
s_theta_modular_inversion | state | theta modular inversion
s_completed_xi_function | state | completed xi function
s_rh_equivalents_robin | theorem | Riemann hypothesis equivalents (Robin, Mertens-type)
s_mertens_and_sigma_translations | state | mertens and sigma translations
s_riemann_von_mangoldt_explicit_formula | theorem | Riemann–von Mangoldt explicit formula
s_perron_for_psi | state | perron for psi
s_de_la_vallee_poussin_zero_free_region | theorem | Zero-free region (de la Vallée Poussin)
s_3_4_cos_trick | state | 3 4 cos trick
s_riemann_pi_x_formula | theorem | Riemann's prime-counting formula
s_j_x_inversion | state | j x inversion
s_ramanujan_partition_congruences | theorem | Ramanujan's congruences
s_partition_generating_modular | state | partition generating modular
s_deligne_tau_bound | theorem | Deligne's bound for τ (Ramanujan conjecture)
s_modular_galois_rep | state | modular galois rep
s_frobenius_eigenvalues_modular | state | frobenius eigenvalues modular
s_arthur_selberg_trace_formula | theorem | Arthur–Selberg trace formula (overview)
s_reductive_algebraic_group | axiom | reductive algebraic group
s_adeles_a | axiom | adeles a
s_automorphic_spectral_side | state | automorphic spectral side
s_geometric_side_decomposition | state | geometric side decomposition
s_langlands_functoriality_known_cases | theorem | Langlands functoriality (transferring from one group to another — known cases, e.g. Arthur for classical groups)
s_automorphic_representation | axiom | automorphic representation
s_l_group_functoriality_setup | state | l group functoriality setup
s_local_langlands_gl_n | theorem | Local Langlands for GL_n
s_local_field_k | axiom | local field k
s_weil_deligne_rep_side | state | weil deligne rep side
s_lafforgue_global_langlands_function_field | theorem | Global Langlands for GL_n over function fields (Lafforgue)
s_function_field_over_fq | axiom | function field over fq
s_shtuka_cohomology_galois_reps | state | shtuka cohomology galois reps
s_carmichael_theorem_orders | theorem | Carmichael's theorem (orders)
s_lucas_sequence | axiom | lucas sequence
s_primitive_divisor_existence | state | primitive divisor existence
s_zsigmondy_theorem | theorem | Zsigmondy's theorem
s_cyclotomic_polynomial_phi_n | axiom | cyclotomic polynomial phi n
s_lifting_exponent_factor_decomposition | state | lifting exponent factor decomposition
s_lame_euclidean_algorithm_bound | theorem | Lamé's theorem on Euclidean algorithm
s_worst_case_consecutive_fibs | state | worst case consecutive fibs
s_midy_theorem | theorem | Midy's theorem
s_decimal_expansion_repetend | axiom | decimal expansion repetend
s_two_halves_of_repetend | state | two halves of repetend
s_beatty_theorem | theorem | Beatty's theorem
s_complementary_irrationals_pair | state | complementary irrationals pair
s_beatty_partition_count_match | state | beatty partition count match
s_davenport_schmidt_theorem | theorem | Davenport–Schmidt theorem
s_quadratic_approximation_bounds | state | quadratic approximation bounds
s_smyth_lower_bound_non_reciprocal | theorem | Lehmer pair / Lehmer's conjecture (Mahler measure) — Smyth's theorem on non-reciprocal polynomials
s_mahler_measure_function | axiom | mahler measure function
s_hilbert_90 | theorem | Hilbert's Theorem 90
s_cyclic_galois_group | axiom | cyclic galois group
s_galois_action_on_units | state | galois action on units
s_local_class_field_theory | theorem | Local class field theory
s_lubin_tate_formal_group | axiom | lubin tate formal group
s_local_reciprocity_isomorphism | state | local reciprocity isomorphism
s_kronecker_weber_theorem | theorem | Kronecker–Weber theorem
s_local_kronecker_weber_components | state | local kronecker weber components
s_chebotarev_effective_grh | theorem | Chebotarev with effective bound under GRH (cite: https://en.wikipedia.org/wiki/Chebotarev%27s_density_theorem#Effective_version) — extends `s_chebotarev_density`
s_grh_zero_density_input | state | grh zero density input
s_weyl_polynomial_equidistribution | theorem | Weyl's polynomial equidistribution
s_weyl_differencing_reduction | state | weyl differencing reduction
s_sarkozy_theorem_squares | theorem | Three-distance / Sárközy-on-squares
s_squares_fourier_input | state | squares fourier input
s_behrend_3_ap_free_set | theorem | Behrend's construction of dense 3-AP-free set
s_high_dimensional_sphere | axiom | high dimensional sphere
s_high_dim_sphere_lattice_layer | state | high dim sphere lattice layer
s_bloom_sisask_roth_bound | theorem | Bloom–Sisask on Roth
s_bohr_density_increment | state | bohr density increment
s_gowers_inverse_theorem | theorem | Gowers norms and inverse theorem
s_finite_abelian_group_zn | axiom | finite abelian group zn
s_function_f_g_to_c | axiom | function f g to c
s_gowers_uk_definition | state | gowers uk definition
s_szemeredi_trotter_incidences | theorem | Szemerédi–Trotter incidence theorem
s_points_in_plane | axiom | points in plane
s_lines_in_plane | axiom | lines in plane
s_cell_decomposition_of_plane | state | cell decomposition of plane
s_guth_katz_distinct_distances | theorem | Erdős distinct-distances (Guth–Katz)
s_lines_in_r3_as_ruled_surfaces | axiom | lines in r3 as ruled surfaces
s_elekes_sharir_lines_in_r3 | state | elekes sharir lines in r3
s_erdos_szemeredi_sum_product | theorem | Erdős–Szemerédi sum–product
s_sum_product_basic_bound | state | sum product basic bound
s_three_distance_theorem | theorem | Three-distance theorem (Steinhaus, distinct from three-gap)
s_circle_orbit_partition | state | circle orbit partition
s_khintchine_diophantine_theorem | theorem | Khintchine's theorem on Diophantine approximation
s_random_alpha_uniform | state | random alpha uniform
s_davenport_erdos_density_multiples | theorem | Davenport–Erdős theorem on multiples
s_density_equality_for_multiples_sets | state | density equality for multiples sets
s_cauchy_lipschitz_local_existence | theorem | Cauchy–Lipschitz local existence
s_lipschitz_vector_field | axiom | lipschitz vector field
s_initial_condition | axiom | initial condition
s_canonical_ivp | state | canonical ivp
s_local_unique_solution | state | local unique solution
s_gronwall_inequality | theorem | Gronwall's inequality (differential form)
s_nonneg_continuous_function | axiom | nonneg continuous function
s_linear_integral_majorant | axiom | linear integral majorant
s_multiplied_inequality_for_integrating_factor | state | multiplied inequality for integrating factor
s_monotone_decreasing_witness | state | monotone decreasing witness
s_ode_comparison_principle | theorem | Comparison principle for scalar ODE
s_scalar_ode_with_monotone_rhs | axiom | scalar ode with monotone rhs
s_two_initial_values_ordered | axiom | two initial values ordered
s_difference_satisfies_linear_majorant | state | difference satisfies linear majorant
s_sign_persistence_of_difference | state | sign persistence of difference
s_sturm_comparison_theorem | theorem | Sturm comparison theorem
s_two_sturm_liouville_problems | axiom | two sturm liouville problems
s_potential_ordering | axiom | potential ordering
s_wronskian_witness | state | wronskian witness
s_interleaving_witness | state | interleaving witness
s_sturm_liouville_eigenvalue_theorem | theorem | Sturm–Liouville eigenvalue theorem
s_regular_sturm_liouville_problem | axiom | regular sturm liouville problem
s_self_adjoint_operator_L | state | self adjoint operator L
s_compact_self_adjoint_inverse | state | compact self adjoint inverse
s_orthonormal_eigenbasis_with_discrete_spectrum | state | orthonormal eigenbasis with discrete spectrum
s_stable_manifold_theorem | theorem | Stable manifold theorem
s_hyperbolic_equilibrium_of_smooth_vector_field | axiom | hyperbolic equilibrium of smooth vector field
s_stable_subspace_E_s | axiom | stable subspace E s
s_dulac_criterion | theorem | Dulac's criterion
s_planar_smooth_flow | axiom | planar smooth flow
s_simply_connected_region | axiom | simply connected region
s_signed_divergence_of_BX | state | signed divergence of BX
s_green_theorem_contradiction | state | green theorem contradiction
s_mather_theorem_twist_maps | theorem | Mather's theorem for twist maps
s_monotone_twist_map_on_cylinder | axiom | monotone twist map on cylinder
s_generating_function | axiom | generating function
s_action_functional_on_configurations | state | action functional on configurations
s_minimal_configurations | state | minimal configurations
s_aubry_mather_invariant_set | state | aubry mather invariant set
s_liouville_volume_preservation | theorem | Liouville's theorem on Hamiltonian flow (volume preservation)
s_hamiltonian_vector_field | axiom | hamiltonian vector field
s_symplectic_form_on_phase_space | axiom | symplectic form on phase space
s_zero_lie_derivative_of_symplectic_form | state | zero lie derivative of symplectic form
s_cauchy_kowalevskaya_theorem | theorem | Cauchy–Kowalevskaya theorem
s_real_analytic_PDE_system | axiom | real analytic PDE system
s_non_characteristic_cauchy_data | axiom | non characteristic cauchy data
s_normal_form_analytic_pde | state | normal form analytic pde
s_formal_power_series_solution | state | formal power series solution
s_convergent_power_series_solution | state | convergent power series solution
s_holmgren_uniqueness_theorem | theorem | Holmgren's uniqueness theorem
s_linear_pde_with_analytic_coefficients | axiom | linear pde with analytic coefficients
s_non_characteristic_hypersurface | axiom | non characteristic hypersurface
s_formal_adjoint_operator_L_star | state | formal adjoint operator L star
s_dense_family_of_analytic_test_functions | state | dense family of analytic test functions
s_ck_holmgren_combined | theorem | Cauchy–Kowalevskaya–Holmgren combined statement
s_analytic_local_existence_witness | state | analytic local existence witness
s_C_infty_uniqueness_witness | state | C infty uniqueness witness
s_hadamard_well_posedness_criterion | theorem | Hadamard well-posedness criterion
s_pde_with_data | axiom | pde with data
s_continuous_dependence_norm | axiom | continuous dependence norm
s_three_axiom_definition_of_well_posedness | state | three axiom definition of well posedness
s_separation_of_well_posed_from_ill_posed_PDE | state | separation of well posed from ill posed PDE
s_backward_heat_ill_posed | theorem | Hadamard ill-posedness of backward heat
s_high_frequency_fourier_modes | axiom | high frequency fourier modes
s_mode_decay_factor_exp_minus_k2_t | state | mode decay factor exp minus k2 t
s_mode_growth_factor_exp_plus_k2_t_backward | state | mode growth factor exp plus k2 t backward
s_weak_maximum_principle_laplace | theorem | Maximum principle for Laplace equation (weak)
s_harmonic_function_on_bounded_domain | axiom | harmonic function on bounded domain
s_continuity_to_boundary | axiom | continuity to boundary
s_strict_subharmonic_perturbation | state | strict subharmonic perturbation
s_no_interior_max_for_perturbation | state | no interior max for perturbation
s_hopf_strong_maximum_principle | theorem | Hopf strong maximum principle
s_elliptic_operator_L | axiom | elliptic operator L
s_connected_domain | axiom | connected domain
s_hopf_barrier_function | state | hopf barrier function
s_propagation_of_max_to_boundary_of_ball | state | propagation of max to boundary of ball
s_hopf_boundary_point_lemma | theorem | Hopf boundary point lemma
s_C2_domain_with_interior_ball_condition | axiom | C2 domain with interior ball condition
s_radial_comparison_function_w | state | radial comparison function w
s_subsolution_inside_ball | state | subsolution inside ball
s_mean_value_property_harmonic | theorem | Mean value property for harmonic functions
s_ball_in_R_n | axiom | ball in R n
s_spherical_average_function_phi_of_r | state | spherical average function phi of r
s_constant_spherical_average | state | constant spherical average
s_liouville_harmonic_theorem | theorem | Liouville's theorem for harmonic functions
s_harmonic_function_on_R_n | axiom | harmonic function on R n
s_uniform_boundedness | axiom | uniform boundedness
s_two_ball_mean_value_comparison | state | two ball mean value comparison
s_zero_difference_between_two_points | state | zero difference between two points
s_harnack_inequality_classical | theorem | Harnack's inequality (classical)
s_nonnegative_harmonic_function | axiom | nonnegative harmonic function
s_compact_subset_of_open_set | axiom | compact subset of open set
s_poisson_kernel_pointwise_bound | state | poisson kernel pointwise bound
s_chain_of_local_ratio_estimates | state | chain of local ratio estimates
s_de_giorgi_nash_moser_regularity | theorem | De Giorgi–Nash–Moser regularity theorem
s_divergence_form_elliptic_with_L_infty_coefficients | axiom | divergence form elliptic with L infty coefficients
s_weak_solution_in_H_1 | axiom | weak solution in H 1
s_caccioppoli_inequality | state | caccioppoli inequality
s_local_boundedness_L_infty | state | local boundedness L infty
s_parabolic_harnack_for_general_coefficients | state | parabolic harnack for general coefficients
s_schauder_estimates | theorem | Schauder estimates
s_elliptic_operator_L_with_C_alpha_coefficients | axiom | elliptic operator L with C alpha coefficients
s_holder_space_C_2_alpha | axiom | holder space C 2 alpha
s_frozen_constant_coefficient_problem | state | frozen constant coefficient problem
s_constant_coefficient_holder_estimate | state | constant coefficient holder estimate
s_local_schauder_estimate | state | local schauder estimate
s_calderon_zygmund_Lp_regularity | theorem | Calderón–Zygmund Lp elliptic regularity
s_elliptic_operator_L_constant_coeff | axiom | elliptic operator L constant coeff
s_singular_integral_operator_with_calderon_zygmund_kernel | axiom | singular integral operator with calderon zygmund kernel
s_L2_boundedness_of_singular_integral | state | L2 boundedness of singular integral
s_cz_good_lambda_decomposition | state | cz good lambda decomposition
s_Lp_boundedness_of_singular_integral | state | Lp boundedness of singular integral
s_krylov_safonov_theorem | theorem | Krylov–Safonov Harnack for nondivergence-form
s_nondivergence_elliptic_with_L_infty_coeff | axiom | nondivergence elliptic with L infty coeff
s_measurable_coefficient_uniform_ellipticity | axiom | measurable coefficient uniform ellipticity
s_abp_maximum_principle | state | abp maximum principle
s_measure_density_lemma_nondivergence | state | measure density lemma nondivergence
s_harnack_inequality_nondivergence | state | harnack inequality nondivergence
s_fredholm_alternative_elliptic | theorem | Fredholm alternative for elliptic operators
s_elliptic_operator_L_self_adjoint | axiom | elliptic operator L self adjoint
s_compact_resolvent | axiom | compact resolvent
s_riesz_schauder_compact_operator_framework | state | riesz schauder compact operator framework
s_orthogonality_range_perp_kernel_adjoint | state | orthogonality range perp kernel adjoint
s_lax_milgram_theorem | theorem | Lax–Milgram theorem
s_coercive_bilinear_form_on_hilbert_space | axiom | coercive bilinear form on hilbert space
s_bounded_linear_functional | axiom | bounded linear functional
s_unique_solution_of_Au_equals_f | state | unique solution of Au equals f
s_weak_solution_dirichlet_existence | theorem | Existence of weak solutions for Dirichlet problem
s_uniformly_elliptic_operator_in_divergence_form | axiom | uniformly elliptic operator in divergence form
s_H_1_0_sobolev_space | axiom | H 1 0 sobolev space
s_bilinear_form_a_continuous_coercive | state | bilinear form a continuous coercive
s_weak_solution_in_H_1_0 | state | weak solution in H 1 0
s_perron_method_for_dirichlet_problem | theorem | Perron's method for Laplace–Dirichlet
s_continuous_boundary_data | axiom | continuous boundary data
s_bounded_domain_in_R_n | axiom | bounded domain in R n
s_family_of_subsolutions_S_phi | state | family of subsolutions S phi
s_perron_harmonic_envelope_u_star | state | perron harmonic envelope u star
s_wiener_criterion | theorem | Wiener criterion for regular boundary points
s_potential_theory_capacity | axiom | potential theory capacity
s_dyadic_capacity_series | state | dyadic capacity series
s_harmonic_measure_representation | theorem | Riesz representation for harmonic measure
s_linear_functional_phi_to_u_phi_at_x | state | linear functional phi to u phi at x
s_hormander_L2_dbar_existence | theorem | Hörmander L² existence for ∂̄
s_pseudoconvex_domain_in_C_n | axiom | pseudoconvex domain in C n
s_dbar_complex | axiom | dbar complex
s_weighted_L2_space_with_phi | state | weighted L2 space with phi
s_a_priori_L2_estimate_for_dbar_star | state | a priori L2 estimate for dbar star
s_solvability_of_dbar_u_equals_f | state | solvability of dbar u equals f
s_hormander_sum_of_squares_theorem | theorem | Hörmander sum-of-squares hypoellipticity
s_smooth_vector_fields_X_1_X_k | axiom | smooth vector fields X 1 X k
s_hormander_bracket_generating_condition | axiom | hormander bracket generating condition
s_subelliptic_a_priori_estimate | state | subelliptic a priori estimate
s_microlocal_smoothing_witness | state | microlocal smoothing witness
s_bony_paradifferential_calculus | theorem | Bony's paradifferential calculus theorem
s_littlewood_paley_decomposition | axiom | littlewood paley decomposition
s_product_of_two_distributions | axiom | product of two distributions
s_paraproduct_decomposition_T_u_T_v_R | state | paraproduct decomposition T u T v R
s_paraproduct_symbol_algebra | state | paraproduct symbol algebra
s_coifman_meyer_theorem | theorem | Coifman–Meyer multilinear multiplier theorem
s_multilinear_fourier_multiplier_symbol | axiom | multilinear fourier multiplier symbol
s_calderon_zygmund_condition_on_symbol | axiom | calderon zygmund condition on symbol
s_multilinear_operator_T_m | state | multilinear operator T m
s_multilinear_calderon_zygmund_kernel_estimate | state | multilinear calderon zygmund kernel estimate
s_gagliardo_nirenberg_interpolation | theorem | Gagliardo–Nirenberg interpolation inequality
s_lebesgue_norms_at_two_exponents | axiom | lebesgue norms at two exponents
s_scaling_invariance_constraint | state | scaling invariance constraint
s_two_parameter_interpolation_inequality | state | two parameter interpolation inequality
s_riesz_potential_kernel | axiom | riesz potential kernel
s_lebesgue_spaces_L_p_L_q | axiom | lebesgue spaces L p L q
s_dyadic_layer_decomposition | state | dyadic layer decomposition
s_weak_type_estimate_at_endpoints | state | weak type estimate at endpoints
s_aubin_talenti_sharp_sobolev | theorem | Aubin–Talenti sharp Sobolev constant
s_radial_decreasing_rearrangement | axiom | radial decreasing rearrangement
s_reduction_to_radial_decreasing_functions | state | reduction to radial decreasing functions
s_talenti_bubble_extremal_functions | state | talenti bubble extremal functions
s_brezis_lieb_lemma | theorem | Brezis–Lieb lemma
s_a_e_convergent_bounded_L_p_sequence | axiom | a e convergent bounded L p sequence
s_lebesgue_space_L_p | axiom | lebesgue space L p
s_pointwise_split_with_w_n_to_0_ae | state | pointwise split with w n to 0 ae
s_norm_convergence_of_difference_term | state | norm convergence of difference term
s_concentration_compactness_principle | theorem | Lions concentration-compactness principle
s_bounded_sequence_in_sobolev_space | axiom | bounded sequence in sobolev space
s_levy_concentration_function | axiom | levy concentration function
s_levy_concentration_function_witness | state | levy concentration function witness
s_concentration_trichotomy | state | concentration trichotomy
s_trace_theorem_sobolev | theorem | Trace theorem for Sobolev spaces
s_sobolev_space_W_1_p_on_domain | axiom | sobolev space W 1 p on domain
s_lipschitz_boundary | axiom | lipschitz boundary
s_trace_on_smooth_dense_subset | state | trace on smooth dense subset
s_trace_operator_to_W_1_minus_1_p_p | state | trace operator to W 1 minus 1 p p
s_poincare_inequality | theorem | Poincaré inequality on bounded domain
s_sobolev_space_W_1_p_zero_boundary_value | axiom | sobolev space W 1 p zero boundary value
s_pointwise_bound_in_terms_of_gradient | state | pointwise bound in terms of gradient
s_korn_inequality | theorem | Korn's inequality
s_vector_valued_sobolev_space | axiom | vector valued sobolev space
s_korn_identity_at_full_space | state | korn identity at full space
s_korn_on_bounded_domain | state | korn on bounded domain
s_strichartz_estimates_schrodinger | theorem | Strichartz estimates (Schrödinger)
s_free_schrodinger_propagator_e_it_delta | axiom | free schrodinger propagator e it delta
s_admissible_pair_q_r | axiom | admissible pair q r
s_dispersive_L_infty_to_L_1_estimate | state | dispersive L infty to L 1 estimate
s_TT_star_bilinear_estimate | state | TT star bilinear estimate
s_space_time_L_q_t_L_r_x_bound | state | space time L q t L r x bound
s_strichartz_estimates_wave | theorem | Strichartz estimates (wave)
s_free_wave_propagator_cos_t_sqrt_minus_delta | axiom | free wave propagator cos t sqrt minus delta
s_wave_admissible_pair | axiom | wave admissible pair
s_wave_dispersive_decay_estimate | state | wave dispersive decay estimate
s_wave_TT_star_bilinear_estimate | state | wave TT star bilinear estimate
s_kato_strichartz_non_trapping | theorem | Kato non-trapping Strichartz
s_non_trapping_metric_perturbation | axiom | non trapping metric perturbation
s_local_smoothing_estimate_kato_type | state | local smoothing estimate kato type
s_perturbed_strichartz | state | perturbed strichartz
s_nls_local_well_posedness | theorem | Local well-posedness of nonlinear Schrödinger (NLS)
s_duhamel_form_of_NLS | state | duhamel form of NLS
s_unique_local_solution_in_strichartz_space | state | unique local solution in strichartz space
s_nls_global_subcritical | theorem | Global existence of energy-subcritical NLS via energy conservation
s_defocusing_subcritical_nonlinearity | axiom | defocusing subcritical nonlinearity
s_energy_conservation_for_NLS | state | energy conservation for NLS
s_a_priori_global_H_1_bound | state | a priori global H 1 bound
s_lax_pair_for_KdV | theorem | Lax pair / integrable structure for KdV
s_schrodinger_operator_L_with_potential_u | axiom | schrodinger operator L with potential u
s_lax_pair_LP | state | lax pair LP
s_isospectral_evolution_of_L | state | isospectral evolution of L
s_inverse_scattering_transform_KdV | theorem | Inverse scattering transform for KdV
s_decaying_initial_data | axiom | decaying initial data
s_time_evolution_of_scattering_data | state | time evolution of scattering data
s_potential_reconstruction_via_GLM | state | potential reconstruction via GLM
s_kdv_local_well_posedness_bourgain | theorem | Local well-posedness of KdV in low regularity (Bourgain spaces)
s_bourgain_space_X_s_b | axiom | bourgain space X s b
s_dyadic_bilinear_estimates_in_X_s_b | state | dyadic bilinear estimates in X s b
s_duhamel_form_in_bourgain_space | state | duhamel form in bourgain space
s_beale_kato_majda_criterion | theorem | Beale–Kato–Majda blow-up criterion
s_local_smooth_solution_of_euler | axiom | local smooth solution of euler
s_vorticity_omega | axiom | vorticity omega
s_log_sobolev_bound_on_velocity_gradient | state | log sobolev bound on velocity gradient
s_double_exponential_growth_bound | state | double exponential growth bound
s_local_existence_euler_3d | theorem | Local existence for incompressible Euler in 3D
s_incompressible_euler_equations | axiom | incompressible euler equations
s_sobolev_space_H_s_with_s_gt_5_2 | axiom | sobolev space H s with s gt 5 2
s_helmholtz_projected_euler_equation | state | helmholtz projected euler equation
s_approximate_smooth_solution | state | approximate smooth solution
s_uniform_H_s_bound_short_time | state | uniform H s bound short time
s_leray_hopf_weak_solutions | theorem | Leray–Hopf weak solutions of Navier–Stokes
s_navier_stokes_equations | axiom | navier stokes equations
s_L2_initial_data | axiom | L2 initial data
s_galerkin_ode_for_first_N_modes | state | galerkin ode for first N modes
s_uniform_energy_bound_on_galerkin | state | uniform energy bound on galerkin
s_convergent_subsequence_to_weak_solution | state | convergent subsequence to weak solution
s_ckn_partial_regularity | theorem | Caffarelli–Kohn–Nirenberg partial regularity for Navier–Stokes
s_suitable_weak_solution | axiom | suitable weak solution
s_local_energy_inequality | state | local energy inequality
s_epsilon_regularity_criterion | state | epsilon regularity criterion
s_vlasov_poisson_global_existence | theorem | Vlasov–Poisson global existence (Pfaffelmoser)
s_vlasov_poisson_system | axiom | vlasov poisson system
s_compactly_supported_smooth_initial_data | axiom | compactly supported smooth initial data
s_velocity_support_evolution_inequality | state | velocity support evolution inequality
s_double_exponential_growth_bound_on_Q | state | double exponential growth bound on Q
s_lions_perthame_vlasov | theorem | Lions–Perthame global existence for Vlasov–Poisson
s_finite_velocity_moment_initial_data | axiom | finite velocity moment initial data
s_evolution_inequality_for_velocity_moments | state | evolution inequality for velocity moments
s_global_bound_on_velocity_moments | state | global bound on velocity moments
s_galerkin_parabolic_existence | theorem | Galerkin existence for parabolic PDE
s_parabolic_pde_in_divergence_form | axiom | parabolic pde in divergence form
s_finite_dim_ode_approximation | state | finite dim ode approximation
s_uniform_a_priori_bounds_in_L2_H_1 | state | uniform a priori bounds in L2 H 1
s_lions_monotone_operator_theorem | theorem | Lions monotone-operator existence theorem
s_monotone_hemicontinuous_coercive_operator | axiom | monotone hemicontinuous coercive operator
s_finite_dim_brouwer_existence | state | finite dim brouwer existence
s_weak_limit_in_reflexive_space | state | weak limit in reflexive space
s_minty_lemma | theorem | Minty's lemma on monotone operators
s_monotone_operator_on_hilbert_space | axiom | monotone operator on hilbert space
s_hemicontinuity | axiom | hemicontinuity
s_pairing_against_test_in_monotone_form | state | pairing against test in monotone form
s_hamilton_short_time_ricci_flow | theorem | Hamilton short-time existence for Ricci flow
s_deturck_strictly_parabolic_system | state | deturck strictly parabolic system
s_short_time_solution_of_deturck_system | state | short time solution of deturck system
s_deturck_trick | theorem | DeTurck trick
s_diffeomorphism_gauge_field | axiom | diffeomorphism gauge field
s_gauge_orbit_of_metrics | state | gauge orbit of metrics
s_deturck_modified_ricci_flow | state | deturck modified ricci flow
s_eells_sampson_harmonic_map_flow | theorem | Eells–Sampson harmonic map heat flow existence
s_smooth_map_between_riemannian_manifolds | axiom | smooth map between riemannian manifolds
s_target_with_nonpositive_sectional_curvature | axiom | target with nonpositive sectional curvature
s_dirichlet_energy_for_maps | state | dirichlet energy for maps
s_bochner_subharmonicity_of_energy_density | state | bochner subharmonicity of energy density
s_long_time_existence_of_harmonic_heat_flow | state | long time existence of harmonic heat flow
s_schoen_uhlenbeck_regularity | theorem | Schoen–Uhlenbeck regularity for harmonic maps
s_minimizing_harmonic_map | axiom | minimizing harmonic map
s_target_riemannian_manifold | axiom | target riemannian manifold
s_tangent_map_homogeneous_minimizer | state | tangent map homogeneous minimizer
s_epsilon_regularity_lemma | state | epsilon regularity lemma
s_mcf_short_time_existence | theorem | Mean curvature flow short-time existence
s_smooth_hypersurface_in_R_n_plus_1 | axiom | smooth hypersurface in R n plus 1
s_mean_curvature_vector_H | axiom | mean curvature vector H
s_graphical_mcf_quasilinear_parabolic_equation | state | graphical mcf quasilinear parabolic equation
s_short_time_graphical_solution | state | short time graphical solution
s_huisken_monotonicity_formula | theorem | Huisken's monotonicity formula for MCF
s_backward_heat_kernel_in_R_n_plus_1 | axiom | backward heat kernel in R n plus 1
s_gaussian_density_function_theta | state | gaussian density function theta
s_schoen_yau_positive_mass | theorem | Schoen–Yau positive mass theorem
s_asymptotically_flat_3_manifold | axiom | asymptotically flat 3 manifold
s_nonnegative_scalar_curvature | axiom | nonnegative scalar curvature
s_minimal_surface_argument | state | minimal surface argument
s_contradiction_via_stable_minimal_surface | state | contradiction via stable minimal surface
s_yang_mills_heat_flow_existence | theorem | Yang–Mills heat flow existence
s_yang_mills_functional | axiom | yang mills functional
s_principal_bundle_over_compact_4_manifold | axiom | principal bundle over compact 4 manifold
s_coulomb_gauge_yang_mills_flow | state | coulomb gauge yang mills flow
s_yang_mills_energy_a_priori_bound | state | yang mills energy a priori bound
s_uhlenbeck_compactness_yang_mills | theorem | Uhlenbeck compactness for Yang–Mills
s_sequence_of_connections_with_L_2_bounded_curvature | axiom | sequence of connections with L 2 bounded curvature
s_coulomb_gauge_representatives | state | coulomb gauge representatives
s_weak_limit_of_connections_modulo_gauge | state | weak limit of connections modulo gauge
s_lewy_unsolvability | theorem | Lewy's example of unsolvable linear PDE
s_lewy_operator_d_dz_minus_2i_z_bar_d_dt | axiom | lewy operator d dz minus 2i z bar d dt
s_smooth_inhomogeneity | axiom | smooth inhomogeneity
s_cr_operator_realization | state | cr operator realization
s_contradiction_via_hartogs_extension | state | contradiction via hartogs extension
s_malgrange_ehrenpreis_theorem | theorem | Malgrange–Ehrenpreis theorem
s_constant_coefficient_linear_pde_operator | axiom | constant coefficient linear pde operator
s_compactly_supported_distribution | axiom | compactly supported distribution
s_division_in_paley_wiener_space | state | division in paley wiener space
s_fundamental_solution_E | state | fundamental solution E
s_poincare_hopf_theorem | theorem | Poincaré–Hopf for vector fields
s_smooth_vector_field_with_isolated_zeros | axiom | smooth vector field with isolated zeros
s_local_index_count | state | local index count
s_global_index_equals_euler_characteristic | state | global index equals euler characteristic
s_direct_method_calculus_of_variations | theorem | Direct method of calculus of variations
s_coercive_lower_semicontinuous_functional | axiom | coercive lower semicontinuous functional
s_bounded_minimizing_sequence | state | bounded minimizing sequence
s_weak_limit_candidate_minimizer | state | weak limit candidate minimizer
s_euler_lagrange_equation | theorem | Euler–Lagrange equation derivation
s_compactly_supported_variation | axiom | compactly supported variation
s_first_variation_formula | state | first variation formula
s_weak_form_with_arbitrary_test_function | state | weak form with arbitrary test function
s_pohozaev_identity | theorem | Pohozaev identity
s_semilinear_elliptic_equation | axiom | semilinear elliptic equation
s_starshaped_bounded_domain | axiom | starshaped bounded domain
s_multiplied_equation_by_x_dot_grad_u | state | multiplied equation by x dot grad u
s_boundary_volume_identity | state | boundary volume identity
s_mountain_pass_theorem | theorem | Mountain pass theorem
s_C_1_functional_with_palais_smale | axiom | C 1 functional with palais smale
s_two_separated_local_minima | axiom | two separated local minima
s_minimax_value_c_equals_inf_max_J_on_gamma | state | minimax value c equals inf max J on gamma
s_palais_smale_sequence_at_level_c | state | palais smale sequence at level c
s_pohozaev_nonexistence_supercritical | theorem | Pohozaev nonexistence for supercritical Yamabe-type equations
s_supercritical_semilinear_elliptic | axiom | supercritical semilinear elliptic
s_signed_boundary_integral | state | signed boundary integral
s_yamabe_problem_solution | theorem | Sobolev–Talenti–Aubin Yamabe problem solution
s_subcritical_minimizers_u_p | state | subcritical minimizers u p
s_gidas_ni_nirenberg_liouville | theorem | Liouville theorem for nonnegative elliptic solutions in R^n
s_semilinear_elliptic_in_R_n | axiom | semilinear elliptic in R n
s_nonnegative_classical_solution | axiom | nonnegative classical solution
s_radial_symmetry_via_moving_planes | state | radial symmetry via moving planes
s_radial_ode_problem | state | radial ode problem
s_gidas_ni_nirenberg_symmetry | theorem | Gidas–Ni–Nirenberg radial symmetry
s_positive_solution_in_ball | axiom | positive solution in ball
s_reflected_solution_u_lambda | state | reflected solution u lambda
s_solution_symmetric_about_critical_plane | state | solution symmetric about critical plane
s_lax_equivalence_theorem | theorem | Lax equivalence theorem
s_well_posed_linear_initial_value_problem | axiom | well posed linear initial value problem
s_consistent_finite_difference_scheme | axiom | consistent finite difference scheme
s_three_property_axiom_definition | state | three property axiom definition
s_stability_implies_uniform_boundedness | state | stability implies uniform boundedness
s_von_neumann_stability_criterion | theorem | Von Neumann stability analysis
s_constant_coefficient_finite_difference_scheme | axiom | constant coefficient finite difference scheme
s_periodic_grid | axiom | periodic grid
s_cfl_criterion | theorem | CFL convergence criterion for hyperbolic schemes
s_explicit_finite_difference_for_hyperbolic_pde | axiom | explicit finite difference for hyperbolic pde
s_characteristic_propagation_cone | axiom | characteristic propagation cone
s_numerical_domain_of_dependence | state | numerical domain of dependence
s_lax_wendroff_convergence | theorem | Lax–Wendroff convergence theorem for conservation laws
s_consistent_conservative_finite_difference_scheme | axiom | consistent conservative finite difference scheme
s_a_priori_BV_bound | axiom | a priori BV bound
s_convergent_subsequence_to_function_v | state | convergent subsequence to function v
s_v_is_weak_entropy_solution | state | v is weak entropy solution
s_kruzhkov_uniqueness_theorem | theorem | Kruzhkov uniqueness for scalar conservation laws
s_kruzhkov_entropy_pair | axiom | kruzhkov entropy pair
s_kruzhkov_doubling_variables_technique | state | kruzhkov doubling variables technique
s_L1_contraction_for_entropy_solutions | state | L1 contraction for entropy solutions
s_vanishing_viscosity_convergence | theorem | Vanishing viscosity convergence
s_viscous_approximation_u_epsilon | axiom | viscous approximation u epsilon
s_BV_uniform_bound | state | BV uniform bound
s_convergent_subsequence | state | convergent subsequence
s_hille_yosida_theorem | theorem | Hille–Yosida theorem for semigroups
s_densely_defined_closed_linear_operator | axiom | densely defined closed linear operator
s_resolvent_bound_on_real_axis | axiom | resolvent bound on real axis
s_generated_C_0_semigroup | state | generated C 0 semigroup
s_lumer_phillips_theorem | theorem | Lumer–Phillips theorem for dissipative operators
s_dissipative_operator_on_hilbert_space | axiom | dissipative operator on hilbert space
s_range_condition_lambda_A_surjective | axiom | range condition lambda A surjective
s_contraction_property_of_resolvent | state | contraction property of resolvent
s_contraction_semigroup_generation | state | contraction semigroup generation
s_trotter_kato_product_formula | theorem | Trotter–Kato product formula
s_two_semigroup_generators_A_B | axiom | two semigroup generators A B
s_sum_A_plus_B_generates_semigroup | axiom | sum A plus B generates semigroup
s_trotter_splitting_approximant | state | trotter splitting approximant
s_kato_rellich_theorem | theorem | Kato–Rellich self-adjointness theorem
s_symmetric_relatively_bounded_perturbation_B | axiom | symmetric relatively bounded perturbation B
s_relatively_bounded_sum | state | relatively bounded sum
s_essentially_self_adjointness_of_sum | state | essentially self adjointness of sum
s_weyl_law | theorem | Weyl's law for eigenvalue counting
s_dirichlet_laplacian_on_bounded_domain | axiom | dirichlet laplacian on bounded domain
s_heat_kernel_trace_asymptotic | state | heat kernel trace asymptotic
s_agmon_decay_theorem | theorem | Agmon decay of eigenfunctions
s_schrodinger_operator_minus_delta_plus_V | axiom | schrodinger operator minus delta plus V
s_eigenfunction_below_essential_spectrum | axiom | eigenfunction below essential spectrum
s_agmon_riemannian_distance | state | agmon riemannian distance
s_weighted_energy_estimate | state | weighted energy estimate
s_lax_phillips_scattering | theorem | Lax–Phillips scattering theory
s_unperturbed_unitary_group | axiom | unperturbed unitary group
s_perturbed_unitary_group_with_compactly_supported_obstacle | axiom | perturbed unitary group with compactly supported obstacle
s_translation_representation_via_D_minus_D_plus | state | translation representation via D minus D plus
s_wave_operators_intertwining | state | wave operators intertwining
s_birman_schwinger_principle | theorem | Birman–Schwinger principle
s_negative_eigenvalue_minus_E | axiom | negative eigenvalue minus E
s_birman_schwinger_operator_K_E | state | birman schwinger operator K E
s_clr_bound | theorem | Cwikel–Lieb–Rozenblum bound on bound states
s_negative_part_V_minus_in_L_n_2 | axiom | negative part V minus in L n 2
s_negative_eigenvalues_via_birman_schwinger | state | negative eigenvalues via birman schwinger
s_trace_norm_bound_on_K_0 | state | trace norm bound on K 0
s_aronson_gaussian_bounds | theorem | Aronson Gaussian bounds for fundamental solution
s_divergence_form_parabolic_with_L_infty_coeff | axiom | divergence form parabolic with L infty coeff
s_holder_regularity_of_fundamental_solution | state | holder regularity of fundamental solution
s_two_sided_gaussian_estimate | state | two sided gaussian estimate
s_li_yau_differential_harnack | theorem | Li–Yau differential Harnack inequality
s_positive_solution_of_heat_equation_on_riemannian_manifold | axiom | positive solution of heat equation on riemannian manifold
s_nonneg_ricci_curvature | axiom | nonneg ricci curvature
s_li_yau_gradient_quantity_F | state | li yau gradient quantity F
s_max_principle_bound_on_F | state | max principle bound on F
s_bakry_emery_criterion | theorem | Bakry–Émery curvature-dimension criterion
s_diffusion_generator_L_with_carre_du_champ | axiom | diffusion generator L with carre du champ
s_gamma_2_operator | state | gamma 2 operator
s_CD_K_N_inequality | state | CD K N inequality
s_otto_gradient_flow_heat | theorem | Otto's gradient flow interpretation of heat equation
s_wasserstein_2_metric_on_probability_measures | axiom | wasserstein 2 metric on probability measures
s_jko_time_discretization | state | jko time discretization
s_continuous_time_gradient_flow | state | continuous time gradient flow
s_viscosity_solutions_theory | theorem | Crandall–Lions viscosity solutions of Hamilton–Jacobi
s_continuous_hamiltonian_H | axiom | continuous hamiltonian H
s_definition_of_viscosity_sub_super_solution | state | definition of viscosity sub super solution
s_doubling_variables_comparison | state | doubling variables comparison
s_evans_krylov_regularity | theorem | Evans–Krylov C^{2,α} regularity for fully nonlinear PDE
s_concave_fully_nonlinear_uniformly_elliptic_pde | axiom | concave fully nonlinear uniformly elliptic pde
s_continuous_viscosity_solution | axiom | continuous viscosity solution
s_second_difference_subsolution_property | state | second difference subsolution property
s_holder_continuity_of_D_2_u | state | holder continuity of D 2 u
s_caffarelli_monge_ampere_regularity | theorem | Caffarelli C^{2,α} regularity for Monge–Ampère
s_strictly_convex_alexandrov_solution | axiom | strictly convex alexandrov solution
s_C_alpha_right_hand_side_f | axiom | C alpha right hand side f
s_normalized_section_of_solution | state | normalized section of solution
s_C2_alpha_estimate_on_sections | state | C2 alpha estimate on sections
s_allard_regularity_theorem | theorem | Allard regularity theorem for varifolds
s_integral_varifold_with_bounded_mean_curvature | axiom | integral varifold with bounded mean curvature
s_small_density_excess_at_point | axiom | small density excess at point
s_flat_tangent_cone | state | flat tangent cone
s_lipschitz_graph_approximation | state | lipschitz graph approximation
s_toponogov_comparison_theorem | theorem | Toponogov triangle comparison
s_complete_riemannian_manifold_with_lower_sectional_curvature_bound | axiom | complete riemannian manifold with lower sectional curvature bound
s_geodesic_triangle | axiom | geodesic triangle
s_geodesic_length_comparison | state | geodesic length comparison
s_cheeger_inequality | theorem | Cheeger isoperimetric–spectral gap inequality
s_coarea_inequality | state | coarea inequality
s_lambda_1_geq_h_squared_over_4 | state | lambda 1 geq h squared over 4
s_cheeger_gromov_compactness | theorem | Cheeger–Gromov compactness theorem
s_sequence_of_riemannian_manifolds_with_curvature_volume_bounds | axiom | sequence of riemannian manifolds with curvature volume bounds
s_pointed_riemannian_manifolds | axiom | pointed riemannian manifolds
s_uniform_harmonic_atlas | state | uniform harmonic atlas
s_subsequential_C_1_alpha_convergence | state | subsequential C 1 alpha convergence
s_hamilton_ricci_flow_compactness | theorem | Hamilton's compactness theorem for Ricci flows
s_sequence_of_pointed_ricci_flows_with_uniform_curvature_bound | axiom | sequence of pointed ricci flows with uniform curvature bound
s_injectivity_radius_lower_bound | axiom | injectivity radius lower bound
s_C_infty_subsequential_metric_limit | state | C infty subsequential metric limit
s_C_infty_smooth_limit_flow | state | C infty smooth limit flow
s_perelman_W_monotonicity | theorem | Perelman entropy monotonicity (W-functional)
s_perelman_W_functional | axiom | perelman W functional
s_coupled_flow_with_conjugate_heat | state | coupled flow with conjugate heat
s_perelman_no_local_collapsing | theorem | Perelman no-local-collapsing theorem
s_W_lower_bound_in_terms_of_initial_data | state | W lower bound in terms of initial data
s_hamilton_tensor_max_principle | theorem | Hamilton tensor maximum principle
s_symmetric_tensor_satisfying_parabolic_inequality | axiom | symmetric tensor satisfying parabolic inequality
s_convex_invariant_K | state | convex invariant K
s_brakke_flow_existence | theorem | Brakke mean-curvature flow weak existence
s_initial_integral_varifold | axiom | initial integral varifold
s_brakke_inequality_for_varifolds | axiom | brakke inequality for varifolds
s_elliptic_regularization_approximation | state | elliptic regularization approximation
s_subsequential_varifold_limit | state | subsequential varifold limit
s_implicit_function_theorem_banach | theorem | Implicit function theorem (Banach version)
s_C1_map_between_banach_spaces | axiom | C1 map between banach spaces
s_invertible_partial_derivative_at_a_point | axiom | invertible partial derivative at a point
s_fixed_point_equation_for_y | state | fixed point equation for y
s_nash_moser_inverse_function_theorem | theorem | Nash–Moser inverse function theorem
s_tame_frechet_space_map | axiom | tame frechet space map
s_tame_estimate_for_inverse_of_linearization | axiom | tame estimate for inverse of linearization
s_smoothing_operator_family | state | smoothing operator family
s_quadratically_convergent_iteration | state | quadratically convergent iteration
s_lasalle_invariance_principle | theorem | Krasovskii–LaSalle invariance principle
s_smooth_dynamical_system | axiom | smooth dynamical system
s_lyapunov_function_V_with_V_dot_le_0 | axiom | lyapunov function V with V dot le 0
s_V_monotonically_nonincreasing | state | V monotonically nonincreasing
s_omega_limit_inside_set_V_dot_equal_0 | state | omega limit inside set V dot equal 0
s_massera_converse_lyapunov | theorem | Massera converse Lyapunov theorem
s_uniformly_asymptotically_stable_equilibrium | axiom | uniformly asymptotically stable equilibrium
s_integral_lyapunov_function_candidate | state | integral lyapunov function candidate
s_smale_horseshoe_theorem | theorem | Smale horseshoe / chaos theorem
s_transverse_homoclinic_orbit | axiom | transverse homoclinic orbit
s_smooth_diffeomorphism | axiom | smooth diffeomorphism
s_horseshoe_invariant_set | state | horseshoe invariant set
s_topological_conjugacy_to_bernoulli_shift | state | topological conjugacy to bernoulli shift
s_arnold_tongues_theorem | theorem | Arnold tongues / mode-locking theorem
s_one_parameter_family_of_circle_maps | axiom | one parameter family of circle maps
s_resonant_rotation_number | axiom | resonant rotation number
s_resonance_tongue_in_parameter_space | state | resonance tongue in parameter space
s_tongue_structure_with_devils_staircase | state | tongue structure with devils staircase
s_weak_law_large_numbers | theorem | Weak Law of Large Numbers (Khintchine)
s_iid_sequence_finite_mean | axiom | iid sequence finite mean
s_characteristic_function_of_sample_mean | state | characteristic function of sample mean
s_phi_n_to_exp_itmu | state | phi n to exp itmu
s_strong_law_large_numbers | theorem | Strong Law of Large Numbers (Kolmogorov)
s_truncated_iid_sequence | state | truncated iid sequence
s_partial_sum_variance_bound | state | partial sum variance bound
s_subsequence_a_s_convergence | state | subsequence a s convergence
s_etemadi_slln | theorem | Etemadi's Strong Law
s_pairwise_iid_finite_mean | axiom | pairwise iid finite mean
s_nonnegative_pairwise_iid | state | nonnegative pairwise iid
s_subseq_partial_sums | state | subseq partial sums
s_subseq_a_s_convergence | state | subseq a s convergence
s_multivariate_clt | theorem | Multivariate CLT
s_iid_random_vectors_finite_cov | axiom | iid random vectors finite cov
s_one_d_projection_iid | state | one d projection iid
s_clt_along_every_direction | state | clt along every direction
s_berry_esseen | theorem | Berry–Esseen Theorem
s_iid_sequence_finite_third_moment | axiom | iid sequence finite third moment
s_characteristic_function_with_third_moment_bound | state | characteristic function with third moment bound
s_phi_minus_gaussian_pointwise_bound | state | phi minus gaussian pointwise bound
s_kolmogorov_distance_bound_C_rho_over_sqrtn | state | kolmogorov distance bound C rho over sqrtn
s_donsker_invariance | theorem | Donsker's Invariance Principle
s_skorokhod_space_DC01 | axiom | skorokhod space DC01
s_polygonal_partial_sum_process | state | polygonal partial sum process
s_finite_dim_marginals_gaussian | state | finite dim marginals gaussian
s_tightness_in_C01 | state | tightness in C01
s_skorokhod_embedding | theorem | Skorokhod Embedding
s_mean_zero_distribution_finite_variance | axiom | mean zero distribution finite variance
s_brownian_motion | axiom | brownian motion
s_two_point_optional_law | state | two point optional law
s_brownian_stopping_time_matches_two_point | state | brownian stopping time matches two point
s_skorokhod_representation | theorem | Skorokhod Representation Theorem
s_weakly_convergent_sequence_of_laws | axiom | weakly convergent sequence of laws
s_inverse_cdf_couplings | state | inverse cdf couplings
s_uniform_coupling_existence | state | uniform coupling existence
s_kolmogorov_three_series | theorem | Kolmogorov Three-Series Theorem
s_independent_real_random_variables | axiom | independent real random variables
s_truncated_independent_series | state | truncated independent series
s_variance_sum_finite_iff_a_s_conv | state | variance sum finite iff a s conv
s_hewitt_savage_zero_one_law | theorem | Hewitt–Savage 0–1 Law
s_iid_sequence | axiom | iid sequence
s_exchangeable_sigma_algebra | axiom | exchangeable sigma algebra
s_exchangeable_events_under_permutation | state | exchangeable events under permutation
s_cylinder_approximation_of_exchangeable_event | state | cylinder approximation of exchangeable event
s_borel_cantelli_first | theorem | Borel–Cantelli Lemma (First)
s_summable_event_probabilities | axiom | summable event probabilities
s_tail_union_bound_goes_to_zero | state | tail union bound goes to zero
s_borel_cantelli_second | theorem | Borel–Cantelli Lemma (Second)
s_independent_events_divergent_sum | axiom | independent events divergent sum
s_product_of_complements_bound | state | product of complements bound
s_finite_intersections_of_complements_to_zero | state | finite intersections of complements to zero
s_conditional_borel_cantelli | theorem | Conditional Borel–Cantelli (Lévy)
s_filtration_F_n | axiom | filtration F n
s_adapted_event_sequence_A_n | axiom | adapted event sequence A n
s_compensated_martingale | state | compensated martingale
s_a_s_limit_of_M_n | state | a s limit of M n
s_doob_martingale_convergence_as | theorem | Doob's Martingale Convergence (a.s.)
s_L1_bounded_martingale | axiom | L1 bounded martingale
s_filtered_probability_space | axiom | filtered probability space
s_upcrossing_inequality | state | upcrossing inequality
s_finite_upcrossings_a_s | state | finite upcrossings a s
s_doob_Lp_convergence | theorem | Doob L^p Martingale Convergence
s_Lp_bounded_martingale | axiom | Lp bounded martingale
s_a_s_limit_X_infty | state | a s limit X infty
s_uniform_integrability_of_Lp_martingale | state | uniform integrability of Lp martingale
s_doob_maximal_inequality | theorem | Doob's Maximal Inequality
s_nonnegative_submartingale | axiom | nonnegative submartingale
s_stopped_submartingale_at_tau | state | stopped submartingale at tau
s_lambda_P_max_le_E_X_n_1_event | state | lambda P max le E X n 1 event
s_doob_meyer_decomposition | theorem | Doob–Meyer Decomposition
s_class_D_submartingale | axiom | class D submartingale
s_discrete_doob_decomposition_X_n | state | discrete doob decomposition X n
s_weak_limit_predictable_A_t | state | weak limit predictable A t
s_optional_stopping_theorem | theorem | Optional Stopping Theorem
s_uniformly_integrable_martingale | axiom | uniformly integrable martingale
s_bounded_stopping_time_tau | axiom | bounded stopping time tau
s_stopped_process_X_tau_wedge_n | state | stopped process X tau wedge n
s_constant_expectation_along_stop | state | constant expectation along stop
s_doob_upcrossing_inequality | theorem | Doob's Upcrossing Inequality
s_submartingale_X_n | axiom | submartingale X n
s_betting_strategy_C_n | state | betting strategy C n
s_gambler_winnings_lower_bound_b_minus_a_times_U | state | gambler winnings lower bound b minus a times U
s_wald_identity | theorem | Wald's Identity
s_stopping_time_finite_mean | axiom | stopping time finite mean
s_centered_random_walk_martingale | state | centered random walk martingale
s_E_S_tau_minus_mu_tau_eq_0 | state | E S tau minus mu tau eq 0
s_wald_variance_identity | theorem | Wald's Second Identity (variance)
s_second_order_martingale | state | second order martingale
s_E_centered_sq_eq_sigma2_E_tau | state | E centered sq eq sigma2 E tau
s_wald_likelihood_ratio_identity | theorem | Wald's Likelihood-Ratio Identity (SPRT)
s_iid_likelihood_ratio_sequence | axiom | iid likelihood ratio sequence
s_stopping_rule_for_sprt | axiom | stopping rule for sprt
s_lr_martingale_under_null | state | lr martingale under null
s_E_lr_at_tau_eq_1 | state | E lr at tau eq 1
s_burkholder_davis_gundy | theorem | Burkholder–Davis–Gundy Inequality
s_local_martingale_M_t | axiom | local martingale M t
s_quadratic_variation_M | axiom | quadratic variation M
s_good_lambda_inequality_pair | state | good lambda inequality pair
s_distribution_function_comparison | state | distribution function comparison
s_khintchine_inequality | theorem | Khintchine Inequality
s_rademacher_random_signs_epsilon_i | axiom | rademacher random signs epsilon i
s_real_sequence_a_i | axiom | real sequence a i
s_phi_sum_eps_i_a_i_explicit | state | phi sum eps i a i explicit
s_subgaussian_moment_bound | state | subgaussian moment bound
s_levy_characterization_bm | theorem | Lévy's Characterization of Brownian Motion
s_continuous_local_martingale_M_t | axiom | continuous local martingale M t
s_quadratic_variation_equal_t | axiom | quadratic variation equal t
s_exponential_local_martingale_e_ixiM | state | exponential local martingale e ixiM
s_phi_t_evolves_as_minus_xi_squared_over_2 | state | phi t evolves as minus xi squared over 2
s_levy_modulus_continuity | theorem | Lévy Modulus of Continuity for BM
s_dyadic_increment_array | state | dyadic increment array
s_borel_cantelli_bound_on_increments | state | borel cantelli bound on increments
s_lil_hartman_wintner | theorem | Law of Iterated Logarithm (Hartman–Wintner)
s_random_walk_as_BM_at_stopping_times | state | random walk as BM at stopping times
s_rescaled_random_walk | state | rescaled random walk
s_lil_for_BM | state | lil for BM
s_ito_isometry | theorem | Itô Isometry
s_progressive_L2_integrand_H | axiom | progressive L2 integrand H
s_simple_predictable_step_process | state | simple predictable step process
s_step_integral_L2_norm_eq_integral_H_squared | state | step integral L2 norm eq integral H squared
s_cameron_martin_theorem | theorem | Cameron–Martin Theorem
s_wiener_measure_on_C0 | axiom | wiener measure on C0
s_cameron_martin_space_H1 | axiom | cameron martin space H1
s_translation_of_wiener_path | state | translation of wiener path
s_radon_nikodym_density_explicit | state | radon nikodym density explicit
s_kolmogorov_backward_equation | theorem | Kolmogorov Backward Equation
s_markov_diffusion_X_t | axiom | markov diffusion X t
s_smooth_terminal_payoff_f | axiom | smooth terminal payoff f
s_expected_payoff_function_u | state | expected payoff function u
s_martingale_in_initial_time | state | martingale in initial time
s_fokker_planck_equation | theorem | Kolmogorov Forward (Fokker–Planck) Equation
s_density_p_t_x | axiom | density p t x
s_adjoint_operator_L_star | state | adjoint operator L star
s_weak_form_for_density | state | weak form for density
s_stratonovich_ito_conversion | theorem | Stratonovich–Itô Conversion
s_ito_integral_definition | axiom | ito integral definition
s_stratonovich_integral_definition | axiom | stratonovich integral definition
s_midpoint_correction_term | state | midpoint correction term
s_half_partial_f_partial_x_times_dX_squared | state | half partial f partial x times dX squared
s_sarkozy_theorem | theorem | Sárközy's Theorem (square-difference)
s_square_step_set | axiom | square step set
s_furstenberg_system_with_n_squared_action | state | furstenberg system with n squared action
s_spectral_resolution_for_polynomial_returns | state | spectral resolution for polynomial returns
s_cramer_large_deviations | theorem | Cramér's Theorem (large deviations)
s_iid_sequence_finite_mgf | axiom | iid sequence finite mgf
s_cumulant_generating_function | state | cumulant generating function
s_rate_function_Lambda_star | state | rate function Lambda star
s_upper_bound_LDP | state | upper bound LDP
s_gartner_ellis_theorem | theorem | Gärtner–Ellis Theorem
s_sequence_random_variables_with_log_mgf_limit | axiom | sequence random variables with log mgf limit
s_essential_smoothness_of_limit | axiom | essential smoothness of limit
s_abstract_cumulant_generator_Lambda | state | abstract cumulant generator Lambda
s_rate_function_Lambda_star_general | state | rate function Lambda star general
s_schilder_theorem | theorem | Schilder's Theorem
s_small_noise_brownian_path | state | small noise brownian path
s_path_density_under_shift | state | path density under shift
s_freidlin_wentzell_theorem | theorem | Freidlin–Wentzell Theorem
s_sde_with_small_noise_eps_sigma | axiom | sde with small noise eps sigma
s_drift_vector_field_b | axiom | drift vector field b
s_brownian_LDP_lifted | state | brownian LDP lifted
s_continuous_image_LDP | state | continuous image LDP
s_sanov_theorem | theorem | Sanov's Theorem
s_iid_sequence_on_polish_space | axiom | iid sequence on polish space
s_empirical_measure_L_n | axiom | empirical measure L n
s_type_class_partition | state | type class partition
s_KL_divergence_emerges_as_rate | state | KL divergence emerges as rate
s_azuma_hoeffding | theorem | Azuma–Hoeffding Inequality
s_martingale_with_bounded_differences | axiom | martingale with bounded differences
s_per_step_subgaussian_bound | state | per step subgaussian bound
s_exp_lambda_M_n_is_supermartingale | state | exp lambda M n is supermartingale
s_mcdiarmid_inequality | theorem | McDiarmid's Inequality
s_independent_inputs_X_i | axiom | independent inputs X i
s_function_with_bounded_differences_c_i | axiom | function with bounded differences c i
s_doob_filtration_martingale | state | doob filtration martingale
s_bounded_differences_for_martingale | state | bounded differences for martingale
s_talagrand_concentration | theorem | Talagrand's Concentration Inequality
s_product_probability_space | axiom | product probability space
s_convex_lipschitz_function_F | axiom | convex lipschitz function F
s_talagrand_convex_distance | state | talagrand convex distance
s_isoperimetric_inequality_for_product | state | isoperimetric inequality for product
s_gaussian_log_sobolev | theorem | Logarithmic Sobolev Inequality (Gaussian)
s_standard_gaussian_measure_gamma | axiom | standard gaussian measure gamma
s_smooth_function_f | axiom | smooth function f
s_ornstein_uhlenbeck_semigroup | state | ornstein uhlenbeck semigroup
s_bakry_emery_CD_inequality_for_OU | state | bakry emery CD inequality for OU
s_erdos_renyi_giant_component | theorem | Erdős–Rényi Giant Component Threshold
s_erdos_renyi_graph_G_n_p | axiom | erdos renyi graph G n p
s_branching_approximation_of_local_neighborhood | state | branching approximation of local neighborhood
s_phase_transition_at_lambda_eq_1 | state | phase transition at lambda eq 1
s_erdos_renyi_connectedness_threshold | theorem | Erdős–Rényi Connectedness Threshold
s_isolated_vertex_count_X_n | state | isolated vertex count X n
s_poisson_limit_for_isolated_vertices | state | poisson limit for isolated vertices
s_bollobas_zero_one_law | theorem | Bollobás 0–1 Law for Random Graphs
s_first_order_graph_property_phi | axiom | first order graph property phi
s_almost_sure_extension_axioms | state | almost sure extension axioms
s_limit_is_complete_theory_of_rado | state | limit is complete theory of rado
s_wigner_semicircle_law | theorem | Wigner Semicircle Law
s_wigner_matrix | axiom | wigner matrix
s_empirical_spectral_distribution | state | empirical spectral distribution
s_moments_match_catalan_numbers | state | moments match catalan numbers
s_marchenko_pastur_law | theorem | Marchenko–Pastur Law
s_iid_rectangular_matrix_X | axiom | iid rectangular matrix X
s_aspect_ratio_lambda | axiom | aspect ratio lambda
s_sample_covariance_matrix_W | state | sample covariance matrix W
s_self_consistent_equation_for_m_z | state | self consistent equation for m z
s_tracy_widom_distribution | theorem | Tracy–Widom Distribution (largest eigenvalue)
s_gaussian_unitary_ensemble | axiom | gaussian unitary ensemble
s_largest_eigenvalue_lambda_max | axiom | largest eigenvalue lambda max
s_determinantal_point_process_at_edge | state | determinantal point process at edge
s_rescaled_edge_point_process | state | rescaled edge point process
s_voiculescu_free_clt | theorem | Voiculescu's Free CLT
s_freely_independent_random_variables | axiom | freely independent random variables
s_free_probability_space | axiom | free probability space
s_R_transform_of_a_i | state | R transform of a i
s_additivity_of_R_under_freeness | state | additivity of R under freeness
s_R_transform_limit_eq_z | state | R transform limit eq z
s_asymptotic_freeness_GUE | theorem | Asymptotic Freeness of Independent GUE
s_independent_GUE_pair_X_N_Y_N | axiom | independent GUE pair X N Y N
s_normalized_trace_tau | axiom | normalized trace tau
s_mixed_moment_diagrams | state | mixed moment diagrams
s_planar_diagrams_only | state | planar diagrams only
s_yamada_watanabe_uniqueness | theorem | Yamada–Watanabe Pathwise Uniqueness Theorem
s_sde_with_1_d_diffusion_coefficient | axiom | sde with 1 d diffusion coefficient
s_holder_half_diffusion_coeff | axiom | holder half diffusion coeff
s_smooth_approximations_to_abs_value | state | smooth approximations to abs value
s_local_time_at_zero_vanishes | state | local time at zero vanishes
s_krylov_estimate | theorem | Krylov's Estimate for SDEs
s_uniformly_elliptic_diffusion | axiom | uniformly elliptic diffusion
s_borel_function_f | axiom | borel function f
s_PDE_solution_to_obstacle_problem | state | PDE solution to obstacle problem
s_expectation_le_norm_f_Ld_bound | state | expectation le norm f Ld bound
s_dynkin_formula | theorem | Dynkin's Formula
s_markov_process_X_t_with_generator_A | axiom | markov process X t with generator A
s_smooth_compactly_supported_f | axiom | smooth compactly supported f
s_dynkin_martingale | state | dynkin martingale
s_E_f_X_tau_minus_E_int_A_f_eq_f_x | state | E f X tau minus E int A f eq f x
s_hunt_theorem_potential | theorem | Hunt's Theorem on Potential
s_transient_markov_process | axiom | transient markov process
s_potential_kernel_U | axiom | potential kernel U
s_capacity_via_hitting_probability | state | capacity via hitting probability
s_dual_potential_kernel_U_hat | state | dual potential kernel U hat
s_spitzer_identity | theorem | Spitzer's Identity
s_iid_step_distribution_F | axiom | iid step distribution F
s_random_walk_S_n | axiom | random walk S n
s_running_maxima_M_n | state | running maxima M n
s_wiener_hopf_factorization | state | wiener hopf factorization
s_polya_recurrence | theorem | Pólya's Recurrence Theorem
s_simple_random_walk_on_Z_d | axiom | simple random walk on Z d
s_characteristic_function_of_step | state | characteristic function of step
s_greens_function_at_origin_integral | state | greens function at origin integral
s_ballot_theorem | theorem | Ballot Theorem (Bertrand)
s_two_candidate_election_count_p_q | axiom | two candidate election count p q
s_sequential_vote_counts | axiom | sequential vote counts
s_ballot_path_encoding | state | ballot path encoding
s_cycle_lemma_count | state | cycle lemma count
s_catalan_via_bertrand | theorem | Catalan Numbers via Bertrand
s_dyck_paths_2n | axiom | dyck paths 2n
s_combinatorial_axioms | axiom | combinatorial axioms
s_set_of_bad_paths | state | set of bad paths
s_bijection_bad_to_unconstrained | state | bijection bad to unconstrained
s_stein_chen_poisson | theorem | Stein–Chen Poisson Approximation
s_sum_of_dependent_indicators_W | axiom | sum of dependent indicators W
s_target_poisson_mean_lambda | axiom | target poisson mean lambda
s_stein_equation_for_poisson | state | stein equation for poisson
s_lipschitz_bounds_on_stein_solution | state | lipschitz bounds on stein solution
s_stein_method_gaussian | theorem | Stein's Method for Gaussian Approximation
s_sum_of_dependent_random_variables_W | axiom | sum of dependent random variables W
s_normal_target_N_0_1 | axiom | normal target N 0 1
s_gaussian_stein_equation | state | gaussian stein equation
s_test_function_bounds | state | test function bounds
s_stein_coupling_yields_error_terms | state | stein coupling yields error terms
s_chebyshev_inequality | theorem | Chebyshev's Inequality
s_random_variable_finite_variance | axiom | random variable finite variance
s_indicator_event_upper_bound | state | indicator event upper bound
s_pointwise_bound_for_indicator | state | pointwise bound for indicator
s_markov_inequality | theorem | Markov's Inequality
s_nonnegative_random_variable | axiom | nonnegative random variable
s_indicator_dominated_by_X_over_a | state | indicator dominated by X over a
s_jensen_inequality | theorem | Jensen's Inequality
s_convex_function_phi | axiom | convex function phi
s_integrable_random_variable_X | axiom | integrable random variable X
s_tangent_line_minorant_at_mean | state | tangent line minorant at mean
s_pointwise_tangent_bound | state | pointwise tangent bound
s_levy_continuity_theorem | theorem | Lévy's Continuity Theorem
s_sequence_of_probability_measures_mu_n | axiom | sequence of probability measures mu n
s_characteristic_functions_phi_n | axiom | characteristic functions phi n
s_pointwise_phi_limit_phi | state | pointwise phi limit phi
s_tightness_of_mu_n | state | tightness of mu n
s_helly_selection_theorem | theorem | Helly's Selection Theorem
s_uniformly_bounded_monotone_function_sequence | axiom | uniformly bounded monotone function sequence
s_pointwise_limit_on_dense_set | state | pointwise limit on dense set
s_prokhorov_theorem | theorem | Prokhorov's Theorem
s_tight_family_of_probability_measures | axiom | tight family of probability measures
s_tightness_compact_set_family | state | tightness compact set family
s_weakly_convergent_subsequence | state | weakly convergent subsequence
s_portmanteau_theorem | theorem | Portmanteau Theorem
s_metric_space | axiom | metric space
s_weak_convergence_definition | axiom | weak convergence definition
s_equivalent_test_classes | state | equivalent test classes
s_limsup_le_open_liminf_ge_closed | state | limsup le open liminf ge closed
s_dkw_inequality | theorem | Dvoretzky–Kiefer–Wolfowitz Inequality
s_iid_real_random_variables_with_cdf_F | axiom | iid real random variables with cdf F
s_empirical_cdf_F_n | axiom | empirical cdf F n
s_kmt_strong_approximation | state | kmt strong approximation
s_subgaussian_tail_for_KS_statistic | state | subgaussian tail for KS statistic
s_kolmogorov_smirnov_law | theorem | Kolmogorov–Smirnov Test Statistic Law
s_empirical_process | state | empirical process
s_brownian_bridge_limit | state | brownian bridge limit
s_cochran_theorem | theorem | Cochran's Theorem
s_iid_gaussian_vector_n | axiom | iid gaussian vector n
s_orthogonal_decomposition_of_quadratic_forms | axiom | orthogonal decomposition of quadratic forms
s_orthogonal_idempotents_summing_to_I | state | orthogonal idempotents summing to I
s_each_quadratic_form_is_chi_squared | state | each quadratic form is chi squared
s_slutsky_theorem | theorem | Slutsky's Theorem
s_X_n_weak_to_X | axiom | X n weak to X
s_Y_n_to_constant_c | axiom | Y n to constant c
s_joint_convergence_pair | state | joint convergence pair
s_apply_continuous_mapping_theorem | state | apply continuous mapping theorem
s_continuous_mapping_theorem | theorem | Continuous Mapping Theorem
s_X_n_to_X_in_some_mode | axiom | X n to X in some mode
s_g_continuous_on_support_of_X | axiom | g continuous on support of X
s_portmanteau_test_function_class | state | portmanteau test function class
s_h_circ_g_bounded_continuous | state | h circ g bounded continuous
s_wiener_existence | theorem | Daniell–Kolmogorov Construction of Brownian Motion
s_gaussian_finite_dim_marginals_with_BM_covariance | axiom | gaussian finite dim marginals with BM covariance
s_continuity_modulus_estimates | axiom | continuity modulus estimates
s_law_on_path_space_without_continuity | state | law on path space without continuity
s_holder_continuous_version | state | holder continuous version
s_markov_chain_ergodic_theorem | theorem | Markov Chain Ergodic Theorem
s_irreducible_aperiodic_positive_recurrent_markov_chain | axiom | irreducible aperiodic positive recurrent markov chain
s_invariant_distribution_pi | axiom | invariant distribution pi
s_regeneration_at_atom_via_nummelin_splitting | state | regeneration at atom via nummelin splitting
s_iid_excursions_yield_LLN | state | iid excursions yield LLN
s_perron_frobenius_stochastic | theorem | Perron–Frobenius for Stochastic Matrices
s_finite_irreducible_stochastic_matrix_P | axiom | finite irreducible stochastic matrix P
s_probability_simplex | axiom | probability simplex
s_invariant_distribution_existence | state | invariant distribution existence
s_largest_eigenvalue_one_simple | state | largest eigenvalue one simple
s_de_finetti_theorem | theorem | De Finetti's Exchangeability Theorem
s_infinite_exchangeable_sequence | axiom | infinite exchangeable sequence
s_polish_state_space | axiom | polish state space
s_symmetric_law_under_permutations | state | symmetric law under permutations
s_random_empirical_limit_M | state | random empirical limit M
s_poisson_process_characterization | theorem | Poisson Process Existence and Characterization
s_rate_measure_lambda_on_sigma_finite_space | axiom | rate measure lambda on sigma finite space
s_independence_and_poisson_count_axioms | axiom | independence and poisson count axioms
s_atomic_partition_construction | state | atomic partition construction
s_finite_dim_distributions_match | state | finite dim distributions match
s_levy_khintchine_representation | theorem | Lévy–Khintchine Representation
s_levy_process_X_t | axiom | levy process X t
s_characteristic_function_phi_t | axiom | characteristic function phi t
s_infinitely_divisible_law | state | infinitely divisible law
s_jump_compensated_decomposition | state | jump compensated decomposition
s_ito_representation_theorem | theorem | Itô Representation Theorem (martingale representation)
s_brownian_filtration_F_t | axiom | brownian filtration F t
s_L2_F_T_random_variable | axiom | L2 F T random variable
s_dense_set_of_explicit_stochastic_integrals | state | dense set of explicit stochastic integrals
s_density_in_L2_F_T | state | density in L2 F T
s_clark_ocone_formula | theorem | Clark–Ocone Formula
s_malliavin_differentiable_random_variable_F | axiom | malliavin differentiable random variable F
s_malliavin_derivative_D_F | state | malliavin derivative D F
s_predictable_projection_of_D_F | state | predictable projection of D F
s_hoeffding_inequality | theorem | Hoeffding's Inequality
s_independent_bounded_random_variables | axiom | independent bounded random variables
s_per_variable_subgaussian_mgf_bound | state | per variable subgaussian mgf bound
s_mgf_of_sum_bound | state | mgf of sum bound
s_bernstein_inequality | theorem | Bernstein's Inequality
s_finite_variance_sum_sigma2 | axiom | finite variance sum sigma2
s_per_variable_bernstein_mgf_bound | state | per variable bernstein mgf bound
s_sum_log_mgf_bound | state | sum log mgf bound
s_chernoff_bound | theorem | Chernoff Bound
s_random_variable_with_mgf | axiom | random variable with mgf
s_exponential_markov_bound | state | exponential markov bound
s_doob_Lp_maximal_inequality | theorem | Doob's L^p Maximal Inequality
s_p_strictly_greater_than_1 | axiom | p strictly greater than 1
s_weak_type_1_1_for_maximum | state | weak type 1 1 for maximum
s_layer_cake_to_Lp_norm | state | layer cake to Lp norm
s_coupling_inequality | theorem | Lindvall's Coupling Inequality
s_two_random_variables_on_common_space | axiom | two random variables on common space
s_coupling_distribution | axiom | coupling distribution
s_coupling_time_tau | state | coupling time tau
s_tv_bound_via_coupling_time | state | tv bound via coupling time
s_kingman_coalescent | theorem | Kingman's Coalescent Existence
s_genealogy_under_neutral_evolution | axiom | genealogy under neutral evolution
s_haploid_population_model_N_haploids | axiom | haploid population model N haploids
s_rescaled_coalescence_time | state | rescaled coalescence time
s_pairwise_rate_in_limit | state | pairwise rate in limit
s_mixing_time_spectral_gap | theorem | Mixing Time and Spectral Gap
s_reversible_finite_markov_chain_P | axiom | reversible finite markov chain P
s_spectral_gap_gamma | axiom | spectral gap gamma
s_spectral_decomposition_of_P_in_L2_pi | state | spectral decomposition of P in L2 pi
s_L2_decay_bound | state | L2 decay bound
s_aldous_hoover_theorem | theorem | Aldous–Hoover Theorem (exchangeable arrays)
s_jointly_exchangeable_random_array | axiom | jointly exchangeable random array
s_double_index_exchangeability | state | double index exchangeability
s_latent_variable_representation | state | latent variable representation
s_strassen_theorem | theorem | Strassen's Theorem (coupling characterization)
s_two_probability_measures_mu_nu | axiom | two probability measures mu nu
s_metric_or_closed_relation | axiom | metric or closed relation
s_lp_duality_for_optimal_transport_form | state | lp duality for optimal transport form
s_optimal_coupling_exists | state | optimal coupling exists
s_kantorovich_rubinstein_duality | theorem | Kantorovich–Rubinstein Duality
s_two_probability_measures_with_finite_first_moment | axiom | two probability measures with finite first moment
s_polish_metric_space | axiom | polish metric space
s_W1_as_primal_optimal_transport | state | W1 as primal optimal transport
s_dual_problem_over_1_lipschitz | state | dual problem over 1 lipschitz
s_ehrhard_inequality | theorem | Brunn–Minkowski for Gaussian (Ehrhard inequality)
s_standard_gaussian_measure_in_R_n | axiom | standard gaussian measure in R n
s_two_borel_sets_A_B | axiom | two borel sets A B
s_gaussian_isoperimetric_profile | state | gaussian isoperimetric profile
s_one_dim_ehrhard_reduction | state | one dim ehrhard reduction
s_sle_existence | theorem | Stochastic Loewner Evolution Existence (SLE)
s_loewner_chordal_ODE | axiom | loewner chordal ODE
s_sle_kappa_driver | state | sle kappa driver
s_conformal_maps_g_t | state | conformal maps g t
s_kolmogorov_two_series_theorem | theorem | Two-Series Theorem (Kolmogorov)
s_independent_centered_random_variables | axiom | independent centered random variables
s_uniform_partial_sum_bound | state | uniform partial sum bound
s_a_s_convergence_via_levy_equivalence | state | a s convergence via levy equivalence
s_levy_equivalence_theorem | theorem | Lévy's Equivalence Theorem
s_sum_of_independent_random_variables | axiom | sum of independent random variables
s_symmetrized_independent_sum | state | symmetrized independent sum
s_max_partial_sum_concentration | state | max partial sum concentration
s_stein_lemma_gaussian | theorem | Stein's Lemma (Gaussian integration by parts)
s_standard_gaussian_Z | axiom | standard gaussian Z
s_absolutely_continuous_function_g | axiom | absolutely continuous function g
s_gaussian_density_ODE_identity | state | gaussian density ODE identity
s_IBP_with_gaussian_density | state | IBP with gaussian density
s_slepian_inequality | theorem | Slepian's Inequality (Gaussian comparison)
s_two_centered_gaussian_vectors_X_Y | axiom | two centered gaussian vectors X Y
s_pairwise_covariance_inequality | axiom | pairwise covariance inequality
s_gaussian_interpolation_path | state | gaussian interpolation path
s_derivative_sign_via_covariance_difference | state | derivative sign via covariance difference
s_borell_tis_inequality | theorem | Borell–TIS Inequality (Gaussian concentration of maxima)
s_centered_gaussian_process_X_t_separable | axiom | centered gaussian process X t separable
s_almost_sure_finite_sup | axiom | almost sure finite sup
s_sup_as_lipschitz_function_of_iid_gaussians | state | sup as lipschitz function of iid gaussians
s_lipschitz_gaussian_concentration_bound | state | lipschitz gaussian concentration bound
s_dudley_entropy_bound | theorem | Dudley's Entropy Bound
s_subgaussian_process_on_metric_space_T | axiom | subgaussian process on metric space T
s_metric_entropy_log_N_eps | axiom | metric entropy log N eps
s_dyadic_chaining_nets | state | dyadic chaining nets
s_chaining_subgaussian_bound | state | chaining subgaussian bound
s_donsker_varadhan_LDP | theorem | Donsker–Varadhan Variational Formula
s_ergodic_markov_process | axiom | ergodic markov process
s_empirical_occupation_measure | axiom | empirical occupation measure
s_variational_rate_function | state | variational rate function
s_DV_rate_function_eq_dirichlet_form | state | DV rate function eq dirichlet form
s_varadhan_lemma | theorem | Varadhan's Lemma
s_family_satisfying_LDP_with_rate_I | axiom | family satisfying LDP with rate I
s_continuous_bounded_function_F | axiom | continuous bounded function F
s_laplace_upper_bound | state | laplace upper bound
s_laplace_lower_bound | state | laplace lower bound
s_contraction_principle_LDP | theorem | Contraction Principle (large deviations)
s_family_satisfying_LDP_with_rate_I_on_X | axiom | family satisfying LDP with rate I on X
s_continuous_map_F_X_to_Y | axiom | continuous map F X to Y
s_push_forward_family_on_Y | state | push forward family on Y
s_LDP_with_inverse_image_rate | state | LDP with inverse image rate
s_blackwell_renewal_theorem | theorem | Renewal Theorem (Blackwell)
s_iid_positive_inter_arrival_times | axiom | iid positive inter arrival times
s_renewal_function_U_t | axiom | renewal function U t
s_stationary_renewal_coupling | state | stationary renewal coupling
s_TV_distance_to_stationary_renewal_zero | state | TV distance to stationary renewal zero
s_kesten_stigum_theorem | theorem | Kesten–Stigum Theorem (Branching processes)
s_supercritical_galton_watson_process | axiom | supercritical galton watson process
s_x_log_x_moment_condition | axiom | x log x moment condition
s_normalized_population_martingale_W_n | state | normalized population martingale W n
s_W_infty_nondegenerate | state | W infty nondegenerate
s_robbins_monro_convergence | theorem | Stochastic Approximation (Robbins–Monro) Convergence
s_root_finding_problem_h_x_eq_0 | axiom | root finding problem h x eq 0
s_step_size_sequence_alpha_n_with_robbins_monro_conditions | axiom | step size sequence alpha n with robbins monro conditions
s_lyapunov_for_robbins_monro | state | lyapunov for robbins monro
s_supermartingale_convergence_for_V_n | state | supermartingale convergence for V n
s_strassen_functional_lil | theorem | Functional Law of the Iterated Logarithm (Strassen)
s_brownian_embedding_of_random_walk | state | brownian embedding of random walk
s_rescaled_brownian_paths | state | rescaled brownian paths
s_cluster_set_is_strassen_ball | state | cluster set is strassen ball
s_karhunen_loeve_expansion | theorem | Karhunen–Loève Expansion
s_L2_centered_stochastic_process_with_continuous_covariance | axiom | L2 centered stochastic process with continuous covariance
s_hilbert_schmidt_kernel | axiom | hilbert schmidt kernel
s_eigenfunction_eigenvalue_decomposition | state | eigenfunction eigenvalue decomposition
s_uncorrelated_coefficient_sequence | state | uncorrelated coefficient sequence
s_urysohn_metrization_theorem | theorem | Urysohn metrization theorem
s_countable_separating_family | state | countable separating family
s_embedding_into_hilbert_cube | state | embedding into hilbert cube
s_nagata_smirnov_metrization | theorem | Nagata–Smirnov metrization theorem
s_regular_hausdorff_space | axiom | regular hausdorff space
s_sigma_locally_finite_basis | axiom | sigma locally finite basis
s_sigma_locally_finite_basis_present | state | sigma locally finite basis present
s_countable_family_of_pseudometrics | state | countable family of pseudometrics
s_compatible_metric | state | compatible metric
s_bing_metrization | theorem | Bing metrization theorem
s_sigma_discrete_basis_present | state | sigma discrete basis present
s_nagata_smirnov_form | state | nagata smirnov form
s_stone_cech_compactification | theorem | Stone–Čech compactification
s_tychonoff_space | axiom | tychonoff space
s_evaluation_into_product_of_intervals | state | evaluation into product of intervals
s_compact_hausdorff_closure_beta_X | state | compact hausdorff closure beta X
s_alexandroff_one_point_compactification | theorem | Alexandroff one-point compactification
s_x_plus_infinity_set | state | x plus infinity set
s_one_point_topology | state | one point topology
s_lebesgue_number_lemma | theorem | Lebesgue number lemma
s_open_cover_of_compact_set | axiom | open cover of compact set
s_finite_subcover_with_radii | state | finite subcover with radii
s_positive_lebesgue_number_delta | state | positive lebesgue number delta
s_pasting_lemma | theorem | Pasting lemma
s_topological_space | axiom | topological space
s_finite_closed_cover_with_continuous_pieces | axiom | finite closed cover with continuous pieces
s_preimage_of_closed_is_closed_per_piece | state | preimage of closed is closed per piece
s_heine_cantor_theorem | theorem | Heine–Cantor theorem (uniform continuity on compacts)
s_continuous_function_on_compact_metric_space | axiom | continuous function on compact metric space
s_epsilon_continuity_finite_subcover | state | epsilon continuity finite subcover
s_alexander_subbase_lemma | theorem | Alexander subbase lemma
s_subbase_of_topology | axiom | subbase of topology
s_maximal_subbase_anticover_via_zorn | state | maximal subbase anticover via zorn
s_michael_selection_theorem | theorem | Michael selection theorem
s_paracompact_hausdorff_space | axiom | paracompact hausdorff space
s_lower_semicontinuous_convex_valued_correspondence | axiom | lower semicontinuous convex valued correspondence
s_epsilon_approximate_selection | state | epsilon approximate selection
s_cauchy_sequence_of_selections | state | cauchy sequence of selections
s_sorgenfrey_not_metrizable | theorem | Sorgenfrey line not metrizable
s_sorgenfrey_line | axiom | sorgenfrey line
s_sorgenfrey_squared_not_lindelof | state | sorgenfrey squared not lindelof
s_hilbert_cube_embedding | theorem | Embedding into Hilbert cube
s_compact_metrizable_space | axiom | compact metrizable space
s_countable_dense_subset_of_X | state | countable dense subset of X
s_distance_to_dense_coords | state | distance to dense coords
s_compact_iff_sequentially_compact_metric | theorem | Compactness ⇔ sequential compactness in metric spaces
s_totally_bounded_and_complete | state | totally bounded and complete
s_cauchy_subsequence_extraction | state | cauchy subsequence extraction
s_cantor_bendixson_theorem | theorem | Cantor–Bendixson theorem
s_cantor_bendixson_derivatives | state | cantor bendixson derivatives
s_perfect_kernel_plus_countable_scattered | state | perfect kernel plus countable scattered
s_lusin_separation_theorem | theorem | Lusin separation theorem
s_pair_of_disjoint_analytic_sets | axiom | pair of disjoint analytic sets
s_souslin_scheme_separation | state | souslin scheme separation
s_borel_separating_set | state | borel separating set
s_sierpinski_continuum_theorem | theorem | Sierpiński theorem on continuum connectedness
s_compact_connected_hausdorff_space | axiom | compact connected hausdorff space
s_countable_closed_partition_hypothesis | state | countable closed partition hypothesis
s_contradiction_with_compactness | state | contradiction with compactness
s_borsuk_ulam_theorem | theorem | Borsuk–Ulam theorem
s_sphere_S_n | axiom | sphere S n
s_continuous_map_S_n_to_R_n | axiom | continuous map S n to R n
s_antipodal_free_map_to_S_n_minus_1 | state | antipodal free map to S n minus 1
s_Z2_equivariant_map_to_lower_sphere | state | Z2 equivariant map to lower sphere
s_ham_sandwich_theorem | theorem | Ham sandwich theorem
s_n_finite_measures_in_R_n | axiom | n finite measures in R n
s_signed_half_volume_map_on_S_n | state | signed half volume map on S n
s_odd_map_S_n_to_R_n | state | odd map S n to R n
s_lyusternik_schnirelmann_theorem | theorem | Lyusternik–Schnirelmann theorem
s_closed_cover_S_n_by_n_plus_1_sets | axiom | closed cover S n by n plus 1 sets
s_antipodal_separated_cover_hypothesis | state | antipodal separated cover hypothesis
s_distance_vector_map_S_n_to_R_n | state | distance vector map S n to R n
s_lefschetz_fixed_point_theorem | theorem | Lefschetz fixed-point theorem
s_compact_polyhedron | axiom | compact polyhedron
s_continuous_self_map_of_polyhedron | axiom | continuous self map of polyhedron
s_induced_map_on_H_star_Q | state | induced map on H star Q
s_lefschetz_number_L_f | state | lefschetz number L f
s_schauder_fixed_point_theorem | theorem | Schauder fixed-point theorem
s_compact_convex_subset_banach_space | axiom | compact convex subset banach space
s_continuous_self_map_on_convex_compact | axiom | continuous self map on convex compact
s_finite_dim_approx_self_map | state | finite dim approx self map
s_finite_dim_approx_fixed_points | state | finite dim approx fixed points
s_kakutani_fixed_point_theorem | theorem | Kakutani fixed-point theorem
s_compact_convex_subset_R_n | axiom | compact convex subset R n
s_upper_hemicontinuous_convex_valued_correspondence | axiom | upper hemicontinuous convex valued correspondence
s_continuous_selection_on_simplex | state | continuous selection on simplex
s_approximate_fixed_points | state | approximate fixed points
s_hairy_ball_theorem | theorem | Hairy ball theorem
s_even_sphere_S_2n | axiom | even sphere S 2n
s_tangent_vector_field | axiom | tangent vector field
s_nonvanishing_field_hypothesis | state | nonvanishing field hypothesis
s_homotopy_id_to_antipodal | state | homotopy id to antipodal
s_jordan_brouwer_separation | theorem | Jordan–Brouwer separation theorem
s_topological_n_minus_1_sphere_in_R_n | axiom | topological n minus 1 sphere in R n
s_n_dim_complement_of_sphere | state | n dim complement of sphere
s_two_components_via_duality | state | two components via duality
s_schoenflies_theorem | theorem | Schoenflies theorem
s_conformal_map_disk_to_inside | state | conformal map disk to inside
s_extension_to_boundary_homeomorphism | state | extension to boundary homeomorphism
s_hex_theorem | theorem | Hex theorem ⇒ Brouwer
s_hex_board_game | axiom | hex board game
s_winning_chain_existence | state | winning chain existence
s_sperner_lemma | theorem | Sperner's lemma
s_simplex_triangulation_with_sperner_labeling | axiom | simplex triangulation with sperner labeling
s_door_counting_argument | state | door counting argument
s_odd_number_of_full_simplices | state | odd number of full simplices
s_kkm_theorem | theorem | KKM theorem (Knaster–Kuratowski–Mazurkiewicz)
s_simplex_with_KKM_cover | axiom | simplex with KKM cover
s_sperner_labeled_triangulation_from_cover | state | sperner labeled triangulation from cover
s_fully_labeled_small_simplex | state | fully labeled small simplex
s_lebesgue_covering_dimension_R_n_equals_n | theorem | Topological dimension equals covering dimension
s_open_cover_with_orders | axiom | open cover with orders
s_nerve_of_cover_simplicial_complex | state | nerve of cover simplicial complex
s_no_refinement_below_order_n | state | no refinement below order n
s_pi_1_S1_equals_Z | theorem | Fundamental group of S¹ is ℤ
s_circle_S_1 | axiom | circle S 1
s_universal_cover_R_to_S_1 | axiom | universal cover R to S 1
s_lifted_path_with_integer_endpoint | state | lifted path with integer endpoint
s_winding_number_map_pi_1_to_Z | state | winding number map pi 1 to Z
s_pi_1_S_n_trivial | theorem | Fundamental group of S^n trivial for n ≥ 2
s_sphere_S_n_n_geq_2 | axiom | sphere S n n geq 2
s_two_hemispheres_cover | state | two hemispheres cover
s_seifert_van_kampen_theorem | theorem | Seifert–van Kampen theorem
s_path_connected_space | axiom | path connected space
s_open_cover_by_path_connected_opens_with_path_connected_intersection | axiom | open cover by path connected opens with path connected intersection
s_pushout_diagram_of_pi_1 | state | pushout diagram of pi 1
s_loop_decomposition_into_segments_in_pieces | state | loop decomposition into segments in pieces
s_universal_cover_existence | theorem | Universal cover existence
s_path_connected_locally_simply_connected_space | axiom | path connected locally simply connected space
s_homotopy_class_path_space | state | homotopy class path space
s_topologized_path_space_with_projection | state | topologized path space with projection
s_covering_space_galois_correspondence | theorem | Galois correspondence of covering spaces
s_fundamental_group | axiom | fundamental group
s_deck_group_action_on_universal_cover | state | deck group action on universal cover
s_classification_of_compact_surfaces | theorem | Classification of surfaces (Smale / Brahana)
s_compact_surface_without_boundary | axiom | compact surface without boundary
s_triangulated_compact_surface | state | triangulated compact surface
s_polygon_identification_form | state | polygon identification form
s_eilenberg_steenrod_uniqueness | theorem | Eilenberg–Steenrod axioms uniqueness
s_cw_complex_category | axiom | cw complex category
s_homotopy_excision_dimension_axioms | axiom | homotopy excision dimension axioms
s_axiom_system_on_finite_cw | state | axiom system on finite cw
s_inductive_determination_by_cells | state | inductive determination by cells
s_mayer_vietoris_sequence | theorem | Mayer–Vietoris sequence
s_open_cover_X_equals_A_union_B | axiom | open cover X equals A union B
s_short_exact_chain_sequence | state | short exact chain sequence
s_long_exact_sequence_in_homology | state | long exact sequence in homology
s_excision_theorem | theorem | Excision theorem
s_topological_pair_X_A | axiom | topological pair X A
s_subset_U_with_closure_in_interior_A | axiom | subset U with closure in interior A
s_small_chains_subdivision | state | small chains subdivision
s_subdivision_operator_chain_homotopy | state | subdivision operator chain homotopy
s_lefschetz_duality | theorem | Lefschetz duality
s_compact_oriented_manifold_with_boundary | axiom | compact oriented manifold with boundary
s_double_M_along_boundary | state | double M along boundary
s_duality_on_double | state | duality on double
s_kunneth_formula | theorem | Künneth formula
s_topological_spaces_X_Y | axiom | topological spaces X Y
s_pid_coefficients | axiom | pid coefficients
s_eilenberg_zilber_cross_product | state | eilenberg zilber cross product
s_tor_short_exact_sequence | state | tor short exact sequence
s_eilenberg_zilber_theorem | theorem | Eilenberg–Zilber theorem
s_AW_EZ_chain_maps | state | AW EZ chain maps
s_chain_homotopy_equivalence | state | chain homotopy equivalence
s_sheaf_de_rham_isomorphism | theorem | Sheaf-theoretic de Rham (Poincaré lemma + sheafification)
s_de_rham_complex | axiom | de rham complex
s_poincare_lemma_local_resolution | state | poincare lemma local resolution
s_sheaf_resolution_of_constant_sheaf | state | sheaf resolution of constant sheaf
s_cup_product_associative_graded_commutative | theorem | Künneth in cohomology / cup product associativity
s_singular_cochain_complex | axiom | singular cochain complex
s_cup_product_definition | state | cup product definition
s_kunneth_for_sheaves | theorem | Künneth for sheaf cohomology
s_compact_spaces_with_field_coefficients | axiom | compact spaces with field coefficients
s_sheafified_external_product | state | sheafified external product
s_cech_singular_iso_good_cover | theorem | Čech-to-singular isomorphism on good covers
s_good_cover_with_contractible_intersections | axiom | good cover with contractible intersections
s_nerve_simplicial_model | state | nerve simplicial model
s_cech_double_complex | state | cech double complex
s_hurewicz_theorem | theorem | Hurewicz theorem
s_n_connected_space_n_geq_1 | axiom | n connected space n geq 1
s_hurewicz_homomorphism | state | hurewicz homomorphism
s_cellular_representatives | state | cellular representatives
s_whitehead_theorem | theorem | Whitehead theorem
s_cw_complexes_X_Y | axiom | cw complexes X Y
s_weak_homotopy_equivalence_f | axiom | weak homotopy equivalence f
s_vanishing_obstructions_in_relative_pi_n | state | vanishing obstructions in relative pi n
s_homotopy_inverse_on_skeleta | state | homotopy inverse on skeleta
s_freudenthal_suspension_theorem | theorem | Freudenthal suspension theorem
s_n_connected_space_X | axiom | n connected space X
s_suspension_map_on_pi_k | state | suspension map on pi k
s_homotopy_excision_range | state | homotopy excision range
s_blakers_massey_theorem | theorem | Blakers–Massey theorem
s_pushout_square_in_spaces | axiom | pushout square in spaces
s_connectivity_hypotheses | axiom | connectivity hypotheses
s_homotopy_pushout_comparison | state | homotopy pushout comparison
s_total_fiber_connectivity_bound | state | total fiber connectivity bound
s_cellular_approximation | theorem | Cellular approximation theorem
s_continuous_map_f | axiom | continuous map f
s_image_meets_finitely_many_cells | state | image meets finitely many cells
s_homotopy_to_lower_skeleton | state | homotopy to lower skeleton
s_simplicial_approximation | theorem | Simplicial approximation theorem
s_simplicial_complexes_K_L | axiom | simplicial complexes K L
s_continuous_map_K_to_L | axiom | continuous map K to L
s_lebesgue_number_for_open_stars | state | lebesgue number for open stars
s_finely_subdivided_K | state | finely subdivided K
s_brown_representability | theorem | Brown representability theorem
s_homotopy_invariant_functor_on_pointed_cw | axiom | homotopy invariant functor on pointed cw
s_wedge_and_mayer_vietoris_axioms | axiom | wedge and mayer vietoris axioms
s_half_exact_functor | state | half exact functor
s_classifying_space_Y_universal_element | state | classifying space Y universal element
s_postnikov_tower_existence | theorem | Postnikov tower existence
s_connected_cw_complex_X | axiom | connected cw complex X
s_X_n_with_pi_k_killed_above_n | state | X n with pi k killed above n
s_k_invariant_obstruction | state | k invariant obstruction
s_K_G_n_represents_H_n_G | theorem | Eilenberg–MacLane spaces represent cohomology
s_abelian_group_G | axiom | abelian group G
s_cw_complex_X | axiom | cw complex X
s_K_G_n_space | state | K G n space
s_serre_spectral_sequence | theorem | Serre spectral sequence
s_serre_fibration | axiom | serre fibration
s_simply_connected_base | axiom | simply connected base
s_skeleton_filtration_of_total_space | state | skeleton filtration of total space
s_exact_couple_E_2_to_E_infinity | state | exact couple E 2 to E infinity
s_leray_hirsch_theorem | theorem | Leray–Hirsch theorem
s_fiber_bundle_F_to_E_to_B | axiom | fiber bundle F to E to B
s_classes_restrict_to_basis_on_fiber | axiom | classes restrict to basis on fiber
s_external_module_map | state | external module map
s_dold_thom_theorem | theorem | Dold–Thom theorem
s_pointed_cw_complex_X | axiom | pointed cw complex X
s_infinite_symmetric_product_SPX | state | infinite symmetric product SPX
s_SP_preserves_quasifibrations | state | SP preserves quasifibrations
s_eilenberg_ganea_theorem | theorem | Eilenberg–Ganea theorem
s_discrete_group_G | axiom | discrete group G
s_cohomological_dimension_n_geq_3 | axiom | cohomological dimension n geq 3
s_low_dim_skeleton_of_K_G_1 | state | low dim skeleton of K G 1
s_vanishing_obstruction_in_high_dim | state | vanishing obstruction in high dim
s_barratt_priddy_theorem | theorem | Barratt–Priddy theorem
s_symmetric_groups_Sigma_n | axiom | symmetric groups Sigma n
s_classifying_space_B_sigma_infty | state | classifying space B sigma infty
s_plus_construction_B_sigma_infty_plus | state | plus construction B sigma infty plus
s_milnor_lim_one_exact_sequence | theorem | Milnor exact sequence (lim¹)
s_inverse_system_of_groups | axiom | inverse system of groups
s_tower_of_fibrations | axiom | tower of fibrations
s_homotopy_inverse_limit_construction | state | homotopy inverse limit construction
s_quillen_theorem_A | theorem | Quillen's theorem A
s_functor_F_between_small_categories | axiom | functor F between small categories
s_comma_categories_contractible | axiom | comma categories contractible
s_bisimplicial_replacement | state | bisimplicial replacement
s_realization_homotopy_equivalence | state | realization homotopy equivalence
s_adams_spectral_sequence | theorem | Adams spectral sequence
s_spectrum_X | axiom | spectrum X
s_mod_p_cohomology_with_steenrod_action | axiom | mod p cohomology with steenrod action
s_adams_resolution_tower | state | adams resolution tower
s_E_2_equals_ext_over_steenrod_algebra | state | E 2 equals ext over steenrod algebra
s_nishida_nilpotence | theorem | Nishida nilpotence theorem
s_stable_homotopy_groups_of_spheres | axiom | stable homotopy groups of spheres
s_positive_degree_element | axiom | positive degree element
s_dyer_lashof_action_on_powers | state | dyer lashof action on powers
s_x_to_the_N_vanishes | state | x to the N vanishes
s_hopf_invariant_one_theorem | theorem | Hopf invariant one (Adams)
s_continuous_map_S_2n_minus_1_to_S_n | axiom | continuous map S 2n minus 1 to S n
s_hopf_invariant_H_f | state | hopf invariant H f
s_secondary_operation_obstruction | state | secondary operation obstruction
s_adams_vector_fields_theorem | theorem | Vector fields on spheres (Adams)
s_sphere_S_n_minus_1 | axiom | sphere S n minus 1
s_max_linearly_independent_vector_fields | axiom | max linearly independent vector fields
s_stiefel_manifold_K_theory | state | stiefel manifold K theory
s_adams_operation_obstruction_psi_k | state | adams operation obstruction psi k
s_adams_conjecture_image_of_J | theorem | J-homomorphism image (Adams conjecture / Quillen)
s_real_vector_bundle_xi | axiom | real vector bundle xi
s_adams_operation_psi_k | axiom | adams operation psi k
s_J_homomorphism_image | state | J homomorphism image
s_psi_k_minus_1_action | state | psi k minus 1 action
s_transversality_theorem | theorem | Transversality theorem
s_smooth_map_f_X_to_Y | axiom | smooth map f X to Y
s_submanifold_Z_of_Y | axiom | submanifold Z of Y
s_parametrized_family_F | state | parametrized family F
s_generic_parameter_makes_f_transverse | state | generic parameter makes f transverse
s_whitney_embedding_theorem | theorem | Whitney embedding theorem
s_smooth_n_manifold_M | axiom | smooth n manifold M
s_initial_immersion_into_high_R_N | state | initial immersion into high R N
s_generic_projection_remains_immersion | state | generic projection remains immersion
s_whitney_immersion_theorem | theorem | Whitney immersion theorem
s_normal_bundle_obstruction_class | state | normal bundle obstruction class
s_obstruction_vanishes | state | obstruction vanishes
s_ehresmann_lemma | theorem | Ehresmann's lemma
s_proper_smooth_submersion_f_M_to_N | axiom | proper smooth submersion f M to N
s_horizontal_distribution | state | horizontal distribution
s_flow_lifts_local_trivialization | state | flow lifts local trivialization
s_morse_lemma | theorem | Morse lemma
s_smooth_function_with_nondegenerate_critical_point | axiom | smooth function with nondegenerate critical point
s_quadratic_part_plus_remainder | state | quadratic part plus remainder
s_canonical_quadratic_form_in_charts | state | canonical quadratic form in charts
s_h_cobordism_theorem | theorem | h-cobordism theorem (Smale)
s_simply_connected_h_cobordism_dim_geq_6 | axiom | simply connected h cobordism dim geq 6
s_handle_decomposition_of_cobordism | state | handle decomposition of cobordism
s_handles_cancelled | state | handles cancelled
s_s_cobordism_theorem | theorem | s-cobordism theorem
s_h_cobordism_with_pi_1_nontrivial | axiom | h cobordism with pi 1 nontrivial
s_whitehead_torsion_tau | axiom | whitehead torsion tau
s_whitehead_torsion_obstruction | state | whitehead torsion obstruction
s_handles_cancellable_when_tau_zero | state | handles cancellable when tau zero
s_annulus_theorem | theorem | Annulus theorem (high-dim)
s_two_locally_flat_embedded_spheres_in_R_n_n_geq_5 | axiom | two locally flat embedded spheres in R n n geq 5
s_region_as_h_cobordism | state | region as h cobordism
s_disc_theorem | theorem | Disc theorem (Palais)
s_connected_smooth_manifold | axiom | connected smooth manifold
s_two_smooth_embeddings_of_disk_D_n | axiom | two smooth embeddings of disk D n
s_isotopy_aligning_basepoints | state | isotopy aligning basepoints
s_pontryagin_thom_iso | theorem | Pontryagin–Thom construction
s_framed_cobordism_class | axiom | framed cobordism class
s_collapse_map_to_thom_space | state | collapse map to thom space
s_stable_map_to_sphere_spectrum | state | stable map to sphere spectrum
s_thom_transversality | theorem | Thom transversality theorem
s_smooth_map_with_jet_extension | axiom | smooth map with jet extension
s_submanifold_W_in_jet_space | axiom | submanifold W in jet space
s_jet_section_construction | state | jet section construction
s_cerf_pseudoisotopy_theorem | theorem | Cerf's theorem (pseudoisotopy in high dim)
s_simply_connected_smooth_manifold_dim_geq_5 | axiom | simply connected smooth manifold dim geq 5
s_pseudoisotopy_F | axiom | pseudoisotopy F
s_cerf_graphic_of_critical_points | state | cerf graphic of critical points
s_simplified_to_no_critical_points | state | simplified to no critical points
s_donaldson_diagonalizability | theorem | Donaldson's theorem (intersection forms of smooth 4-manifolds)
s_closed_simply_connected_smooth_4_manifold | axiom | closed simply connected smooth 4 manifold
s_instanton_moduli_space_M | state | instanton moduli space M
s_compactified_5_dim_cobordism | state | compactified 5 dim cobordism
s_exotic_R_4_existence | theorem | Exotic R^4 (Freedman+Donaldson)
s_freedman_topological_4_manifolds | axiom | freedman topological 4 manifolds
s_topological_E8_manifold | state | topological E8 manifold
s_no_smooth_E8_via_donaldson | state | no smooth E8 via donaldson
s_rokhlin_theorem | theorem | Rokhlin's theorem
s_smooth_closed_oriented_spin_4_manifold | axiom | smooth closed oriented spin 4 manifold
s_dirac_operator_index_A_hat | state | dirac operator index A hat
s_index_equals_signature_over_16_with_quaternion_correction | state | index equals signature over 16 with quaternion correction
s_hirzebruch_signature_theorem | theorem | Hirzebruch signature theorem
s_closed_oriented_4k_manifold | axiom | closed oriented 4k manifold
s_signature_operator_on_forms | state | signature operator on forms
s_index_equals_L_genus_pairing | state | index equals L genus pairing
s_atiyah_bott_fixed_point | theorem | Atiyah–Bott fixed-point theorem
s_elliptic_complex | axiom | elliptic complex
s_smooth_endomorphism_with_simple_fixed_points | axiom | smooth endomorphism with simple fixed points
s_elliptic_lefschetz_number | state | elliptic lefschetz number
s_local_contribution_at_fixed_points | state | local contribution at fixed points
s_atiyah_hirzebruch_spectral_sequence | theorem | Atiyah–Hirzebruch spectral sequence
s_generalized_cohomology_theory_E | axiom | generalized cohomology theory E
s_skeleton_filtration_E_cohomology | state | skeleton filtration E cohomology
s_exact_couple_for_E | state | exact couple for E
s_bott_periodicity_complex | theorem | Bott periodicity (complex K-theory)
s_unitary_group_U_infinity | axiom | unitary group U infinity
s_morse_theory_on_loop_space_U | state | morse theory on loop space U
s_handle_decomposition_BU_times_Z | state | handle decomposition BU times Z
s_bott_periodicity_real | theorem | Bott periodicity (real K-theory, period 8)
s_orthogonal_group_O_infinity | axiom | orthogonal group O infinity
s_clifford_algebra_Cl_k | axiom | clifford algebra Cl k
s_clifford_8_periodicity | state | clifford 8 periodicity
s_homotopy_of_loop_O | state | homotopy of loop O
s_atiyah_segal_completion | theorem | Atiyah–Segal completion theorem
s_representation_ring_R_G | axiom | representation ring R G
s_equivariant_K_theory_EG | state | equivariant K theory EG
s_I_adic_completion_K_G | state | I adic completion K G
s_kuiper_theorem | theorem | Kuiper's theorem
s_infinite_dimensional_hilbert_space | axiom | infinite dimensional hilbert space
s_unitary_group_U_H | axiom | unitary group U H
s_eilenberg_swindle_homotopy | state | eilenberg swindle homotopy
s_U_H_contractible | state | U H contractible
s_loop_theorem | theorem | Loop theorem (Papakyriakopoulos)
s_3_manifold_M_with_boundary | axiom | 3 manifold M with boundary
s_essential_loop_in_boundary | axiom | essential loop in boundary
s_tower_of_branched_covers | state | tower of branched covers
s_terminal_tower_with_embedded_disk | state | terminal tower with embedded disk
s_sphere_theorem_3_manifolds | theorem | Sphere theorem (3-manifolds)
s_orientable_3_manifold_M_with_pi_2_nontrivial | axiom | orientable 3 manifold M with pi 2 nontrivial
s_least_area_sphere_representative | state | least area sphere representative
s_embedded_essential_sphere | state | embedded essential sphere
s_lickorish_wallace_theorem | theorem | Lickorish–Wallace theorem
s_closed_orientable_3_manifold | axiom | closed orientable 3 manifold
s_heegaard_splitting | state | heegaard splitting
s_dehn_surgery_description | state | dehn surgery description
s_gordon_luecke_theorem | theorem | Gordon–Luecke theorem (knot complements determine knots)
s_knot_K_in_S_3 | axiom | knot K in S 3
s_knot_complement_S_3_minus_K | axiom | knot complement S 3 minus K
s_hypothetical_nontrivial_surgery_giving_S_3 | state | hypothetical nontrivial surgery giving S 3
s_planar_surface_obstruction | state | planar surface obstruction
s_nielsen_thurston_classification | theorem | Nielsen–Thurston classification
s_mapping_class_of_closed_surface_genus_geq_2 | axiom | mapping class of closed surface genus geq 2
s_action_on_teichmuller_space | state | action on teichmuller space
s_trichotomy_of_actions | state | trichotomy of actions
s_smale_diff_S_2_homotopy_equiv_O_3 | theorem | Smale's theorem on 2-sphere diffeomorphisms
s_diffeomorphism_group_of_S_2 | axiom | diffeomorphism group of S 2
s_diff_S2_as_fiber_bundle | state | diff S2 as fiber bundle
s_hatcher_smale_conjecture_S_3 | theorem | Smale conjecture for S³ (Hatcher)
s_diffeomorphism_group_of_S_3 | axiom | diffeomorphism group of S 3
s_space_of_unknotted_2_spheres | state | space of unknotted 2 spheres
s_contractibility_of_space_of_spheres | state | contractibility of space of spheres
s_double_suspension_theorem | theorem | Double suspension theorem (Edwards–Cannon)
s_homology_3_sphere_H | axiom | homology 3 sphere H
s_single_suspension_with_two_singular_points | state | single suspension with two singular points
s_double_suspension_S_5_topologically | state | double suspension S 5 topologically
s_bing_recognition_theorem | theorem | Bing's recognition theorem for S³
s_compact_3_manifold_with_simply_connected_sphere_property | axiom | compact 3 manifold with simply connected sphere property
s_curve_in_3_cell_condition | state | curve in 3 cell condition
s_exhaustion_by_3_cells | state | exhaustion by 3 cells
s_fary_milnor_theorem | theorem | Fáry–Milnor theorem
s_smooth_closed_curve_in_R_3_with_total_curvature_leq_4pi | axiom | smooth closed curve in R 3 with total curvature leq 4pi
s_average_number_of_crossings_per_projection | state | average number of crossings per projection
s_projection_with_few_extrema | state | projection with few extrema
s_reeb_sphere_theorem | theorem | Reeb sphere theorem
s_closed_n_manifold_with_morse_function_with_exactly_2_critical_points | axiom | closed n manifold with morse function with exactly 2 critical points
s_two_disk_decomposition | state | two disk decomposition
s_phragmen_brouwer_theorem | theorem | Phragmén–Brouwer theorem
s_connected_locally_connected_normal_space | axiom | connected locally connected normal space
s_two_separating_continua | axiom | two separating continua
s_irreducible_separator | state | irreducible separator
s_contradicting_pair_of_continua | state | contradicting pair of continua
s_thurston_geometrization_haken | theorem | Geometrization / Thurston for Haken manifolds
s_haken_3_manifold | axiom | haken 3 manifold
s_jsj_decomposition | state | jsj decomposition
s_hyperbolic_structures_on_atoroidal_pieces | state | hyperbolic structures on atoroidal pieces
s_smith_conjecture | theorem | Smith conjecture
s_finite_cyclic_smooth_action_on_S_3_with_nonempty_fixed_set | axiom | finite cyclic smooth action on S 3 with nonempty fixed set
s_fixed_circle_K | state | fixed circle K
s_quotient_orbifold_geometric | state | quotient orbifold geometric
s_tameness_theorem | theorem | Tameness theorem (Agol–Calegari–Gabai)
s_hyperbolic_3_manifold_with_finitely_generated_pi_1 | axiom | hyperbolic 3 manifold with finitely generated pi 1
s_shrinkwrapped_minimal_surfaces | state | shrinkwrapped minimal surfaces
s_exhaustion_by_compact_cores | state | exhaustion by compact cores
s_kahn_markovic_surface_subgroup | theorem | Surface subgroup theorem (Kahn–Markovic)
s_closed_hyperbolic_3_manifold | axiom | closed hyperbolic 3 manifold
s_collection_of_pants | state | collection of pants
s_pants_pair_gluing_compatible | state | pants pair gluing compatible
s_virtually_haken_theorem | theorem | Virtually Haken theorem (Agol)
s_dual_cube_complex_action | state | dual cube complex action
s_finite_index_special_cover | state | finite index special cover
s_sullivan_minimal_model_theorem | theorem | Sullivan's structure theorem on rational homotopy
s_simply_connected_space_X_finite_type | axiom | simply connected space X finite type
s_PL_de_rham_algebra | state | PL de rham algebra
s_sullivan_minimal_model | state | sullivan minimal model
s_andreotti_frankel_theorem | theorem | Andreotti–Frankel theorem
s_smooth_affine_complex_variety_complex_dimension_n | axiom | smooth affine complex variety complex dimension n
s_real_distance_squared_morse_function | state | real distance squared morse function
s_index_bound_on_critical_points | state | index bound on critical points
s_hodge_decomposition_theorem | theorem | Hodge decomposition theorem
s_compact_kahler_manifold | axiom | compact kahler manifold
s_harmonic_forms_kernel_of_Laplacian | state | harmonic forms kernel of Laplacian
s_hodge_orthogonal_decomposition | state | hodge orthogonal decomposition
s_hodge_index_theorem | theorem | Hodge index theorem
s_smooth_projective_surface_X | axiom | smooth projective surface X
s_intersection_pairing_on_H_2 | state | intersection pairing on H 2
s_signature_of_intersection_pairing | state | signature of intersection pairing
s_smale_2_manifold_classification | theorem | Smale's classification of compact 2-manifolds (gradient flow)
s_closed_surface_with_morse_function | axiom | closed surface with morse function
s_handle_decomposition_of_surface | state | handle decomposition of surface
s_canonical_genus_handle_form | state | canonical genus handle form
s_quillen_plus_construction_K_theory | theorem | Quillen +-construction and algebraic K-theory
s_associative_ring_R | axiom | associative ring R
s_classifying_space_BGL_R | axiom | classifying space BGL R
s_BGL_R_plus_simply_connected_in_pi_1_quotient | state | BGL R plus simply connected in pi 1 quotient
s_group_completion_yields_K_R_spectrum | state | group completion yields K R spectrum
s_snaith_theorem | theorem | Snaith's theorem
s_infinite_loop_space_CP_infinity | axiom | infinite loop space CP infinity
s_bott_element_beta_in_K_0 | axiom | bott element beta in K 0
s_sigma_infty_CP_infty_bott_inverted | state | sigma infty CP infty bott inverted
s_anderson_kadec_theorem | theorem | Anderson–Kadec theorem
s_infinite_dimensional_separable_frechet_space | axiom | infinite dimensional separable frechet space
s_coordinate_maps_to_R_infty | state | coordinate maps to R infty
s_kuratowski_14_set_theorem | theorem | Kuratowski 14-set theorem
s_subset_A_of_X | axiom | subset A of X
s_free_monoid_closure_complement | state | free monoid closure complement
s_relations_collapse_to_14 | state | relations collapse to 14
s_topological_invariance_of_dimension | theorem | Topological invariance of Lebesgue covering dimension
s_homeomorphic_topological_spaces | axiom | homeomorphic topological spaces
s_covering_dimension_definition | state | covering dimension definition
s_dim_preserved_under_homeomorphism | state | dim preserved under homeomorphism
s_suspension_isomorphism_H_tilde | theorem | Suspension isomorphism (reduced homology)
s_pointed_space_X | axiom | pointed space X
s_reduced_suspension_sigma_X | axiom | reduced suspension sigma X
s_LES_cone_pair | state | LES cone pair
s_eilenberg_swindle_K_theory_vanishing | theorem | Eilenberg swindle / vanishing of K-theory of "infinite sum" rings
s_ring_admitting_countable_infinite_direct_sums | axiom | ring admitting countable infinite direct sums
s_swindle_isomorphism | state | swindle isomorphism
s_K_0_vanishes_via_swindle | state | K 0 vanishes via swindle
s_whitney_graustein_theorem | theorem | Whitney–Graustein theorem
s_immersion_of_circle_in_plane | axiom | immersion of circle in plane
s_winding_number_of_tangent | axiom | winding number of tangent
s_rotation_number_of_immersion | state | rotation number of immersion
s_complete_invariant_for_regular_homotopy | state | complete invariant for regular homotopy
s_smale_sphere_eversion | theorem | Smale's sphere eversion
s_immersion_S_2_in_R_3 | axiom | immersion S 2 in R 3
s_immersions_classified_by_homotopy_of_tangent_map | state | immersions classified by homotopy of tangent map
s_no_obstruction_to_homotopy_id_to_minus_id | state | no obstruction to homotopy id to minus id
s_hopf_classification_theorem | theorem | Hopf classification theorem
s_compact_connected_oriented_n_manifold | axiom | compact connected oriented n manifold
s_continuous_map_to_S_n | axiom | continuous map to S n
s_degree_of_map | state | degree of map
s_degree_complete_invariant | state | degree complete invariant
s_preimage_theorem | theorem | Preimage theorem (regular value theorem)
s_smooth_map_f_M_to_N | axiom | smooth map f M to N
s_regular_value_q_in_N | axiom | regular value q in N
s_local_submersion_form | state | local submersion form
s_tubular_neighborhood_theorem | theorem | Tubular neighborhood theorem
s_smooth_submanifold_S_of_M | axiom | smooth submanifold S of M
s_normal_bundle_NS | axiom | normal bundle NS
s_exp_normal_bundle_map | state | exp normal bundle map
s_local_diffeomorphism_on_small_disk_bundle | state | local diffeomorphism on small disk bundle
s_stiefel_whitney_classes_axioms | theorem | Stiefel–Whitney class characterization (axioms)
s_axiom_list_for_w_i | state | axiom list for w i
s_classes_in_H_i_M_F_2 | state | classes in H i M F 2
s_chern_classes_axioms | theorem | Chern class characterization (axioms)
s_complex_vector_bundle_eta | axiom | complex vector bundle eta
s_axiom_list_for_c_i | state | axiom list for c i
s_classes_in_H_2i_M_Z | state | classes in H 2i M Z
s_pontryagin_classes_definition_theorem | theorem | Pontryagin classes from Chern
s_complexification_xi_otimes_C | axiom | complexification xi otimes C
s_chern_classes_of_complexification | state | chern classes of complexification
s_pontryagin_class_p_k | state | pontryagin class p k
s_chern_weil_theorem | theorem | Chern–Weil theorem
s_principal_G_bundle_with_connection | axiom | principal G bundle with connection
s_invariant_polynomial_in_curvature | state | invariant polynomial in curvature
s_closed_de_rham_class_of_invariant_polynomial | state | closed de rham class of invariant polynomial
s_gauss_bonnet_chern_theorem | theorem | Gauss–Bonnet–Chern theorem
s_closed_oriented_riemannian_2n_manifold | axiom | closed oriented riemannian 2n manifold
s_pfaffian_of_curvature | state | pfaffian of curvature
s_pfaffian_represents_euler_class | state | pfaffian represents euler class
s_unoriented_bordism_via_thom_spectrum_MO | theorem | Bordism groups via Pontryagin–Thom
s_smooth_n_manifolds_modulo_cobordism | axiom | smooth n manifolds modulo cobordism
s_collapse_to_MO_k_thom_space | state | collapse to MO k thom space
s_stable_map_to_MO_spectrum | state | stable map to MO spectrum
s_thom_unoriented_bordism_calculation | theorem | Thom's calculation of unoriented bordism
s_MO_splits_as_HF_2_wedges | state | MO splits as HF 2 wedges
s_adem_relations_steenrod_algebra | theorem | Adem relations / Steenrod algebra structure
s_mod_p_cohomology_with_steenrod_operations | axiom | mod p cohomology with steenrod operations
s_sq_i_via_equivariant_diagonal | state | sq i via equivariant diagonal
s_relations_from_double_cyclic_structure | state | relations from double cyclic structure
s_spanier_whitehead_duality | theorem | Spanier–Whitehead duality
s_finite_cw_complex_X_embedded_in_S_n | axiom | finite cw complex X embedded in S n
s_spanier_whitehead_dual_DX | state | spanier whitehead dual DX
s_thom_isomorphism | theorem | Thom isomorphism theorem
s_oriented_real_vector_bundle_xi_rank_n | axiom | oriented real vector bundle xi rank n
s_thom_class_U | state | thom class U
s_cup_with_U_iso | state | cup with U iso
s_gysin_sequence | theorem | Gysin sequence
s_oriented_sphere_bundle_S_n_to_E_to_B | axiom | oriented sphere bundle S n to E to B
s_disk_sphere_pair_with_thom_iso | state | disk sphere pair with thom iso
s_LES_of_pair | state | LES of pair
s_wang_sequence | theorem | Wang sequence
s_fiber_bundle_with_base_S_n | axiom | fiber bundle with base S n
s_two_hemispheres_MV | state | two hemispheres MV
s_borel_equivariant_cohomology | theorem | Borel construction / equivariant cohomology
s_topological_group_G_acting_on_X | axiom | topological group G acting on X
s_borel_construction_EG_times_X | state | borel construction EG times X
s_equivariant_cohomology_is_module_over_H_BG | state | equivariant cohomology is module over H BG
s_lefschetz_coincidence_formula | theorem | Lefschetz coincidence / fixed-point trace formula (general)
s_continuous_map_pair_f_g_M_to_N | axiom | continuous map pair f g M to N
s_coincidence_intersection_number | state | coincidence intersection number
s_cauchy_riemann_characterization | theorem | Cauchy–Riemann equations characterize holomorphy
s_differentiable_complex_function | axiom | differentiable complex function
s_real_C1_function_pair_u_v | axiom | real C1 function pair u v
s_two_directional_limit_equality | state | two directional limit equality
s_cauchy_riemann_system | state | cauchy riemann system
s_cauchy_integral_theorem_general | theorem | Cauchy's theorem on simply connected domains (Goursat)
s_nested_shrinking_triangle_with_integral_bound | state | nested shrinking triangle with integral bound
s_integral_over_triangle_vanishes | state | integral over triangle vanishes
s_liouville_theorem | theorem | Liouville's theorem
s_bounded_entire_function | axiom | bounded entire function
s_cauchy_estimate_f_prime_bounded_by_M_over_R | state | cauchy estimate f prime bounded by M over R
s_f_prime_vanishes_everywhere | state | f prime vanishes everywhere
s_identity_theorem | theorem | Identity theorem
s_set_with_accumulation_point_in_domain | axiom | set with accumulation point in domain
s_power_series_with_zero_coefficients | state | power series with zero coefficients
s_open_subset_where_f_zero | state | open subset where f zero
s_maximum_modulus_principle | theorem | Maximum modulus principle
s_connected_open_subset_of_C | axiom | connected open subset of C
s_mean_value_property_for_f | state | mean value property for f
s_constant_on_neighborhood_of_max | state | constant on neighborhood of max
s_complex_open_mapping_theorem | theorem | Open mapping theorem (complex analysis)
s_nonconstant_holomorphic_function | axiom | nonconstant holomorphic function
s_local_normal_form_n_to_1 | state | local normal form n to 1
s_winding_number_n_for_small_disk_image | state | winding number n for small disk image
s_schwarz_pick_theorem | theorem | Schwarz–Pick theorem
s_holomorphic_self_map_of_unit_disk | axiom | holomorphic self map of unit disk
s_hyperbolic_metric_on_disk | axiom | hyperbolic metric on disk
s_normalized_problem_at_origin | state | normalized problem at origin
s_bound_at_origin | state | bound at origin
s_casorati_weierstrass | theorem | Casorati–Weierstrass theorem
s_holomorphic_function_with_essential_singularity | axiom | holomorphic function with essential singularity
s_punctured_disk_neighborhood | axiom | punctured disk neighborhood
s_1_over_f_minus_w_bounded_near_singularity | state | 1 over f minus w bounded near singularity
s_pole_or_removable_contradiction | state | pole or removable contradiction
s_jensen_formula | theorem | Jensen's formula
s_holomorphic_function_on_closed_disk | axiom | holomorphic function on closed disk
s_zeros_inside_disk | axiom | zeros inside disk
s_zero_free_factor_g | state | zero free factor g
s_mean_value_for_log_abs_g | state | mean value for log abs g
s_hadamard_three_lines_theorem | theorem | Hadamard three-lines theorem
s_holomorphic_function_on_vertical_strip | axiom | holomorphic function on vertical strip
s_bounded_function | axiom | bounded function
s_modified_function_with_uniform_bound_on_boundary | state | modified function with uniform bound on boundary
s_max_modulus_bound_on_strip | state | max modulus bound on strip
s_hadamard_three_circles_theorem | theorem | Hadamard three-circles theorem
s_holomorphic_function_on_annulus | axiom | holomorphic function on annulus
s_log_radius_substitution | axiom | log radius substitution
s_strip_version_of_problem | state | strip version of problem
s_borel_caratheodory_theorem | theorem | Borel–Carathéodory theorem
s_real_part_bound | axiom | real part bound
s_holomorphic_self_map_of_disk | state | holomorphic self map of disk
s_bound_on_phi_implies_bound_on_f | state | bound on phi implies bound on f
s_hadamard_factorization_theorem | theorem | Hadamard factorization theorem
s_entire_function_of_finite_order | axiom | entire function of finite order
s_zero_distribution_with_genus_p | axiom | zero distribution with genus p
s_canonical_product_with_zero_set | state | canonical product with zero set
s_zero_free_entire_with_growth_order | state | zero free entire with growth order
s_log_f_is_polynomial_of_degree_leq_rho | state | log f is polynomial of degree leq rho
s_runge_theorem | theorem | Runge's theorem
s_holomorphic_function_on_open_set | axiom | holomorphic function on open set
s_compact_subset_with_connected_complement | axiom | compact subset with connected complement
s_integral_representation_with_pole_at_each_point | state | integral representation with pole at each point
s_approximation_by_rational_functions_with_poles_in_complement | state | approximation by rational functions with poles in complement
s_koebe_quarter_theorem | theorem | Koebe 1/4 theorem
s_univalent_function_on_unit_disk | axiom | univalent function on unit disk
s_auxiliary_univalent_g_in_class_S | state | auxiliary univalent g in class S
s_coefficient_bound_abs_a2_leq_2 | state | coefficient bound abs a2 leq 2
s_schwarz_reflection_principle | theorem | Schwarz reflection principle
s_holomorphic_on_upper_half_disk_continuous_to_real_axis | axiom | holomorphic on upper half disk continuous to real axis
s_real_boundary_values | axiom | real boundary values
s_piecewise_function_on_full_disk | state | piecewise function on full disk
s_holomorphic_F_on_full_disk | state | holomorphic F on full disk
s_schwarz_christoffel_mapping | theorem | Schwarz–Christoffel mapping
s_simply_connected_polygonal_domain | axiom | simply connected polygonal domain
s_upper_half_plane | axiom | upper half plane
s_abstract_conformal_map_to_polygon | state | abstract conformal map to polygon
s_derivative_form_product_w_minus_xk_alpha_k | state | derivative form product w minus xk alpha k
s_de_branges_theorem | theorem | Bieberbach conjecture / de Branges theorem
s_univalent_function_in_class_S | axiom | univalent function in class S
s_taylor_coefficients_a_n | axiom | taylor coefficients a n
s_Loewner_ODE_for_f_t | state | Loewner ODE for f t
s_milin_inequality_on_log_derivatives | state | milin inequality on log derivatives
s_riemann_removable_singularity | theorem | Riemann's removable singularity theorem
s_holomorphic_function_on_punctured_disk | axiom | holomorphic function on punctured disk
s_bounded_near_puncture | axiom | bounded near puncture
s_holomorphic_extension_h_with_zero_of_order_2 | state | holomorphic extension h with zero of order 2
s_holomorphic_recovery_of_f | state | holomorphic recovery of f
s_riemann_lebesgue_lemma | theorem | Riemann–Lebesgue lemma
s_l1_function_on_R | axiom | l1 function on R
s_fourier_transform_definition | axiom | fourier transform definition
s_decay_for_indicator_of_interval | state | decay for indicator of interval
s_decay_for_step_functions | state | decay for step functions
s_fejer_theorem | theorem | Fejér's theorem
s_continuous_function_on_torus | axiom | continuous function on torus
s_cesaro_means_of_fourier_series | axiom | cesaro means of fourier series
s_positive_approximate_identity | state | positive approximate identity
s_cesaro_average_as_convolution | state | cesaro average as convolution
s_dirichlet_pointwise_convergence | theorem | Dirichlet's theorem on pointwise Fourier convergence
s_piecewise_C1_function_on_torus | axiom | piecewise C1 function on torus
s_partial_sums_of_fourier_series | axiom | partial sums of fourier series
s_integral_representation_S_N_f | state | integral representation S N f
s_localized_integrand_with_smooth_factor | state | localized integrand with smooth factor
s_hausdorff_young_inequality | theorem | Hausdorff–Young inequality
s_lp_function_on_R | axiom | lp function on R
s_L1_L_inf_endpoint | state | L1 L inf endpoint
s_L2_L2_endpoint | state | L2 L2 endpoint
s_riesz_thorin_theorem | theorem | Riesz–Thorin interpolation theorem
s_linear_operator_bounded_at_endpoints | axiom | linear operator bounded at endpoints
s_lp_lq_endpoint_data | axiom | lp lq endpoint data
s_holomorphic_in_strip_with_endpoint_bounds | state | holomorphic in strip with endpoint bounds
s_log_convex_bound_on_strip | state | log convex bound on strip
s_paley_wiener_theorem | theorem | Paley–Wiener theorem
s_L2_function_with_compactly_supported_fourier_transform | axiom | L2 function with compactly supported fourier transform
s_entire_function_of_exponential_type | axiom | entire function of exponential type
s_holomorphic_extension_of_f | state | holomorphic extension of f
s_exponential_type_bound | state | exponential type bound
s_poisson_summation_formula | theorem | Poisson summation formula
s_schwartz_function_on_R | axiom | schwartz function on R
s_integer_lattice_in_R | axiom | integer lattice in R
s_1_periodic_smooth_function_F | state | 1 periodic smooth function F
s_fourier_coefficients_of_F_equal_hat_f_at_integers | state | fourier coefficients of F equal hat f at integers
s_heisenberg_uncertainty_principle | theorem | Heisenberg uncertainty principle
s_L2_normalization | axiom | L2 normalization
s_canonical_commutator_identity | state | canonical commutator identity
s_product_of_variances_bound | state | product of variances bound
s_hardy_inequality | theorem | Hardy's inequality
s_nonneg_function_on_positive_reals | axiom | nonneg function on positive reals
s_lp_norm_definition | axiom | lp norm definition
s_hardy_averaging_operator | state | hardy averaging operator
s_lp_norm_inequality_for_A_f | state | lp norm inequality for A f
s_hilbert_inequality | theorem | Hilbert's inequality
s_nonneg_sequences_a_m_b_n_in_l2 | axiom | nonneg sequences a m b n in l2
s_double_sum_kernel_1_over_m_plus_n | axiom | double sum kernel 1 over m plus n
s_bilinear_form_with_homogeneous_kernel | state | bilinear form with homogeneous kernel
s_Schur_test_satisfied | state | Schur test satisfied
s_hilbert_transform_Lp_bound | theorem | Hilbert transform L^p boundedness
s_principal_value_convolution_with_1_over_x | axiom | principal value convolution with 1 over x
s_L2_isometry_up_to_sign | state | L2 isometry up to sign
s_CZ_kernel_class_membership | state | CZ kernel class membership
s_littlewood_paley_square_function_theorem | theorem | Littlewood–Paley decomposition
s_dyadic_frequency_annulus_decomposition | axiom | dyadic frequency annulus decomposition
s_dyadic_blocks_Delta_k_f | state | dyadic blocks Delta k f
s_square_function_S_f | state | square function S f
s_lp_norm_equivalence_with_square_function | state | lp norm equivalence with square function
s_stone_theorem_unitary_groups | theorem | Stone's theorem on one-parameter unitary groups
s_strongly_continuous_one_parameter_unitary_group | axiom | strongly continuous one parameter unitary group
s_dense_domain_with_skew_adjoint_generator | state | dense domain with skew adjoint generator
s_spectral_resolution_of_iA | state | spectral resolution of iA
s_spectral_theorem_bounded_self_adjoint | theorem | Spectral theorem for bounded self-adjoint operators
s_bounded_self_adjoint_operator_on_hilbert_space | axiom | bounded self adjoint operator on hilbert space
s_polynomial_calculus_with_C_star_norm | state | polynomial calculus with C star norm
s_continuous_functional_calculus_extension | state | continuous functional calculus extension
s_projection_valued_spectral_measure | state | projection valued spectral measure
s_spectral_theorem_compact_self_adjoint | theorem | Spectral theorem for compact self-adjoint operators
s_compact_self_adjoint_operator_on_hilbert_space | axiom | compact self adjoint operator on hilbert space
s_largest_eigenvalue_lambda_1_attained | state | largest eigenvalue lambda 1 attained
s_recurse_on_invariant_subspace | state | recurse on invariant subspace
s_darboux_theorem_symplectic | theorem | Darboux's theorem (symplectic)
s_symplectic_form_omega | state | symplectic form omega
s_linear_symplectic_basis_at_p | state | linear symplectic basis at p
s_local_diffeomorphism_to_standard_omega | state | local diffeomorphism to standard omega
s_moser_trick_theorem | theorem | Moser's trick
s_homotopy_of_forms_omega_t | state | homotopy of forms omega t
s_moser_vector_field_ODE | state | moser vector field ODE
s_isotopy_phi_t | state | isotopy phi t
s_weinstein_neighbourhood | theorem | Weinstein neighbourhood theorem
s_lagrangian_submanifold | state | lagrangian submanifold
s_normal_bundle_identification | state | normal bundle identification
s_gromov_non_squeezing | theorem | Gromov non-squeezing theorem
s_alleged_squeezing_phi_B_to_Z | state | alleged squeezing phi B to Z
s_j_holomorphic_disk_existence | state | j holomorphic disk existence
s_area_bound_pi_r2_leq_pi_R2 | state | area bound pi r2 leq pi R2
s_gromov_compactness_j_holomorphic | theorem | Gromov compactness for J-holomorphic curves
s_moduli_bounded_energy_curves | state | moduli bounded energy curves
s_bubble_tree_at_singular_points | state | bubble tree at singular points
s_compactified_moduli_of_stable_maps | state | compactified moduli of stable maps
s_arnold_conjecture | theorem | Arnold conjecture (Floer)
s_hamiltonian_diffeomorphism_phi_H | state | hamiltonian diffeomorphism phi H
s_floer_pde_for_chords | state | floer pde for chords
s_floer_homology_HF_M | state | floer homology HF M
s_fixed_point_count_geq_Betti_sum | state | fixed point count geq Betti sum
s_delzant_theorem | theorem | Delzant's theorem
s_toric_symplectic_manifold | state | toric symplectic manifold
s_delzant_polytope | state | delzant polytope
s_duistermaat_heckman | theorem | Duistermaat–Heckman formula
s_hamiltonian_circle_action | state | hamiltonian circle action
s_atiyah_bott_localization_formula | state | atiyah bott localization formula
s_eliashberg_contact_3_classification | theorem | Eliashberg classification of contact 3-manifolds
s_tight_overtwisted_dichotomy | state | tight overtwisted dichotomy
s_overtwisted_disk_class | state | overtwisted disk class
s_h_principle_classification_of_overtwisted | state | h principle classification of overtwisted
s_weinstein_conjecture_3d | theorem | Weinstein conjecture (3-dim, Taubes)
s_reeb_vector_field | state | reeb vector field
s_sw_to_closed_reeb_orbit | state | sw to closed reeb orbit
s_existence_of_closed_reeb_orbit | state | existence of closed reeb orbit
s_conley_conjecture | theorem | Conley conjecture
s_aspherical_symplectic_manifold | state | aspherical symplectic manifold
s_hypothetical_finite_periodic_phi | state | hypothetical finite periodic phi
s_local_floer_homology_obstruction | state | local floer homology obstruction
s_lagrangian_intersection_floer | theorem | Lagrangian intersection theorem (Floer)
s_lagrangian_pair_L0_L1 | state | lagrangian pair L0 L1
s_floer_strips_between_L0_L1 | state | floer strips between L0 L1
s_lagrangian_floer_homology | state | lagrangian floer homology
s_caratheodory_jacobi_lie | theorem | Carathéodory–Jacobi–Lie theorem
s_involutive_function_system | state | involutive function system
s_commuting_hamiltonian_flows | state | commuting hamiltonian flows
s_liouville_arnold | theorem | Liouville–Arnold integrability theorem
s_completely_integrable_hamiltonian_system | state | completely integrable hamiltonian system
s_compact_connected_level_set | state | compact connected level set
s_lagrangian_torus_fibration | state | lagrangian torus fibration
s_marsden_weinstein_reduction | theorem | Marsden–Weinstein symplectic reduction
s_hamiltonian_G_action_with_mu | state | hamiltonian G action with mu
s_zero_level_set_mu | state | zero level set mu
s_reduced_space_M_sslash_G | state | reduced space M sslash G
s_atiyah_guillemin_sternberg | theorem | Atiyah–Guillemin–Sternberg convexity theorem
s_compact_torus_hamiltonian_action | state | compact torus hamiltonian action
s_connected_fibers_of_moment_map | state | connected fibers of moment map
s_newlander_nirenberg | theorem | Newlander–Nirenberg theorem
s_almost_complex_structure | state | almost complex structure
s_nijenhuis_obstruction | state | nijenhuis obstruction
s_dbar_system_solvability | state | dbar system solvability
s_kahler_hodge_decomposition | theorem | Hodge decomposition for compact Kähler manifolds
s_kahler_manifold | state | kahler manifold
s_kahler_identities | state | kahler identities
s_p_q_decomposition_of_harmonic_forms | state | p q decomposition of harmonic forms
s_kodaira_embedding | theorem | Kodaira embedding theorem
s_hodge_manifold | state | hodge manifold
s_sections_of_high_tensor_power | state | sections of high tensor power
s_base_point_free_separating_jets | state | base point free separating jets
s_kodaira_vanishing | theorem | Kodaira vanishing theorem
s_positive_line_bundle_L | state | positive line bundle L
s_bochner_kodaira_identity | state | bochner kodaira identity
s_pointwise_positivity_of_curvature_operator | state | pointwise positivity of curvature operator
s_akizuki_nakano | theorem | Akizuki–Nakano vanishing theorem
s_bochner_kodaira_nakano_identity_extended | state | bochner kodaira nakano identity extended
s_positive_curvature_pairing_on_p_q_forms | state | positive curvature pairing on p q forms
s_yau_calabi | theorem | Calabi conjecture (Yau's theorem)
s_kahler_class_with_target_Ricci | state | kahler class with target Ricci
s_complex_monge_ampere_pde | state | complex monge ampere pde
s_unique_smooth_solution_phi | state | unique smooth solution phi
s_aubin_yau_KE | theorem | Aubin–Yau theorem (negative curvature Kähler–Einstein)
s_kahler_manifold_with_negative_c1 | state | kahler manifold with negative c1
s_monge_ampere_negative_c1 | state | monge ampere negative c1
s_a_priori_estimates_for_KE_metric | state | a priori estimates for KE metric
s_donaldson_uhlenbeck_yau | theorem | Donaldson–Uhlenbeck–Yau theorem
s_stable_holomorphic_bundle | state | stable holomorphic bundle
s_hermitian_yang_mills_eqn | state | hermitian yang mills eqn
s_long_time_existence_of_metric_flow | state | long time existence of metric flow
s_seiberg_witten_invariants | theorem | Seiberg–Witten invariants
s_spin_c_4_manifold | state | spin c 4 manifold
s_sw_equations | state | sw equations
s_compact_sw_moduli_space | state | compact sw moduli space
s_bertini_smoothness | theorem | Bertini's theorem on smoothness
s_smooth_subvariety_in_PN | state | smooth subvariety in PN
s_hyperplane_family_in_dual_PN | state | hyperplane family in dual PN
s_generic_H_meets_X_transversely | state | generic H meets X transversely
s_bertini_irreducibility | theorem | Bertini irreducibility theorem
s_irreducible_variety_dim_ge_2 | state | irreducible variety dim ge 2
s_incidence_variety_construction | state | incidence variety construction
s_generic_section_connected | state | generic section connected
s_riemann_roch_surfaces | theorem | Riemann–Roch theorem for surfaces
s_smooth_projective_surface_with_divisor | state | smooth projective surface with divisor
s_euler_characteristic_of_O_D | state | euler characteristic of O D
s_intersection_with_canonical_class | state | intersection with canonical class
s_hirzebruch_riemann_roch | theorem | Hirzebruch–Riemann–Roch theorem
s_holomorphic_bundle_on_X | state | holomorphic bundle on X
s_ch_and_todd_classes | state | ch and todd classes
s_index_of_dbar_operator | state | index of dbar operator
s_lefschetz_hyperplane | theorem | Lefschetz hyperplane theorem
s_smooth_projective_with_hyperplane_section | state | smooth projective with hyperplane section
s_morse_function_with_indices_geq_n | state | morse function with indices geq n
s_Y_obtained_from_X_by_cells_of_dim_geq_n | state | Y obtained from X by cells of dim geq n
s_hard_lefschetz | theorem | Hard Lefschetz theorem
s_compact_kahler_with_class | state | compact kahler with class
s_sl2_action_on_H_star_X | state | sl2 action on H star X
s_primitive_decomposition | state | primitive decomposition
s_lefschetz_1_1_classes | theorem | Lefschetz (1,1)-classes theorem
s_integral_1_1_class_on_kahler | state | integral 1 1 class on kahler
s_exponential_exact_sequence_pic | state | exponential exact sequence pic
s_pic_to_NS_via_chern_class | state | pic to NS via chern class
s_minkowski_inequality | theorem | Minkowski's inequality
s_lp_function_pair | axiom | lp function pair
s_pointwise_split_of_pth_power | state | pointwise split of pth power
s_two_holder_bounds_on_each_term | state | two holder bounds on each term
s_young_convolution_inequality | theorem | Young's convolution inequality
s_three_factor_decomposition_of_kernel | state | three factor decomposition of kernel
s_pointwise_bound_for_convolution | state | pointwise bound for convolution
s_hardy_inequality_continuous | theorem | Hardy's inequality (continuous)
s_lebesgue_measure_on_R_plus | axiom | lebesgue measure on R plus
s_lp_function_on_R_plus | axiom | lp function on R plus
s_scale_invariant_kernel | state | scale invariant kernel
s_hardy_inequality_discrete | theorem | Hardy's inequality (discrete)
s_lp_sequence | axiom | lp sequence
s_natural_numbers | axiom | natural numbers
s_discrete_averaging_sequence | state | discrete averaging sequence
s_dual_pairing_with_test_sequence | state | dual pairing with test sequence
s_carleman_inequality | theorem | Carleman's inequality
s_l1_positive_sequence | axiom | l1 positive sequence
s_weighted_substitution_for_GM | state | weighted substitution for GM
s_term_by_term_AM_GM_bound | state | term by term AM GM bound
s_riesz_thorin_interpolation | theorem | Riesz–Thorin interpolation theorem
s_linear_operator_T_bounded_at_endpoints | axiom | linear operator T bounded at endpoints
s_complex_strip_0_leq_Re_z_leq_1 | axiom | complex strip 0 leq Re z leq 1
s_analytic_function_on_complex_strip | state | analytic function on complex strip
s_log_norm_convex_in_z | state | log norm convex in z
s_stein_interpolation | theorem | Stein interpolation theorem
s_analytic_family_of_operators_T_z | axiom | analytic family of operators T z
s_lp_lq_pair_endpoints | axiom | lp lq pair endpoints
s_analytic_F_with_growth_control | state | analytic F with growth control
s_hahn_decomposition | theorem | Hahn decomposition theorem
s_signed_measure_on_measurable_space | axiom | signed measure on measurable space
s_measurable_space | axiom | measurable space
s_supremum_of_positive_set_values | state | supremum of positive set values
s_limit_positive_set_P_star | state | limit positive set P star
s_jordan_decomposition_measure | theorem | Jordan decomposition of a measure
s_two_positive_measures_mutually_singular | state | two positive measures mutually singular
s_lebesgue_decomposition | theorem | Lebesgue decomposition theorem
s_pair_of_sigma_finite_measures_mu_nu | axiom | pair of sigma finite measures mu nu
s_radon_nikodym_density_against_sum | state | radon nikodym density against sum
s_singular_plus_absolutely_continuous_split | state | singular plus absolutely continuous split
s_riesz_representation_C_K | theorem | Riesz representation theorem for C(K)
s_bounded_linear_functional_on_C_K | axiom | bounded linear functional on C K
s_pair_of_positive_functionals | state | pair of positive functionals
s_pair_of_radon_measures | state | pair of radon measures
s_lebesgue_stieltjes_existence | theorem | Lebesgue–Stieltjes integral existence
s_right_continuous_nondecreasing_function_on_R | axiom | right continuous nondecreasing function on R
s_borel_sigma_algebra_on_R | axiom | borel sigma algebra on R
s_premeasure_on_half_open_intervals | state | premeasure on half open intervals
s_borel_cantelli_lemmas | theorem | Borel–Cantelli lemmas
s_sequence_of_measurable_events | axiom | sequence of measurable events
s_limsup_event_definition | state | limsup event definition
s_first_borel_cantelli | state | first borel cantelli
s_lebesgue_density_theorem | theorem | Lebesgue density theorem
s_lebesgue_measurable_set_in_Rn | axiom | lebesgue measurable set in Rn
s_indicator_function_of_E | state | indicator function of E
s_ae_average_indicator_converges_to_indicator | state | ae average indicator converges to indicator
s_cauchy_riemann_equations | theorem | Cauchy–Riemann equations
s_complex_differentiable_function | axiom | complex differentiable function
s_open_domain_in_C | axiom | open domain in C
s_real_imaginary_split | state | real imaginary split
s_two_directional_derivatives_match | state | two directional derivatives match
s_looman_menchoff_theorem | theorem | Looman–Menchoff theorem
s_continuous_function_satisfying_CR_ae | axiom | continuous function satisfying CR ae
s_vanishing_rectangle_integral | state | vanishing rectangle integral
s_morera_hypothesis_satisfied | state | morera hypothesis satisfied
s_cauchy_goursat_theorem | theorem | Cauchy's theorem (rectangle / Goursat)
s_rectangle_in_C | axiom | rectangle in C
s_nested_quartered_rectangle_with_largest_integral | state | nested quartered rectangle with largest integral
s_point_with_local_linear_approximation | state | point with local linear approximation
s_cauchy_theorem_homotopy | theorem | Cauchy's theorem (homotopy version)
s_two_homotopic_closed_curves | axiom | two homotopic closed curves
s_homotopy_grid_partition | state | homotopy grid partition
s_finite_grid_with_goursat_per_cell | state | finite grid with goursat per cell
s_liouville_theorem_complex | theorem | Liouville's theorem
s_cauchy_estimate_for_derivative | state | cauchy estimate for derivative
s_f_prime_identically_zero | state | f prime identically zero
s_open_mapping_theorem_complex | theorem | Open mapping theorem (complex)
s_shifted_function_with_zero_at_z_0 | state | shifted function with zero at z 0
s_neighborhood_of_w_0_in_image | state | neighborhood of w 0 in image
s_phragmen_lindelof | theorem | Phragmén–Lindelöf principle
s_holomorphic_function_on_sector_with_growth_bound | axiom | holomorphic function on sector with growth bound
s_unbounded_sector_in_C | axiom | unbounded sector in C
s_auxiliary_decaying_function_g | state | auxiliary decaying function g
s_g_bounded_on_bounded_part | state | g bounded on bounded part
s_hadamard_three_lines | theorem | Hadamard three-lines theorem
s_holomorphic_bounded_function_on_strip | axiom | holomorphic bounded function on strip
s_vertical_strip_0_leq_Re_z_leq_1 | axiom | vertical strip 0 leq Re z leq 1
s_normalized_function_with_boundary_modulus_one | state | normalized function with boundary modulus one
s_modulus_g_leq_1_in_strip | state | modulus g leq 1 in strip
s_hadamard_three_circles | theorem | Hadamard three-circles theorem
s_pullback_to_vertical_strip | state | pullback to vertical strip
s_log_max_convex_in_log_r | state | log max convex in log r
s_hadamard_factorization | theorem | Hadamard factorization theorem
s_zero_sequence_with_density_bound | state | zero sequence with density bound
s_canonical_weierstrass_product | state | canonical weierstrass product
s_weierstrass_factorization | theorem | Weierstrass factorization theorem
s_sequence_a_n_to_infinity_with_multiplicities | axiom | sequence a n to infinity with multiplicities
s_elementary_factors_with_convergence_exponents | state | elementary factors with convergence exponents
s_convergent_infinite_product | state | convergent infinite product
s_vitali_convergence_complex | theorem | Vitali's convergence theorem (complex)
s_locally_uniformly_bounded_holomorphic_sequence | axiom | locally uniformly bounded holomorphic sequence
s_open_connected_domain_in_C | axiom | open connected domain in C
s_normal_family_with_subsequence_limit | state | normal family with subsequence limit
s_limit_function_determined_by_pointwise_set | state | limit function determined by pointwise set
s_mergelyan_theorem | theorem | Mergelyan's theorem
s_continuous_function_on_compact_K_holomorphic_in_interior | axiom | continuous function on compact K holomorphic in interior
s_compact_subset_K_of_C_with_connected_complement | axiom | compact subset K of C with connected complement
s_smooth_approximant_of_f | state | smooth approximant of f
s_rational_approximation_uniform_on_K | state | rational approximation uniform on K
s_carleman_approximation | theorem | Carleman approximation theorem
s_continuous_function_on_real_line | axiom | continuous function on real line
s_positive_continuous_error_function_eps | axiom | positive continuous error function eps
s_sequence_of_entire_partial_approximants | state | sequence of entire partial approximants
s_uniform_approximation_with_error_eps_on_R | state | uniform approximation with error eps on R
s_bernstein_theorem_monotonic | theorem | Bernstein's theorem on absolutely monotonic functions
s_completely_monotonic_function_on_R_plus | axiom | completely monotonic function on R plus
s_positive_real_line | axiom | positive real line
s_nonneg_finite_differences_at_grid | state | nonneg finite differences at grid
s_limit_measure_dmu_on_R_plus | state | limit measure dmu on R plus
s_kahane_katznelson_de_leeuw | theorem | Kahane–Katznelson–de Leeuw theorem
s_continuous_function_on_circle | axiom | continuous function on circle
s_l2_fourier_coefficient_sequence | axiom | l2 fourier coefficient sequence
s_random_sign_modification_of_coefficients | state | random sign modification of coefficients
s_existence_of_realization_with_bounded_sup_norm | state | existence of realization with bounded sup norm
s_hormander_mikhlin_multiplier | theorem | Hörmander–Mikhlin multiplier theorem
s_symbol_m_smooth_with_homogeneous_derivative_bounds | axiom | symbol m smooth with homogeneous derivative bounds
s_dyadic_decomposition_of_symbol | state | dyadic decomposition of symbol
s_kernel_satisfies_CZ_conditions | state | kernel satisfies CZ conditions
s_cotlar_stein_lemma | theorem | Cotlar–Stein lemma
s_family_of_bounded_operators_T_j_on_Hilbert_space | axiom | family of bounded operators T j on Hilbert space
s_almost_orthogonality_estimates | axiom | almost orthogonality estimates
s_pair_of_positive_operator_sums | state | pair of positive operator sums
s_norm_bound_on_sum_via_schur | state | norm bound on sum via schur
s_T_one_theorem | theorem | T(1) theorem (David–Journé)
s_singular_integral_operator_with_CZ_kernel | axiom | singular integral operator with CZ kernel
s_modified_operator_with_T_one_zero | state | modified operator with T one zero
s_L2_bounded_modified_operator | state | L2 bounded modified operator
s_rellich_kondrachov_compactness | theorem | Rellich–Kondrachov compactness theorem
s_bounded_lipschitz_domain_in_Rn | axiom | bounded lipschitz domain in Rn
s_sobolev_space_W_kp_on_domain | axiom | sobolev space W kp on domain
s_mollification_with_uniform_Lp_control | state | mollification with uniform Lp control
s_relatively_compact_in_Lp_after_mollification | state | relatively compact in Lp after mollification
s_symmetrized_radial_problem | state | symmetrized radial problem
s_scale_critical_setup | state | scale critical setup
s_gagliardo_nirenberg_inequality | theorem | Gagliardo–Nirenberg interpolation inequality
s_smooth_compactly_supported_function_on_Rn | axiom | smooth compactly supported function on Rn
s_sobolev_norms_with_mixed_orders | axiom | sobolev norms with mixed orders
s_dyadic_pieces_with_localized_frequency | state | dyadic pieces with localized frequency
s_pointwise_interpolation_inequality | state | pointwise interpolation inequality
s_john_nirenberg_inequality | theorem | John–Nirenberg inequality
s_bmo_function_on_Rn | axiom | bmo function on Rn
s_cube_in_Rn | axiom | cube in Rn
s_stopping_cubes_at_threshold_lambda | state | stopping cubes at threshold lambda
s_exponential_decay_of_distribution | state | exponential decay of distribution
s_fefferman_duality_H1_BMO | theorem | Fefferman duality H¹–BMO
s_hardy_space_H1_on_Rn | axiom | hardy space H1 on Rn
s_bmo_space_on_Rn | axiom | bmo space on Rn
s_atomic_decomposition_of_H1 | state | atomic decomposition of H1
s_bounded_bilinear_pairing_BMO_times_H1 | state | bounded bilinear pairing BMO times H1
s_cauchy_schwarz_inequality | theorem | Cauchy–Schwarz inequality
s_two_vectors_in_space | axiom | two vectors in space
s_quadratic_in_t_nonneg | state | quadratic in t nonneg
s_discriminant_inequality | state | discriminant inequality
s_gagliardo_nirenberg_sobolev_inequality | theorem | Gagliardo–Nirenberg–Sobolev inequality
s_lp_norm_interpolation_relation | axiom | lp norm interpolation relation
s_one_derivative_L_n_over_n_minus_1_bound | state | one derivative L n over n minus 1 bound
s_full_GN_interpolation_chain | state | full GN interpolation chain
s_sobolev_trace_theorem | theorem | Trace theorem for Sobolev spaces
s_W_1_p_function_on_lipschitz_domain | axiom | W 1 p function on lipschitz domain
s_boundary_of_domain | axiom | boundary of domain
s_half_space_local_chart | state | half space local chart
s_pointwise_boundary_value_bound | state | pointwise boundary value bound
s_riesz_potential_kernel_abs_x_minus_lambda | axiom | riesz potential kernel abs x minus lambda
s_dyadic_kernel_pieces | state | dyadic kernel pieces
s_bound_via_maximal_function_majorant | state | bound via maximal function majorant
s_choquet_representation_theorem | theorem | Choquet representation theorem
s_point_in_convex_set | axiom | point in convex set
s_positive_linear_functional_on_continuous_functions | state | positive linear functional on continuous functions
s_probability_measure_on_K | state | probability measure on K
s_riesz_lemma | theorem | Riesz lemma (almost-perpendicular vector)
s_normed_space_with_proper_closed_subspace_Y | axiom | normed space with proper closed subspace Y
s_quantity_alpha_in_0_1 | axiom | quantity alpha in 0 1
s_near_extremal_x_outside_Y | state | near extremal x outside Y
s_unit_vector_alpha_far_from_Y | state | unit vector alpha far from Y
s_riesz_compactness_unit_ball | theorem | Riesz's theorem on compactness of unit ball
s_normed_space | axiom | normed space
s_unit_ball_compactness | axiom | unit ball compactness
s_compact_unit_ball_assumption | state | compact unit ball assumption
s_sequence_with_pairwise_distance_geq_half | state | sequence with pairwise distance geq half
s_hahn_decomposition_theorem | theorem | Hahn decomposition theorem
s_supremum_achieved_via_chain | state | supremum achieved via chain
s_positive_set_P_with_sup | state | positive set P with sup
s_jordan_decomposition_theorem | theorem | Jordan decomposition theorem
s_hahn_decomposition_pair | axiom | hahn decomposition pair
s_two_positive_measures_nu_plus_nu_minus | state | two positive measures nu plus nu minus
s_mutually_singular_decomposition | state | mutually singular decomposition
s_lebesgue_decomposition_theorem | theorem | Lebesgue decomposition theorem
s_sigma_finite_signed_measure | axiom | sigma finite signed measure
s_reference_sigma_finite_measure | axiom | reference sigma finite measure
s_maximal_singular_carrier_A | state | maximal singular carrier A
s_two_part_decomposition | state | two part decomposition
s_riesz_representation_hilbert | theorem | Riesz representation theorem on Hilbert space
s_bounded_linear_functional_on_hilbert | axiom | bounded linear functional on hilbert
s_closed_subspace_kernel_of_L | state | closed subspace kernel of L
s_orthogonal_complement_spanned_by_z_0 | state | orthogonal complement spanned by z 0
s_mertens_cauchy_product_theorem | theorem | Mertens's theorem on convolution of series
s_two_real_series_one_absolutely_convergent | axiom | two real series one absolutely convergent
s_cauchy_product | axiom | cauchy product
s_partial_sum_split_main_tail | state | partial sum split main tail
s_tail_estimate_uniform | state | tail estimate uniform
s_abel_summation_theorem | theorem | Abel's summation theorem
s_real_power_series_with_radius_of_convergence_1 | axiom | real power series with radius of convergence 1
s_convergent_series_at_endpoint_x_eq_1 | axiom | convergent series at endpoint x eq 1
s_partial_summation_identity | state | partial summation identity
s_uniform_convergence_on_segment_to_1 | state | uniform convergence on segment to 1
s_hardy_littlewood_tauberian_theorem | theorem | Tauberian theorem (Hardy–Littlewood)
s_series_Abel_summable_to_s | axiom | series Abel summable to s
s_tauberian_growth_condition_on_a_n | axiom | tauberian growth condition on a n
s_polynomial_smoothed_partial_sum | state | polynomial smoothed partial sum
s_indicator_approximation_via_polynomial | state | indicator approximation via polynomial
s_wiener_tauberian_theorem | theorem | Wiener Tauberian theorem
s_l1_function_on_R_with_nonvanishing_fourier_transform | axiom | l1 function on R with nonvanishing fourier transform
s_translation_invariant_subspace_of_L1 | axiom | translation invariant subspace of L1
s_closed_ideal_in_Wiener_algebra | state | closed ideal in Wiener algebra
s_no_common_zero_in_spectrum | state | no common zero in spectrum
s_polynomial_continuity_theorem | theorem | Bolzano's theorem on continuity of polynomials
s_polynomial_function_on_R | axiom | polynomial function on R
s_epsilon_delta_definition | axiom | epsilon delta definition
s_monomial_continuity | state | monomial continuity
s_stone_weierstrass_complex | theorem | Stone–Weierstrass complex version
s_self_adjoint_separating_unital_subalgebra_of_complex_C_X | axiom | self adjoint separating unital subalgebra of complex C X
s_real_subalgebra_separating | state | real subalgebra separating
s_real_subalgebra_dense | state | real subalgebra dense
s_gelfand_representation_theorem | theorem | Gelfand representation theorem
s_space_of_characters_with_weak_star_topology | axiom | space of characters with weak star topology
s_compact_hausdorff_character_space | state | compact hausdorff character space
s_continuous_algebra_homomorphism_A_to_C_Delta_A | state | continuous algebra homomorphism A to C Delta A
s_eberlein_smulian_theorem | theorem | Eberlein–Šmulian theorem
s_subset_of_banach_space | axiom | subset of banach space
s_net_based_weak_compactness | state | net based weak compactness
s_metrizable_weak_topology_on_separable_part | state | metrizable weak topology on separable part
s_mackey_arens_theorem | theorem | Mackey–Arens theorem
s_dual_pair_of_vector_spaces | axiom | dual pair of vector spaces
s_locally_convex_topology_compatible_with_duality | axiom | locally convex topology compatible with duality
s_polar_topology_family | state | polar topology family
s_compatible_topology_characterization | state | compatible topology characterization
s_banach_mazur_theorem | theorem | Banach–Mazur theorem
s_separable_banach_space | axiom | separable banach space
s_continuous_function_space_C_0_1 | axiom | continuous function space C 0 1
s_countable_dense_sequence_in_unit_ball | state | countable dense sequence in unit ball
s_isometric_embedding_via_Auerbach_type_basis | state | isometric embedding via Auerbach type basis
s_krein_smulian_theorem | theorem | Krein–Šmulian theorem
s_convex_subset_of_dual_with_weak_star_closed_intersections_with_bounded_sets | axiom | convex subset of dual with weak star closed intersections with bounded sets
s_compatible_topology_setup | state | compatible topology setup
s_weak_star_closed_globally | state | weak star closed globally
s_rearrangement_inequality | theorem | Hardy–Littlewood–Polya rearrangement inequality
s_two_finite_sequences_of_real_numbers | axiom | two finite sequences of real numbers
s_permutation_group_on_indices | axiom | permutation group on indices
s_pairwise_swap_decreases_sum_if_inversion | state | pairwise swap decreases sum if inversion
s_sorted_pairing_extremal | state | sorted pairing extremal
s_brunn_minkowski_inequality | theorem | Brunn–Minkowski inequality
s_two_nonempty_measurable_subsets_of_Rn | axiom | two nonempty measurable subsets of Rn
s_Minkowski_sum_A_plus_B | axiom | Minkowski sum A plus B
s_brunn_minkowski_for_boxes_via_AM_GM | state | brunn minkowski for boxes via AM GM
s_brunn_minkowski_for_finite_unions_of_boxes | state | brunn minkowski for finite unions of boxes
s_isoperimetric_inequality_Rn | theorem | Isoperimetric inequality in R^n
s_bounded_open_subset_of_Rn_with_smooth_boundary | axiom | bounded open subset of Rn with smooth boundary
s_volume_and_surface_area | axiom | volume and surface area
s_surface_area_as_eps_limit_of_volumes | state | surface area as eps limit of volumes
s_first_variation_inequality | state | first variation inequality
s_prekopa_leindler_inequality | theorem | Prékopa–Leindler inequality
s_three_nonneg_measurable_functions_with_log_concave_constraint | axiom | three nonneg measurable functions with log concave constraint
s_indicator_case_equivalent_to_brunn_minkowski | state | indicator case equivalent to brunn minkowski
s_layer_cake_lift_to_general_functions | state | layer cake lift to general functions
s_loomis_whitney_inequality | theorem | Loomis–Whitney inequality
s_measurable_subset_of_Rn | axiom | measurable subset of Rn
s_n_coordinate_projections | axiom | n coordinate projections
s_iterated_holder_on_indicator | state | iterated holder on indicator
s_projection_product_bound | state | projection product bound
s_hausdorff_moment_problem | theorem | Hausdorff moment problem
s_real_sequence_m_n | axiom | real sequence m n
s_uniqueness_of_measure_on_unit_interval | axiom | uniqueness of measure on unit interval
s_complete_monotonicity_criterion | state | complete monotonicity criterion
s_positive_functional_on_C_0_1_via_density | state | positive functional on C 0 1 via density
s_bernstein_widder_theorem | theorem | Bernstein–Widder theorem on completely monotone functions
s_completely_monotone_function_on_positive_reals | axiom | completely monotone function on positive reals
s_laplace_transform_of_positive_measure | axiom | laplace transform of positive measure
s_finite_difference_quadrature_approximation | state | finite difference quadrature approximation
s_representing_measures_on_intervals_converging | state | representing measures on intervals converging
s_herglotz_representation | theorem | Riesz–Herglotz representation
s_holomorphic_function_on_unit_disk_with_nonneg_real_part | axiom | holomorphic function on unit disk with nonneg real part
s_positive_borel_measure_on_circle | axiom | positive borel measure on circle
s_family_of_finite_positive_measures_mu_r | state | family of finite positive measures mu r
s_weak_star_limit_measure_mu | state | weak star limit measure mu
s_vitali_convergence_theorem | theorem | Vitali convergence theorem
s_uniformly_integrable_sequence | axiom | uniformly integrable sequence
s_ae_convergent_sequence_on_finite_measure_space | axiom | ae convergent sequence on finite measure space
s_uniform_tail_smallness_in_n | state | uniform tail smallness in n
s_split_integral_main_plus_tail | state | split integral main plus tail
s_dunford_pettis_theorem | theorem | Eberlein theorem on weak compactness in L^1 (Dunford–Pettis)
s_subset_of_L1_finite_measure_space | axiom | subset of L1 finite measure space
s_weak_compactness_in_L1 | axiom | weak compactness in L1
s_uniform_integrability_criterion | state | uniform integrability criterion
s_weak_compactness_via_alaoglu_and_eberlein | state | weak compactness via alaoglu and eberlein
s_markov_kakutani_fpt | theorem | Markov–Kakutani fixed-point theorem
s_commuting_family_of_continuous_affine_self_maps | axiom | commuting family of continuous affine self maps
s_compact_convex_subset_of_locally_convex_space | axiom | compact convex subset of locally convex space
s_cesaro_average_iterates_in_K | state | cesaro average iterates in K
s_limit_point_almost_fixed_for_T | state | limit point almost fixed for T
s_bv_sobolev_endpoint_inequality | theorem | Sobolev–Gagliardo–Nirenberg inequality endpoint (BV–L^{n/(n-1)})
s_BV_function_on_Rn | axiom | BV function on Rn
s_total_variation_and_perimeter | axiom | total variation and perimeter
s_coarea_formula_for_BV | state | coarea formula for BV
s_per_level_isoperimetric_bound | state | per level isoperimetric bound
s_levy_khintchine_formula | theorem | Lévy–Khintchine formula
s_infinitely_divisible_distribution_on_R | axiom | infinitely divisible distribution on R
s_characteristic_function_log_psi | axiom | characteristic function log psi
s_decomposition_drift_plus_brownian_plus_jumps | state | decomposition drift plus brownian plus jumps
s_levy_measure_integral_existence | state | levy measure integral existence
s_kolmogorov_three_series_theorem | theorem | Three-series theorem (Kolmogorov)
s_sequence_of_independent_random_variables_X_n | axiom | sequence of independent random variables X n
s_truncation_at_level_A | axiom | truncation at level A
s_three_series_summability_condition | state | three series summability condition
s_ae_convergence_of_centered_truncations | state | ae convergence of centered truncations
s_strong_law_of_large_numbers | theorem | Strong law of large numbers (Kolmogorov)
s_iid_sequence_with_finite_mean | axiom | iid sequence with finite mean
s_centered_truncated_sequence | state | centered truncated sequence
s_partial_sum_concentration | state | partial sum concentration
s_donsker_invariance_principle | theorem | Donsker's invariance principle
s_iid_sequence_with_finite_variance_unit_normalized | axiom | iid sequence with finite variance unit normalized
s_brownian_motion_on_0_1 | axiom | brownian motion on 0 1
s_polygonal_random_walk_process | state | polygonal random walk process
s_finite_dim_marginals_converge_to_BM | state | finite dim marginals converge to BM
s_wiener_inversion_theorem | theorem | Wiener's theorem on Fourier series (absolute convergence)
s_function_with_absolutely_convergent_fourier_series_nonvanishing | axiom | function with absolutely convergent fourier series nonvanishing
s_inverse_function_1_over_f | axiom | inverse function 1 over f
s_banach_algebra_A_T | state | banach algebra A T
s_spectrum_of_f_in_A_T_equals_image | state | spectrum of f in A T equals image
s_helly_bray_theorem | theorem | Helly–Bray theorem (weak convergence of distribution functions)
s_bounded_continuous_function_on_R | axiom | bounded continuous function on R
s_weakly_convergent_sequence_of_distribution_functions | axiom | weakly convergent sequence of distribution functions
s_truncated_integrand_on_compact | state | truncated integrand on compact
s_convergence_on_compact_window | state | convergence on compact window
s_helly_theorem_convex | theorem | Helly's theorem on convex sets
s_finite_family_of_convex_subsets_of_Rd | axiom | finite family of convex subsets of Rd
s_d_plus_1_intersection_condition | axiom | d plus 1 intersection condition
s_induction_base_at_d_plus_1 | state | induction base at d plus 1
s_radon_partition_giving_common_intersection | state | radon partition giving common intersection
s_caratheodory_convex_hull_theorem | theorem | Carathéodory's convex hull theorem
s_subset_of_Rd | axiom | subset of Rd
s_convex_hull_with_convex_combinations | axiom | convex hull with convex combinations
s_redundant_convex_combination_assumption | state | redundant convex combination assumption
s_linear_dependence_extracts_smaller_combination | state | linear dependence extracts smaller combination
s_lindelof_principle_subharmonic | theorem | Lindelöf principle (subharmonic version)
s_subharmonic_function_on_unbounded_domain | axiom | subharmonic function on unbounded domain
s_growth_bound_at_infinity | axiom | growth bound at infinity
s_harmonic_comparison_function | state | harmonic comparison function
s_bound_on_truncations | state | bound on truncations
s_perron_method_dirichlet | theorem | Perron method for Dirichlet problem
s_bounded_domain_with_boundary_data_in_C_boundary | axiom | bounded domain with boundary data in C boundary
s_family_of_subharmonic_subsolutions | axiom | family of subharmonic subsolutions
s_perron_solution_candidate_u | state | perron solution candidate u
s_harmonic_interior_function | state | harmonic interior function
s_harmonic_mvp_characterization | theorem | Mean value property characterizes harmonic functions
s_continuous_function_on_open_subset_of_Rn | axiom | continuous function on open subset of Rn
s_mean_value_property_on_balls | axiom | mean value property on balls
s_smooth_mollification_equals_f | state | smooth mollification equals f
s_laplacian_zero_almost_everywhere | state | laplacian zero almost everywhere
s_harnack_inequality | theorem | Harnack's inequality
s_nonneg_harmonic_function_on_ball | axiom | nonneg harmonic function on ball
s_compactly_contained_subdomain | axiom | compactly contained subdomain
s_poisson_kernel_explicit_bounds | state | poisson kernel explicit bounds
s_two_point_ratio_inequality | state | two point ratio inequality
s_carleson_embedding_theorem | theorem | Carleson measure characterization (Carleson embedding)
s_positive_measure_mu_on_upper_half_plane | axiom | positive measure mu on upper half plane
s_lp_function_on_real_line | axiom | lp function on real line
s_Poisson_extension_to_half_plane | state | Poisson extension to half plane
s_tent_decomposition_estimate | state | tent decomposition estimate
s_fefferman_BMO_H1_duality | theorem | BMO–H^1 duality (Fefferman)
s_real_hardy_space_H1_on_Rn | axiom | real hardy space H1 on Rn
s_BMO_space_on_Rn | axiom | BMO space on Rn
s_atomic_decomposition_for_H1 | state | atomic decomposition for H1
s_dual_pairing_estimate_per_atom | state | dual pairing estimate per atom
s_muckenhoupt_Ap_theorem | theorem | A_p weight theorem (Muckenhoupt)
s_locally_integrable_nonneg_weight_w_on_Rn | axiom | locally integrable nonneg weight w on Rn
s_Hardy_Littlewood_maximal_operator | axiom | Hardy Littlewood maximal operator
s_Ap_class_definition | state | Ap class definition
s_weak_p_p_bound_on_M_with_weight_w | state | weak p p bound on M with weight w
s_stein_interpolation_theorem | theorem | Stein interpolation theorem
s_analytic_family_of_operators_T_z_on_strip | axiom | analytic family of operators T z on strip
s_lp_lq_endpoint_bounds_with_admissible_growth | axiom | lp lq endpoint bounds with admissible growth
s_holomorphic_in_strip_with_admissible_growth | state | holomorphic in strip with admissible growth
s_three_lines_with_admissible_growth | state | three lines with admissible growth
s_fractional_sobolev_embedding | theorem | Sobolev–Slobodeckij embedding (fractional Sobolev)
s_fractional_sobolev_space_W_s_p_Rn | axiom | fractional sobolev space W s p Rn
s_lq_target_with_critical_exponent | axiom | lq target with critical exponent
s_HLS_lp_to_lq_bound | state | HLS lp to lq bound
s_polya_szego_rearrangement_inequality | theorem | Pólya–Szegő rearrangement inequality
s_nonneg_measurable_function_on_Rn | axiom | nonneg measurable function on Rn
s_symmetric_decreasing_rearrangement_f_star | axiom | symmetric decreasing rearrangement f star
s_layer_cake_representation_of_f | state | layer cake representation of f
s_level_set_isoperimetric_bound | state | level set isoperimetric bound
s_strichartz_estimate | theorem | Strichartz estimates
s_schrodinger_propagator_e_it_delta | axiom | schrodinger propagator e it delta
s_dispersive_decay_estimate | state | dispersive decay estimate
s_marcinkiewicz_zygmund_inequality | theorem | Riesz–Thorin endpoint Marcinkiewicz–Zygmund
s_iid_random_variable_sequence | axiom | iid random variable sequence
s_lp_norm_of_partial_sums | axiom | lp norm of partial sums
s_symmetrized_partial_sum_via_random_signs | state | symmetrized partial sum via random signs
s_conditional_khintchine_bound | state | conditional khintchine bound
s_bochner_riesz_lp_bound | theorem | Bochner–Riesz mean theorem
s_bochner_riesz_multiplier_1_minus_abs_xi_2_to_alpha | axiom | bochner riesz multiplier 1 minus abs xi 2 to alpha
s_decomposition_into_annular_pieces | state | decomposition into annular pieces
s_oscillatory_kernel_estimate | state | oscillatory kernel estimate
s_tomas_stein_restriction_theorem | theorem | Tomas–Stein restriction theorem
s_smooth_curved_hypersurface_in_Rn | axiom | smooth curved hypersurface in Rn
s_lp_function_with_fourier_restriction | axiom | lp function with fourier restriction
s_extension_operator_setup | state | extension operator setup
s_surface_measure_decay_estimate | state | surface measure decay estimate
s_kakeya_maximal_inequality_planar | theorem | Kakeya maximal inequality (n=2)
s_family_of_unit_segments_in_R2_with_delta_separated_directions | axiom | family of unit segments in R2 with delta separated directions
s_kakeya_maximal_operator | axiom | kakeya maximal operator
s_combinatorial_incidence_setup | state | combinatorial incidence setup
s_log_loss_bound_on_kakeya_maximal | state | log loss bound on kakeya maximal
s_calderon_reproducing_formula | theorem | Calderón reproducing formula (continuous wavelet)
s_admissible_mother_wavelet_psi_on_R | axiom | admissible mother wavelet psi on R
s_L2_function_f | axiom | L2 function f
s_admissibility_constant_C_psi_finite | state | admissibility constant C psi finite
s_parseval_identity_for_wavelet_pairing | state | parseval identity for wavelet pairing
s_frostman_lemma | theorem | Frostman's lemma
s_compact_subset_K_of_Rn | axiom | compact subset K of Rn
s_hausdorff_dimension_s_lower_bound | axiom | hausdorff dimension s lower bound
s_balanced_dyadic_mass | state | balanced dyadic mass
s_borel_measure_with_growth_bound | state | borel measure with growth bound
s_besicovitch_covering_theorem | theorem | Besicovitch covering theorem
s_collection_of_balls_in_Rn_with_bounded_radii_centered_at_set_A | axiom | collection of balls in Rn with bounded radii centered at set A
s_sorted_ball_sequence | state | sorted ball sequence
s_finite_number_of_disjoint_subfamilies_covering_A | state | finite number of disjoint subfamilies covering A
s_whitney_extension_theorem | theorem | Whitney extension theorem
s_closed_subset_of_Rn_with_jet_data_satisfying_compatibility | axiom | closed subset of Rn with jet data satisfying compatibility
s_target_class_C_m | axiom | target class C m
s_whitney_decomposition_of_complement | state | whitney decomposition of complement
s_locally_polynomial_extension | state | locally polynomial extension
s_borel_theorem_taylor_series | theorem | Borel's theorem on Taylor series
s_arbitrary_real_sequence_a_n | axiom | arbitrary real sequence a n
s_smooth_function_on_R | axiom | smooth function on R
s_scaled_bump_functions_with_decreasing_support | state | scaled bump functions with decreasing support
s_smooth_function_with_prescribed_jet_at_0 | state | smooth function with prescribed jet at 0
s_whitney_approximation_theorem | theorem | Whitney C^k approximation theorem
s_continuous_function_on_smooth_manifold | axiom | continuous function on smooth manifold
s_target_smooth_function_class | axiom | target smooth function class
s_partition_of_unity_chart_decomposition | state | partition of unity chart decomposition
s_locally_smooth_approximation | state | locally smooth approximation
s_torelli_curves | theorem | Torelli theorem for curves
s_smooth_curve_C | state | smooth curve C
s_polarized_jacobian | state | polarized jacobian
s_theta_divisor_singularities_recover_C | state | theta divisor singularities recover C
s_torelli_K3 | theorem | Torelli theorem for K3 surfaces
s_K3_surface_with_hodge_structure | state | K3 surface with hodge structure
s_period_map_for_K3 | state | period map for K3
s_fourier_mukai_transform | theorem | Fourier–Mukai transform
s_abelian_variety_with_dual | state | abelian variety with dual
s_poincare_bundle | state | poincare bundle
s_integral_transform_on_derived_category | state | integral transform on derived category
s_castelnuovo_mumford_regularity | theorem | Castelnuovo–Mumford regularity
s_coherent_sheaf_F_on_PN | state | coherent sheaf F on PN
s_regularity_definition | state | regularity definition
s_vanishing_propagation | state | vanishing propagation
s_castelnuovo_contractibility | theorem | Castelnuovo's contractibility criterion
s_minus_one_curve_on_surface | state | minus one curve on surface
s_morphism_contracting_E | state | morphism contracting E
s_mori_cone_theorem | theorem | Mori cone theorem
s_smooth_X_with_canonical | state | smooth X with canonical
s_cone_NE_decomposition | state | cone NE decomposition
s_extremal_rays_spanned_by_rational_curves | state | extremal rays spanned by rational curves
s_mori_extremal_contraction | theorem | Mori extremal contraction
s_extremal_ray_R | state | extremal ray R
s_supporting_nef_divisor | state | supporting nef divisor
s_semi_ample_supporting_divisor | state | semi ample supporting divisor
s_base_point_free_theorem | theorem | Kawamata–Shokurov base-point-free theorem
s_nef_big_divisor_setup | state | nef big divisor setup
s_KV_vanishing_application | state | KV vanishing application
s_inductive_section_lifting | state | inductive section lifting
s_kawamata_viehweg_vanishing | theorem | Kawamata–Viehweg vanishing theorem
s_big_nef_q_divisor | state | big nef q divisor
s_cyclic_cover_construction | state | cyclic cover construction
s_descended_vanishing_statement | state | descended vanishing statement
s_cartan_A_B | theorem | Cartan's theorems A and B
s_stein_manifold_with_coherent_F | state | stein manifold with coherent F
s_psh_exhaustion_of_X | state | psh exhaustion of X
s_H_i_X_F_vanish_for_i_geq_1 | state | H i X F vanish for i geq 1
s_hurwitz_automorphisms | theorem | Hurwitz's automorphisms theorem
s_curve_with_automorphism_group | state | curve with automorphism group
s_belyi_theorem | theorem | Belyi's theorem
s_curve_over_algebraic_closure_of_Q | state | curve over algebraic closure of Q
s_belyi_map_construction | state | belyi map construction
s_ramification_over_3_points_only | state | ramification over 3 points only
s_gaga_theorem | theorem | GAGA principle (Serre)
s_proper_X_with_coherent_sheaf | state | proper X with coherent sheaf
s_analytification_functor | state | analytification functor
s_cohomology_isomorphism | state | cohomology isomorphism
s_zariski_connectedness | theorem | Zariski's connectedness theorem
s_proper_birational_to_normal | state | proper birational to normal
s_pushforward_of_structure_sheaf_iso | state | pushforward of structure sheaf iso
s_tarski_seidenberg | theorem | Tarski–Seidenberg theorem
s_semialgebraic_set | state | semialgebraic set
s_CAD_of_S | state | CAD of S
s_projection_is_semialgebraic | state | projection is semialgebraic
s_chevalley_constructibility | theorem | Chevalley constructibility theorem
s_finite_type_morphism | state | finite type morphism
s_noetherian_induction_setup | state | noetherian induction setup
s_borel_fixed_point | theorem | Borel fixed-point theorem
s_solvable_action_on_proper | state | solvable action on proper
s_minimal_orbit_closure | state | minimal orbit closure
s_fixed_point_exists | state | fixed point exists
s_chow_lemma | theorem | Chow's lemma
s_proper_morphism_finite_type | state | proper morphism finite type
s_blow_up_yielding_projective_birational_X_tilde | state | blow up yielding projective birational X tilde
s_nagata_compactification | theorem | Nagata compactification theorem
s_separated_finite_type | state | separated finite type
s_local_compactifications | state | local compactifications
s_glued_proper_overscheme_X_bar | state | glued proper overscheme X bar
s_modularity_theorem | theorem | Modularity theorem (Taylor–Wiles–Breuil–Conrad–Diamond)
s_galois_rep_of_elliptic_curve | state | galois rep of elliptic curve
s_deformation_ring_R_to_T_map | state | deformation ring R to T map
s_R_equals_T_for_E | state | R equals T for E
s_tate_isogeny | theorem | Tate isogeny theorem
s_abelian_variety_over_Fq | state | abelian variety over Fq
s_tate_module_with_frobenius | state | tate module with frobenius
s_hom_iso_via_galois_action | state | hom iso via galois action
s_chow_rashevskii | theorem | Chow–Rashevskii theorem
s_bracket_generating_distribution | state | bracket generating distribution
s_lie_bracket_filtration | state | lie bracket filtration
s_horizontal_path_reaches_neighborhood | state | horizontal path reaches neighborhood
s_pansu_differentiation | theorem | Pansu differentiation theorem
s_carnot_group_setting | state | carnot group setting
s_blowup_limit_of_lipschitz | state | blowup limit of lipschitz
s_cartan_dieudonne | theorem | Cartan–Dieudonné theorem
s_orthogonal_transformation | state | orthogonal transformation
s_canonical_reflection | state | canonical reflection
s_reduction_to_identity_after_n_reflections | state | reduction to identity after n reflections
s_beltrami_theorem | theorem | Beltrami theorem
s_projectively_flat_metric | state | projectively flat metric
s_constant_sectional_curvature | state | constant sectional curvature
s_schur_riemannian | theorem | Schur's theorem (Riemannian, isotropic implies constant curvature)
s_isotropic_pointwise_curvature | state | isotropic pointwise curvature
s_bianchi_identity_constraint | state | bianchi identity constraint
s_schwarz_ahlfors_pick | theorem | Schwarz–Ahlfors–Pick theorem
s_holomorphic_map_of_hyperbolic_surfaces | state | holomorphic map of hyperbolic surfaces
s_poincare_metric_pair | state | poincare metric pair
s_distance_decreasing_property | state | distance decreasing property
s_williamson_normal_form | theorem | Williamson normal form theorem
s_positive_quadratic_form_on_symplectic_space | state | positive quadratic form on symplectic space
s_symplectic_spectrum | state | symplectic spectrum
s_birkhoff_grothendieck | theorem | Birkhoff–Grothendieck theorem
s_holomorphic_bundle_on_P1 | state | holomorphic bundle on P1
s_maximal_subbundle | state | maximal subbundle
s_inductive_splitting_into_line_bundles | state | inductive splitting into line bundles
s_clifford_theorem | theorem | Clifford's theorem on special divisors
s_special_divisor_setup | state | special divisor setup
s_multiplication_map_of_linear_systems | state | multiplication map of linear systems
s_clifford_inequality | state | clifford inequality
s_bernoulli_number | state | Bernoulli number
s_brauer_group | state | Brauer group
s_composition_series | state | Composition series
s_dedekind_domain | axiom | Dedekind domain
s_derivation | axiom | Derivation (algebra)
s_determinant | state | Determinant
s_discriminant | state | Discriminant
s_endomorphism_ring | state | Endomorphism ring
s_ext_functor | state | Ext functor
s_exterior_algebra | axiom | Exterior algebra
s_flat_module | axiom | Flat module
s_free_group | axiom | Free group
s_free_module | axiom | Free module
s_frobenius_endomorphism | state | Frobenius endomorphism
s_global_field | axiom | Global field
s_graded_ring | axiom | Graded ring
s_grobner_basis | state | Gröbner basis
s_grothendieck_group | state | Grothendieck group
s_group_algebra | axiom | Group algebra
s_hensels_lemma | theorem | Hensel's lemma
s_hilbert_class_field | state | Hilbert class field
s_hopf_algebra | axiom | Hopf algebra
s_ideal_class_group | state | Ideal class group
s_injective_module | axiom | Injective module
s_integral_closure | state | Integral closure
s_integral_domain | axiom | Integral domain
s_koszul_complex | state | Koszul complex
s_local_ring | axiom | Local ring
s_localization | state | Localization (ring theory)
s_morita_equivalence | theorem | Morita equivalence
s_nilpotent_group | axiom | Nilpotent group
s_p_adic_number | axiom | p-adic number
s_permutation_group | state | Permutation group
s_picard_group | state | Picard group
s_radical | state | Radical (Jacobson)
s_ring_of_integers | state | Ring of integers
s_root_system | axiom | Root system
s_semi_direct_product | state | Semi-direct product
s_solvable_group | axiom | Solvable group
s_tor_functor | state | Tor functor
s_transcendence_degree | state | Transcendence degree
s_universal_enveloping_algebra | state | Universal enveloping algebra
s_witt_vectors | state | Witt vectors
s_adele_ring | axiom | Adèle ring
s_idele_group | state | Idèle group
s_noether_normalization_lemma | theorem | Noether normalization lemma
s_cohen_macaulay_ring | axiom | Cohen–Macaulay ring
s_dynkin_diagram | axiom | Dynkin diagram
s_weyl_group | state | Weyl group
s_cartan_subalgebra | axiom | Cartan subalgebra
s_algebra | axiom | σ-algebra
s_borel_algebra | axiom | Borel σ-algebra
s_hausdorff_measure | axiom | Hausdorff measure
s_radon_measure | axiom | Radon measure
s_frechet_space | axiom | Fréchet space
s_dual_space | state | Dual space
s_c_algebra | axiom | C*-algebra
s_spectrum | state | Spectrum (operator)
s_schwartz_space | axiom | Schwartz space
s_greens_function | state | Green's function
s_gamma_function | state | Gamma function
s_analytic_continuation | state | Analytic continuation
s_normal_distribution | axiom | Normal distribution
s_bessel_function | state | Bessel function
s_legendre_polynomial | state | Legendre polynomial
s_hermite_polynomial | state | Hermite polynomial
s_orthogonal_polynomial | state | Orthogonal polynomial
s_bernstein_polynomial | state | Bernstein polynomial
s_stirlings_formula | theorem | Stirling's formula
s_elliptic_integral | state | Elliptic integral
s_normal_space | axiom | Normal space
s_metrization_theorem | theorem | Metrization theorem (Nagata–Smirnov)
s_quotient_space | state | Quotient space (topology)
s_cofibration | axiom | Cofibration
s_fibration | axiom | Fibration
s_cellular_homology | state | Cellular homology
s_cohomology_ring | state | Cohomology ring
s_cech_cohomology | state | Čech cohomology
s_classifying_space | state | Classifying space
s_characteristic_class | state | Characteristic class
s_obstruction_theory | state | Obstruction theory
s_k_theory | state | K-theory (topological)
s_cobordism | state | Cobordism
s_knot_theory | axiom | Knot theory
s_jones_polynomial | state | Jones polynomial
s_connection | axiom | Connection (differential geometry)
s_levi_civita_connection_hzw | state | Levi-Civita connection
s_holonomy_group | state | Holonomy group
s_riemann_curvature_tensor | state | Riemann curvature tensor
s_exponential_map | state | Exponential map (Riemannian)
s_jacobi_field | state | Jacobi field
s_stokes_theorem_hzw | theorem | Stokes' theorem
s_exterior_derivative | axiom | Exterior derivative
s_hodge_theory | state | Hodge theory
s_hodge_star_operator | state | Hodge star operator
s_immersion | axiom | Immersion
s_submersion | axiom | Submersion
s_transversality | axiom | Transversality
s_tangent_bundle | state | Tangent bundle
s_cotangent_bundle | state | Cotangent bundle
s_foliation | axiom | Foliation
s_algebraic_variety | axiom | Algebraic variety
s_zariski_topology | axiom | Zariski topology
s_ample_line_bundle | axiom | Ample line bundle
s_etale_cohomology_hzw | state | Étale cohomology
s_intersection_theory | state | Intersection theory
s_chow_group | state | Chow group
s_kodaira_dimension | state | Kodaira dimension
s_canonical_bundle | state | Canonical bundle
s_calabi_yau_manifold | axiom | Calabi–Yau manifold
t_maximum_likelihood_estimation | technique | Maximum likelihood estimation
s_neyman_pearson_lemma | theorem | Neyman–Pearson lemma
s_fisher_information | state | Fisher information
s_cramer_rao_inequality | theorem | Cramér–Rao inequality
s_conditional_expectation | axiom | Conditional expectation
s_kolmogorov_axioms | axiom | Kolmogorov axioms
s_wiener_process | axiom | Wiener process
t_it_calculus | technique | Itô calculus
s_stochastic_differential_equation | axiom | Stochastic differential equation
s_levy_process | axiom | Lévy process
t_kalman_filter | technique | Kalman filter
s_propositional_calculus | axiom | Propositional calculus
s_predicate_calculus | axiom | Predicate calculus
t_transfinite_induction | technique | Transfinite induction
s_cardinal_number | axiom | Cardinal number
s_ordinal_number | axiom | Ordinal number
s_zorns_lemma | theorem | Zorn's lemma
s_wave_equation | axiom | Wave equation
t_finite_element_method | technique | Finite element method
t_runge_kutta_method | technique | Runge–Kutta method
t_galerkin_method | technique | Galerkin method
s_pontryagin_maximum_principle | theorem | Pontryagin maximum principle
s_linear_programming | axiom | Linear programming
t_simplex_method | technique | Simplex method
s_hamilton_jacobi_equation | axiom | Hamilton–Jacobi equation
s_catalan_number | state | Catalan number
s_monad | axiom | Monad (category theory)
s_limit | axiom | Limit (category theory)
s_kan_extension | state | Kan extension
s_equivalence_of_categories | axiom | Equivalence of categories
s_triangulated_category | axiom | Triangulated category
s_group_cohomology | state | Group cohomology
s_partially_ordered_set | axiom | Partially ordered set
t_proof_by_contradiction | technique | Proof by Contradiction
t_proof_by_induction | technique | Proof by Induction
s_total_order | axiom | Total Order
s_cantor_diagonal_argument | theorem | Cantor Diagonal Argument
s_subfield | axiom | Subfield
s_finite_field_f_p | state | Finite Field F_p
s_basis | axiom | Basis
s_dimension | axiom | Dimension
s_ring | axiom | Ring
s_ring_with_unity | axiom | Ring with Unity
s_ring_homomorphism | axiom | Ring Homomorphism
s_field_of_fractions | state | Field of Fractions
s_open_ball | axiom | Open Ball
s_open_set_metric | axiom | Open Set (metric)
s_open_set_topological | axiom | Open Set (topological)
s_closed_set | axiom | Closed Set
s_neighborhood | axiom | Neighborhood
s_homeomorphism | axiom | Homeomorphism
s_closure | state | Closure
s_interior | state | Interior
s_boundary | state | Boundary
s_basis_for_a_topology | axiom | Basis for a Topology
s_measure | axiom | Measure
s_measurable_set | axiom | Measurable Set
s_orbit | state | Orbit
s_stabilizer | state | Stabilizer
s_limit_of_a_sequence | axiom | Limit of a Sequence
s_continuity | axiom | Continuity (ε-δ)
s_derivative | axiom | Derivative
s_gradient | state | Gradient
s_series | state | Series
s_event | axiom | Event
s_expectation | state | Expectation
s_variance | state | Variance
s_independence_events | axiom | Independence (events)
s_probability_distribution | state | Probability Distribution
s_martingale | state | Martingale
s_chart | axiom | Chart
s_atlas | axiom | Atlas
s_kform | state | k-Form
s_geodesic | axiom | geodesic
s_complex_manifold | axiom | complex manifold
s_volume_form | state | Volume Form
s_embedding | axiom | Embedding
s_genus | state | Genus
s_torus_complex | state | Torus (complex)
s_orbifold | axiom | Orbifold
s_orbifold_point | state | Orbifold Point
s_fiber | axiom | Fiber
s_section | axiom | Section
s_curvature_of_a_connection | state | Curvature of a Connection
s_flat_connection | state | Flat Connection
s_covariant_derivative | axiom | Covariant Derivative
s_reynolds_number | state | Reynolds Number
s_knot | axiom | Knot
s_knot_invariant | axiom | Knot Invariant
s_unknot | state | Unknot
s_trefoil_knot | state | Trefoil Knot
s_knot_diagram | state | Knot Diagram
s_link | state | Link
s_reidemeister_moves | axiom | Reidemeister Moves
s_matroid | axiom | Matroid
s_matroid_circuit | state | Matroid Circuit
s_dual_matroid | state | Dual Matroid
s_graphic_matroid | state | Graphic Matroid
s_representable_matroid | state | Representable Matroid
s_morphism | axiom | Morphism
s_isomorphism_category | state | Isomorphism (category)
s_product_categorical | state | Product (categorical)
s_coproduct | state | Coproduct
s_colimit | state | Colimit
s_schwartz_distribution | axiom | Schwartz Distribution
s_dirac_delta_distribution | state | Dirac Delta Distribution
s_distributional_derivative | state | Distributional Derivative
s_convolution | state | Convolution
s_tensor_product_of_modules | state | Tensor Product of Modules
s_tensor | axiom | Tensor
s_tensor_field | state | Tensor Field
s_spinor | state | Spinor
s_quaternions | state | Quaternions ℍ
s_grassmannian_grkn | state | Grassmannian Gr(k,n)
s_plucker_embedding | state | Plücker Embedding
s_mirror_symmetry_calabiyau | state | Mirror Symmetry (Calabi-Yau)
s_modular_arithmetic | axiom | Modular Arithmetic
s_quadratic_residue | axiom | Quadratic Residue
s_jinvariant | state | j-Invariant
s_lattice_in | axiom | Lattice (in ℂ)
s_weierstrass_function | state | Weierstrass ℘-function
s_mandelbrot_set | state | Mandelbrot set
s_julia_set | state | Julia set
s_filled_julia_set_k_c | state | Filled Julia Set K_c
s_ordinary_differential_equation_ode | axiom | Ordinary Differential Equation (ODE)
s_linear_ode | state | Linear ODE
s_autonomous_ode | state | Autonomous ODE
s_flow | state | Flow
s_equilibrium_point | state | Equilibrium Point
s_partial_differential_equation_pde | axiom | Partial Differential Equation (PDE)
s_elliptic_pde | state | Elliptic PDE
s_parabolic_pde | state | Parabolic PDE
s_hyperbolic_pde | state | Hyperbolic PDE
t_lagrange_multipliers | technique | Lagrange Multipliers
t_convex_optimization | technique | Convex Optimization
s_bernoulli_distribution | state | Bernoulli Distribution
s_binomial_distribution | state | Binomial Distribution
s_poisson_distribution | state | Poisson Distribution
s_geometric_distribution | state | Geometric Distribution
s_chisquared_distribution | state | Chi-Squared Distribution
s_students_tdistribution | state | Student's t-Distribution
s_multivariate_normal_distribution | state | Multivariate Normal Distribution
s_prior_distribution | axiom | Prior Distribution
s_posterior_distribution | state | Posterior Distribution
t_bayesian_inference | technique | Bayesian Inference
s_conjugate_prior | state | Conjugate Prior
s_wavelet | state | Wavelet
s_haar_wavelet | state | Haar Wavelet
s_multiresolution_analysis | state | Multiresolution Analysis
s_daubechies_wavelet | state | Daubechies Wavelet
s_block_design | axiom | Block Design
s_tdesign | state | t-Design
s_steiner_system_stkn | state | Steiner System S(t,k,n)
s_linear_code | state | Linear Code
s_hamming_distance | state | Hamming Distance
s_hamming_code | state | Hamming Code
s_minimum_distance | axiom | Minimum Distance
s_natural_logarithm | state | Natural Logarithm
s_matrix_exponential | state | Matrix Exponential
s_spherical_harmonics_y_lm | state | Spherical Harmonics Y_l^m
s_stirlings_approximation | theorem | Stirling's Approximation
s_transcendental_number | axiom | Transcendental Number
s_irrationality_of_2 | theorem | Irrationality of √2
s_transcendence_of | theorem | Transcendence of π
s_transcendence_of_e | theorem | Transcendence of e
s_liouville_numbers | state | Liouville Numbers
s_degree_of_a_field_extension | state | Degree of a Field Extension
s_maximal_ideal | axiom | Maximal Ideal
s_radical_of_an_ideal | state | Radical of an Ideal
s_affine_scheme_specr | state | Affine Scheme Spec(R)
s_scheme | axiom | scheme
s_sheaf | axiom | sheaf
s_morphism_of_schemes | state | Morphism of Schemes
s_homotopy | state | Homotopy
s_eigenvalue | state | Eigenvalue
s_eigenvector | state | Eigenvector
s_number_field | axiom | number_field
s_field_extension_degree | axiom | field_extension_degree
s_trace_of_element | axiom | trace_of_element
s_discriminant_of_number_field | axiom | discriminant_of_number_field
s_prime_ideal | axiom | prime_ideal
s_fractional_ideal | axiom | fractional_ideal
s_ideal_class_group_2 | axiom | ideal_class_group
s_class_number_2 | axiom | class_number
s_root_of_unity | axiom | root_of_unity
s_real_embedding | axiom | real_embedding
s_complex_embedding | axiom | complex_embedding
s_signature_of_number_field | axiom | signature_of_number_field
s_ramification_index | axiom | ramification_index
s_inertia_degree | axiom | inertia_degree
s_ramified_prime | axiom | ramified_prime
s_unramified_prime | axiom | unramified_prime
s_split_prime | axiom | split_prime
s_inert_prime | axiom | inert_prime
s_decomposition_group | axiom | decomposition_group
s_inertia_group | axiom | inertia_group
s_frobenius_element | axiom | frobenius_element
s_p_adic_numbers | axiom | p_adic_numbers
s_p_adic_valuation | axiom | p_adic_valuation
s_p_adic_absolute_value | axiom | p_adic_absolute_value
s_p_adic_integers | axiom | p_adic_integers
s_completion_of_number_field | axiom | completion_of_number_field
s_place_of_number_field | axiom | place_of_number_field
s_adele_ring_2 | axiom | adele_ring
s_idele_group_2 | axiom | idele_group
s_idele_class_group | state | idele_class_group
s_absolute_value_product_formula | axiom | absolute_value_product_formula
s_integral_basis | state | integral_basis
s_regulator_of_number_field | state | regulator_of_number_field
s_fundamental_units | state | fundamental_units
s_class_field | state | class_field
s_ray_class_group | state | ray_class_group
s_lattice_in_r_n | state | lattice_in_R_n
s_class_number_formula_2 | theorem | class_number_formula
s_ostrowski_theorem | theorem | ostrowski_theorem
s_hensels_lemma_2 | theorem | hensels_lemma
s_fundamental_identity_efg | theorem | fundamental_identity_efg
t_geometry_of_numbers | technique | geometry_of_numbers
t_localization_at_prime | technique | localization_at_prime
t_ideal_arithmetic | technique | ideal_arithmetic
s_von_mangoldt_function | axiom | von_mangoldt_function
s_chebyshev_functions | axiom | chebyshev_functions
s_mobius_function | axiom | mobius_function
s_principal_character | state | principal_character
s_conductor_of_character | axiom | conductor_of_character
s_primitive_character | axiom | primitive_character
s_trivial_zeros_of_zeta | axiom | trivial_zeros_of_zeta
s_critical_strip | axiom | critical_strip
s_critical_line | axiom | critical_line
s_arithmetic_function | axiom | arithmetic_function
s_multiplicative_function | axiom | multiplicative_function
s_dirichlet_convolution | axiom | dirichlet_convolution
s_sieve_of_eratosthenes | axiom | sieve_of_eratosthenes
s_siegel_zero | state | siegel_zero
s_li_function | state | li_function
s_prime_gaps | state | prime_gaps
s_major_arcs | state | major_arcs
s_minor_arcs | state | minor_arcs
s_gaussian_sum | state | gaussian_sum
s_kloosterman_sum | state | kloosterman_sum
s_ramanujan_sum | state | ramanujan_sum
s_density_estimate_for_zeros | state | density_estimate_for_zeros
s_prime_number_theorem_2 | theorem | prime_number_theorem
s_chebyshev_estimates | theorem | chebyshev_estimates
s_dirichlet_l_nonvanishing_at_1 | theorem | dirichlet_l_nonvanishing_at_1
s_vinogradovs_theorem | theorem | vinogradovs_theorem
s_warings_problem | theorem | warings_problem
s_landau_prime_ideal_theorem | theorem | landau_prime_ideal_theorem
t_exponential_sum_technique | technique | exponential_sum_technique
t_selberg_sieve | technique | selberg_sieve
t_bruns_sieve | technique | bruns_sieve
t_perron_formula | technique | perron_formula
t_mobius_inversion | technique | mobius_inversion
t_contour_integration_in_ant | technique | contour_integration_in_ant
s_computational_complexity_class | axiom | computational_complexity_class
s_probabilistic_algorithm | axiom | probabilistic_algorithm
s_deterministic_algorithm | axiom | deterministic_algorithm
s_integer_factorization_problem | axiom | integer_factorization_problem
s_primality_testing_problem | axiom | primality_testing_problem
s_discrete_logarithm_problem | axiom | discrete_logarithm_problem
s_rsa_cryptosystem | axiom | rsa_cryptosystem
s_lattice_in_z_n | axiom | lattice_in_Z_n
s_smooth_number | axiom | smooth_number
s_b_smooth | axiom | b_smooth
s_miller_rabin_witness | state | miller_rabin_witness
s_pseudoprime | state | pseudoprime
s_carmichael_number | state | carmichael_number
s_factor_base | state | factor_base
s_closest_vector_problem | state | closest_vector_problem
s_aks_primality_theorem_2 | theorem | aks_primality_theorem
s_miller_rabin_correctness | theorem | miller_rabin_correctness
s_lenstra_ecm_expected_time | theorem | lenstra_ecm_expected_time
s_number_field_sieve_complexity | theorem | number_field_sieve_complexity
s_lll_algorithm_properties | theorem | lll_algorithm_properties
t_trial_division | technique | trial_division
t_miller_rabin_test | technique | miller_rabin_test
t_aks_algorithm | technique | aks_algorithm
t_sieve_of_eratosthenes_computational | technique | sieve_of_eratosthenes_computational
t_quadratic_sieve | technique | quadratic_sieve
t_number_field_sieve | technique | number_field_sieve
t_pollard_rho_method | technique | pollard_rho_method
t_pollard_p_minus_1 | technique | pollard_p_minus_1
t_baby_step_giant_step | technique | baby_step_giant_step
t_index_calculus | technique | index_calculus
t_lll_algorithm | technique | lll_algorithm
t_schoof_algorithm | technique | schoof_algorithm
s_projective_space | axiom | projective_space
s_coordinate_ring | axiom | coordinate_ring
s_function_field | axiom | function_field
s_morphism_of_varieties | axiom | morphism_of_varieties
s_rational_map | axiom | rational_map
s_birational_equivalence | axiom | birational_equivalence
s_affine_scheme | axiom | affine_scheme
s_stalk_of_sheaf | axiom | stalk_of_sheaf
s_morphism_of_schemes_2 | axiom | morphism_of_schemes
s_fiber_of_morphism | axiom | fiber_of_morphism
s_base_change | axiom | base_change
s_reduced_scheme | axiom | reduced_scheme
s_integral_scheme | axiom | integral_scheme
s_noetherian_scheme | axiom | noetherian_scheme
s_locally_free_sheaf | axiom | locally_free_sheaf
s_divisor | axiom | divisor
s_weil_divisor | axiom | weil_divisor
s_cartier_divisor | axiom | cartier_divisor
s_linear_equivalence_of_divisors | axiom | linear_equivalence_of_divisors
s_canonical_divisor | axiom | canonical_divisor
s_cotangent_sheaf | axiom | cotangent_sheaf
s_tangent_sheaf | axiom | tangent_sheaf
s_cech_cohomology_2 | state | cech_cohomology
s_chow_ring | state | chow_ring
s_hilbert_scheme | state | hilbert_scheme
s_grassmannian | state | grassmannian
s_blowup | state | blowup
s_exceptional_divisor | state | exceptional_divisor
s_nef_divisor | state | nef_divisor
s_canonical_ring | state | canonical_ring
s_variety_of_general_type | state | variety_of_general_type
s_fano_variety | state | fano_variety
s_calabi_yau_variety | state | calabi_yau_variety
s_jacobian_of_curve | state | jacobian_of_curve
s_neron_model | state | neron_model
s_serres_gaga | theorem | serres_gaga
s_bezouts_theorem | theorem | bezouts_theorem
s_finite_generation_of_canonical_ring | theorem | finite_generation_of_canonical_ring
s_existence_of_hilbert_schemes | theorem | existence_of_hilbert_schemes
s_deligne_mumford_irreducibility | theorem | deligne_mumford_irreducibility
t_birational_geometry_program | technique | birational_geometry_program
t_grothendieck_cohomology_methods | technique | grothendieck_cohomology_methods
t_deformation_theory | technique | deformation_theory
t_grobner_basis_computation | technique | grobner_basis_computation
s_rational_point | axiom | rational_point
s_diophantine_equation | axiom | diophantine_equation
s_height_of_rational_point | axiom | height_of_rational_point
s_weil_height | axiom | weil_height
s_neron_tate_height | axiom | neron_tate_height
s_good_reduction | axiom | good_reduction
s_bad_reduction | axiom | bad_reduction
s_etale_topology | axiom | etale_topology
s_etale_morphism | axiom | etale_morphism
s_etale_fundamental_group | axiom | etale_fundamental_group
s_shimura_variety | axiom | shimura_variety
s_newform | axiom | newform
s_automorphic_form | axiom | automorphic_form
s_selmer_group | axiom | selmer_group
s_bsd_order_of_vanishing | state | bsd_order_of_vanishing
s_motive | state | motive
s_l_function_of_motive | state | l_function_of_motive
s_brauer_manin_obstruction | state | brauer_manin_obstruction
s_tamagawa_number | state | tamagawa_number
s_birch_swinnerton_dyer_conjecture | theorem | birch_swinnerton_dyer_conjecture
s_tate_conjecture | theorem | tate_conjecture
s_finiteness_of_sha_conjecture | theorem | finiteness_of_sha_conjecture
t_descent_method | technique | descent_method
t_height_machine | technique | height_machine
t_deformation_of_galois_representations | technique | deformation_of_galois_representations
t_iwasawa_theory | technique | iwasawa_theory
s_group_representation | axiom | group_representation
s_degree_of_representation | axiom | degree_of_representation
s_subrepresentation | axiom | subrepresentation
s_direct_sum_of_representations | axiom | direct_sum_of_representations
s_tensor_product_of_representations | axiom | tensor_product_of_representations
s_character_of_representation | axiom | character_of_representation
s_class_function | axiom | class_function
s_trivial_representation | axiom | trivial_representation
s_representation_ring | axiom | representation_ring
s_induced_representation | axiom | induced_representation
s_restricted_representation | axiom | restricted_representation
s_lie_algebra_representation | axiom | lie_algebra_representation
s_weight_of_representation | axiom | weight_of_representation
s_weight_space | axiom | weight_space
s_fundamental_weight | axiom | fundamental_weight
s_positive_root | axiom | positive_root
s_simple_root | axiom | simple_root
s_category_o | axiom | category_O
s_isotypic_component | state | isotypic_component
s_multiplicity_of_irreducible | state | multiplicity_of_irreducible
s_weyl_character | state | weyl_character
s_formal_character | state | formal_character
s_jordan_holder_multiplicity | state | jordan_holder_multiplicity
s_kazhdan_lusztig_polynomial | state | kazhdan_lusztig_polynomial
s_bgg_resolution | state | bgg_resolution
s_harish_chandra_module | state | harish_chandra_module
s_langlands_parameter | state | langlands_parameter
s_schurs_lemma_2 | theorem | schurs_lemma
s_maschkes_theorem | theorem | maschkes_theorem
s_number_of_irreducibles_equals_conjugacy_classes | theorem | number_of_irreducibles_equals_conjugacy_classes
s_weyl_character_formula | theorem | weyl_character_formula
s_kazhdan_lusztig_conjecture | theorem | kazhdan_lusztig_conjecture
t_character_methods | technique | character_methods
t_induction_restriction_technique | technique | induction_restriction_technique
t_geometric_representation_theory | technique | geometric_representation_theory
t_crystal_base_method | technique | crystal_base_method
s_presentation_of_group | axiom | presentation_of_group
s_word_metric | axiom | word_metric
s_conjugacy_problem | axiom | conjugacy_problem
s_isomorphism_problem | axiom | isomorphism_problem
s_free_group_2 | state | free_group
s_free_product | axiom | free_product
s_amalgamated_free_product | axiom | amalgamated_free_product
s_hnn_extension | axiom | hnn_extension
t_quasi_isometry | technique | quasi_isometry
s_quasi_isometry_invariant | axiom | quasi_isometry_invariant
s_dehn_function | axiom | dehn_function
s_hyperbolic_group | axiom | hyperbolic_group
s_boundary_at_infinity | axiom | boundary_at_infinity
s_cat_0_space | axiom | cat_0_space
s_cat_0_group | axiom | cat_0_group
s_amenable_group | axiom | amenable_group
s_residually_finite_group | axiom | residually_finite_group
s_linear_group | axiom | linear_group
s_automatic_group | axiom | automatic_group
s_growth_rate | state | growth_rate
s_ends_of_group | state | ends_of_group
s_dehn_presentation | state | dehn_presentation
s_rips_complex | state | rips_complex
s_hyperbolic_dehn_function_is_linear | theorem | hyperbolic_dehn_function_is_linear
s_milnor_svarc_lemma | theorem | milnor_svarc_lemma
s_flat_torus_theorem | theorem | flat_torus_theorem
t_quasi_isometric_classification | technique | quasi_isometric_classification
t_van_kampen_diagram | technique | van_kampen_diagram
t_small_cancellation_theory | technique | small_cancellation_theory
t_geometric_group_actions | technique | geometric_group_actions
s_christoffel_symbols | axiom | christoffel_symbols
s_exponential_map_diff_geom | axiom | exponential_map_diff_geom
s_parallel_transport | axiom | parallel_transport
s_covariant_derivative_2 | axiom | covariant_derivative
s_differential_form_2 | axiom | differential_form
s_de_rham_cohomology | axiom | de_rham_cohomology
s_connection_form | axiom | connection_form
s_curvature_form | axiom | curvature_form
s_todd_class | state | todd_class
s_cut_locus | state | cut_locus
s_volume_form_2 | state | volume_form
s_laplacian_on_manifold | state | laplacian_on_manifold
s_harmonic_form | state | harmonic_form
s_einstein_metric | state | einstein_metric
s_kahler_metric | state | kahler_metric
s_bochner_weitzenboeck_formula | theorem | bochner_weitzenboeck_formula
s_berger_classification | theorem | berger_classification
s_hamilton_three_manifolds | theorem | hamilton_three_manifolds
t_comparison_geometry | technique | comparison_geometry
t_bochner_technique | technique | bochner_technique
t_moving_frames_method | technique | moving_frames_method
s_trigonometric_system | axiom | trigonometric system
s_lp_space_on_the_circle | axiom | L^p space on the circle
s_dirichlet_kernel | state | Dirichlet kernel
s_fejer_kernel | state | Fejér kernel
s_plancherel_identity | state | Plancherel identity
s_maximal_partial_sum_operator | state | maximal partial sum operator
s_riesz_transform | state | Riesz transform
s_multiplier_operator | state | multiplier operator
s_stationary_phase_point | state | stationary phase point
s_kakeya_set | state | Kakeya set
s_bmo_space | state | BMO space
s_a_p_weight | state | A_p weight
s_stationary_phase_method | theorem | stationary phase method
s_van_der_corput_lemma | theorem | van der Corput lemma
s_h1bmo_duality | theorem | H^1-BMO duality
s_kakeya_conjecture | theorem | Kakeya conjecture
t_tt_method | technique | TT* method
s_weak_derivative | state | weak derivative
s_neumann_problem | state | Neumann problem
s_conservation_law | state | conservation law
s_shock_wave | state | shock wave
s_rankinehugoniot_condition | state | Rankine-Hugoniot condition
s_characteristics_of_a_pde | state | characteristics of a PDE
s_riemann_problem | state | Riemann problem
s_leray_existence_of_weak_solutions_for_navierstokes | theorem | Leray existence of weak solutions for Navier-Stokes
s_hopf_maximum_principle_boundary_point_lemma | theorem | Hopf maximum principle / boundary point lemma
t_energy_method_for_pde | technique | energy method for PDE
t_method_of_characteristics | technique | method of characteristics
t_galerkin_approximation | technique | Galerkin approximation
s_stressenergy_tensor | axiom | stress-energy tensor
s_causal_structure | axiom | causal structure
s_einstein_tensor | state | Einstein tensor
s_schwarzschild_solution | state | Schwarzschild solution
s_kerr_solution | state | Kerr solution
s_friedmannlematrerobertsonwalker_metric | state | Friedmann-Lemaître-Robertson-Walker metric
s_event_horizon | state | event horizon
s_singularity_in_spacetime | state | singularity in spacetime
s_penrose_diagram | state | Penrose diagram
s_adm_mass | state | ADM mass
s_bondi_mass | state | Bondi mass
s_cauchy_surface | state | Cauchy surface
s_weyl_tensor | state | Weyl tensor
s_penrose_inequality | theorem | Penrose inequality
s_orbit_of_a_point | state | orbit of a point
s_attractor | state | attractor
s_strange_attractor | state | strange attractor
s_basin_of_attraction | state | basin of attraction
s_sensitive_dependence_on_initial_conditions | state | sensitive dependence on initial conditions
s_symbolic_dynamics | state | symbolic dynamics
s_hyperbolic_set | state | hyperbolic set
s_fatou_set | state | Fatou set
s_bifurcation | state | bifurcation
s_logistic_map | state | logistic map
s_henon_map | state | Hénon map
s_lorenz_attractor | state | Lorenz attractor
s_kam_torus | state | KAM torus
s_jakobsons_theorem | theorem | Jakobson's theorem
t_poincare_section | technique | Poincaré section
s_involution_on_an_algebra | axiom | involution on an algebra
s_spectrum_of_an_element | state | spectrum of an element
s_gns_representation | state | GNS representation
s_factor | state | factor
s_type_i_factor | state | Type I factor
s_type_ii_1_factor | state | Type II_1 factor
s_type_ii_factor | state | Type II_∞ factor
s_type_iii_factor | state | Type III factor
s_hyperfinite_ii_1_factor_r | state | hyperfinite II_1 factor R
s_kms_condition | state | KMS condition
s_subfactor | state | subfactor
s_jones_index_mn | state | Jones index [M:N]
s_standard_invariant_of_a_subfactor | state | standard invariant of a subfactor
s_k_0_group_of_a_calgebra | state | K_0 group of a C*-algebra
s_k_1_group_of_a_calgebra | state | K_1 group of a C*-algebra
s_gns_theorem | theorem | GNS theorem
s_jones_index_theorem | theorem | Jones index theorem
s_elliott_classification_of_af_algebras | theorem | Elliott classification of AF algebras
s_takesaki_duality_theorem | theorem | Takesaki duality theorem
t_gns_construction_method | technique | GNS construction method
t_jones_basic_construction | technique | Jones basic construction
s_string_theory_compactification | axiom | string theory compactification
s_hodge_numbers_hpq | state | Hodge numbers h^{p,q}
s_hodge_diamond | state | Hodge diamond
s_period_integral | state | period integral
s_yukawa_coupling | state | Yukawa coupling
s_gromovwitten_invariant | state | Gromov-Witten invariant
s_quantum_cohomology | state | quantum cohomology
s_prepotential | state | prepotential
s_fukaya_category | state | Fukaya category
s_amodel | state | A-model
s_bmodel | state | B-model
s_picardfuchs_equation | state | Picard-Fuchs equation
s_mirror_symmetry_for_the_quintic_candelasde_la_ossagreenparkes | theorem | mirror symmetry for the quintic (Candelas-de la Ossa-Green-Parkes)
s_kontsevich_homological_mirror_symmetry_conjecture | theorem | Kontsevich homological mirror symmetry conjecture
s_syz_conjecture_stromingeryauzaslow | theorem | SYZ conjecture (Strominger-Yau-Zaslow)
s_batyrev_mirror_construction | theorem | Batyrev mirror construction
t_localization_on_moduli_spaces | technique | localization on moduli spaces
s_vertex_algebra | axiom | vertex algebra
s_vertex_operator | state | vertex operator
s_virasoro_algebra | state | Virasoro algebra
s_conformal_vertex_algebra | state | conformal vertex algebra
s_monster_group_m | state | monster group M
s_moonshine_module_v | state | moonshine module V^♮
s_jfunction | state | j-function
s_mckaythompson_series | state | McKay-Thompson series
s_monstrous_moonshine | state | monstrous moonshine
s_affine_lie_algebra_kacmoody | state | affine Lie algebra (Kac-Moody)
s_wzw_model | state | WZW model
s_rational_vertex_algebra | state | rational vertex algebra
s_conformal_field_theory_2d | state | conformal field theory (2D)
s_operator_product_expansion | state | operator product expansion
s_modular_tensor_category | state | modular tensor category
s_borcherds_theorem_monstrous_moonshine_conjecture | theorem | Borcherds' theorem (monstrous moonshine conjecture)
s_frenkellepowskymeurman_construction | theorem | Frenkel-Lepowsky-Meurman construction
s_goddardkentolive_coset_construction | theorem | Goddard-Kent-Olive coset construction
s_singular_homology_h_nxz | state | singular homology H_n(X;Z)
s_singular_cohomology_hnxr | state | singular cohomology H^n(X;R)
s_cup_product | state | cup product
s_relative_homology_h_nxa | state | relative homology H_n(X,A)
s_excision_in_homology | state | excision in homology
s_reduced_ktheory_kx | state | reduced K-theory K̃(X)
s_cobordism_group_n | state | cobordism group Ω_n
s_thom_spectrum | state | Thom spectrum
s_splitting_principle | theorem | splitting principle
s_smooth_map | axiom | smooth map
s_smooth_structure | state | smooth structure
s_surgery_on_a_manifold | state | surgery on a manifold
s_connected_sum | state | connected sum
s_exotic_sphere | state | exotic sphere
s_milnors_exotic_spheres | theorem | Milnor's exotic spheres
s_kervairemilnor_classification | theorem | Kervaire-Milnor classification
s_exotic_r4 | theorem | exotic R^4
s_rohlins_theorem | theorem | Rohlin's theorem
t_surgery_theory | technique | surgery theory
s_delignemumford_compactification_m_g | state | Deligne-Mumford compactification M̄_g
s_stable_curve | state | stable curve
s_teichmuller_space_t_g | state | Teichmüller space T_g
s_moduli_of_vector_bundles_on_a_curve | state | moduli of vector bundles on a curve
s_sequivalence | state | S-equivalence
s_period_matrix | state | period matrix
s_torelli_map | state | Torelli map
s_stability_in_git | state | stability in GIT
s_algebraic_stack | state | algebraic stack
s_delignemumford_stack | state | Deligne-Mumford stack
s_tautological_classes_on_m_g | state | tautological classes on M̄_g
s_mumfords_git_construction | theorem | Mumford's GIT construction
s_narasimhanseshadri_theorem | theorem | Narasimhan-Seshadri theorem
s_harer_stability | theorem | Harer stability
s_witten_conjecture_kontsevichs_theorem | theorem | Witten conjecture (Kontsevich's theorem)
s_partition_of_an_integer | axiom | partition of an integer
s_young_diagram | axiom | Young diagram
s_symmetric_function | state | symmetric function
s_schur_polynomial_s | state | Schur polynomial s_λ
s_kostka_number_k | state | Kostka number K_{λ,μ}
s_littlewoodrichardson_coefficient_c | state | Littlewood-Richardson coefficient c^ν_{λ,μ}
s_littlewoodrichardson_rule | state | Littlewood-Richardson rule
s_robinsonschensted_correspondence | state | Robinson-Schensted correspondence
s_mobius_function_of_a_poset | state | Möbius function of a poset
s_mobius_inversion_on_posets | state | Möbius inversion on posets
s_zeta_polynomial_of_a_poset | state | zeta polynomial of a poset
s_incidence_algebra_of_a_poset | state | incidence algebra of a poset
s_ppartition | state | P-partition
s_jacobitrudi_formula | state | Jacobi-Trudi formula
s_dual_jacobitrudi_formula | state | dual Jacobi-Trudi formula
s_cauchy_identity | state | Cauchy identity
s_pieris_rule | state | Pieri's rule
s_frobenius_characteristic_map | state | Frobenius characteristic map
s_cauchy_identity_proof_and_generalizations | theorem | Cauchy identity proof and generalizations
s_simple_graph | axiom | simple graph
s_turan_graph_tnr | state | Turán graph T(n,r)
s_graph_density | state | graph density
s_clique_number_g | state | clique number ω(G)
s_independence_number_g | state | independence number α(G)
s_regularity_partition | state | regularity partition
s_regular_pair | state | ε-regular pair
s_arithmetic_progression_of_length_k | state | arithmetic progression of length k
s_graph_ramsey_theory | state | graph Ramsey theory
s_sunflower_system | state | sunflower (Δ-system)
s_extremal_number_exnh | state | extremal number ex(n,H)
s_sunflower_lemma_erdsko | theorem | Sunflower lemma (Erdős-Ko)
t_probabilistic_method | technique | probabilistic method
t_regularity_method | technique | regularity method
t_alteration_method | technique | alteration method
s_alphabet_and_strings | axiom | alphabet and strings
s_boolean_circuit | axiom | Boolean circuit
s_decision_problem_language | state | decision problem / language
s_complexity_class_p | state | complexity class P
s_complexity_class_np | state | complexity class NP
s_npcompleteness | state | NP-completeness
s_sat_boolean_satisfiability | state | SAT (Boolean satisfiability)
s_3sat | state | 3-SAT
s_complexity_class_conp | state | complexity class coNP
s_complexity_class_pspace | state | complexity class PSPACE
s_complexity_class_l_logspace | state | complexity class L (logspace)
s_complexity_class_nl_nondeterministic_logspace | state | complexity class NL (nondeterministic logspace)
s_complexity_class_bpp | state | complexity class BPP
s_complexity_class_rp | state | complexity class RP
s_complexity_class_ip | state | complexity class IP
s_circuit_complexity | state | circuit complexity
s_circuit_class_nc | state | circuit class NC
s_circuit_class_ac0 | state | circuit class AC^0
s_polynomial_hierarchy_ph | state | polynomial hierarchy PH
s_space_complexity_class | state | space complexity class
s_approximation_algorithm | state | approximation algorithm
s_pcp_verifier | state | PCP verifier
s_p_np_conjecture | theorem | P ≠ NP conjecture
s_ip_pspace | theorem | IP = PSPACE
s_pcp_theorem_and_inapproximability | theorem | PCP theorem and inapproximability
s_npcompleteness_of_3sat | theorem | NP-completeness of 3-SAT
s_bakergillsolovay_relativization_barrier | theorem | Baker-Gill-Solovay relativization barrier
t_diagonalization | technique | diagonalization
t_arithmetization | technique | arithmetization
s_floatingpoint_number_system | axiom | floating-point number system
s_mesh_grid | axiom | mesh / grid
s_interpolation | state | interpolation
s_polynomial_interpolation | state | polynomial interpolation
s_lagrange_interpolation | state | Lagrange interpolation
s_newton_interpolation | state | Newton interpolation
s_spline_interpolation | state | spline interpolation
s_numerical_quadrature | state | numerical quadrature
s_gaussian_quadrature | state | Gaussian quadrature
s_newtoncotes_formula | state | Newton-Cotes formula
s_condition_number_a | state | condition number κ(A)
s_finite_element_space | state | finite element space
s_stiffness_matrix | state | stiffness matrix
s_spectral_method | state | spectral method
s_linear_multistep_method | state | linear multistep method
s_truncation_error | state | truncation error
s_stability_of_a_numerical_method | state | stability of a numerical method
s_convergence_of_a_numerical_method | state | convergence of a numerical method
s_cfl_condition | state | CFL condition
s_conjugate_gradient_method | state | conjugate gradient method
s_multigrid_method | state | multigrid method
s_krylov_subspace | state | Krylov subspace
s_ceas_lemma | theorem | Céa's lemma
s_bramblehilbert_lemma | theorem | Bramble-Hilbert lemma
s_runge_phenomenon | theorem | Runge phenomenon
s_laxrichtmyer_theorem | theorem | Lax-Richtmyer theorem
s_dahlquist_barrier_theorems | theorem | Dahlquist barrier theorems
t_richardson_extrapolation | technique | Richardson extrapolation
t_adaptive_mesh_refinement | technique | adaptive mesh refinement
s_axiom_of_extensionality | axiom | axiom of extensionality
s_axiom_of_infinity | axiom | axiom of infinity
s_axiom_of_replacement | axiom | axiom of replacement
s_axiom_of_power_set | axiom | axiom of power set
s_axiom_of_regularity_foundation | axiom | axiom of regularity / foundation
s_successor_ordinal | state | successor ordinal
s_limit_ordinal | state | limit ordinal
s_aleph_numbers | state | aleph numbers ℵ_α
s_v_l_axiom_of_constructibility | state | V = L (axiom of constructibility)
s_forcing | state | forcing
s_generic_filter | state | generic filter
s_forcing_extension_mg | state | forcing extension M[G]
s_booleanvalued_model | state | Boolean-valued model
s_large_cardinal | state | large cardinal
s_supercompact_cardinal | state | supercompact cardinal
s_woodin_cardinal | state | Woodin cardinal
s_determinacy_ad | state | determinacy (AD)
s_borel_set | state | Borel set
s_analytic_1_1_set | state | analytic / Σ^1_1 set
s_descriptive_set_theory | state | descriptive set theory
s_godels_consistency_of_ch | theorem | Gödel's consistency of CH
s_cohens_independence_of_ch | theorem | Cohen's independence of CH
s_large_cardinal_consistency_hierarchy | theorem | large cardinal consistency hierarchy
t_forcing_technique | technique | forcing technique
t_iterated_forcing | technique | iterated forcing
s_model_of_a_theory | state | model of a theory
s_complete_theory | state | complete theory
s_type_over_a | state | type over A
s_definable_set | state | definable set
s_ultrafilter | state | ultrafilter
s_ultraproduct | state | ultraproduct
s_saturated_model | state | saturated model
s_homogeneous_model | state | homogeneous model
s_model_completeness | state | model completeness
s_morley_rank | state | Morley rank
s_ominimal_structure | state | o-minimal structure
s_categoricity_in_power | state | categoricity in power κ
s_baldwinlachlan_theorem | theorem | Baldwin-Lachlan theorem
s_shelahs_classification_theory | theorem | Shelah's classification theory
t_ultraproduct_construction | technique | ultraproduct construction
t_backandforth_method | technique | back-and-forth method
s_filtration_f_t | axiom | filtration {F_t}
s_adapted_process | axiom | adapted process
s_stationary_distribution | state | stationary distribution
s_supermartingale | state | supermartingale
s_it_integral | state | Itô integral
s_it_process | state | Itô process
s_its_formula | state | Itô's formula
s_gaussian_free_field | state | Gaussian free field
s_percolation_model | state | percolation model
s_critical_probability_p_c | state | critical probability p_c
s_harrisfkg_inequality | theorem | Harris-FKG inequality
t_coupling_of_markov_chains | technique | coupling of Markov chains
s_levys_lemma_on_the_sphere | state | Lévy's lemma on the sphere
s_johnsonlindenstrauss_projection | state | Johnson-Lindenstrauss projection
s_rademacher_average | state | Rademacher average
s_volume_ratio | state | volume ratio
s_entropy_and_covering_numbers | state | entropy and covering numbers
s_sudakov_minoration | state | Sudakov minoration
s_restricted_invertibility | state | restricted invertibility
s_johnsonlindenstrauss_lemma | theorem | Johnson-Lindenstrauss lemma
s_gaussian_isoperimetric_theorem_borell_sudakovtsirelson | theorem | Gaussian isoperimetric theorem (Borell, Sudakov-Tsirelson)
s_bourgainmilman_inverse_santal_inequality | theorem | Bourgain-Milman inverse Santaló inequality
s_talagrands_majorizing_measures_theorem | theorem | Talagrand's majorizing measures theorem
s_schechtmans_theorem_on_random_sections | theorem | Schechtman's theorem on random sections
s_bourgaintzafriri_restricted_invertibility_theorem | theorem | Bourgain-Tzafriri restricted invertibility theorem
t_random_projection_sketching | technique | random projection / sketching
t_generic_chaining | technique | generic chaining
s_radical_of_integer | state | radical_of_integer
s_abc_triple | state | abc_triple
s_abc_conjecture | state | abc_conjecture
s_quality_of_abc_triple | state | quality_of_abc_triple
s_mason_stothers_theorem | theorem | mason_stothers_theorem
s_szpiros_conjecture | state | szpiros_conjecture
s_halls_conjecture | state | halls_conjecture
s_erdos_woods_conjecture_consequence | theorem | erdos_woods_conjecture_consequence
s_weak_abc_conjecture | state | weak_abc_conjecture
s_abc_implies_fermat_asymptotic | theorem | abc_implies_fermat_asymptotic
s_abc_implies_mordell | state | abc_implies_mordell
s_langevin_result | theorem | langevin_result
s_analytic_index | state | analytic_index
s_k_theory_2 | state | k_theory
t_k_theory_proof | technique | k_theory_proof
s_families_index_theorem | state | families_index_theorem
s_eta_invariant | state | eta_invariant
s_hausdorff_paradox | theorem | hausdorff_paradox
s_tarski_alternative | theorem | tarski_alternative
s_finitely_additive_measure | state | finitely_additive_measure
s_von_neumann_conjecture | state | von_neumann_conjecture
s_no_paradox_in_plane | theorem | no_paradox_in_plane
s_equidecomposable | state | equidecomposable
s_algebraic_rank | state | algebraic_rank
s_analytic_rank | state | analytic_rank
s_bsd_conjecture | state | bsd_conjecture
s_bsd_leading_coefficient | state | bsd_leading_coefficient
s_neron_tate_height_pairing | state | neron_tate_height_pairing
s_carlesons_theorem | theorem | carlesons_theorem
s_carleson_operator | state | carleson_operator
t_time_frequency_analysis | technique | time_frequency_analysis
s_luzin_conjecture | theorem | luzin_conjecture
s_du_bois_reymond_example | state | du_bois_reymond_example
s_normal_distribution_2 | state | normal_distribution
s_lindeberg_clt | theorem | lindeberg_clt
s_lindeberg_condition | state | lindeberg_condition
s_lyapunov_clt | theorem | lyapunov_clt
s_multivariate_clt_2 | state | multivariate_clt
s_simple_group | state | simple_group
s_alternating_group | state | alternating_group
s_groups_of_lie_type | state | groups_of_lie_type
s_chevalley_groups | state | chevalley_groups
s_sporadic_groups | state | sporadic_groups
s_monster_group | state | monster_group
s_mathieu_groups | state | mathieu_groups
s_baby_monster | state | baby_monster
s_thompson_classification | theorem | thompson_classification
s_bn_pair | state | bn_pair
t_local_analysis | technique | local_analysis
t_character_theory_in_cfsg | technique | character_theory_in_cfsg
s_quasithin_groups | state | quasithin_groups
s_monstrous_moonshine_2 | state | monstrous_moonshine
s_griess_construction | theorem | griess_construction
s_arithmetic_progression | state | arithmetic_progression
s_non_vanishing_of_l1_chi | theorem | non_vanishing_of_l1_chi
t_orthogonality_of_characters | technique | orthogonality_of_characters
s_dirichlet_density | state | dirichlet_density
s_equidistribution_of_primes | theorem | equidistribution_of_primes
t_logarithmic_derivative_of_l | technique | logarithmic_derivative_of_l
s_furstenberg_correspondence_principle | theorem | furstenberg_correspondence_principle
s_furstenberg_szemeredi | theorem | furstenberg_szemeredi
s_mixing_transformation | state | mixing_transformation
s_upper_density | state | upper_density
s_ribets_theorem | theorem | ribets_theorem
s_taniyama_shimura_conjecture | state | taniyama_shimura_conjecture
s_hecke_algebra | state | hecke_algebra
s_r_equals_t | theorem | r_equals_t
s_langlands_tunnell | theorem | langlands_tunnell
s_contraction_mapping | state | contraction_mapping
t_picard_iteration | technique | picard_iteration
s_knaster_tarski | theorem | knaster_tarski
s_poincare_birkhoff | theorem | poincare_birkhoff
t_discharging_method | technique | discharging_method
s_unavoidable_set | state | unavoidable_set
s_fundamental_theorem_of_algebra_2 | theorem | fundamental_theorem_of_algebra
s_fta_topological_proof | theorem | fta_topological_proof
s_fta_analytic_proof | theorem | fta_analytic_proof
s_splitting_of_polynomials | state | splitting_of_polynomials
s_fta_algebraic_proof | theorem | fta_algebraic_proof
s_euclids_lemma | theorem | euclids_lemma
s_bezouts_identity | theorem | bezouts_identity
s_formal_system | state | formal_system
s_goedel_numbering | state | goedel_numbering
s_omega_consistency | state | omega_consistency
s_rossers_theorem | theorem | rossers_theorem
t_diagonalization_in_arithmetic | technique | diagonalization_in_arithmetic
s_primitive_recursive_function | state | primitive_recursive_function
s_representability | state | representability
s_consistency_statement | state | consistency_statement
s_bass_guivarch_formula | theorem | bass_guivarch_formula
s_montgomery_zippin | theorem | montgomery_zippin
s_vanishing_ideal | state | vanishing_ideal
s_weak_nullstellensatz | theorem | weak_nullstellensatz
s_strong_nullstellensatz | theorem | strong_nullstellensatz
s_radical_ideal | state | radical_ideal
t_rabinowitsch_trick | technique | rabinowitsch_trick
s_zariski_correspondence | state | zariski_correspondence
s_maximal_ideals_are_points | state | maximal_ideals_are_points
s_zfc_axioms_2 | axiom | zfc_axioms
s_goedel_constructibility | theorem | goedel_constructibility
s_generic_filter_2 | state | generic_filter
s_cohen_independence_of_ch | theorem | cohen_independence_of_ch
s_partial_order_for_forcing | state | partial_order_for_forcing
s_boolean_valued_models | state | boolean_valued_models
s_large_cardinal_axioms | state | large_cardinal_axioms
s_am_gm_inequality | theorem | am_gm_inequality
s_holders_inequality | theorem | holders_inequality
s_minkowskis_inequality | theorem | minkowskis_inequality
s_jensens_inequality | theorem | jensens_inequality
s_gronwalls_inequality | theorem | gronwalls_inequality
s_schur_inequality | theorem | schur_inequality
s_poincare_inequality_2 | theorem | poincare_inequality
s_halting_problem | state | halting_problem
t_diagonalization_argument | technique | diagonalization_argument
s_decidable_set | state | decidable_set
s_computably_enumerable_set | state | computably_enumerable_set
s_rices_theorem | theorem | rices_theorem
t_reduction_between_problems | technique | reduction_between_problems
s_halting_not_co_ce | theorem | halting_not_co_ce
s_polynomial_equation_by_radicals | state | polynomial_equation_by_radicals
s_galois_criterion | theorem | galois_criterion
s_sn_not_solvable | theorem | sn_not_solvable
s_quadratic_cubic_quartic_formulas | theorem | quadratic_cubic_quartic_formulas
s_transcendental_number_2 | state | transcendental_number
s_liouville_number | state | liouville_number
s_liouville_transcendence | theorem | liouville_transcendence
s_irrationality_measure | state | irrationality_measure
t_thue_siegel_method | technique | thue_siegel_method
s_lattice_in_lie_group | state | lattice_in_lie_group
s_margulis_superrigidity | theorem | margulis_superrigidity
s_no_rigidity_in_dimension_2 | state | no_rigidity_in_dimension_2
s_complexity_class_p_2 | state | complexity_class_p
s_complexity_class_np_2 | state | complexity_class_np
s_p_vs_np_problem | state | p_vs_np_problem
s_boolean_satisfiability | state | boolean_satisfiability
s_np_hard | state | np_hard
s_co_np | state | co_np
s_ladners_theorem | theorem | ladners_theorem
s_three_sat | state | three_sat
s_traveling_salesman_problem | state | traveling_salesman_problem
s_graph_isomorphism | state | graph_isomorphism
s_poincare_conjecture_statement | state | poincare_conjecture_statement
s_poincare_conjecture_2 | theorem | poincare_conjecture
t_hamiltons_program | technique | hamiltons_program
s_perelman_geometrization | theorem | perelman_geometrization
s_neckpinch_singularity | state | neckpinch_singularity
s_freedman_4d_poincare | theorem | freedman_4d_poincare
s_non_trivial_zeros | state | non_trivial_zeros
s_logarithmic_integral | state | logarithmic_integral
s_goldbach_conjecture | state | goldbach_conjecture
s_schnirelmann_density | state | schnirelmann_density
s_sumset | state | sumset
s_freimans_theorem | theorem | freimans_theorem
s_szemeredis_theorem | theorem | szemeredis_theorem
s_goldbach_weak_conjecture | state | goldbach_weak_conjecture
s_conductor_of_abelian_extension | state | conductor_of_abelian_extension
s_langlands_program | state | langlands_program
s_mordell_conjecture_statement | state | mordell_conjecture_statement
t_chabauty_method | technique | chabauty_method
t_heights_in_number_theory | technique | heights_in_number_theory
s_tate_conjecture_for_abelian_varieties | theorem | tate_conjecture_for_abelian_varieties
s_siegel_finiteness | theorem | siegel_finiteness
s_hironaka_theorem | theorem | hironaka_theorem
t_blowing_up | technique | blowing_up
s_embedded_resolution | state | embedded_resolution
s_normal_crossings_divisor | state | normal_crossings_divisor
s_resolution_of_curves | theorem | resolution_of_curves
s_resolution_of_surfaces | theorem | resolution_of_surfaces
s_nash_problem | state | nash_problem
s_resolution_in_positive_characteristic | state | resolution_in_positive_characteristic
t_induction_on_invariants | technique | induction_on_invariants
s_linear_system_of_divisor | state | linear_system_of_divisor
s_riemann_inequality | theorem | riemann_inequality
s_minor_closed_class | state | minor_closed_class
s_wagners_conjecture | state | wagners_conjecture
s_kuratowski_wagner | theorem | kuratowski_wagner
s_excluded_minor_characterization | state | excluded_minor_characterization
s_n_body_problem | state | n_body_problem
s_poincare_nonintegrability | theorem | poincare_nonintegrability
s_sundmans_theorem | theorem | sundmans_theorem
s_kam_theorem_2 | theorem | kam_theorem
s_lagrange_points | state | lagrange_points
s_restricted_three_body_problem | state | restricted_three_body_problem
t_perturbation_theory | technique | perturbation_theory
s_poincare_homoclinic_tangle | state | poincare_homoclinic_tangle
s_chaos_in_celestial_mechanics | state | chaos_in_celestial_mechanics
s_invariant_torus | state | invariant_torus
s_koebe_uniformization | theorem | koebe_uniformization
s_fuchsian_group | state | fuchsian_group
s_weil_conjecture_rationality | theorem | weil_conjecture_rationality
s_deligne_proof | theorem | deligne_proof
s_grothendieck_standard_conjectures | state | grothendieck_standard_conjectures
s_l_adic_cohomology | state | l_adic_cohomology
s_weak_mixing | axiom | Weak mixing
s_k_system | axiom | K-system (Kolmogorov system)
s_shannon_entropy_of_partition | state | Shannon entropy of a partition
s_pesin_stable_unstable_manifold_theorem | theorem | Pesin stable/unstable manifold theorem
s_arnold_diffusion | state | Arnold diffusion
s_nekhoroshev_theorem | theorem | Nekhoroshev theorem
s_pitchfork_bifurcation | state | Pitchfork bifurcation
t_birkhoff_normal_form | technique | Birkhoff normal form
s_rossler_attractor | state | Rössler attractor
s_benedicks_carleson_theorem | theorem | Benedicks–Carleson theorem
t_thermodynamic_formalism | technique | Thermodynamic formalism
s_ruelle_perron_frobenius_theorem | theorem | Ruelle–Perron–Frobenius theorem
s_domain_of_holomorphy | axiom | Domain of holomorphy
s_holomorphically_convex | axiom | Holomorphically convex domain
s_plurisubharmonic_function | axiom | Plurisubharmonic function
s_levi_problem | theorem | Levi problem (solution)
s_cousin_problem | state | Cousin problem
s_oka_cartan_theorem | theorem | Oka–Cartan theorem
s_grauert_direct_image_theorem | theorem | Grauert direct image theorem
s_remmert_proper_mapping_theorem | theorem | Remmert proper mapping theorem
s_bergman_kernel | state | Bergman kernel
s_szego_kernel | state | Szegő kernel
s_fefferman_mapping_theorem | theorem | Fefferman mapping theorem
s_tangential_cauchy_riemann_equation | axiom | Tangential Cauchy–Riemann equation
s_baouendi_treves_approximation | theorem | Baouendi–Trèves approximation theorem
s_bochner_martinelli_kernel | state | Bochner–Martinelli kernel
t_henkin_ramirez_integral | technique | Henkin–Ramirez integral representation
s_koppelman_formula | theorem | Koppelman formula
s_kohn_laplacian | state | Kohn Laplacian □_b
s_edge_of_the_wedge_theorem | theorem | Edge-of-the-wedge theorem
s_analytic_sheaf | axiom | Analytic sheaf
s_pluripotential_theory | state | Pluripotential theory
s_proper_holomorphic_mapping_theorem | theorem | Proper holomorphic mapping theorem
s_oka_principle | theorem | Oka principle
s_weierstrass_preparation_theorem_scv | theorem | Weierstrass preparation theorem (several variables)
s_fiber_product_of_schemes | state | Fiber product of schemes
s_smooth_morphism | axiom | Smooth morphism
s_verdier_duality | theorem | Verdier duality
s_perverse_sheaf | axiom | Perverse sheaf
s_bbd_decomposition_theorem | theorem | BBD decomposition theorem
s_mixed_hodge_structure | axiom | Mixed Hodge structure
s_higher_algebraic_k_theory | state | Higher algebraic K-theory (Quillen)
s_flip | state | Flip (algebraic geometry)
s_abundance_conjecture | state | Abundance conjecture
s_zariski_decomposition | state | Zariski decomposition
s_bloch_higher_chow_group | state | Bloch higher Chow group
s_quillen_q_construction | state | Quillen Q-construction
s_riemann_roch_chern_character | theorem | Riemann–Roch via Chern character
s_rigid_analytic_space | axiom | Rigid analytic space
s_nakai_moishezon_criterion | theorem | Nakai–Moishezon criterion
s_quantum_group | state | Quantum group
s_gorenstein_ring | axiom | Gorenstein ring
s_local_cohomology | state | Local cohomology
s_associated_prime | axiom | Associated prime
s_hilbert_samuel_multiplicity | state | Hilbert–Samuel multiplicity
s_hilbert_polynomial | state | Hilbert polynomial
s_discrete_valuation_ring | axiom | Discrete valuation ring
s_i_adic_completion | state | I-adic completion of a ring
s_global_class_field_theory | theorem | Global class field theory
s_b_spline | state | B-spline
t_calderon_zygmund_operator | technique | Calderón–Zygmund singular integral operator
s_wave_front_set | state | Wave front set
s_killing_form | state | Killing form
a_abstract_root_system | axiom | Abstract root system
s_classification_of_simple_lie_algebras | theorem | Classification of simple Lie algebras
s_weyl_dimension_formula | theorem | Weyl dimension formula
s_cartan_decomposition | theorem | Cartan decomposition
a_symmetric_space | axiom | Symmetric space
s_cartan_classification_of_symmetric_spaces | theorem | Cartan classification of symmetric spaces
s_invariant_theory_hilbert_finiteness | theorem | Invariant theory (Hilbert finiteness)
s_lie_group_exponential_map | state | Lie group exponential map
s_adjoint_representation | state | Adjoint representation
s_harish_chandra_isomorphism | theorem | Harish-Chandra isomorphism
s_bruhat_decomposition | theorem | Bruhat decomposition
s_r_matrix_yang_baxter | state | R-matrix (Yang-Baxter)
s_iwasawa_decomposition | theorem | Iwasawa decomposition
a_quantum_group_drinfeld_jimbo | axiom | Quantum group (Drinfeld-Jimbo)
s_demazure_character_formula | theorem | Demazure character formula
s_duflo_theorem | theorem | Duflo theorem
s_weyl_integration_formula | theorem | Weyl integration formula
s_harish_chandra_character_formula | theorem | Harish-Chandra character formula
s_plancherel_for_semisimple_groups | theorem | Plancherel theorem for semisimple groups
s_langlands_classification | theorem | Langlands classification
s_steinberg_representation | state | Steinberg representation
s_jackson_theorem | theorem | Jackson's theorem
s_bernstein_inverse_theorem | theorem | Bernstein inverse theorem
s_best_approximation_chebyshev | state | Best approximation (Chebyshev)
s_chebyshev_equioscillation_theorem | theorem | Chebyshev equioscillation theorem
a_spline_function | axiom | Spline function
s_schoenberg_whitney_theorem | theorem | Schoenberg-Whitney theorem
s_scaling_equation | state | Scaling equation
s_calderon_zygmund_singular_integral_operator | state | Calderón-Zygmund singular integral operator
t_calderon_zygmund_decomposition_technique | technique | Calderón-Zygmund decomposition (technique)
s_parametrix | state | Parametrix
s_fourier_integral_operator | state | Fourier integral operator
s_hormander_propagation_of_singularities | theorem | Hörmander propagation of singularities
a_besov_space | axiom | Besov space
s_mikhlin_multiplier_theorem | theorem | Mikhlin multiplier theorem
s_kolmogorov_superposition_theorem | theorem | Kolmogorov superposition theorem
s_shannon_sampling_theorem | theorem | Shannon sampling theorem
t_kotelnikov_shannon_whittaker_interpolation | technique | Kotelnikov-Shannon-Whittaker interpolation
t_symbol_calculus | technique | Symbol calculus
s_leray_spectral_sequence | theorem | Leray spectral sequence
s_grothendieck_spectral_sequence | theorem | Grothendieck spectral sequence
s_riemann_hilbert_correspondence | theorem | Riemann-Hilbert correspondence
s_wall_groups | state | Wall groups L_n(π)
t_surgery_exact_sequence | technique | Surgery exact sequence
s_surgery_obstruction | state | Surgery obstruction
s_normal_invariant | state | Normal invariant
s_poincare_duality_complex | state | Poincaré duality complex
s_spivak_normal_fibration | state | Spivak normal fibration
t_assembly_map | technique | Assembly map
s_novikov_conjecture | axiom | Novikov conjecture
s_borel_conjecture | axiom | Borel conjecture
s_oriented_cobordism_ring | state | Oriented cobordism ring Ω^SO_*
s_complex_cobordism_ring | state | Complex cobordism ring MU_*
s_thom_cobordism_theorem | theorem | Thom's cobordism theorem
s_formal_group_law_of_MU | state | Formal group law of MU
s_quillen_theorem_MU_lazard | theorem | Quillen's theorem (MU_* ≅ Lazard ring)
t_landweber_exact_functor | technique | Landweber exact functor theorem
s_adams_e_invariant | state | Adams e-invariant
s_chromatic_homotopy_theory | state | Chromatic homotopy theory
s_morava_k_theory | state | Morava K-theory K(n)
s_nilpotence_theorem | theorem | Nilpotence theorem (Devinatz–Hopkins–Smith)
s_periodicity_theorem | theorem | Periodicity theorem (Hopkins–Smith)
s_thick_subcategory_theorem | theorem | Thick subcategory theorem (Hopkins–Smith)
s_smash_product_theorem | theorem | Smash product theorem (Hopkins–Ravenel)
s_segal_conjecture | theorem | Segal conjecture (Carlsson's theorem)
s_eta_invariant_ems | state | Atiyah–Patodi–Singer eta invariant
s_aps_index_theorem | theorem | Atiyah–Patodi–Singer index theorem
s_families_index_theorem_ems | theorem | Families index theorem
s_chern_character | state | Chern character
s_l_class | state | Hirzebruch L-class
s_a_hat_genus | state | Â-genus
s_associated_bundle | state | Associated bundle
t_milnor_construction | technique | Milnor's construction of classifying spaces
t_clutching_construction | technique | Clutching construction
s_homflypt_polynomial | state | HOMFLY-PT polynomial
s_vassiliev_invariants | state | Vassiliev (finite-type) invariants
t_kontsevich_integral | technique | Kontsevich integral
s_kauffman_bracket | state | Kauffman bracket
s_alexander_polynomial | state | Alexander polynomial
s_seifert_surface | state | Seifert surface
s_khovanov_homology | state | Khovanov homology
s_eight_thurston_geometries | state | Eight Thurston geometries
s_kneser_milnor_theorem | theorem | Kneser–Milnor decomposition
s_mapping_class_group | state | Mapping class group
s_dehn_twist | state | Dehn twist
t_kirby_calculus | technique | Kirby calculus
s_cuntz_algebra | state | Cuntz algebra O_n
s_toeplitz_algebra | state | Toeplitz algebra
s_irrational_rotation_algebra | state | Irrational rotation algebra A_θ
s_modular_automorphism_group | state | Modular automorphism group
s_kirchberg_phillips_theorem | theorem | Kirchberg–Phillips theorem
s_spectral_triple | state | Spectral triple (Connes)
s_cyclic_cohomology | state | Cyclic cohomology
s_connes_chern_character | state | Connes–Chern character
s_hochschild_cohomology | state | Hochschild cohomology
s_noncommutative_torus | state | Noncommutative torus
s_connes_index_foliations | theorem | Connes index theorem for foliations
s_kasparov_kk_theory | state | Kasparov KK-theory
t_kasparov_product | technique | Kasparov product
s_uct_for_kk | theorem | Universal coefficient theorem for KK-theory
s_baum_connes_conjecture | axiom | Baum–Connes conjecture
s_planar_algebra | state | Planar algebra
s_free_independence | axiom | Free independence
s_r_transform | state | R-transform (Voiculescu)
s_free_entropy | state | Free entropy (Voiculescu)
s_s_transform | state | S-transform (Voiculescu)
s_kpz_equation | state | KPZ equation
s_kpz_universality | state | KPZ universality class
s_stochastic_heat_equation | state | Stochastic heat equation
s_hairer_regularity_structures | state | Hairer's regularity structures
s_paracontrolled_distributions | state | Paracontrolled distributions
s_random_matrix_theory | state | Random matrix theory
s_sine_kernel_universality | theorem | Sine kernel universality (bulk)
s_dyson_brownian_motion | state | Dyson Brownian motion
s_rsw_estimates | theorem | Russo–Seymour–Welsh estimates
s_smirnov_percolation_theorem | theorem | Smirnov's theorem (critical site percolation)
s_contact_process | state | Contact process
s_voter_model | state | Voter model
s_exclusion_process | state | Exclusion process
s_hydrodynamic_limit | state | Hydrodynamic limit
s_liggett_ergodic_theorem | theorem | Liggett ergodic theorem for IPS
s_malliavin_hormander_proof | theorem | Malliavin's probabilistic proof of Hörmander's theorem
s_rough_path_theory | state | Rough path theory (Lyons)
t_lyons_universal_limit_theorem | theorem | Lyons universal limit theorem
t_lyons_extension_theorem | theorem | Lyons extension theorem
t_gubinelli_controlled_rough_paths | technique | Gubinelli controlled rough paths
t_sewing_lemma | technique | Sewing lemma
s_sle_phases | state | SLE phases (κ ≤ 4, 4 < κ < 8, κ ≥ 8)
s_sle_cft_correspondence | state | SLE/CFT correspondence
s_yang_mills_connection | state | Yang–Mills connection
s_bpst_instanton | state | BPST instanton
s_uhlenbeck_removable_singularity | theorem | Uhlenbeck removable singularity theorem
s_chern_simons_functional | state | Chern–Simons functional
s_sugawara_construction | state | Sugawara construction
s_modular_functor | state | Modular functor
s_kdv_soliton | state | KdV soliton
s_zakharov_shabat_system | state | Zakharov–Shabat system
s_hitchin_system | state | Hitchin system
s_kp_hierarchy | state | KP hierarchy
s_drinfeld_jimbo_quantum_group | state | Drinfeld–Jimbo quantum group U_q(𝔤)
s_quantum_r_matrix | state | Quantum R-matrix
t_reshetikhin_turaev_invariants | technique | Reshetikhin–Turaev invariants
s_atiyah_tqft_axioms | axiom | Atiyah's TQFT axioms
s_witten_chern_simons_tqft | state | Witten–Chern–Simons TQFT
s_turaev_viro_invariant | state | Turaev–Viro invariant
s_dijkgraaf_witten_invariant | state | Dijkgraaf–Witten invariant
s_ising_model | state | Ising model
s_gopakumar_vafa_invariants | state | Gopakumar–Vafa invariants
s_donaldson_thomas_invariants | state | Donaldson–Thomas invariants
s_supermanifold | state | Supermanifold
s_super_lie_algebra | state | Super Lie algebra
s_spectral_zeta_function | state | Spectral zeta function
s_heat_kernel_expansion | state | Heat kernel asymptotic expansion
s_spectral_invariants | state | Spectral invariants
s_scattering_matrix | state | Scattering matrix (PDE)
s_moller_wave_operators | state | Møller wave operators
s_asymptotic_completeness | theorem | Asymptotic completeness
s_limiting_absorption_principle | theorem | Limiting absorption principle
s_mourre_estimate | theorem | Mourre estimate
t_compensated_compactness | technique | Compensated compactness
s_div_curl_lemma | theorem | Div-curl lemma
s_crandall_ishii_lemma | theorem | Crandall–Ishii lemma
s_stefan_problem | state | Stefan problem
s_caffarelli_free_boundary_regularity | theorem | Caffarelli free boundary regularity
s_alt_caffarelli_friedman | theorem | Alt–Caffarelli–Friedman monotonicity formula
s_homogenization_theorem | theorem | Homogenization theorem (elliptic)
s_cell_problem | state | Cell problem (homogenization)
s_two_scale_convergence | state | Two-scale convergence
s_gamma_convergence | state | Γ-convergence
s_matroid_rank_function | state | Matroid rank function
s_tutte_polynomial | state | Tutte polynomial
s_matroid_union_theorem | theorem | Matroid union theorem
s_rados_theorem | theorem | Rado's theorem (matroid transversals)
s_matroid_duality | state | Matroid duality
s_mds_codes | state | Maximum distance separable (MDS) codes
s_bch_code | state | BCH code
s_turbo_ldpc_codes | state | Turbo and LDPC codes
s_elliptic_curve_cryptography | state | Elliptic curve cryptography
s_lattice_based_cryptography | state | Lattice-based cryptography
s_lwe | state | Learning with errors (LWE)
s_bibd | state | Balanced incomplete block design (BIBD)
s_fisher_inequality | theorem | Fisher's inequality
s_latin_square | state | Latin square
s_witt_design | state | Witt design
s_graphon | state | Graphon
s_lovasz_szegedy_theorem | theorem | Lovász–Szegedy theorem
s_density_hales_jewett | theorem | Density Hales–Jewett theorem
s_intersection_cohomology | state | Intersection cohomology
s_bbdg_decomposition_theorem | theorem | BBD decomposition theorem
s_d_module | state | D-module
s_beilinson_bernstein_localization | theorem | Beilinson–Bernstein localization
s_springer_correspondence | state | Springer correspondence
s_geometric_satake | theorem | Geometric Satake equivalence
s_weyl_module | state | Weyl module
s_lusztig_conjecture | axiom | Lusztig conjecture
s_steinberg_tensor_product | theorem | Steinberg tensor product theorem
s_jantzen_filtration | state | Jantzen filtration
s_langlands_dual_group | state | Langlands dual group G^∨
s_langlands_l_function | state | Langlands L-function
s_arthur_endoscopic_classification | theorem | Arthur's endoscopic classification
s_fundamental_lemma_ngo | theorem | Fundamental lemma (Ngô)
s_geometric_langlands | axiom | Geometric Langlands conjecture
s_hecke_eigensheaf | state | Hecke eigensheaf
s_reynolds_operator | state | Reynolds operator
s_molien_formula | theorem | Molien's formula
s_chevalley_shephard_todd | theorem | Chevalley–Shephard–Todd theorem
s_hilbert_mumford_criterion | theorem | Hilbert–Mumford criterion
s_first_fundamental_theorem | theorem | First fundamental theorem of invariant theory
s_second_fundamental_theorem | theorem | Second fundamental theorem of invariant theory
s_pontryagin_thom_framed | state | Pontryagin–Thom (framed cobordism)
s_cobordism_hypothesis | axiom | Cobordism hypothesis (Baez–Dolan)
s_nuclear_c_star_algebra | state | Nuclear C*-algebra
s_exact_c_star_algebra | state | Exact C*-algebra
s_formal_language | axiom | Formal language
s_quantifier | axiom | Quantifier (universal, existential)
s_axiom_scheme_of_substitution | axiom | Axiom scheme of substitution (logic)
t_modus_ponens | technique | Modus ponens
s_deductive_system | axiom | Deductive system
s_hilbert_epsilon_operator | axiom | Hilbert epsilon operator (tau-operator)
s_axiom_of_the_empty_set | axiom | Axiom of the empty set
s_axiom_of_pairing | axiom | Axiom of pairing
s_axiom_of_separation | axiom | Axiom scheme of separation (comprehension)
s_axiom_of_union | axiom | Axiom of union
s_set_membership | axiom | Set membership (element-of relation)
s_subset_inclusion | axiom | Subset (inclusion)
s_set_intersection | state | Set intersection
s_set_union | state | Set union (binary)
s_set_complement | state | Set complement (difference)
s_symmetric_difference | state | Symmetric difference
s_ordered_pair | axiom | Ordered pair
s_cartesian_product | state | Cartesian product
s_relation | axiom | Relation (graph of a relation)
s_function | axiom | Function (map, mapping)
s_injection | axiom | Injection (one-to-one map)
s_surjection | axiom | Surjection (onto map)
s_bijection | axiom | Bijection
s_composition_of_functions | state | Composition of functions
s_inverse_function | state | Inverse function
s_image_and_preimage | state | Image and preimage
s_restriction_of_function | state | Restriction and extension of a function
s_family_of_sets | axiom | Family of sets (indexed family)
s_generalized_union_intersection | state | Generalized union and intersection
s_generalized_cartesian_product | state | Generalized Cartesian product (product of a family)
s_equivalence_relation | axiom | Equivalence relation
s_equivalence_class | state | Equivalence class
s_quotient_set | state | Quotient set
s_canonical_decomposition_of_map | theorem | Canonical decomposition of a map
s_partition_of_a_set | state | Partition of a set
s_diagonal_of_a_product | state | Diagonal of a product
s_projection_map | state | Projection map
s_preorder | axiom | Preorder (quasi-order)
s_strict_partial_order | state | Strict partial order
s_greatest_least_element | state | Greatest element and least element
s_supremum_infimum | state | Supremum and infimum
s_lattice | axiom | Lattice
s_complete_lattice | axiom | Complete lattice
s_directed_set | axiom | Directed set
s_order_preserving_map | axiom | Increasing (order-preserving) map
s_order_isomorphism | axiom | Order isomorphism
s_chain | state | Chain (totally ordered subset)
s_well_ordered_set | axiom | Well-ordered set
s_hausdorff_maximality_principle | theorem | Hausdorff maximality principle
s_initial_segment | state | Initial segment (section)
s_comparability_well_ordered | theorem | Comparability of well-ordered sets
s_cantor_theorem_power_set | theorem | Cantor's theorem
s_equipotence | state | Equipotence (equinumerosity)
s_finite_set | axiom | Finite set
s_countable_set | axiom | Countable set (denumerable)
s_disjoint_union | state | Disjoint union (coproduct of sets)
s_absorption_infinite_cardinals | theorem | Absorption law for infinite cardinals
t_mathematical_induction | technique | Mathematical induction
s_arithmetic_of_naturals | state | Arithmetic of natural numbers
s_order_on_naturals | state | Order on natural numbers
s_species_of_structures | axiom | Species of structures
s_echelon_construction_type | state | Echelon construction type
s_isomorphism_of_structures | axiom | Isomorphism of structures
t_transport_of_structures | technique | Transport of structures
s_induced_structure | state | Induced structure (substructure)
s_product_structure | state | Product structure
s_quotient_structure | state | Quotient structure
s_initial_structure | state | Initial structure
s_final_structure | state | Final structure
s_morphism_of_structures | axiom | Morphism (of a species of structures)
s_neighborhood_filter | state | Neighborhood filter
s_filter | axiom | Filter
s_filter_base | axiom | Filter base (basis of a filter)
s_convergence_of_filter | axiom | Convergence of a filter
s_cluster_point_of_filter | state | Cluster point of a filter
s_continuous_map | axiom | Continuous map
s_open_map | axiom | Open map and closed map
s_subspace_topology | axiom | Subspace topology (induced topology)
s_initial_topology | axiom | Initial topology
s_final_topology | axiom | Final topology
s_hausdorff_space | axiom | Hausdorff space (T2)
s_t1_space | axiom | T1 space
s_regular_space | axiom | Regular space (T3)
s_completely_regular_space | axiom | Completely regular space (Tychonoff, T3.5)
s_limit_of_function | axiom | Limit of a function
s_net | axiom | Net (Moore-Smith sequence)
s_connected_space | axiom | Connected space
s_connected_component | state | Connected component
s_locally_connected_space | axiom | Locally connected space
s_second_countable_space | axiom | Second-countable space
s_first_countable_space | axiom | First-countable space
s_separable_space | axiom | Separable space
s_uniform_space | axiom | Uniform space (uniformity)
s_entourage | axiom | Entourage (surrounding)
s_uniform_topology | state | Uniform topology
s_uniformly_continuous_map | axiom | Uniformly continuous map
s_cauchy_filter | axiom | Cauchy filter
s_complete_uniform_space | axiom | Complete uniform space
s_completion_of_uniform_space | state | Completion of a uniform space
s_uniform_isomorphism | axiom | Uniform isomorphism
s_product_uniformity | state | Product uniformity
s_induced_uniformity | state | Induced uniformity (subspace uniformity)
s_minimal_cauchy_filter | state | Minimal Cauchy filter
s_left_uniform_structure_topgrp | state | Left uniform structure on a topological group
s_subgroup_topology | state | Subgroup topology
s_quotient_group_topology | state | Quotient group topology
s_closure_of_subgroup | theorem | Closure of a subgroup
s_neighborhood_base_identity | state | Neighborhood base at the identity
s_open_subgroup_theorem | theorem | Open subgroup theorem
s_completion_of_topological_group | state | Completion of a topological group
s_completeness_of_R | axiom | Completeness of R (least upper bound property)
s_archimedean_property | theorem | Archimedean property
s_density_of_Q_in_R | theorem | Density of Q in R
s_order_topology_on_R | state | Order topology on R
s_extended_real_line | state | Extended real line
s_absolute_value_on_R | state | Absolute value (on R)
s_monotone_convergence_sequences | theorem | Monotone convergence for sequences in R
s_one_sided_limits | state | Right and left limits
s_power_function | state | Power function
s_euclidean_topology_Rn | axiom | Euclidean topology on R^n
s_normed_vector_space | axiom | Normed vector space (over R)
s_metric_uniformity | state | Metric uniformity
s_equivalent_norms_theorem | theorem | Equivalent norms on finite-dimensional spaces
s_projective_limit | state | Projective limit (inverse limit)
s_inductive_limit | state | Inductive limit (direct limit)
s_solenoid | state | Solenoid
s_profinite_group | state | Profinite group
s_quasi_compact_space | axiom | Quasi-compact space
s_finite_intersection_property | theorem | Finite intersection property characterization of compactness
s_continuous_image_compact | theorem | Continuous image of compact is compact
s_closed_subset_compact | theorem | Closed subsets of compact spaces are compact
s_compact_hausdorff_normal | theorem | Compact Hausdorff implies normal
s_compact_to_hausdorff_homeomorphism | theorem | Continuous bijection from compact to Hausdorff is homeomorphism
s_proper_map | axiom | Proper map
s_locally_finite_family | axiom | Locally finite family
s_refinement_of_cover | axiom | Refinement of a cover
s_metrizable_paracompact | theorem | Paracompactness of metrizable spaces (Stone's theorem)
s_nowhere_dense_set | axiom | Nowhere dense set
s_meager_set | state | Meager set (first category)
s_semicontinuity_liminf_limsup | theorem | Characterization of semicontinuity via liminf/limsup
s_sup_lsc_functions | theorem | Supremum of lower semicontinuous functions
s_paracompact_partition_unity | theorem | Paracompactness and partitions of unity
s_lindelof_space | axiom | Lindelof space
s_second_countable_lindelof | theorem | Second-countable implies Lindelof
s_compact_open_topology | axiom | Compact-open topology
s_topology_uniform_convergence | axiom | Topology of uniform convergence
s_topology_compact_convergence | state | Topology of compact convergence
s_pointwise_convergence_topology | state | Pointwise convergence topology
s_evaluation_map | state | Evaluation map
s_exponential_law_function_spaces | theorem | Exponential law for function spaces
s_uniform_limit_continuity | theorem | Uniform limit of continuous functions is continuous
s_completeness_function_spaces | theorem | Completeness of function spaces
s_filter_generated_by_base | state | Filter generated by a base
s_finer_coarser_topologies | axiom | Finer and coarser topologies
s_discrete_indiscrete_topology | state | Discrete and indiscrete topologies
s_clopen_set | state | Clopen set
s_sigma_compact_space | axiom | Sigma-compact space
s_cauchy_sequence | state | Cauchy sequence
s_pseudometric_uniformity | state | Pseudometric uniformity
s_continuity_via_filters | theorem | Continuity via filters
s_universal_property_completion | theorem | Universal property of completion
s_compact_complete_totally_bounded | theorem | Compactness equals complete and totally bounded
s_lebesgue_covering_lemma | theorem | Lebesgue covering lemma
s_topological_embedding | axiom | Topological embedding
s_monoid | axiom | Monoid
s_group | axiom | Group
s_characteristic_of_a_ring | state | Characteristic of a ring
s_opposite_ring | state | Opposite ring
s_left_right_module | axiom | Left module / right module
s_basis_of_a_module | axiom | Basis of a module
s_rank_of_a_linear_map | state | Rank of a linear map
s_direct_sum_decomposition | state | Direct sum decomposition
s_noetherian_module | axiom | Noetherian module
s_artinian_module | axiom | Artinian module
s_annihilator | state | Annihilator
s_multilinear_map | axiom | Multilinear map
s_right_exactness_of_tensor_product | theorem | Right exactness of tensor product
s_tensor_algebra | state | Tensor algebra
s_exterior_product | state | Exterior product (wedge product)
s_exterior_power | state | Exterior power
s_symmetric_algebra | state | Symmetric algebra
s_symmetric_power | state | Symmetric power
s_graded_algebra | axiom | Graded algebra
t_extension_of_scalars | technique | Extension of scalars
s_polynomial_ring_several_variables | state | Polynomial ring in several variables
s_euclidean_division_polynomials | theorem | Euclidean division for polynomials
s_root_of_a_polynomial | axiom | Root of a polynomial
s_factor_theorem | theorem | Factor theorem
s_irreducible_polynomial | axiom | Irreducible polynomial
s_unique_factorization_in_kx | theorem | Unique factorization in K[X]
s_gcd_of_polynomials | state | Greatest common divisor of polynomials
t_partial_fraction_decomposition | technique | Partial fraction decomposition
s_algebraic_element | axiom | Algebraic element
s_transcendental_element | axiom | Transcendental element
s_simple_algebraic_extension | state | Simple algebraic extension
s_tower_law | theorem | Tower law (multiplicativity of degrees)
s_existence_uniqueness_algebraic_closure | theorem | Existence and uniqueness of algebraic closure
s_separable_polynomial | axiom | Separable polynomial
s_inseparable_extension | axiom | Inseparable extension / purely inseparable extension
s_perfect_field | axiom | Perfect field
s_normal_extension | axiom | Normal extension
s_transcendence_basis | state | Transcendence basis
s_norm_and_trace_of_field_extension | state | Norm and trace of a field extension
s_artin_theorem_on_characters | theorem | Artin's theorem on characters
s_totally_ordered_group | axiom | Totally ordered group
s_archimedean_ordered_group | axiom | Archimedean ordered group
s_krull_valuation | axiom | Valuation (Krull valuation)
s_absolute_value_on_a_field | axiom | Absolute value on a field
s_convex_subgroup | axiom | Convex subgroup
s_elementary_divisors | state | Elementary divisors
s_characteristic_polynomial | state | Characteristic polynomial
s_minimal_polynomial_of_endomorphism | state | Minimal polynomial of an endomorphism
s_torsion_submodule | state | Torsion submodule
s_torsion_free_module | axiom | Torsion-free module
s_semisimple_module | axiom | Semisimple module
s_semisimple_ring | axiom | Semisimple ring
s_simple_ring | axiom | Simple ring
s_symmetric_bilinear_form | axiom | Symmetric bilinear form
s_alternating_bilinear_form | axiom | Alternating bilinear form
s_sesquilinear_form | axiom | Sesquilinear form
s_hermitian_form | axiom | Hermitian form
s_non_degenerate_form | axiom | Non-degenerate form
s_radical_of_a_form | state | Radical of a form
s_orthogonal_complement | state | Orthogonal complement
s_witt_theorem | theorem | Witt's theorem
s_witt_group | state | Witt group
s_hyperbolic_plane_quadratic | state | Hyperbolic plane (quadratic forms)
s_symplectic_group | state | Symplectic group
s_spin_group | state | Spin group
s_spinor_norm | state | Spinor norm
s_associative_algebra | axiom | Associative algebra
s_lie_subalgebra | axiom | Lie subalgebra
s_ideal_of_lie_algebra | axiom | Ideal of a Lie algebra
s_commutator_lie_algebra | state | Commutator Lie algebra
s_poincare_birkhoff_witt_theorem | theorem | Poincare-Birkhoff-Witt theorem
s_free_lie_algebra | state | Free Lie algebra
s_nilpotent_lie_algebra | axiom | Nilpotent Lie algebra
s_solvable_lie_algebra | axiom | Solvable Lie algebra
s_abelian_lie_algebra | axiom | Abelian Lie algebra
s_faithfully_flat_module | axiom | Faithfully flat module
t_faithfully_flat_descent | technique | Faithfully flat descent
t_local_global_principle | technique | Local-global principle
s_support_of_a_module | state | Support of a module
s_associated_graded | state | Associated graded ring/module
s_filtered_ring | axiom | Filtered ring
s_i_adic_filtration | state | I-adic filtration
s_i_adic_topology | state | I-adic topology
s_hilbert_function | state | Hilbert function / Hilbert-Samuel function
s_primary_ideal | axiom | Primary ideal
s_uniqueness_primary_decomposition | theorem | Uniqueness theorems for primary decomposition
s_integral_element | axiom | Integral element
s_lying_over_theorem | theorem | Lying-over theorem
s_valuation_ring | axiom | Valuation ring
s_place_of_a_valuation | state | Place
s_uniformizer | state | Uniformizer
s_rank_of_a_valuation | state | Rank of a valuation
s_henselian_ring | axiom | Henselian ring
s_ramification_index_and_residue_degree | state | Ramification index and residue degree
s_extension_of_valuations | theorem | Extension of valuations
s_principal_divisor | state | Principal divisor
s_factorial_ring | axiom | Factorial ring (UFD)
s_invertible_ideal | state | Invertible ideal
s_unique_factorization_ideals_dedekind | theorem | Unique factorization of ideals in Dedekind domains
s_approximation_theorem_valuations | theorem | Approximation theorem for valuations
s_system_of_parameters | state | System of parameters
s_dimension_theorem_noetherian_local | theorem | Dimension theorem for Noetherian local rings
s_dimension_of_polynomial_rings | theorem | Dimension of polynomial rings
s_catenary_ring | axiom | Catenary ring
s_universally_catenary_ring | axiom | Universally catenary ring
s_unmixedness_theorem | theorem | Unmixedness theorem
s_multiplicity_of_local_ring | state | Multiplicity of a local ring
s_coefficient_field | state | Coefficient field / coefficient ring
s_weierstrass_division_theorem | theorem | Weierstrass division theorem
s_nagata_ring | axiom | Nagata ring (Japanese ring)
s_excellent_ring | axiom | Excellent ring
s_depth | state | Depth
s_regular_sequence | axiom | Regular sequence
s_depth_lemma | theorem | Depth lemma
s_injective_hull | state | Injective hull (injective envelope)
s_matlis_duality | theorem | Matlis duality
s_dualizing_module | state | Dualizing module (canonical module)
s_local_duality_theorem | theorem | Local duality theorem
s_regular_rings_finite_global_dimension | theorem | Regular rings have finite global dimension
s_bass_theorem | theorem | Bass's theorem
s_derivative_of_a_vectorvalued_function | axiom | Derivative of a vector-valued function
s_differentiability | axiom | Differentiability
s_linearity_of_differentiation | theorem | Linearity of differentiation
s_product_rule | theorem | Product rule (Leibniz rule)
s_chain_rule | theorem | Chain rule
s_derivative_of_the_inverse_function | theorem | Derivative of the inverse function
s_monotonicity_criterion_via_derivative | theorem | Monotonicity criterion via derivative
s_convexity_criterion_via_second_derivative | theorem | Convexity criterion via second derivative
s_higherorder_derivative | axiom | Higher-order derivative
s_leibniz_formula_for_nth_derivative_of_a_product | theorem | Leibniz formula for n-th derivative of a product
s_regulated_function | axiom | Regulated function
s_step_function | axiom | Step function
s_integral_of_a_regulated_function | axiom | Integral of a regulated function
s_primitive | axiom | Primitive (antiderivative)
s_integration_by_parts | theorem | Integration by parts
s_change_of_variables_in_integrals | theorem | Change of variables in integrals
s_integral_remainder_for_taylors_formula | theorem | Integral remainder for Taylor's formula
s_uniform_convergence_and_integration | theorem | Uniform convergence and integration
s_improper_integral | axiom | Improper integral (convergent integral on unbounded interval)
s_absolute_convergence_of_improper_integrals | theorem | Absolute convergence of improper integrals
s_power_function_x | axiom | Power function x^α
s_number | axiom | Number π
s_inverse_trigonometric_functions | axiom | Inverse trigonometric functions (arcsin, arccos, arctan)
s_hyperbolic_functions | axiom | Hyperbolic functions (sinh, cosh, tanh)
s_inverse_hyperbolic_functions | axiom | Inverse hyperbolic functions
s_analyticity_of_elementary_functions | theorem | Analyticity of elementary functions
s_growth_comparison_of_elementary_functions | theorem | Growth comparison of elementary functions
s_lipschitz_condition | axiom | Lipschitz condition
s_integral_equation_formulation_of_ode | state | Integral equation formulation of ODE
s_maximal_solution_of_an_ode | axiom | Maximal solution of an ODE
s_blowup_criterion | theorem | Blow-up criterion (escape to boundary)
s_linear_ordinary_differential_equation | axiom | Linear ordinary differential equation
s_homogeneous_linear_ode | axiom | Homogeneous linear ODE
s_fundamental_matrix | state | Fundamental matrix (resolvent)
t_variation_of_constants | technique | Variation of constants (variation of parameters)
s_linear_ode_with_constant_coefficients | state | Linear ODE with constant coefficients
s_differentiable_dependence_on_parameters | theorem | Differentiable dependence on parameters
s_lagrange_remainder | state | Lagrange remainder
s_asymptotic_expansion | axiom | Asymptotic expansion
s_asymptotic_scale | axiom | Asymptotic scale
s_littleo_and_bigo_notation | axiom | Little-o and big-O notation
t_asymptotic_comparison_of_functions | technique | Asymptotic comparison of functions
s_bernoulli_numbers | axiom | Bernoulli numbers
s_bernoulli_polynomials | axiom | Bernoulli polynomials
s_eulermaclaurin_summation_formula | theorem | Euler-Maclaurin summation formula
t_summation_of_divergent_series_by_eulermaclaurin | technique | Summation of divergent series by Euler-Maclaurin
s_asymptotic_expansion_of_the_gamma_function | state | Asymptotic expansion of the gamma function
s_beta_function | state | Beta function
s_digamma_function | state | Digamma function
s_abscissa_of_convergence | axiom | Abscissa of convergence
s_abscissa_of_absolute_convergence | axiom | Abscissa of absolute convergence
s_laplace_transform_of_derivatives | theorem | Laplace transform of derivatives
s_laplace_transform_of_convolutions | theorem | Laplace transform of convolutions
s_inversion_formula_for_laplace_transform | theorem | Inversion formula for Laplace transform
s_uniqueness_theorem_for_laplace_transform | theorem | Uniqueness theorem for Laplace transform
t_application_of_laplace_transform_to_linear_ode | technique | Application of Laplace transform to linear ODE
s_valued_field | axiom | Valued field
s_neighborhood_of_zero | axiom | Neighborhood of zero (fundamental system)
s_absorbing_set | axiom | Absorbing set
s_balanced_set | axiom | Balanced set
s_bounded_set_in_a_tvs | axiom | Bounded set in a TVS
s_metrizable_tvs | state | Metrizable TVS
s_normable_tvs | state | Normable TVS
s_completeness_of_a_tvs | axiom | Completeness of a TVS
s_topological_direct_sum_of_tvs | state | Topological direct sum of TVS
s_quotient_tvs | state | Quotient TVS
s_product_of_tvs | state | Product of TVS
s_minkowski_functional | axiom | Minkowski functional (gauge)
s_strict_separation_theorem | theorem | Strict separation theorem
s_supporting_hyperplane_theorem | theorem | Supporting hyperplane theorem
s_bipolar_theorem | theorem | Bipolar theorem
s_space_of_continuous_linear_maps_l | axiom | Space of continuous linear maps L(E, F)
s_topology_of_bounded_convergence | axiom | Topology of bounded convergence (strong topology on L(E,F))
s_topology_of_pointwise_convergence | axiom | Topology of pointwise convergence (weak topology on L(E,F))
s_barrelled_space | axiom | Barrelled space
s_barrel | axiom | Barrel
s_weak_topology | axiom | Weak-* topology σ(E', E)
s_mackey_topology | axiom | Mackey topology τ(E, F)
s_semireflexive_space | axiom | Semi-reflexive space
s_strong_topology | axiom | Strong topology β(E', E)
s_grothendieck_completion_theorem | theorem | Grothendieck completion theorem
s_transpose_of_a_continuous_linear_map | state | Transpose (adjoint) of a continuous linear map
s_weak_compactness_and_completeness | theorem | Weak compactness and completeness
s_parallelogram_law | theorem | Parallelogram law
s_orthogonality | axiom | Orthogonality
s_orthogonal_projection_theorem | theorem | Orthogonal projection theorem
s_orthonormal_system | axiom | Orthonormal system
s_normal_operator | axiom | Normal operator
s_direct_sum_of_hilbert_spaces | state | Direct sum of Hilbert spaces
s_youngs_inequality_for_products | theorem | Young's inequality for products
s_conjugate_exponents | axiom | Conjugate exponents
s_logconvexity_of_lp_norms | theorem | Log-convexity of Lp norms (Riesz interpolation inequality)
s_bounded_radon_measure | axiom | Bounded Radon measure
s_support_of_a_measure | axiom | Support of a measure
s_total_variation_measure | state | Total variation measure
s_vague_topology_on_measures | axiom | Vague topology on measures
s_dirac_measure | axiom | Dirac measure
s_image_measure_pushforward | state | Image measure (pushforward measure)
s_regularity_of_radon_measures | theorem | Regularity of Radon measures
s_upper_integral | state | Upper integral
s_mu_measurable_function_bourbaki | axiom | Measurable function (Bourbaki, mu-measurable)
s_integrable_function_lebesgue | axiom | Integrable function (Lebesgue integrable)
s_negligible_set | axiom | Negligible set (mu-null set)
s_lp_space | axiom | Lp space (1 <= p < infinity)
s_l_infinity_space | axiom | L-infinity space
s_dense_subspaces_of_lp | theorem | Dense subspaces of Lp
s_essential_supremum | axiom | Essential supremum
s_convergence_in_measure | axiom | Convergence in measure
s_product_measure | axiom | Product measure
s_tonellis_theorem | theorem | Tonelli's theorem
s_change_of_variables_formula_rn | theorem | Change of variables formula (integration on Rn)
s_bochner_integral | axiom | Bochner integral
s_strongly_measurable_function | axiom | Strongly measurable function
s_vector_valued_measure | axiom | Vector-valued measure (vector measure)
s_radon_nikodym_derivative | state | Radon-Nikodym derivative (density)
s_left_invariance_of_haar_measure | axiom | Left invariance of Haar measure
s_right_haar_measure | state | Right Haar measure
s_modular_function_of_locally_compact_group | axiom | Modular function
s_unimodular_group | axiom | Unimodular group
s_haar_measure_on_compact_groups | state | Haar measure on compact groups
s_integration_formula_for_quotient_groups | theorem | Integration formula for quotient groups
s_weils_formula | theorem | Weil's formula (quotient integral formula)
s_convolution_of_measures | axiom | Convolution of measures
s_convolution_of_functions_on_a_group | axiom | Convolution of functions on a group
s_approximate_identity_in_group_algebra | state | Approximate identity in L1(G)
s_character_of_a_representation | axiom | Character of a representation
s_convolution_algebra_of_measures | state | Convolution algebra of measures M(G)
s_integrated_representation | state | Integrated representation
s_fourier_transform_on_lca_groups | state | Fourier transform on locally compact abelian groups
s_fourier_inversion_formula_lca | theorem | Fourier inversion formula (LCA groups)
s_tight_measure | axiom | Tight measure
s_weak_convergence_of_measures | axiom | Weak convergence of measures (narrow convergence)
s_regular_borel_measure | axiom | Regular Borel measure
s_inner_regularity | axiom | Inner regularity
s_outer_regularity | axiom | Outer regularity
s_suslin_space | axiom | Suslin space (analytic space)
s_choquet_capacitability_theorem | theorem | Choquet capacitability theorem
t_daniell_integral_approach | technique | Daniell integral approach
t_approximation_by_continuous_functions | technique | Approximation by continuous functions
s_inductive_limit_topology | axiom | Inductive limit topology
s_abels_liouville_formula | theorem | Abel-Liouville formula (Wronskian formula)
s_pontryagin_dual_group | axiom | Pontryagin dual group
s_spectral_radius_formula | theorem | Spectral radius formula
s_resolvent_of_an_operator | state | Resolvent of an operator
s_pettis_integral | axiom | Pettis integral (weak integral)
s_convolution_semigroup | state | Convolution semigroup
s_lie_algebra_homomorphism | axiom | Lie algebra homomorphism
s_quotient_lie_algebra | state | Quotient Lie algebra
s_center_of_a_lie_algebra | state | Center of a Lie algebra
s_derived_subalgebra | state | Derived subalgebra (commutator subalgebra)
s_derived_series | state | Derived series
s_radical_of_a_lie_algebra | state | Radical of a Lie algebra
s_simple_lie_algebra | axiom | Simple Lie algebra
s_lie_s_theorem | theorem | Lie's theorem
s_cartan_s_criterion_for_solvability | theorem | Cartan's criterion for solvability
s_cartan_s_criterion_for_semisimplicity | theorem | Cartan's criterion for semisimplicity
s_levi_decomposition | theorem | Levi decomposition
s_poincar_birkhoff_witt_theorem | theorem | Poincaré-Birkhoff-Witt theorem
s_normalizer_of_a_subalgebra | state | Normalizer of a subalgebra
s_centralizer_in_a_lie_algebra | state | Centralizer in a Lie algebra
s_semidirect_product_of_lie_algebras | state | Semidirect product of Lie algebras
s_derivation_of_a_lie_algebra | axiom | Derivation of a Lie algebra
s_whitehead_first_lemma | theorem | Whitehead's lemma (first)
s_whitehead_second_lemma | theorem | Whitehead's lemma (second)
s_jordan_decomposition_in_a_lie_algebra | state | Jordan decomposition in a Lie algebra
s_invariant_bilinear_form | axiom | Invariant bilinear form
s_witt_formula | theorem | Witt formula
s_baker_campbell_hausdorff_formula | theorem | Baker-Campbell-Hausdorff formula
s_lie_group_homomorphism | axiom | Lie group homomorphism
s_lie_algebra_of_a_lie_group | state | Lie algebra of a Lie group
s_lie_subgroup | state | Lie subgroup
s_closed_subgroup_theorem | theorem | Closed subgroup theorem (Cartan's theorem)
s_quotient_of_a_lie_group | state | Quotient of a Lie group
t_lie_functor | technique | Lie functor
t_integration_of_lie_algebra_homomorphisms | technique | Integration of Lie algebra homomorphisms
s_coxeter_group | axiom | Coxeter group
s_coxeter_matrix | axiom | Coxeter matrix
s_coxeter_graph | state | Coxeter graph
s_exchange_property | theorem | Exchange property
s_deletion_property | theorem | Deletion property
s_tits_system | axiom | Tits system (BN-pair)
s_parabolic_subgroup_of_a_coxeter_group | state | Parabolic subgroup of a Coxeter group
s_coxeter_complex | state | Coxeter complex
s_matsumoto_s_theorem | theorem | Matsumoto's theorem
s_strong_exchange_property | theorem | Strong exchange property
s_classification_of_finite_coxeter_groups | theorem | Classification of finite Coxeter groups
s_coroot | state | Coroot
s_cartan_matrix | state | Cartan matrix
s_classification_of_irreducible_root_systems | theorem | Classification of irreducible root systems
s_weyl_chamber | state | Weyl chamber
s_root_lattice | state | Root lattice
s_fundamental_weights | state | Fundamental weights
s_extended_dynkin_diagram | state | Extended (affine) Dynkin diagram
s_affine_root_system | state | Affine root system
s_affine_weyl_group | state | Affine Weyl group
s_alcove | state | Alcove
s_root_space_decomposition | state | Root space decomposition
s_chevalley_basis | state | Chevalley basis
s_conjugacy_of_cartan_subalgebras | theorem | Conjugacy of Cartan subalgebras
s_sl2_triple | state | sl₂-triple
s_jacobson_morozov_theorem | theorem | Jacobson-Morozov theorem
s_split_semisimple_lie_algebra | axiom | Split semisimple Lie algebra
s_parabolic_subalgebra | state | Parabolic subalgebra
s_kostant_multiplicity_formula | theorem | Kostant multiplicity formula
s_kostant_partition_function | state | Kostant partition function
s_freudenthal_s_formula | theorem | Freudenthal's formula
s_existence_theorem_for_semisimple_lie_algebras | theorem | Existence theorem for semisimple Lie algebras
s_serre_s_presentation | theorem | Serre's presentation (Serre relations)
s_maximal_torus | state | Maximal torus
s_conjugacy_of_maximal_tori | theorem | Conjugacy of maximal tori
s_compact_real_form | state | Compact real form
t_weyl_s_unitary_trick | technique | Weyl's unitary trick
s_banach_algebra | axiom | Banach algebra
s_resolvent_identity | theorem | Resolvent identity
s_character | axiom | Character (multiplicative linear functional)
s_gelfand_spectrum | state | Gelfand spectrum (maximal ideal space)
s_c_star_identity | axiom | C*-identity
s_state_on_c_star_algebra | axiom | State on a C*-algebra
s_cyclic_vector | axiom | Cyclic vector
s_spectral_mapping_theorem | theorem | Spectral mapping theorem
s_unitization_of_c_star_algebra | state | Unitization of a C*-algebra
s_locally_compact_group | axiom | Locally compact group
s_convolution_product | state | Convolution product
s_group_c_star_algebra | state | Group C*-algebra C*(G)
s_spectral_theorem_for_bounded_normal_operators | theorem | Spectral theorem for bounded normal operators
s_spectral_theorem_for_commuting_normal_operators | theorem | Spectral theorem for commuting normal operators
s_snag_theorem | theorem | SNAG theorem (Stone-Naimark-Ambrose-Godement)
s_type_decomposition_of_von_neumann_algebras | theorem | Type decomposition of von Neumann algebras
s_holomorphic_functional_calculus | theorem | Holomorphic functional calculus
s_fell_topology | state | Fell topology
s_spectral_permanence | theorem | Spectral permanence
s_mellin_transform | axiom | Mellin Transform
s_faa_di_bruno_formula | theorem | Faa di Bruno's Formula
s_leibniz_rule_integral | theorem | Leibniz Integral Rule (Differentiation Under the Sign)
s_poincare_asymptotic_expansion | axiom | Poincare Asymptotic Expansion
s_watsons_lemma | theorem | Watson's Lemma
s_stokes_phenomenon | state | Stokes Phenomenon
s_abel_plana_summation_formula | theorem | Abel-Plana Summation Formula
t_laplaces_method | technique | Laplace's Method
t_method_of_steepest_descents | technique | Method of Steepest Descents
t_bleisteins_method | technique | Bleistein's Method
t_mellin_transform_asymptotics | technique | Mellin Transform Asymptotics
t_liouville_green_wkb_approximation | technique | Liouville-Green (WKB) Approximation
t_fuchs_frobenius_theory | technique | Fuchs-Frobenius Theory
t_darboux_method | technique | Darboux's Method
t_exponentially_improved_asymptotics | technique | Exponentially Improved Asymptotics
t_hyperasymptotics | technique | Hyperasymptotics
s_borel_summation | axiom | Borel Summation
t_clenshaw_curtis_quadrature | technique | Clenshaw-Curtis Quadrature
t_romberg_integration | technique | Romberg Integration
t_pade_approximation | technique | Pade Approximation
t_chebyshev_expansion | technique | Chebyshev Expansion
t_minimax_rational_approximation | technique | Minimax Rational Approximation
t_aitken_delta_squared_process | technique | Aitken's Delta-Squared Process
t_wynn_epsilon_algorithm | technique | Wynn's Epsilon Algorithm
t_levin_transformation | technique | Levin's Transformation
t_gauss_legendre_quadrature | technique | Gauss-Legendre Quadrature
t_gauss_laguerre_quadrature | technique | Gauss-Laguerre Quadrature
t_gauss_hermite_quadrature | technique | Gauss-Hermite Quadrature
s_lambert_w_function | axiom | Lambert W-Function
s_gudermannian_function | state | Gudermannian Function gd(x)
s_euler_mascheroni_constant | axiom | Euler-Mascheroni Constant
s_pochhammer_symbol | axiom | Pochhammer Symbol (Rising Factorial)
s_gamma_recurrence_relation | theorem | Gamma Function Recurrence Relation
s_euler_reflection_formula | theorem | Euler's Reflection Formula
s_legendre_duplication_formula | theorem | Legendre Duplication Formula
s_gauss_multiplication_formula | theorem | Gauss Multiplication Formula
s_bohr_mollerup_theorem | theorem | Bohr-Mollerup Theorem
s_hankel_contour_integral_reciprocal_gamma | theorem | Hankel's Contour Integral for 1/Gamma(z)
s_binet_formula_log_gamma | theorem | Binet's Formula for ln Gamma
s_polygamma_functions | state | Polygamma Functions
s_polygamma_recurrence_relation | theorem | Polygamma Recurrence Relation
s_polygamma_reflection_formula | theorem | Polygamma Reflection Formula
s_barnes_g_function | state | Barnes G-Function
s_q_gamma_function | state | q-Gamma Function
s_reciprocal_gamma_entire | state | Reciprocal Gamma as Entire Function
s_beta_integral_representation | theorem | Beta Function Integral Representation
s_exponential_integral_E1 | axiom | Exponential Integral E_1(z)
s_exponential_integral_Ei | axiom | Exponential Integral Ei(x)
s_sine_integral | axiom | Sine Integral Si(z)
s_cosine_integral | axiom | Cosine Integral Ci(z)
s_error_function | axiom | Error Function erf(z)
s_complementary_error_function | axiom | Complementary Error Function erfc(z)
s_faddeeva_function | state | Faddeeva Function w(z)
s_dawsons_integral | axiom | Dawson's Integral F(z)
s_fresnel_cosine_integral | axiom | Fresnel Cosine Integral C(z)
s_fresnel_sine_integral | axiom | Fresnel Sine Integral S(z)
s_voigt_profile | state | Voigt Profile
s_lower_incomplete_gamma | axiom | Lower Incomplete Gamma Function gamma(a,z)
s_upper_incomplete_gamma | axiom | Upper Incomplete Gamma Function Gamma(a,z)
s_regularized_incomplete_gamma | state | Regularized Incomplete Gamma P(a,z) and Q(a,z)
s_incomplete_beta_function | axiom | Incomplete Beta Function B_x(a,b)
s_generalized_exponential_integral | state | Generalized Exponential Integral E_p(z)
s_airy_differential_equation | axiom | Airy Differential Equation
s_airy_function_ai | axiom | Airy Function Ai(z)
s_airy_function_bi | axiom | Airy Function Bi(z)
s_scorer_function_gi | state | Scorer Function Gi(z)
s_scorer_function_hi | state | Scorer Function Hi(z)
s_airy_integral_representation | theorem | Airy Integral Representation
s_bessel_differential_equation | axiom | Bessel Differential Equation
s_bessel_function_j_nu | axiom | Bessel Function J_nu(z)
s_bessel_function_y_nu | axiom | Bessel Function Y_nu(z)
s_hankel_functions | state | Hankel Functions H_nu^{(1)}(z) and H_nu^{(2)}(z)
s_modified_bessel_function_i_nu | axiom | Modified Bessel Function I_nu(z)
s_modified_bessel_function_k_nu | axiom | Modified Bessel Function K_nu(z)
s_spherical_bessel_functions | state | Spherical Bessel Functions
s_kelvin_functions | state | Kelvin Functions ber, bei, ker, kei
s_mittag_leffler_function | state | Mittag-Leffler Function E_{a,b}(z)
t_hankel_transform | technique | Hankel Transform
s_bessel_addition_theorem | theorem | Bessel Addition Theorem
s_nicholson_integral | theorem | Nicholson's Integral
s_bessel_integral_representation_poisson | theorem | Poisson Integral for Bessel Functions
s_struve_function_h_nu | axiom | Struve Function H_nu(z)
s_modified_struve_function_l_nu | state | Modified Struve Function L_nu(z)
s_lommel_function_s | axiom | Lommel Functions s_{mu,nu} and S_{mu,nu}
s_anger_function | axiom | Anger Function
s_weber_function_e_nu | axiom | Weber Function E_nu(z)
s_parabolic_cylinder_equation | axiom | Parabolic Cylinder Differential Equation
s_parabolic_cylinder_function_u | axiom | Parabolic Cylinder Function U(a,z)
s_parabolic_cylinder_function_v | axiom | Parabolic Cylinder Function V(a,z)
s_weber_parabolic_cylinder_d_nu | state | Weber Parabolic Cylinder Function D_nu(z)
s_kummer_confluent_equation | axiom | Kummer's Confluent Hypergeometric Equation
s_kummer_function_m | axiom | Kummer Function M(a,b,z)
s_tricomi_function_u | axiom | Tricomi Function U(a,b,z)
s_kummer_transformation | theorem | Kummer Transformation
s_whittaker_differential_equation | axiom | Whittaker Differential Equation
s_whittaker_function_m_kappa_mu | state | Whittaker Function M_{kappa,mu}(z)
s_whittaker_function_w_kappa_mu | state | Whittaker Function W_{kappa,mu}(z)
s_confluent_hypergeometric_limit | theorem | Confluent Hypergeometric Limit
s_associated_legendre_equation | axiom | Associated Legendre Differential Equation
s_ferrers_function_p | axiom | Ferrers Function P_nu^mu(x)
s_toroidal_functions | state | Toroidal (Ring) Functions
s_conical_mehler_functions | state | Conical (Mehler) Functions
t_mehler_fock_transform | technique | Mehler-Fock Transform
s_legendre_function_second_kind | axiom | Legendre Function of the Second Kind Q_nu(z)
s_associated_legendre_addition_theorem | theorem | Addition Theorem for Legendre Functions
s_gauss_hypergeometric_function | axiom | Gauss Hypergeometric Function 2F1(a,b;c;z)
s_hypergeometric_differential_equation | axiom | Hypergeometric Differential Equation
s_gauss_summation_theorem | theorem | Gauss Summation Theorem
s_chu_vandermonde_identity | theorem | Chu-Vandermonde Identity
s_euler_transformation_hypergeometric | theorem | Euler Transformation for 2F1
s_pfaff_transformation | theorem | Pfaff Transformation
s_riemann_p_symbol | axiom | Riemann Differential Equation and P-symbol
s_euler_integral_representation_2f1 | theorem | Euler Integral Representation for 2F1
s_kummer_24_solutions | theorem | Kummer's 24 Solutions
s_generalized_hypergeometric_pfq | axiom | Generalized Hypergeometric Function pFq
s_pfaff_saalschutz_theorem | theorem | Pfaff-Saalschutz Summation Theorem
s_dixon_summation_theorem | theorem | Dixon's Summation Theorem
s_dougall_summation_theorem | theorem | Dougall Summation Theorem
s_meijer_g_function | axiom | Meijer G-Function
s_appell_function_f1 | axiom | Appell Function F_1
s_appell_functions_f2_f3_f4 | axiom | Appell Functions F_2, F_3, F_4
s_fox_h_function | axiom | Fox H-Function
s_bilateral_hypergeometric_series | axiom | Bilateral Hypergeometric Series
s_clausen_formula | theorem | Clausen's Formula
s_whipple_transformation | theorem | Whipple's Transformation
s_q_binomial_coefficient | axiom | q-Binomial Coefficient (Gaussian Binomial)
s_q_derivative_operator | axiom | q-Derivative Operator D_q
s_q_integral_jackson | axiom | q-Integral (Jackson Integral)
s_q_binomial_theorem | theorem | q-Binomial Theorem
s_basic_hypergeometric_function | axiom | Basic Hypergeometric Function r+1-phi-s
s_ramanujan_1_psi_1 | theorem | Ramanujan's 1-psi-1 Bilateral Summation
t_bailey_pairs_chain | technique | Bailey Pairs and Bailey Chain
s_q_exponential_functions | state | q-Exponential Functions e_q and E_q
s_rogers_ramanujan_identities | theorem | Rogers-Ramanujan Identities
s_jacobi_polynomial | axiom | Jacobi Polynomial P_n^{alpha,beta}(x)
s_gegenbauer_polynomial | state | Gegenbauer (Ultraspherical) Polynomial C_n^lambda(x)
s_laguerre_polynomial | axiom | Laguerre Polynomial L_n^alpha(x)
s_christoffel_darboux_formula | theorem | Christoffel-Darboux Formula
s_favard_theorem | theorem | Favard's Theorem
s_hahn_polynomial | axiom | Hahn Polynomial
s_wilson_polynomial | axiom | Wilson Polynomial
s_racah_polynomial | axiom | Racah Polynomial
s_askey_scheme | state | Askey Scheme
s_askey_wilson_polynomial | axiom | Askey-Wilson Polynomial
s_krawtchouk_polynomial | state | Krawtchouk Polynomial
s_meixner_polynomial | state | Meixner Polynomial
s_charlier_polynomial | state | Charlier Polynomial
s_chebyshev_polynomial_first_kind | axiom | Chebyshev Polynomial of the First Kind T_n(x)
s_chebyshev_polynomial_second_kind | axiom | Chebyshev Polynomial of the Second Kind U_n(x)
s_meixner_pollaczek_polynomial | state | Meixner-Pollaczek Polynomial P_n^lambda(x;phi)
s_continuous_hahn_polynomial | state | Continuous Hahn Polynomial
s_continuous_dual_hahn_polynomial | state | Continuous Dual Hahn Polynomial
s_dual_hahn_polynomial | state | Dual Hahn Polynomial
s_rodrigues_formula_general | theorem | Rodrigues Formula (General)
s_three_term_recurrence | theorem | Three-Term Recurrence Relation
s_complete_elliptic_integral_k | axiom | Complete Elliptic Integral K(k)
s_complete_elliptic_integral_e | axiom | Complete Elliptic Integral E(k)
s_complete_elliptic_integral_pi | axiom | Complete Elliptic Integral Pi(n,k)
s_carlson_symmetric_rf | axiom | Carlson Symmetric R_F(x,y,z)
s_carlson_symmetric_rj | axiom | Carlson Symmetric R_J(x,y,z,p)
s_carlson_symmetric_rd | axiom | Carlson Symmetric R_D(x,y,z)
s_carlson_symmetric_rc | axiom | Carlson Symmetric R_C(x,y)
t_arithmetic_geometric_mean | technique | Arithmetic-Geometric Mean M(a,g)
t_landen_transformation | technique | Landen Transformation
s_legendre_relation_elliptic | theorem | Legendre's Relation for Elliptic Integrals
s_jacobi_theta_1 | axiom | Jacobi Theta Function theta_1(z,q)
s_jacobi_theta_2 | axiom | Jacobi Theta Function theta_2(z,q)
s_jacobi_theta_3 | axiom | Jacobi Theta Function theta_3(z,q)
s_jacobi_theta_4 | axiom | Jacobi Theta Function theta_4(z,q)
s_theta_function_heat_equation | theorem | Theta Function Heat Equation
s_jacobi_imaginary_transformation | theorem | Jacobi Imaginary Transformation
s_riemann_theta_function | axiom | Riemann Theta Function
s_siegel_modular_group | state | Siegel Modular Group Sp(2g,Z)
s_jacobian_sn | axiom | Jacobian Elliptic Function sn(z,k)
s_jacobian_cn | axiom | Jacobian Elliptic Function cn(z,k)
s_jacobian_dn | axiom | Jacobian Elliptic Function dn(z,k)
s_jacobi_amplitude_function | axiom | Jacobi Amplitude Function am(u,k)
s_jacobi_twelve_elliptic_functions | state | Twelve Jacobian Elliptic Functions
s_jacobi_elliptic_addition_theorems | theorem | Jacobi Elliptic Addition Theorems
s_weierstrass_p_function | axiom | Weierstrass Elliptic Function wp(z)
s_weierstrass_sigma_function | axiom | Weierstrass Sigma Function sigma(z)
s_weierstrass_zeta_function | axiom | Weierstrass Zeta Function zeta(z)
s_weierstrass_ode | axiom | Weierstrass Differential Equation
s_modular_discriminant_delta | axiom | Modular Discriminant Delta(tau)
s_klein_j_invariant | axiom | Klein j-Invariant j(tau)
s_eisenstein_series_e2k | axiom | Eisenstein Series E_{2k}(tau)
s_elliptic_modular_function_lambda | axiom | Elliptic Modular Function lambda(tau)
s_bernoulli_polynomial | axiom | Bernoulli Polynomial B_n(x)
s_euler_polynomial | axiom | Euler Polynomial E_n(x)
s_genocchi_numbers | state | Genocchi Numbers
s_bernoulli_euler_complement_formula | theorem | Bernoulli-Euler Complement Formula
s_euler_maclaurin_formula | theorem | Euler-Maclaurin Summation Formula
s_hurwitz_zeta_function | axiom | Hurwitz Zeta Function zeta(s,a)
s_polylogarithm | axiom | Polylogarithm Li_s(z)
s_lerch_transcendent | axiom | Lerch Transcendent Phi(z,s,a)
s_fermi_dirac_integral | state | Fermi-Dirac Integral
s_bose_einstein_integral | state | Bose-Einstein Integral
s_stieltjes_constants | state | Stieltjes Constants
s_dirichlet_beta_function | axiom | Dirichlet Beta Function beta(s)
s_clausen_function | state | Clausen Function Cl_2(theta)
s_bell_number | axiom | Bell Number B_n
s_bell_polynomial | axiom | Bell Polynomial B_{n,k}
s_plane_partition | axiom | Plane Partition
s_macmahon_box_formula | theorem | MacMahon Box Formula
t_twelvefold_way | technique | Twelvefold Way
s_derangement_number | axiom | Derangement Number D_n
s_narayana_number | axiom | Narayana Number N(n,k)
s_jordan_totient_function | state | Jordan's Totient Function J_k(n)
s_ramanujan_tau_function | axiom | Ramanujan Tau Function
t_lambert_series | technique | Lambert Series
s_liouville_lambda_function | state | Liouville Lambda Function lambda(n)
s_dedekind_sum | state | Dedekind Sum s(h,k)
s_mathieu_equation | axiom | Mathieu Equation
s_mathieu_characteristic_values | axiom | Mathieu Characteristic Values a_n(q), b_n(q)
s_mathieu_function_ce | axiom | Mathieu Function ce_n(z,q)
s_mathieu_function_se | axiom | Mathieu Function se_n(z,q)
s_modified_mathieu_equation | axiom | Modified Mathieu Equation
s_hill_equation | axiom | Hill's Equation
s_hill_discriminant | state | Hill Discriminant
s_mathieu_floquet_solution | state | Mathieu-Floquet Solution
s_lame_equation | axiom | Lame Equation
s_lame_polynomials | axiom | Lame Polynomials
s_ellipsoidal_harmonics | state | Ellipsoidal Harmonics
s_lame_eigenvalues | state | Lame Eigenvalues
s_spheroidal_equation | axiom | Spheroidal Differential Equation
s_angular_spheroidal_wave_function | axiom | Angular Spheroidal Wave Function Ps_n^m
s_radial_spheroidal_wave_functions | axiom | Radial Spheroidal Wave Functions
s_spheroidal_eigenvalues | state | Spheroidal Eigenvalues lambda_n^m(gamma^2)
s_prolate_spheroidal_function | state | Prolate Spheroidal Wave Function
s_heun_equation | axiom | Heun's Equation
s_heun_function | axiom | Heun Function Hl
s_accessory_parameter | axiom | Accessory Parameter
s_confluent_heun_equation | axiom | Confluent Heun Equation
s_biconfluent_heun_equation | state | Biconfluent Heun Equation
s_triconfluent_heun_equation | state | Triconfluent Heun Equation
s_heun_192_local_solutions | state | Heun's 192 Local Solutions
s_painleve_equation_p1 | axiom | First Painleve Equation P_I
s_painleve_equation_p2 | axiom | Second Painleve Equation P_II
s_painleve_equation_p3 | axiom | Third Painleve Equation P_III
s_painleve_equation_p4 | axiom | Fourth Painleve Equation P_IV
s_painleve_equation_p5 | axiom | Fifth Painleve Equation P_V
s_painleve_equation_p6 | axiom | Sixth Painleve Equation P_VI
s_painleve_coalescence_cascade | theorem | Painleve Coalescence Cascade
t_backlund_transformations_painleve | technique | Backlund Transformations for Painleve
s_isomonodromic_deformation_connection | theorem | Isomonodromic Deformation Connection
s_painleve_property | axiom | Painleve Property
s_coulomb_wave_equation | axiom | Coulomb Wave Equation
s_regular_coulomb_function | axiom | Regular Coulomb Function F_l(eta,rho)
s_irregular_coulomb_functions | axiom | Irregular Coulomb Functions G_l, H_l^{+/-}
s_gamow_sommerfeld_factor | state | Gamow-Sommerfeld Factor
s_coulomb_phase_shift | state | Coulomb Phase Shift sigma_l(eta)
s_wigner_3j_symbol | axiom | Wigner 3j Symbol
s_6j_symbol | axiom | 6j Symbol (Racah Coefficient)
s_9j_symbol | axiom | 9j Symbol
s_gaunt_coefficient | state | Gaunt Coefficient
s_biedenharn_elliott_identity | theorem | Biedenharn-Elliott Identity
s_clebsch_gordan_coefficient | axiom | Clebsch-Gordan Coefficient
s_wigner_d_matrix | axiom | Wigner D-Matrix D^j_{m'm}(alpha,beta,gamma)
s_multivariate_gamma_function | axiom | Multivariate Gamma Function Gamma_m(a)
s_zonal_polynomials | axiom | Zonal Polynomials Z_kappa(T)
s_hypergeometric_matrix_argument | axiom | Hypergeometric Function of Matrix Argument
s_wishart_distribution | state | Wishart Distribution
s_jack_polynomial | state | Jack Polynomial P_kappa^{(alpha)}
s_cuspoid_canonical_integral | axiom | Cuspoid Canonical Integral Psi_K
s_pearcey_integral | state | Pearcey Integral
s_swallowtail_integral | state | Swallowtail Canonical Integral
s_elliptic_umbilic_catastrophe | axiom | Elliptic Umbilic Catastrophe
s_hyperbolic_umbilic_catastrophe | axiom | Hyperbolic Umbilic Catastrophe
s_butterfly_canonical_integral | state | Butterfly Canonical Integral
t_hilbert_transform | technique | Hilbert Transform
t_abel_transform | technique | Abel Transform
t_hankel_contour_method | technique | Hankel Contour Method
s_bateman_function | state | Bateman's Function k_nu(x)
s_wright_function | state | Wright Function W_{lambda,mu}(z)
s_fox_wright_psi_function | state | Fox-Wright Psi Function
s_dedekind_eta_function | axiom | Dedekind Eta Function eta(tau)
s_weber_modular_function | state | Weber Modular Functions f, f_1, f_2
s_darboux_christoffel_kernel | state | Darboux-Christoffel Kernel
s_turán_inequality | theorem | Turan Inequality for Orthogonal Polynomials
s_dirichlet_character | axiom | Dirichlet Character chi mod q
s_gauss_sum | axiom | Gauss Sum g(chi)
s_jacobi_sum | axiom | Jacobi Sum J(chi_1, chi_2)
s_riemann_xi_function | axiom | Riemann Xi Function xi(s)
s_functional_equation_hurwitz_zeta | theorem | Functional Equation for Hurwitz Zeta
s_partition_function_stat_mech | state | Partition Function (Statistical Mechanics) Z
s_hamiltonians_painleve | state | Painleve Hamiltonians
s_painleve_tau_function | state | Painleve Tau Function
t_separation_of_variables | technique | Separation of Variables in Coordinate Systems
s_connection_formulas_hypergeometric | theorem | Connection Formulas for 2F1
s_lipschitz_summation_formula | theorem | Lipschitz Summation Formula
s_ramanujan_master_theorem | theorem | Ramanujan's Master Theorem
s_big_q_jacobi_polynomial | state | Big q-Jacobi Polynomial
s_little_q_jacobi_polynomial | state | Little q-Jacobi Polynomial
s_al_salam_chihara_polynomial | state | Al-Salam-Chihara Polynomial
s_q_racah_polynomial | state | q-Racah Polynomial
s_q_hahn_polynomial | state | q-Hahn Polynomial
s_volterra_integral_equation | axiom | Volterra Integral Equation
s_fredholm_integral_equation | axiom | Fredholm Integral Equation
s_appell_sequence | state | Appell Sequence
s_sheffer_sequence | state | Sheffer Sequence
s_bessel_zeros | state | Zeros of Bessel Functions
s_debye_expansion_bessel | theorem | Debye Asymptotic Expansion for Bessel Functions
s_olver_uniform_asymptotics_bessel | theorem | Olver Uniform Asymptotic Expansion
s_weierstrass_addition_theorem | theorem | Weierstrass Addition Theorem
s_abel_theorem_elliptic | theorem | Abel's Theorem for Elliptic Functions
t_stationary_phase_method | technique | Method of Stationary Phase
t_watson_transformation | technique | Watson Transformation
s_perron_formula | theorem | Perron's Formula
s_marcenko_pastur_distribution | state | Marchenko-Pastur Distribution
s_motzkin_number | axiom | Motzkin Number M_n
s_schroder_number | axiom | Schroder Number
s_greatest_common_divisor | axiom | Greatest common divisor
s_least_common_multiple | axiom | Least common multiple
s_gcd_lcm_product_relation | theorem | GCD–LCM product relation
s_prime_number_definition | axiom | Prime number
s_composite_number | axiom | Composite number
s_canonical_prime_factorization | axiom | Canonical prime factorization
s_divisor_characterization_via_factorization | theorem | Divisor characterization via prime factorization
s_gcd_lcm_via_prime_factorization | theorem | GCD and LCM via prime factorization
s_integrality_of_binomial_coefficients | theorem | Integrality of binomial coefficients
s_prime_divisibility_of_binomial_coefficients | theorem | Prime divisibility of binomial coefficients
s_completely_multiplicative_function | axiom | Completely multiplicative function
s_multiplicative_determination_by_prime_powers | theorem | Multiplicative function determination by prime powers
s_mobius_sum_identity | theorem | Möbius function sum identity
s_euler_totient_divisor_sum | theorem | Euler totient divisor sum
s_euler_totient_product_formula | theorem | Euler totient product formula
s_dirichlet_identity_function_epsilon | axiom | Dirichlet identity function ε
s_dirichlet_inverse_existence | theorem | Dirichlet inverse existence
s_dirichlet_product_abelian_group | theorem | Abelian group of arithmetic functions under Dirichlet product
s_unit_function | axiom | Unit function u(n) = 1
s_number_theoretic_identity_function_N | axiom | Number-theoretic identity function N(n) = n
s_mobius_dirichlet_inverse_of_unit | theorem | Möbius function is Dirichlet inverse of unit function
s_von_mangoldt_divisor_sum_identity | theorem | Von Mangoldt divisor sum identity
s_liouville_completely_multiplicative | theorem | Liouville function is completely multiplicative
s_liouville_divisor_sum | theorem | Liouville function divisor sum
s_divisor_function_sigma_alpha | axiom | Generalized divisor function σ_α
s_number_of_divisors_function | axiom | Number of divisors function d(n)
s_sum_of_divisors_function | axiom | Sum of divisors function σ(n)
s_divisor_function_multiplicativity | theorem | Divisor function multiplicativity
s_divisor_function_prime_power_formula | theorem | Divisor function prime power formula
s_product_multiplicative_is_multiplicative | theorem | Dirichlet product of multiplicative functions is multiplicative
s_dirichlet_inverse_multiplicative_is_multiplicative | theorem | Dirichlet inverse of multiplicative function is multiplicative
s_bell_series | axiom | Bell series
s_bell_series_multiplication | theorem | Bell series multiplication theorem
s_euler_totient_via_mobius | theorem | Euler totient via Möbius inversion
s_euler_totient_multiplicativity | theorem | Euler totient multiplicativity
s_dirichlet_ring_of_arithmetic_functions | axiom | Dirichlet ring of arithmetic functions
s_generalized_mobius_inversion | theorem | Generalized Möbius inversion
s_completely_multiplicative_inverse_formula | theorem | Completely multiplicative Dirichlet inverse formula
s_asymptotic_equality | axiom | Asymptotic equality
s_average_order_definition | axiom | Average order of an arithmetic function
s_euler_summation_formula | theorem | Euler summation formula
s_average_order_of_d_n | theorem | Average order of d(n)
t_dirichlet_hyperbola_method | technique | Dirichlet hyperbola method
s_average_order_of_sigma_n | theorem | Average order of σ(n)
s_average_order_of_phi_n | theorem | Average order of φ(n)
s_harmonic_sum_estimate | theorem | Harmonic sum estimate
s_average_order_of_mu | theorem | Average order of μ(n) is zero
s_mertens_function | axiom | Mertens function
t_lattice_point_counting | technique | Lattice point counting
s_asymptotic_reciprocal_phi | theorem | Asymptotic average of 1/φ(n)
s_prime_counting_function | axiom | Prime counting function π(x)
s_theta_psi_relation | theorem | Relation between Chebyshev θ and ψ
s_abel_summation_pi_theta | theorem | Abel summation relating π(x) and θ(x)
s_equivalence_of_pnt_forms | theorem | Equivalence of PNT forms
s_shapiro_tauberian_theorem | theorem | Shapiro's Tauberian theorem
s_selberg_identity | theorem | Selberg's identity
s_nth_prime_asymptotic | theorem | Asymptotic formula for the n-th prime
s_sum_prime_reciprocals_diverges | theorem | Divergence of the sum of prime reciprocals
s_residue_class | axiom | Residue class
s_complete_residue_system | axiom | Complete residue system
s_reduced_residue_system | axiom | Reduced residue system
s_linear_congruence | axiom | Linear congruence
s_linear_congruence_solvability | theorem | Linear congruence solvability
s_multiplicative_inverse_mod_m | theorem | Multiplicative inverse modulo m
s_polynomial_congruence | axiom | Polynomial congruence
s_lagrange_polynomial_congruence_mod_p | theorem | Lagrange's theorem on polynomial congruences mod p
s_order_of_group_element | axiom | Order of a group element
s_order_divides_group_order | theorem | Order of element divides group order
s_cyclic_group | axiom | Cyclic group
s_subgroups_of_cyclic_are_cyclic | theorem | Subgroups of cyclic groups are cyclic
s_direct_product_of_groups | axiom | Direct product of groups
s_group_character | axiom | Group character (finite abelian)
s_character_group_isomorphic_to_G | theorem | Character group is isomorphic to G
s_orthogonality_dirichlet_characters | theorem | Orthogonality relations for Dirichlet characters
t_extraction_ap_via_characters | technique | Extraction of arithmetic progressions via characters
s_euler_product_for_l_function | theorem | Euler product for L(s,χ)
s_l_chi0_relation_to_zeta | state | L(s,χ₀) relation to Riemann zeta
s_log_l_series | state | Logarithm of L(s,χ) series
s_primes_in_ap_dirichlet_series | state | Dirichlet series for primes in arithmetic progressions
s_l_chi0_pole_at_1 | theorem | Pole of L(s,χ₀) at s = 1
t_product_all_l_functions_mod_m | technique | Product of all L-functions mod m
t_nonvanishing_real_chi_via_landau | technique | Nonvanishing of L(1,χ) for real χ via Landau's theorem
s_equidistribution_primes_residue_classes | theorem | Equidistribution of primes in residue classes
s_convergence_l_nonprincipal_s_gt_0 | theorem | Convergence of L(s,χ) for nonprincipal χ at s > 0
s_periodic_arithmetical_function | axiom | Periodic arithmetical function
s_finite_fourier_expansion | theorem | Finite Fourier expansion of periodic functions
s_ramanujan_sum_mobius_formula | theorem | Ramanujan sum via Möbius function
s_ramanujan_sum_multiplicativity | theorem | Multiplicativity of Ramanujan sums
t_ramanujan_expansion | technique | Ramanujan expansion of arithmetical functions
s_separability_of_gauss_sums | axiom | Separability of Gauss sums
s_primitive_characters_are_separable | theorem | Primitive characters are separable
s_gauss_sum_modulus_primitive | theorem | Modulus of Gauss sum for primitive characters
s_quadratic_gauss_sum | state | Quadratic Gauss sum
s_sign_of_quadratic_gauss_sum | theorem | Sign of the quadratic Gauss sum
s_chi_via_gauss_sum | theorem | Representation of χ(n) via Gauss sums
s_gauss_sum_product_identity | theorem | Gauss sum product identity G(1,χ)G(1,χ̄) = χ(−1)m
s_von_sterneck_formula | theorem | Von Sterneck's formula for Ramanujan sums
s_half_residues_are_qr | theorem | Half of reduced residues are quadratic residues
s_euler_criterion | theorem | Euler's criterion
s_legendre_symbol_completely_multiplicative | theorem | Legendre symbol is completely multiplicative
s_first_supplement_qr | theorem | First supplement to quadratic reciprocity
s_gauss_lemma_legendre | theorem | Gauss's lemma for the Legendre symbol
s_second_supplement_qr | theorem | Second supplement to quadratic reciprocity
s_jacobi_symbol | axiom | Jacobi symbol
s_jacobi_reciprocity | theorem | Quadratic reciprocity for Jacobi symbol
s_jacobi_first_supplement | theorem | First supplement for Jacobi symbol
s_jacobi_second_supplement | theorem | Second supplement for Jacobi symbol
s_eisenstein_lemma | theorem | Eisenstein's lemma
t_qr_proof_via_gauss_sums | technique | Proof of quadratic reciprocity via Gauss sums
t_lattice_point_qr_proof | technique | Lattice point counting proof of quadratic reciprocity
s_number_solutions_x2_eq_a_mod_p | theorem | Number of solutions of x² ≡ a (mod p)
s_order_of_integer_mod_m | axiom | Order of an integer modulo m
s_order_divides_iff | theorem | Order divides exponent iff congruence holds
s_order_of_powers | theorem | Order of a power of an element
s_number_of_primitive_roots | theorem | Number of primitive roots modulo m
s_primitive_roots_mod_prime_powers | theorem | Existence of primitive roots modulo p^k
s_primitive_roots_mod_2pk | theorem | Existence of primitive roots modulo 2p^k
s_no_primitive_roots_mod_2k | theorem | Nonexistence of primitive roots modulo 2^k for k ≥ 3
s_classification_moduli_primitive_roots | theorem | Classification of moduli having primitive roots
s_discrete_logarithm_index | axiom | Index (discrete logarithm) modulo m
s_index_homomorphism | theorem | Homomorphism properties of the index function
t_solving_congruences_via_indices | technique | Solving polynomial congruences via indices
s_power_residue_criterion | theorem | Power residue criterion via indices
s_characters_mod_p_via_primitive_roots | theorem | Characters mod p via primitive roots
s_abscissa_relation | theorem | Abscissa of convergence vs absolute convergence relation
s_analyticity_dirichlet_series | theorem | Analyticity of Dirichlet series
s_uniqueness_dirichlet_series | theorem | Uniqueness theorem for Dirichlet series
s_euler_product_multiplicative | theorem | Euler product for multiplicative functions
s_euler_product_completely_multiplicative | theorem | Euler product for completely multiplicative functions
s_dirichlet_series_1_over_zeta | state | Dirichlet series for 1/ζ(s)
s_dirichlet_series_zeta_prime_over_zeta | state | Dirichlet series for −ζ′(s)/ζ(s)
s_l_function_via_hurwitz_zeta | theorem | L(s,χ) as linear combination of Hurwitz zeta functions
s_analytic_continuation_zeta_re_gt_0 | theorem | Analytic continuation of ζ(s) to Re(s) > 0
s_analytic_continuation_l_re_gt_0 | theorem | Analytic continuation of L(s,χ) to Re(s) > 0
s_gamma_analytic_continuation | theorem | Analytic continuation of Γ to meromorphic function on ℂ
s_bernoulli_polynomial_difference | theorem | Difference equation for Bernoulli polynomials
s_fourier_expansion_bernoulli_polynomials | theorem | Fourier expansion of Bernoulli polynomials
s_contour_integral_hurwitz_zeta | theorem | Contour integral representation of Hurwitz zeta
s_hurwitz_zeta_continuation_all_c | theorem | Analytic continuation of Hurwitz zeta to all of ℂ
s_hurwitz_zeta_negative_integers | theorem | Values of ζ(s,a) at negative integers
s_zeta_at_negative_integers | theorem | Values of ζ(s) at negative integers
s_evaluation_zeta_2n | theorem | Evaluation of ζ(2n) via Bernoulli numbers
s_functional_equation_l_primitive | theorem | Functional equation of L(s,χ) for primitive characters
s_analytic_continuation_l_all_c | theorem | Analytic continuation of L(s,χ) to all of ℂ
s_integral_representation_zeta_gamma | theorem | Integral representation Γ(s)ζ(s) via Mellin transform
s_periodic_zeta_function | state | Periodic zeta function F(a,s)
s_pole_of_zeta_at_1 | theorem | Simple pole of ζ(s) at s = 1
t_mertens_3_4_1_inequality | technique | Mertens 3-4-1 inequality
s_meromorphic_zeta_prime_over_zeta | theorem | Meromorphic structure of ζ′(s)/ζ(s)
t_pnt_from_wiener_ikehara | technique | Derivation of PNT from Wiener–Ikehara theorem
t_newman_tauberian_pnt | technique | Newman's short Tauberian proof of PNT
s_integral_psi_via_zeta | theorem | Integral representation of ψ(x) via ζ′/ζ
t_contour_integration_pnt | technique | Contour integration method for PNT
s_pnt_error_term | theorem | PNT with error term (de la Vallée-Poussin)
s_pnt_equivalent_to_nonvanishing | theorem | PNT equivalent to nonvanishing of ζ on Re(s) = 1
s_convexity_bound_zeta | theorem | Convexity bound for ζ(s) in the critical strip
s_dirichlet_eta_zeta_relation | state | Dirichlet eta function relation to ζ(s)
s_zeta_squared_divisor_series | state | ζ(s)² = ∑ d(n)/nˢ
s_zeta_product_sigma_series | state | ζ(s)ζ(s−1) = ∑ σ(n)/nˢ
s_landau_theorem_nonneg_coefficients | theorem | Landau's theorem on Dirichlet series with nonneg coefficients
s_psi_pi_relation | theorem | Relation between ψ(x) and π(x) via Abel summation
s_euler_totient_function | axiom | Euler totient function φ(n)
s_asymptotic_euler_totient_sum | theorem | Asymptotic for the summatory Euler totient
s_density_of_squarefree_integers | theorem | Density of squarefree integers
s_big_omega_function | axiom | Big-omega function Ω(n)
s_mean_value_of_omega | theorem | Mean value of ω(n)
s_hardy_ramanujan_normal_order | theorem | Hardy–Ramanujan theorem on normal order of ω(n)
s_turan_inequality_number_theory | theorem | Turán's inequality for ω(n)
s_legendre_sieve_formula | state | Legendre sieve formula
s_twin_prime_sieve_estimate | state | Twin prime sieve estimate
s_additive_character_mod_q | axiom | Additive character mod q
s_orthogonality_additive_characters | theorem | Orthogonality of additive characters
s_truncated_perron_formula | theorem | Truncated Perron formula
s_cesaro_riesz_means | axiom | Cesàro summability and Riesz means
s_landau_coefficient_formula | theorem | Landau's formula for Dirichlet series coefficients
s_three_four_one_inequality | state | 3-4-1 trigonometric inequality
s_smooth_number_definition | axiom | y-smooth integer (friable number)
s_dickman_rho_function | axiom | Dickman rho function ρ(u)
s_smooth_number_asymptotic | theorem | Smooth number asymptotic (de Bruijn / Hildebrand–Tenenbaum)
s_buchstab_function | axiom | Buchstab function ω(u)
s_primes_in_short_intervals | theorem | Primes in short intervals
s_sathe_selberg_theorem | theorem | Sathe–Selberg theorem on integers with k prime factors
s_beurling_generalized_primes | axiom | Beurling generalized primes
s_beurling_pnt | theorem | Beurling PNT for generalized primes
s_induced_dirichlet_character | axiom | Induced Dirichlet character
s_quadratic_dirichlet_character | axiom | Quadratic Dirichlet character
s_riemann_hypothesis | state | Riemann hypothesis
s_generalized_riemann_hypothesis | state | Generalized Riemann hypothesis (GRH)
s_hadamard_product_for_xi | theorem | Hadamard product for ξ(s)
s_zero_counting_function_N_T | state | Zero-counting function N(T)
s_zero_free_region_l_functions | theorem | Classical zero-free region for L-functions
s_siegel_theorem_l_functions | theorem | Siegel's theorem on L(1,χ)
s_page_theorem | theorem | Page's theorem
s_pnt_arithmetic_progressions | theorem | Prime number theorem for arithmetic progressions
s_siegel_walfisz_theorem | theorem | Siegel–Walfisz theorem
t_vaughan_identity | technique | Vaughan's identity
s_explicit_formula_psi_q_a | theorem | Explicit formula for ψ(x;q,a)
s_weil_explicit_formula | theorem | Weil's explicit formula
s_guinand_weil_formula | theorem | Guinand–Weil explicit formula
s_von_koch_rh_prime_estimate | theorem | Von Koch's theorem: RH implies sharp prime counting
s_rh_sharp_error_ap | theorem | RH implies sharp error in arithmetic progressions
s_lindelof_hypothesis | state | Lindelöf hypothesis
s_conditional_bounds_on_zeta | theorem | Conditional bounds on ζ(s) under RH
s_conditional_estimate_mertens_function | theorem | Conditional estimate for M(x) under RH
s_conditional_prime_gaps | theorem | Conditional bounds on gaps between primes under RH
s_conditional_pi_large_q | theorem | Conditional estimate for π(x;q,a) with large q under GRH
s_zero_density_estimate | theorem | Zero density estimate N(σ,T)
s_ingham_density_estimate | theorem | Ingham density estimate
s_hardy_theorem_zeros_critical_line | theorem | Hardy's theorem on zeros on the critical line
s_proportion_zeros_critical_line | theorem | Positive proportion of zeros on the critical line
s_density_hypothesis | state | Density hypothesis
s_landau_oscillation_theorem | theorem | Landau's oscillation theorem
s_omega_plus_minus_notation | axiom | Ω₊/Ω₋ notation for oscillation
s_littlewood_omega_theorem | theorem | Littlewood's omega theorem for π(x) − Li(x)
s_oscillation_psi_minus_x | theorem | Oscillation of ψ(x) − x
s_omega_result_mertens_function | theorem | Omega result for M(x)
s_ingham_omega_theorem | theorem | Ingham's theorem on omega results from zeros
s_skewes_number | state | Skewes number and sign changes of π(x) − Li(x)
s_riemann_stieltjes_integral | axiom | Riemann–Stieltjes integral
t_integration_by_parts_stieltjes | technique | Integration by parts for Stieltjes integrals
s_zeta_at_positive_even_integers | theorem | Values of ζ(s) at positive even integers
s_weierstrass_product_gamma | theorem | Weierstrass product for 1/Γ(z)
s_theta_function_number_theory | axiom | Theta function (number-theoretic)
s_theta_transformation_formula | theorem | Theta function transformation formula
t_rankin_trick | technique | Rankin's trick
s_dirichlet_divisor_problem | state | Dirichlet's divisor problem
s_selberg_class | axiom | Selberg class
s_vinogradov_korobov_zero_free_region | theorem | Vinogradov–Korobov zero-free region
s_mertens_conjecture_disproved | state | Mertens conjecture (disproved)
s_twin_prime_conjecture | state | Twin prime conjecture
s_hardy_littlewood_twin_prime_conjecture | state | Hardy–Littlewood twin prime conjecture
s_dirichlet_series_liouville | state | Dirichlet series for the Liouville function
s_ramanujan_sum_formula_sigma | theorem | Ramanujan's sum formula for σ_s(n)
s_vinogradov_korobov_pnt_error | theorem | Vinogradov–Korobov error term in PNT
s_montgomery_pair_correlation | state | Montgomery's pair correlation conjecture
t_explicit_formula_test_function | technique | Explicit formula with general test function
s_halasz_theorem | theorem | Halász's theorem
s_generalized_von_mangoldt_lambda_k | axiom | Generalized von Mangoldt function Λ_k
s_grand_riemann_hypothesis | state | Grand Riemann hypothesis
s_effective_vs_ineffective_constants | axiom | Effective vs ineffective constants
t_riemann_stieltjes_partial_summation | technique | Riemann–Stieltjes partial summation technique
s_dirichlet_series_multiplication_theorem | theorem | Dirichlet series multiplication theorem
s_explicit_zero_free_region_improvements | state | Explicit zero-free region constant improvements
s_completely_additive_function | axiom | Completely additive function ;; totally additive function
s_generalized_divisor_function_tau_k | axiom | Generalized divisor function τ_k(n) ;; Piltz divisor function ;; d_k(n) ;; τ_k
t_selberg_delange_method | technique | Selberg–Delange method ;; Selberg-Delange method
s_smoothed_perron_formula | theorem | Smoothed Perron formula ;; smooth Perron formula
t_dyadic_decomposition | technique | Dyadic decomposition ;; dyadic subdivision
t_smooth_partition_of_unity | technique | Smooth partition of unity ;; smooth dyadic partition
t_heath_brown_identity | technique | Heath-Brown's identity ;; Heath-Brown decomposition
t_linnik_identity | technique | Linnik's identity ;; Linnik decomposition
s_type_i_sum | axiom | Type I sum ;; Type I estimate ;; linear sum
s_type_ii_sum | axiom | Type II sum ;; Type II estimate ;; bilinear sum
s_wirsing_theorem | theorem | Wirsing's theorem ;; Wirsing mean value theorem
s_shiu_bound | theorem | Shiu's bound ;; Shiu's theorem
s_general_l_function_ik | axiom | General L-function (Iwaniec–Kowalski framework) ;; standard L-function ;; IK axiomatic L-function
s_completed_l_function | axiom | Completed L-function ;; Lambda function ;; Λ-function
s_root_number | axiom | Root number ;; epsilon factor ;; sign of functional equation
s_gamma_factors_functional_equation | axiom | Gamma factors in functional equations ;; archimedean gamma factor
s_conductor_l_function | axiom | Conductor of an L-function ;; arithmetic conductor ;; q(π)
s_analytic_conductor | axiom | Analytic conductor ;; C(π) ;; effective conductor
s_degree_l_function | axiom | Degree of an L-function ;; d(π)
s_convexity_bound_l_functions | theorem | Convexity bound for L-functions ;; convexity estimate
s_subconvexity_bound | theorem | Subconvexity bound ;; subconvex estimate ;; breaking convexity
s_grand_lindelof_hypothesis | axiom | Grand Lindelöf hypothesis ;; GLH ;; generalized Lindelöf hypothesis
s_approximate_functional_equation | theorem | Approximate functional equation ;; AFE
s_mean_value_theorem_dirichlet_polynomials | theorem | Mean value theorem for Dirichlet polynomials ;; Montgomery-Vaughan mean value theorem
s_montgomery_large_values | theorem | Montgomery's mean value theorem for large values ;; large values estimate
s_huxley_jutila_large_values | theorem | Huxley–Jutila large values estimate ;; Jutila large values
s_fourth_moment_zeta | theorem | Fourth moment of the Riemann zeta function ;; Ingham fourth moment
s_second_moment_l_functions_characters | theorem | Second moment of L-functions over characters ;; mean square of L(1/2,χ)
s_moments_l_functions_conjectural | axiom | Moments of L-functions (conjectural asymptotics) ;; CFKRS moment conjecture ;; Keating-Snaith conjecture
s_hecke_l_function_grossencharakter | axiom | Hecke L-function ;; Hecke Grössencharakter L-function
s_rankin_selberg_l_function | axiom | Rankin–Selberg L-function ;; Rankin-Selberg convolution
s_kronecker_limit_formula | theorem | Kronecker's limit formula ;; first Kronecker limit formula
s_sifting_function | axiom | Sifting function S(A, P, z) ;; sieve sum ;; S(A,z)
s_sieve_dimension | axiom | Sieve dimension ;; sifting density
s_buchstab_identity | theorem | Buchstab's identity ;; Buchstab recursion
s_linear_sieve | theorem | Linear sieve (Iwaniec's formulation) ;; Iwaniec linear sieve
s_sieve_limit_functions | axiom | Sieve upper/lower bound functions F_κ(s), f_κ(s) ;; sieve adjoint functions
t_rosser_iwaniec_sieve | technique | Rosser–Iwaniec sieve (β-sieve) ;; beta-sieve ;; β-sieve
s_parity_problem_sieve | axiom | Parity problem in sieve theory ;; parity barrier ;; Selberg parity phenomenon
t_semilinear_sieve | technique | Semilinear sieve (κ = 1/2) ;; half-dimensional sieve
t_bombieri_asymptotic_sieve | technique | Bombieri's asymptotic sieve ;; Bombieri sieve
t_asymptotic_sieve_for_primes | technique | Asymptotic sieve for primes (Bombieri–Friedlander–Iwaniec) ;; BFI sieve
s_level_of_distribution | axiom | Level of distribution ;; exponent of distribution
s_well_factorable_function | axiom | Well-factorable function ;; well-factorable weights
t_switching_principle_sieve | technique | Switching principle in sieve theory ;; Buchstab switching
t_van_der_corput_a_process | technique | Van der Corput's A-process ;; van der Corput second derivative test ;; A-process
t_van_der_corput_b_process | technique | Van der Corput's B-process ;; B-process ;; Poisson transform for exponential sums
t_method_of_exponent_pairs | technique | Method of exponent pairs ;; exponent pair method
s_exponent_pair | axiom | Exponent pair (k, ℓ) ;; van der Corput exponent pair
t_vinogradov_method_exponential_sums | technique | Vinogradov's method of exponential sums ;; Vinogradov exponential sum method
s_salie_sum | axiom | Salié sum ;; Salié-Kloosterman sum
s_hyper_kloosterman_sum | axiom | Hyper-Kloosterman sum ;; generalized Kloosterman sum
t_stepanov_method | technique | Stepanov's method ;; Stepanov auxiliary polynomial method
s_bombieri_proof_rh_curves | theorem | Bombieri's proof of RH for curves ;; Bombieri-Stepanov proof
t_square_sieve | technique | Square sieve (Heath-Brown) ;; Heath-Brown square sieve
t_bombieri_iwaniec_method | technique | Bombieri–Iwaniec method ;; Bombieri-Iwaniec first spacing method
s_duality_principle_bilinear | theorem | Duality principle for bilinear forms ;; bilinear duality ;; norm duality
s_dual_large_sieve_inequality | theorem | Dual large sieve inequality ;; dual form of the large sieve
s_large_sieve_characters | theorem | Large sieve inequality for Dirichlet characters ;; multiplicative large sieve
t_deshouillers_iwaniec_method | technique | Deshouillers–Iwaniec method ;; Deshouillers-Iwaniec spectral method
t_linnik_dispersion_method | technique | Linnik's dispersion method ;; dispersion method
s_halasz_montgomery_inequality | theorem | Halász–Montgomery inequality ;; Montgomery-Halász inequality
s_motohashi_formula | theorem | Motohashi's formula ;; Motohashi spectral formula
s_gallagher_larger_sieve | theorem | Gallagher's larger sieve ;; Gallagher sieve ;; the larger sieve
s_barban_davenport_halberstam_theorem | theorem | Barban–Davenport–Halberstam theorem ;; BDH theorem
s_elliott_halberstam_conjecture | axiom | Elliott–Halberstam conjecture ;; EH conjecture
s_modular_group_sl2z | axiom | Modular group SL₂(ℤ) ;; SL(2,Z) ;; full modular group
s_congruence_subgroup_gamma0 | axiom | Congruence subgroup Γ₀(N) ;; Gamma_0(N) ;; Hecke congruence subgroup
s_congruence_subgroup_gamma1 | axiom | Congruence subgroup Γ₁(N) ;; Gamma_1(N)
s_principal_congruence_subgroup | axiom | Principal congruence subgroup Γ(N) ;; Gamma(N)
s_nebentypus_character | axiom | Nebentypus character ;; Nebentypus
s_fundamental_domain_sl2z | axiom | Fundamental domain for SL₂(ℤ) ;; standard fundamental domain
s_cusps_of_congruence_subgroup | axiom | Cusps of a congruence subgroup ;; cusps ;; parabolic vertices
s_hyperbolic_laplacian | axiom | Hyperbolic Laplacian on H ;; Laplace-Beltrami operator on H
s_cusp_form | axiom | Cusp form ;; cuspidal modular form ;; Spitzenform
s_slash_operator | axiom | Slash operator |_k ;; slash operator ;; weight-k action
s_poincare_series_modular | axiom | Poincaré series P_m(z, k, Γ) ;; Poincaré series for modular forms
s_petersson_inner_product | axiom | Petersson inner product ;; Petersson scalar product
s_fourier_expansion_at_cusps | axiom | Fourier expansion at cusps ;; q-expansion
s_atkin_lehner_theory | theorem | Atkin-Lehner theory ;; newform-oldform decomposition
s_atkin_lehner_involution | axiom | Atkin-Lehner involution W_Q ;; Atkin-Lehner operator
s_fricke_involution | axiom | Fricke involution W_N ;; Fricke operator
s_oldform | axiom | Oldform ;; old form
s_conductor_of_modular_form | axiom | Conductor of a modular form ;; level of a newform
s_dimension_formula_mk_sl2z | theorem | Dimension formula for M_k(SL₂(ℤ)) ;; dimension of modular forms for SL₂(ℤ)
s_dimension_formula_sk_gamma0 | theorem | Dimension formula for S_k(Γ₀(N)) ;; Riemann-Roch dimension formula for cusp forms
s_theta_function_of_quadratic_form | axiom | Theta function of a quadratic form ;; theta series of a lattice
s_maass_form | axiom | Maass form ;; Maass wave form
s_maass_cusp_form | axiom | Maass cusp form ;; Maass cuspidal form
s_spectral_parameter_maass | axiom | Spectral parameter r ;; Maass spectral parameter
s_ramanujan_petersson_conjecture | theorem | Ramanujan-Petersson conjecture ;; Ramanujan conjecture for modular forms
s_real_analytic_eisenstein_series | axiom | Real-analytic Eisenstein series E(z,s) ;; non-holomorphic Eisenstein series
s_scattering_matrix_automorphic | axiom | Scattering matrix for Γ\H ;; scattering matrix ;; phi function
s_spectral_decomposition_l2_gamma_h | theorem | Spectral decomposition of L²(Γ\H) ;; Roelcke-Selberg spectral decomposition
s_discrete_spectrum_gamma_h | axiom | Discrete spectrum of Γ\H ;; cuspidal plus residual spectrum
s_continuous_spectrum_gamma_h | axiom | Continuous spectrum of Γ\H ;; Eisenstein contribution
s_pre_trace_formula | theorem | Pre-trace formula ;; automorphic kernel identity
s_selberg_zeta_function | axiom | Selberg zeta function Z_Γ(s) ;; Selberg zeta function
s_maass_selberg_relations | theorem | Maass-Selberg relations ;; Maass-Selberg formula
s_petersson_trace_formula | theorem | Petersson trace formula ;; Petersson-Kuznetsov formula
s_eichler_selberg_trace_formula | theorem | Eichler-Selberg trace formula ;; Selberg trace formula for Hecke operators
s_kuznetsov_trace_formula | theorem | Kuznetsov trace formula ;; Kuznetsov sum formula ;; Bruggeman-Kuznetsov formula
s_voronoi_summation_formula | theorem | Voronoi summation formula (GL(2)) ;; Voronoi formula
s_l_function_of_modular_form | axiom | L-function of a modular form ;; standard L-function of f
s_euler_product_hecke_eigenform | theorem | Euler product for Hecke eigenform L-function ;; Hecke Euler product
t_rankin_selberg_method | technique | Rankin-Selberg method ;; Rankin-Selberg unfolding
s_symmetric_square_l_function | axiom | Symmetric square L-function L(sym²f, s) ;; symmetric square lift
s_jacquet_langlands_correspondence | theorem | Jacquet-Langlands correspondence ;; Jacquet-Langlands transfer
s_gelbart_jacquet_lift | theorem | Gelbart-Jacquet lift ;; symmetric square lift to GL(3)
s_gl_n_automorphic_form | axiom | GL(n) automorphic form ;; automorphic form on GL(n)
s_kirillov_model | axiom | Kirillov model ;; Kirillov space
s_strong_multiplicity_one_gl2 | theorem | Strong multiplicity one for GL(2) ;; strong multiplicity one theorem
s_twisted_l_function | axiom | Twisted L-function L(f⊗χ, s) ;; character twist of L-function
s_satake_parameters | axiom | Satake parameters ;; Langlands parameters at unramified primes
s_hecke_multiplicativity | theorem | Hecke multiplicativity relations ;; Hecke recursion
t_amplification_method | technique | Amplification method ;; amplification technique
s_spectral_large_sieve | theorem | Spectral large sieve inequality ;; large sieve for automorphic forms
s_subconvexity_gl2 | theorem | Subconvexity for GL(2) L-functions ;; GL(2) subconvexity
t_spectral_mean_value_estimates | technique | Mean value estimates via spectral methods ;; spectral moment method
s_spectral_reciprocity | theorem | Spectral reciprocity formula ;; reciprocity formula
s_nonvanishing_l_functions_at_center | theorem | Nonvanishing results for central L-values ;; nonvanishing at the central point
s_singular_integral_circle_method | axiom | Singular integral (circle method) ;; singular integral in Waring's problem
s_hua_inequality | theorem | Hua's inequality ;; Hua's lemma
s_hardy_littlewood_asymptotic_waring | theorem | Hardy-Littlewood asymptotic for Waring's problem ;; Waring asymptotic formula
t_farey_dissection | technique | Farey dissection ;; Farey arc decomposition
t_kloosterman_refinement | technique | Kloosterman refinement (HLK method) ;; HLK method
t_delta_method | technique | Delta method (Duke-Friedlander-Iwaniec) ;; DFI delta method
s_huxley_zero_density_estimate | theorem | Huxley's zero density estimate ;; Huxley density estimate
s_jutila_zero_density_estimate | theorem | Jutila's zero density estimate ;; Jutila density estimate
s_log_free_zero_density_estimate | theorem | Log-free zero density estimate ;; log-free density estimate
t_zero_detecting_polynomial | technique | Zero-detecting polynomial ;; zero detector
t_turan_power_sum_method | technique | Turán's power sum method ;; power sum method
t_halasz_montgomery_method | technique | Halász-Montgomery method ;; Halász-Montgomery zero density method
s_gue_hypothesis | axiom | GUE hypothesis for ζ(s) zeros ;; random matrix theory connection ;; Katz-Sarnak philosophy
t_mollifier_method | technique | Mollifier method ;; mollification technique
t_levinson_method | technique | Levinson's method ;; Levinson's theorem on zeros
s_deuring_heilbronn_repulsion | theorem | Deuring-Heilbronn repulsion phenomenon ;; Deuring-Heilbronn phenomenon ;; zero repulsion
s_linnik_constant | axiom | Linnik's constant ;; Linnik constant L
s_fouvry_theorem | theorem | Fouvry's theorem ;; Fouvry-Iwaniec level of distribution
s_duke_equidistribution | theorem | Duke's theorem (equidistribution of CM points) ;; Duke's equidistribution theorem
s_subconvexity_implies_equidistribution | theorem | Subconvexity implies equidistribution ;; subconvexity-equidistribution paradigm
s_erdos_turan_inequality | theorem | Erdős-Turán inequality ;; Erdős-Turán discrepancy inequality
s_discrepancy_of_sequence | axiom | Discrepancy of a sequence ;; discrepancy ;; star discrepancy
s_friedlander_iwaniec_theorem | theorem | Friedlander-Iwaniec theorem ;; primes of the form a²+b⁴
t_half_dimensional_sieve | technique | Iwaniec's half-dimensional sieve ;; half-dimensional sieve
s_shifted_convolution_sums | axiom | Shifted convolution sums ;; shifted convolution problem
s_additive_divisor_problem | axiom | Additive divisor problem ;; binary additive divisor problem
s_hecke_converse_theorem | theorem | Hecke's converse theorem ;; Hecke's inverse Mellin theorem
s_weil_converse_theorem | theorem | Weil's converse theorem ;; Weil's converse theorem for GL(2)
s_holomorphic_modular_form_formal | axiom | Holomorphic modular form (weight k, level N) ;; classical modular form
s_eisenstein_series_nebentypus | axiom | Eisenstein series with Nebentypus ;; Eisenstein series with character
s_ramanujan_conjecture_gl2 | axiom | Ramanujan conjecture for GL(2) ;; Ramanujan-Petersson conjecture
s_extremal_order_definition | axiom | Extremal order of an arithmetic function
s_highly_composite_number | axiom | Highly composite number
s_largest_prime_factor | axiom | Largest prime factor P⁺(n)
s_smallest_prime_factor_definition | axiom | Smallest prime factor P⁻(n)
s_integers_free_of_small_prime_factors | axiom | Rough integers Φ(x, y)
s_natural_density | axiom | Natural density (asymptotic density)
s_logarithmic_density | axiom | Logarithmic density
s_slowly_varying_function | axiom | Slowly varying function
s_generalized_divisor_function_d_z | axiom | Generalized divisor function d_z(n)
s_local_zeta_function_y_smooth | axiom | Local zeta function ζ(s, y)
s_pretentious_distance_function | axiom | Pretentious distance D(f, g; x)
s_hooley_delta_function | axiom | Hooley's Δ-function
s_distribution_function_arithmetic_function | axiom | Distribution function of an arithmetic function
s_limiting_distribution_arithmetic_function | axiom | Limiting distribution of an arithmetic function
s_summatory_liouville_function | state | Summatory Liouville function L(x)
s_dickman_function_differential_delay_equation | state | Dickman function differential-delay equation
s_buchstab_function_differential_delay_equation | state | Buchstab function differential-delay equation
s_saddle_point_alpha_x_y | state | Saddle-point parameter α(x, y)
s_robin_inequality | state | Robin's inequality
s_mertens_first_theorem | theorem | Mertens' first theorem
s_mertens_second_theorem | theorem | Mertens' second theorem
s_mertens_third_theorem | theorem | Mertens' third theorem
s_average_order_omega_n | theorem | Average order of ω(n)
s_average_order_big_omega_n | theorem | Average order of Ω(n)
s_squarefree_counting_asymptotic | theorem | Squarefree counting asymptotic
s_maximal_order_divisor_function | theorem | Maximal order of d(n)
s_gronwall_theorem_sigma | theorem | Gronwall's theorem (maximal order of σ(n)/n)
s_minimal_order_euler_totient_ratio | theorem | Minimal order of φ(n)/n
s_mean_value_zeta_critical_line | theorem | Mean value of |ζ(1/2+it)|²
s_dickman_theorem | theorem | Dickman's theorem
s_karamata_tauberian_theorem | theorem | Karamata's Tauberian theorem
s_delange_tauberian_theorem | theorem | Delange's Tauberian theorem
s_landau_page_theorem | theorem | Landau–Page theorem
s_dual_turan_kubilius_inequality | theorem | Dual Turán–Kubilius inequality
s_normal_order_log_divisor_function | theorem | Normal order of log d(n)
s_delange_mean_value_multiplicative | theorem | Delange's theorem on mean values of multiplicative functions
s_equivalence_pnt_mobius_mean_zero | theorem | Equivalence of PNT and ∑μ(n) = o(x)
s_elliott_multiplicative_in_progressions | theorem | Elliott's theorem on multiplicative functions in progressions
s_hildebrand_dickman_extension | theorem | Hildebrand's extension of Dickman's theorem
s_hildebrand_tenenbaum_saddle_point_estimate | theorem | Hildebrand–Tenenbaum saddle-point estimate for Ψ(x, y)
s_canfield_erdos_pomerance_theorem | theorem | Canfield–Erdős–Pomerance theorem
s_alladi_erdos_largest_prime_factor_average | theorem | Alladi–Erdős result on average of log P⁺(n)
s_smooth_numbers_short_intervals | theorem | Smooth numbers in short intervals
s_asymptotic_rough_integers_phi | theorem | Asymptotic for rough integers Φ(x, y)
s_friable_integers_arithmetic_progressions | theorem | Friable integers in arithmetic progressions
s_distribution_smallest_prime_factor | theorem | Distribution of the smallest prime factor
s_wintner_random_multiplicative_theorem | theorem | Wintner's theorem on random multiplicative functions
s_maier_theorem_short_intervals | theorem | Maier's theorem on short intervals
s_daboussi_theorem | theorem | Daboussi's theorem
s_schnirelman_additive_basis_theorem | theorem | Schnirelman's additive basis theorem
s_tenenbaum_divisor_distribution | theorem | Tenenbaum's theorem on divisor distribution
s_generalized_divisor_d_z_asymptotic | theorem | Asymptotic for ∑d_z(n) (Selberg–Delange)
t_laplace_method_integrals | technique | Laplace method for integrals
t_kubilius_probabilistic_model | technique | Kubilius probabilistic model
t_turan_second_moment_method | technique | Turán's second moment method
t_pretentious_distance_method | technique | Pretentious distance method (Halász)
t_effective_tauberian_method | technique | Effective Tauberian method
t_counting_primes_smooth_rough_decomposition | technique | Smooth–rough decomposition for counting primes
t_granville_soundararajan_pretentious | technique | Granville–Soundararajan pretentious number theory
t_renyi_turan_power_sum_method | technique | Rényi–Turán power sum method
t_twisting_by_dirichlet_characters | technique | Twisting by Dirichlet characters
s_logarithmic_density_primes_in_ap | state | Logarithmic density of primes in arithmetic progressions
s_gaussian_periods | state | Gaussian periods
s_gauss_constructible_regular_polygons | theorem | Gauss's theorem on constructible regular polygons
s_fundamental_discriminant | axiom | Fundamental discriminant
s_binary_quadratic_form | axiom | Binary quadratic form
s_finite_expression_l1_chi | state | Finite expression for L(1, χ)
s_riemann_approximate_formula_for_N_T | state | Riemann's approximate formula for N(T)
t_contour_integration_arithmetic_functions | technique | Contour integration for arithmetic functions
s_truncated_explicit_formula_psi | state | Truncated explicit formula for ψ(x)
s_chebyshev_psi_twisted_by_chi | state | Chebyshev ψ-function twisted by χ
t_character_sum_completion | technique | Character sum completion technique
s_further_prime_number_sums | state | Further prime number sums
s_generalized_xi_function | state | Generalized xi-function ξ(s, χ)
s_root_number_w_chi | state | Root number W(χ)
s_counting_zeros_entire_order_1 | state | Counting zeros of entire functions of order 1
s_hadamard_product_for_xi_chi | theorem | Hadamard product for ξ(s, χ)
s_partial_fraction_l_prime_over_l | state | Partial fraction for −L′/L
t_zero_counting_argument_principle | technique | Zero-counting via argument principle
s_log_zeta_bound_near_sigma_1 | state | log ζ(s) bound near σ = 1
s_repulsion_of_zeros_by_exceptional_zero | theorem | Repulsion of zeros by exceptional zero
s_s_t_argument_function | state | S(T) argument function
s_n_t_chi_zero_counting | state | N(T, χ) zero-counting function
s_asymptotic_n_t_chi | theorem | Asymptotic for N(T, χ)
s_exponential_sum_over_primes | state | Exponential sum over primes
s_vinogradov_estimate_prime_exponential_sums | theorem | Vinogradov's estimate for prime exponential sums
s_singular_series_three_primes | state | Singular series for three primes
t_major_arc_approximation_via_l_functions | technique | Major arc approximation via L-functions
s_large_sieve_as_mean_value_estimate | state | Large sieve as mean-value estimate
t_vaughan_proof_bombieri_vinogradov | technique | Vaughan's proof of Bombieri–Vinogradov
t_reduction_to_primitive_characters | technique | Reduction to primitive characters
s_bombieri_vinogradov_as_grh_substitute | state | Bombieri–Vinogradov as substitute for GRH
s_application_titchmarsh_divisor_problem | state | Application to the Titchmarsh divisor problem
t_de_la_vallee_poussin_proof_strategy | technique | De la Vallée-Poussin's proof strategy
s_zero_free_region_real_characters_near_t_0 | theorem | Zero-free region for real characters near t = 0
s_lower_bound_l1_chi_via_class_number | state | Lower bound for L(1, χ) via class number
s_siegel_lemma_l_function_products | theorem | Siegel's lemma on L-function products
s_ineffectivity_in_siegel_theorem | state | Ineffectivity in Siegel's theorem
s_symmetric_pairing_of_zeros | state | Symmetric pairing of zeros
s_density_of_zeros_near_sigma_1 | state | Density of zeros near σ = 1
s_contribution_of_major_arcs | state | Contribution of major arcs
s_contribution_of_minor_arcs | state | Contribution of minor arcs
s_dirichlet_polynomial | state | Dirichlet polynomial
s_character_table_of_z_qz_star | state | Character table of (ℤ/qℤ)*
s_factorization_characters_via_crt | state | Factorization of characters via CRT
s_dirichlet_l1_evaluation_quadratic_chi | theorem | Dirichlet's L(1, χ) evaluation for quadratic χ
s_estimate_sum_chi_p_over_p | state | Estimate for ∑ χ(p)/p
s_mertens_type_estimate_in_ap | state | Mertens-type estimate in arithmetic progressions
s_cyclotomic_polynomial_irreducibility | theorem | Cyclotomic polynomial irreducibility over ℚ
s_algebraic_integer_neukirch | axiom | Algebraic integer ;; integral element over ℤ
s_norm_of_ideal | axiom | Norm of an ideal N(𝔞)
s_minkowski_embedding | axiom | Minkowski embedding (canonical embedding)
s_finiteness_of_class_number | theorem | Finiteness of the class number
s_transitivity_of_different_discriminant | theorem | Transitivity of different and discriminant
s_complementary_module | axiom | Complementary module (codifferent)
s_different_discriminant_relation | theorem | Different–discriminant relation
s_exponent_of_different_at_prime | theorem | Exponent of the different at a prime
s_tame_and_wild_ramification | axiom | Tame and wild ramification
s_discriminant_of_cyclotomic_field | theorem | Discriminant of cyclotomic fields
s_crt_dedekind_domains | theorem | Chinese Remainder Theorem for Dedekind domains
s_finiteness_of_ramification | theorem | Finiteness of ramification
s_discrete_valuation | axiom | Discrete valuation
s_newton_polygon | axiom | Newton polygon
s_newton_polygon_theorem | theorem | Newton polygon theorem
s_krasners_lemma | theorem | Krasner's lemma
s_classification_of_local_fields | theorem | Classification of local fields
s_unramified_extension_local | axiom | Unramified extension of a local field
s_totally_ramified_extension_local | axiom | Totally ramified extension of a local field
s_eisenstein_polynomial | axiom | Eisenstein polynomial
s_structure_of_unramified_extensions | theorem | Structure of unramified extensions
s_unramified_totally_ramified_decomposition | theorem | Decomposition into unramified and totally ramified parts
s_teichmuller_representatives | axiom | Teichmüller representatives
s_product_formula_valuations | theorem | Product formula for valuations
s_replete_ideal | axiom | Replete ideal (Arakelov ideal)
s_replete_divisor | axiom | Replete divisor
s_arakelov_class_group | state | Arakelov class group (replete divisor class group)
s_metrized_line_bundle | axiom | Metrized line bundle on Spec(O_K)
s_degree_map_replete | state | Degree map on replete divisors
s_euler_minkowski_characteristic | state | Euler–Minkowski characteristic
s_riemann_roch_number_fields | theorem | Riemann–Roch theorem for number fields
s_ideal_lattice | state | Ideal lattice
s_tate_cohomology_groups | axiom | Tate cohomology groups Ĥⁿ(G,A)
s_herbrand_quotient | state | Herbrand quotient h(G,A)
s_class_formation | axiom | Class formation
s_formation_module | axiom | Formation module
s_invariant_map | state | Invariant map inv_{L/K}
s_fundamental_class | state | Fundamental class u_{L/K}
s_nakayama_tate_duality | theorem | Nakayama–Tate duality theorem
s_abstract_reciprocity_map | theorem | Abstract reciprocity map
s_abstract_existence_theorem | theorem | Abstract existence theorem
s_inflation_map | state | Inflation map in group cohomology
s_restriction_map_cohomology | state | Restriction map in group cohomology
s_corestriction_map | state | Corestriction (transfer) map in group cohomology
s_inflation_restriction_exact_sequence | theorem | Inflation–restriction exact sequence
s_shapiros_lemma | theorem | Shapiro's lemma
s_cohomologically_trivial_module | axiom | Cohomologically trivial G-module
s_tate_theorem_cohom_trivial | theorem | Tate's theorem on cohomologically trivial modules
s_norm_map_g_modules | state | Norm map N_G for G-modules
s_augmentation_ideal | axiom | Augmentation ideal I_G
s_two_periodicity_tate | theorem | 2-periodicity of Tate cohomology for cyclic groups
s_profinite_completion | state | Profinite completion
s_g_module_discrete | axiom | G-module (discrete)
s_cohomological_dimension | axiom | Cohomological dimension
s_brauer_group_as_h2 | theorem | Brauer group as H²
s_long_exact_cohomology_sequence | theorem | Long exact cohomology sequence (Tate)
s_hochschild_serre_exact_sequence | theorem | Hochschild–Serre five-term exact sequence
s_hilbert_90_abstract | theorem | Hilbert 90 (abstract version)
s_higher_unit_groups | axiom | Higher unit groups U^{(n)}
s_formal_group_law | axiom | Formal group law
s_lubin_tate_polynomial | axiom | Lubin–Tate polynomial
s_lubin_tate_extension | state | Lubin–Tate extension
s_maximal_abelian_extension_local | state | Maximal abelian extension of a local field
s_local_reciprocity_law | theorem | Local reciprocity law
s_local_existence_theorem | theorem | Local existence theorem
s_hilbert_symbol | axiom | Hilbert symbol
s_ramification_upper_numbering | axiom | Ramification filtration (upper numbering)
s_herbrand_function | state | Herbrand function
s_maximal_unramified_extension | state | Maximal unramified extension
s_local_norm_residue_symbol | state | Local norm residue symbol (local Artin map)
s_local_kronecker_weber | theorem | Local Kronecker–Weber theorem
s_strong_approximation_theorem | theorem | Strong approximation theorem
s_global_artin_map | state | Global Artin map
s_artin_reciprocity_global | theorem | Artin reciprocity law (global)
s_global_existence_theorem | theorem | Global existence theorem
s_ray_class_field | state | Ray class field
s_modulus_cycle | axiom | Modulus (cycle)
s_conductor_of_extension | state | Conductor of an abelian extension
s_principal_ideal_theorem_cft | theorem | Principal ideal theorem (Hauptidealsatz, class field theory)
s_hasse_norm_theorem | theorem | Hasse norm theorem
s_grunwald_wang_theorem | theorem | Grunwald–Wang theorem
s_class_field_tower | state | Class field tower
s_norm_limitation_theorem | theorem | Norm limitation theorem
s_artin_symbol | state | Artin symbol (Frobenius at unramified primes)
s_idelic_norm_map | state | Idelic norm map
s_conductor_discriminant_formula | theorem | Conductor–discriminant formula
s_decomposition_law_abelian | theorem | Decomposition law in abelian extensions
s_content_idele_norm | state | Content (idèle norm)
s_analytic_continuation_dedekind_zeta | theorem | Analytic continuation of the Dedekind zeta function
s_functional_equation_dedekind_zeta | theorem | Functional equation for the Dedekind zeta function
s_hecke_character | axiom | Hecke character (Größencharakter)
s_artin_conductor | state | Artin conductor
s_artin_conjecture | state | Artin conjecture
s_meromorphic_continuation_artin | theorem | Meromorphic continuation of Artin L-functions
s_l_function_nonvanishing_s1 | theorem | L-function non-vanishing at s = 1
s_schwartz_bruhat_function | axiom | Schwartz–Bruhat function
s_local_zeta_integral | state | Local zeta integral (Tate)
s_local_epsilon_factor | state | Local epsilon factor
s_adelic_poisson_summation | theorem | Adelic Poisson summation
s_self_duality_adeles | state | Self-duality of 𝔸_K/K
s_abelian_l_equals_hecke | theorem | Abelian Artin L-function equals Hecke L-function
s_inductivity_artin_l | theorem | Inductivity of Artin L-functions
s_factorization_dedekind_zeta | theorem | Factorization of the Dedekind zeta function
s_functional_equation_hecke_l | theorem | Functional equation for Hecke L-functions
s_functional_equation_artin_l | theorem | Functional equation for Artin L-functions
s_analytic_continuation_hecke_l | theorem | Analytic continuation of Hecke L-functions
s_haar_measure_ideles | axiom | Haar measure on idèles
s_inertia_invariant_subspace | state | Inertia-invariant subspace V^{I_𝔭}
s_dirichlet_theorem_primes_number_fields | theorem | Dirichlet's theorem on primes in number fields
t_logarithmic_embedding | technique | Logarithmic embedding (Dirichlet map)
t_minkowski_lattice_point_argument | technique | Minkowski's lattice point argument
t_ideal_norm_reduction | technique | Ideal norm reduction
t_hensel_lifting | technique | Hensel lifting
t_dimension_shifting_cohomology | technique | Dimension shifting in group cohomology
t_herbrand_quotient_computation | technique | Herbrand quotient computation
t_cup_product_technique | technique | Cup product pairing technique
t_reduction_to_sylow | technique | Reduction to Sylow subgroups in cohomology
t_lubin_tate_construction | technique | Lubin–Tate formal group construction
t_brauer_induction | technique | Brauer induction
t_tate_thesis_adelic | technique | Tate's thesis (adelic zeta integrals)
t_transfer_homomorphism | technique | Transfer (Verlagerung) homomorphism
t_poisson_summation_number_field | technique | Poisson summation in number field setting
s_pro_p_group | axiom | Pro-p group ;; pro-p-group ;; p-profinite group
s_cm_field | axiom | Complex multiplication (CM) field ;; CM field
s_linear_algebraic_group | axiom | Linear algebraic group ;; algebraic group
s_semi_simple_algebraic_group | axiom | Semi-simple algebraic group ;; semisimple algebraic group
s_simply_connected_algebraic_group | axiom | Simply connected algebraic group ;; simply connected semisimple group
s_restricted_topological_product | axiom | Restricted topological product (restricted direct product) ;; restricted direct product ;; restricted product
s_group_homology | state | Group homology H_n(G, M) ;; homology of groups
s_absolute_galois_group | state | Absolute Galois group Gal(K^sep/K) ;; absolute Galois group
s_supernatural_number | state | Supernatural number ;; Steinitz number
s_free_profinite_group | state | Free profinite group
s_continuous_cohomology_profinite | state | Continuous cohomology of profinite groups ;; profinite group cohomology ;; Galois cohomology
s_cm_type | state | CM type
s_abelian_variety_with_cm | state | Abelian variety with complex multiplication ;; CM abelian variety
s_reflex_field | state | Reflex field (reflex CM type) ;; reflex CM type ;; dual CM type
s_algebraic_hecke_character | state | Algebraic Hecke character (Grössencharakter of type A₀) ;; algebraic Grössencharakter ;; type A₀ Hecke character
s_l_extension | state | ℓ-extension ;; l-extension ;; p-extension
s_z_l_extension | state | ℤ_ℓ-extension ;; Z_l-extension ;; Z_p-extension
s_global_zeta_integral_tate | state | Global zeta integral (Tate) ;; Tate's global zeta function
s_quasi_character | state | Quasi-character of K_v* ;; quasi-character
s_self_dual_measure | state | Self-dual Haar measure ;; self-dual measure
s_kroneckers_jugendtraum | state | Kronecker's Jugendtraum ;; Kronecker's youth dream ;; Hilbert's 12th problem (partial)
s_kneser_tits_conjecture | state | Kneser–Tits conjecture ;; Kneser–Tits problem
s_golod_shafarevich_theorem | theorem | Golod–Shafarevich theorem ;; Golod–Šafarevič theorem ;; Golod–Shafarevich inequality
s_existence_infinite_class_field_towers | theorem | Existence of infinite class field towers
s_hasse_principle_algebraic_groups | theorem | Hasse principle for algebraic groups (Kneser–Harder) ;; Kneser–Harder theorem
s_strong_approximation_algebraic_groups | theorem | Strong approximation for algebraic groups
s_main_theorem_complex_multiplication | theorem | Main theorem of complex multiplication ;; Shimura–Taniyama reciprocity for CM
s_serre_tate_good_reduction_cm | theorem | Serre–Tate theorem (good reduction of CM abelian varieties)
s_iwasawa_class_number_theorem | theorem | Iwasawa's theorem on class numbers in ℤ_p-extensions ;; Iwasawa class number formula
s_local_functional_equation_tate | theorem | Local functional equation (Tate)
s_global_functional_equation_tate | theorem | Global functional equation (Tate's thesis)
s_l_function_euler_product_local | theorem | Decomposition of L-functions into local zeta integrals ;; Euler product from local factors
s_global_local_compatibility_reciprocity | theorem | Global–local compatibility of reciprocity maps ;; compatibility of local and global Artin maps
s_hilbert_different_formula | theorem | Hilbert's different formula ;; Hilbert's formula for the different
s_first_inequality_cft | theorem | First inequality (global class field theory) ;; first inequality
s_second_inequality_cft | theorem | Second inequality (global class field theory) ;; second inequality
s_krull_theorem_infinite_galois | theorem | Krull's theorem (infinite Galois theory) ;; fundamental theorem of infinite Galois theory
s_herbrand_theorem_upper_numbering | theorem | Herbrand's theorem (upper numbering of ramification groups) ;; Herbrand's upper numbering theorem
t_bar_resolution | technique | Standard complex (bar resolution)
t_cohomology_of_ideles | technique | Cohomology of idèles (local-to-global reduction)
t_lyndon_hochschild_serre_spectral_sequence | technique | Lyndon–Hochschild–Serre spectral sequence
s_projective_modules_over_dedekind_domains | theorem | Projective modules over Dedekind domains
s_non_archimedean_absolute_value | axiom | Non-archimedean absolute value
s_trace_form | axiom | Trace form
s_different_ramification_theorem | theorem | Different–ramification theorem
s_discriminant_ramification_theorem | theorem | Discriminant–ramification theorem
s_ring_of_integers_of_cyclotomic_field | theorem | Ring of integers of cyclotomic fields
s_ramification_in_cyclotomic_fields | theorem | Ramification in cyclotomic fields
s_gauss_sum_norm_formula | theorem | Gauss sum norm formula
s_stickelberger_element | state | Stickelberger element
t_volume_computation_fundamental_domain | technique | Volume computation of fundamental domain
s_counting_ideals_of_bounded_norm | theorem | Counting ideals of bounded norm
t_lattice_points_in_expanding_domains | technique | Lattice points in homogeneously expanding domains
s_content_map | state | Content (idèle norm) map
s_compactness_of_c_k_1 | theorem | Compactness of C_K^1
s_idele_class_group_ray_class_group_relationship | theorem | Relationship between idèle class group and ray class groups
s_galois_action_on_ideles | state | Galois action on idèles
s_convergence_and_euler_product_for_dedekind_zeta | theorem | Convergence and Euler product for ζ_K
s_faltings_finiteness_theorem | theorem | Faltings’ finiteness theorem (Shafarevich conjecture)
s_local_norm_group | state | Local norm group
s_local_norm_index_theorem | theorem | Local norm index theorem
t_p_adic_exponential_and_logarithm | technique | p-adic exponential and logarithm
s_structure_of_local_units | theorem | Structure of local units
s_global_cyclic_norm_index | theorem | Global cyclic norm index inequality
s_takagi_existence_theorem | theorem | Takagi’s existence theorem
s_conductor_class_field_relationship | theorem | Conductor–class field relationship
s_existence_theorem_of_cft | theorem | Existence theorem of class field theory
t_reduction_to_kummer_extensions | technique | Reduction to Kummer extensions
s_complete_splitting_theorem | theorem | Complete splitting theorem
s_ramification_theorem_cft | theorem | Ramification theorem (class field theory)
s_infinite_divisibility_of_universal_norms | theorem | Infinite divisibility of universal norms
s_induced_character_l_series_factorization | theorem | Induced character and L-series factorization
s_local_additive_duality | theorem | Local additive duality (self-duality of K_v)
t_dirichlet_integral_technique | technique | Dirichlet integral
s_tauberian_theorem_for_dirichlet_series | theorem | Tauberian theorem for Dirichlet series
s_nonvanishing_l1_hecke_characters | theorem | Non-vanishing of L(1, χ) for Hecke characters
s_dirichlet_as_special_case_of_chebotarev | theorem | Dirichlet’s theorem as special case of Chebotarev
s_upper_estimate_residue_zeta_k | theorem | Upper estimate for residue of ζ_K at s = 1
s_siegel_lower_bound_residue | theorem | Siegel’s lower bound for the residue
t_comparison_of_residues_in_normal_extensions | technique | Comparison of residues in normal extensions
s_weierstrass_product_for_completed_l_function | state | Weierstrass product for completed L-function
t_logarithmic_derivative_xi | technique | Logarithmic derivative ξ’/ξ
s_estimate_for_xi_prime_over_xi | theorem | Estimate for ξ’/ξ
s_weil_positivity_criterion | theorem | Weil’s positivity criterion
t_evaluation_of_basic_sum_part_1 | technique | Evaluation of basic sum (Part 1)
t_evaluation_of_basic_sum_part_2 | technique | Evaluation of basic sum (Part 2)
s_characteristic_of_a_field | axiom | Characteristic of a field
s_cyclicity_of_fq_star | theorem | Cyclicity of F_q*
t_character_sum_counting_solutions | technique | Character sum method for counting solutions
s_p_adic_units | axiom | Units of Z_p
s_compactness_of_z_p | theorem | Compactness of Z_p
s_structure_of_qp_star | theorem | Structure of Q_p*
s_squares_in_qp_star | theorem | Squares in Q_p*
s_density_of_q_in_qp | theorem | Density of Q in Q_p
s_hilbert_symbol_at_infinity | axiom | Hilbert symbol at the archimedean place
s_hilbert_symbol_properties | theorem | Bilinearity and non-degeneracy of the Hilbert symbol
s_hilbert_symbol_computation_odd_p | theorem | Hilbert symbol computation for odd p
s_existence_theorem_prescribed_hilbert_symbols | theorem | Existence theorem for prescribed Hilbert symbols
s_discriminant_of_quadratic_form | axiom | Discriminant of a quadratic form
s_hasse_witt_invariant | axiom | Hasse–Witt invariant
s_isotropic_quadratic_form | axiom | Isotropic quadratic form
s_witt_decomposition | theorem | Witt decomposition
s_classification_quadratic_forms_qp | theorem | Classification of quadratic forms over Q_p
s_representation_theorem_qp | theorem | Representation theorem for quadratic forms over Q_p
s_classification_quadratic_forms_q | theorem | Classification of quadratic forms over Q
t_hasse_invariant_computation | technique | Hasse invariant computation
s_unimodular_lattice | axiom | Unimodular lattice
s_even_type_ii_lattice | axiom | Even (type II) lattice
s_odd_type_i_lattice | axiom | Odd (type I) lattice
s_signature_of_real_quadratic_form | axiom | Signature of a real quadratic form
s_e8_lattice | axiom | E_8 lattice
s_classification_even_unimodular_lattices | theorem | Classification of even unimodular lattices
s_classification_indefinite_unimodular_lattices | theorem | Classification of indefinite unimodular lattices
s_genus_of_integral_quadratic_form | axiom | Genus of an integral quadratic form
s_genus_determines_rational_equivalence | theorem | Genus determines rational equivalence
t_local_to_global_lattices | technique | Local-to-global passage for lattices
s_character_of_finite_abelian_group | axiom | Character of a finite abelian group
s_modular_function | axiom | Modular function
s_q_expansion_modular_form | axiom | Fourier expansion (q-expansion) of a modular form
s_valence_formula_modular_forms | theorem | Valence formula for modular forms
s_decomposition_mk_eisenstein_cusp | theorem | Decomposition M_k = C·E_k ⊕ S_k
s_algebra_of_modular_forms | theorem | Structure of the algebra of modular forms
s_delta_eisenstein_relation | theorem | Relation Δ = (E_4³ − E_6²)/1728
s_hecke_operators_self_adjoint | theorem | Hecke operators are self-adjoint
s_ramanujan_tau_multiplicativity | theorem | Multiplicativity of the Ramanujan tau function
s_theta_function_is_modular_form | theorem | Theta function of a quadratic form is a modular form
s_representation_numbers_via_modular_forms | axiom | Representation numbers via modular forms
s_eight_squares_formula | theorem | Eight squares formula
s_modular_form_fourier_coefficient_bound | theorem | Growth bound for Fourier coefficients of modular forms
t_contour_integration_valence | technique | Contour integration on the fundamental domain
t_poisson_summation_for_theta | technique | Poisson summation for theta functions
t_hecke_eigenform_method | technique | Hecke eigenform method
s_farey_sequence | axiom | Farey sequence F_n ;; Farey series
s_mediant | axiom | Mediant of two fractions ;; Farey mediant ;; freshman sum
s_normal_number | axiom | Normal number
s_simply_normal_number | axiom | Simply normal number
s_absolutely_normal_number | axiom | Absolutely normal number
s_badly_approximable_number | axiom | Badly approximable number ;; bounded partial quotients
s_durfee_square | axiom | Durfee square
s_stern_brocot_tree | axiom | Stern–Brocot tree ;; Stern-Brocot tree ;; Calkin-Wilf tree
s_visible_lattice_point_density | state | Density of visible lattice points ;; coprimality probability
s_fibonacci_golden_ratio_convergents | state | Fibonacci numbers as convergents of the golden ratio
s_failure_of_ufd_in_z_sqrt_neg5 | state | Failure of unique factorization in ℤ[√(−5)]
s_hurwitz_sharpness_sqrt5 | state | Sharpness of √5 in Hurwitz's theorem
s_waring_g_k_lower_bound | state | Lower bound for g(k) in Waring's problem
s_waring_G_k_lower_bounds | state | Lower bounds for G(k) in Waring's problem
s_farey_neighbor_theorem | theorem | Farey neighbor property
s_farey_mediant_theorem | theorem | Farey mediant property
s_borel_normal_number_theorem | theorem | Borel's normal number theorem ;; Borel's theorem on normal numbers
s_kempner_series_convergence | theorem | Kempner series convergence ;; Kempner's series
s_lagrange_periodic_cf_theorem | theorem | Lagrange's periodic continued fraction theorem ;; Lagrange's theorem on continued fractions
s_galois_purely_periodic_cf_theorem | theorem | Galois's theorem on purely periodic continued fractions
s_leudesdorf_theorem | theorem | Leudesdorf's theorem
s_bauer_identical_congruence | theorem | Bauer's identical congruence
s_kronecker_approximation_theorem | theorem | Kronecker's approximation theorem ;; Kronecker's theorem
s_kronecker_simultaneous_approximation | theorem | Kronecker's simultaneous approximation theorem ;; Kronecker's theorem in k dimensions
s_rogers_ramanujan_continued_fraction | theorem | Rogers–Ramanujan continued fraction ;; Ramanujan's continued fraction
s_euclidean_quadratic_fields_classification | theorem | Classification of Euclidean quadratic fields
s_minkowski_theorem_linear_forms | theorem | Minkowski's theorem on linear forms
s_tchebotaref_inhomogeneous_form_theorem | theorem | Tchebotaref's theorem on inhomogeneous forms ;; Chebotarev's theorem on forms
t_quaternion_norm_descent | technique | Quaternion norm descent
t_durfee_square_decomposition | technique | Durfee square decomposition
t_franklin_involution | technique | Franklin's involution
t_lettenmeyer_kronecker_proof | technique | Lettenmeyer's proof of Kronecker's theorem ;; Estermann's proof
t_elliptic_curve_chord_tangent_addition | technique | Chord-and-tangent addition on elliptic curves ;; elliptic curve group law
s_polygonal_number | axiom | Polygonal number p_m(k)
s_cauchy_polygonal_number_theorem | theorem | Cauchy’s polygonal number theorem
s_cauchy_lemma_polygonal | theorem | Cauchy’s lemma (polygonal numbers)
s_pepin_five_pentagonal_theorem | theorem | Pépin’s theorem on five pentagonal numbers
s_waring_g_function | axiom | Waring’s function g(k)
s_waring_G_function | axiom | Asymptotic Waring function G(k)
s_wieferich_kempner_theorem | theorem | Wieferich–Kempner theorem (g(3) = 9)
s_linnik_seven_cubes | theorem | Linnik’s theorem on sums of seven cubes (G(3) ≤ 7)
s_sums_two_cubes_impossibility | state | Impossibility of representing all integers as sums of few cubes
s_ideal_waring_conjecture | state | Ideal Waring conjecture
t_hilbert_multilinear_identity_proof | technique | Hilbert’s proof via multilinear identities
t_schur_trick | technique | Schur’s trick
s_stridsberg_improvement | theorem | Stridsberg’s improvement of Hilbert’s bound
t_dress_simplification | technique | Dress’s simplification of Hilbert’s proof
s_hurwitz_identity_sums_of_squares | state | Hurwitz identity for sums of squares
s_weyl_inequality | theorem | Weyl’s inequality
t_hua_iterative_squaring | technique | Hua’s lemma proof by iterative squaring
s_singular_series_waring | state | Singular series S(n) for Waring’s problem
s_singular_integral_waring | axiom | Singular integral J(n) for Waring’s problem
s_positivity_singular_series | theorem | Positivity of the singular series for Waring’s problem
s_complete_exponential_sum_mod_q | axiom | Complete exponential sum mod q
t_pruning | technique | Pruning (circle method)
s_local_solubility_condition | state | Local solubility condition for Waring’s problem
s_three_prime_asymptotic_formula | theorem | Asymptotic formula for three-prime representations
t_major_arc_approximation_prime_sums | technique | Major arc approximation for prime exponential sums
s_goldbach_exceptional_set | state | Goldbach exceptional set
s_vinogradov_bound_G_k | theorem | Vinogradov’s bound on G(k)
s_easy_waring_problem | axiom | Easy Waring problem
s_segre_mahler_wright_theorem | theorem | Segre–Mahler–Wright theorem
t_efficient_differencing | technique | Efficient differencing (Vinogradov–Wooley)
s_davenport_G4_equals_15 | theorem | Davenport’s theorem G(4) = 15
s_G2_equals_4 | theorem | G(2) = 4
s_wooley_refinement | theorem | Wooley’s refinement of Vinogradov’s mean value theorem
s_asymptotic_basis | axiom | Asymptotic basis of order h
s_schnirelmann_theorem_primes | theorem | Schnirelmann’s theorem (primes form an additive basis)
s_schnirelmann_inequality | theorem | Schnirelmann’s inequality
s_mann_theorem | theorem | Mann’s theorem
s_khinchin_addition_theorem | theorem | Khinchin’s addition theorem
s_essential_component | axiom | Essential component
s_primes_essential_component | theorem | Primes are an essential component
s_khinchin_essential_components_theorem | theorem | Khinchin’s theorem on essential components
s_erdos_essential_components_theorem | theorem | Erdős’s theorem on essential components
t_mann_sequential_addition | technique | Mann’s proof via sequential addition
s_selberg_sieve_goldbach | state | Selberg sieve applied to Goldbach
s_romanov_theorem | theorem | Romanov’s theorem
s_dickson_pillai_result | theorem | Dickson–Pillai theorem on g(k)
s_balasubramanian_deshouillers_dress_theorem | theorem | Balasubramanian–Deshouillers–Dress theorem (g(4) = 19)
s_multiplicativity_singular_series | state | Euler product for the singular series
s_hua_theorem_cubes_of_primes | theorem | Hua’s theorem on sums of cubes of primes
s_waring_problem_for_primes | state | Waring’s problem for primes (Waring–Goldbach problem)
t_davenport_heilbronn_method | technique | Davenport–Heilbronn method
s_representation_function_polygonal | axiom | Representation function for polygonal numbers
s_representation_function_waring | axiom | Representation function r_{s,k}(n)
s_liouville_identity_four_squares | state | Liouville’s identity (sums of squares)
s_quadratic_forms_and_squares | state | Connection between quadratic forms and sums of squares
s_h_fold_sumset | axiom | h-fold sumset hA
s_h_fold_representation_function | axiom | h-fold representation function r_{A,h}(n)
s_counting_function_A_n | axiom | Counting function A(n)
t_schnirelmann_original_density_argument | technique | Schnirelmann’s original density argument
s_generating_function_waring | axiom | Generating function for Waring’s problem
s_p_adic_solubility | state | p-adic solubility for Waring’s problem
s_minor_arc_estimate_waring | state | Minor arc estimate for Waring’s problem
s_minor_arc_estimate_three_primes | state | Minor arc estimate for the three-primes problem
s_major_arc_contribution_waring | state | Major arc contribution for Waring’s problem
s_four_cubes_impossibility | state | Four cubes impossibility modulo 9
s_23_and_239_require_9_cubes | state | 23 and 239 require nine cubes
t_descent_four_squares | technique | Descent method for four squares
t_linnik_elementary_method | technique | Linnik’s elementary method for sums of cubes
s_subgroup | axiom | Subgroup
s_normal_subgroup | axiom | Normal subgroup
s_quotient_group | axiom | Quotient group
s_index_of_a_subgroup | axiom | Index of a subgroup
s_center_of_a_group | axiom | Center of a group
s_centralizer | axiom | Centralizer
s_normalizer | axiom | Normalizer
s_commutator_subgroup | axiom | Commutator subgroup
s_pgroup | axiom | p-group
s_sylow_psubgroup | axiom | Sylow p-subgroup
s_group_presentation | axiom | Group presentation
s_orbit_of_a_group_action | axiom | Orbit of a group action
s_stabilizer_of_a_group_action | axiom | Stabilizer of a group action
s_conjugacy_class | axiom | Conjugacy class
s_exponent_of_a_group | axiom | Exponent of a group
s_lower_central_series | state | Lower central series
s_upper_central_series | state | Upper central series
s_fitting_subgroup | state | Fitting subgroup
s_socle_of_a_group | state | Socle of a group
s_automorphism_group_of_a_group | state | Automorphism group of a group
s_inner_automorphism_group | state | Inner automorphism group
s_outer_automorphism_group | state | Outer automorphism group
s_lagranges_theorem_group_theory | theorem | Lagrange's theorem (group theory)
s_second_isomorphism_theorem_for_groups | theorem | Second isomorphism theorem for groups
s_third_isomorphism_theorem_for_groups | theorem | Third isomorphism theorem for groups
s_lattice_isomorphism_theorem_for_groups | theorem | Lattice isomorphism theorem for groups
s_sylows_first_theorem | theorem | Sylow's first theorem
s_sylows_second_theorem | theorem | Sylow's second theorem
s_sylows_third_theorem | theorem | Sylow's third theorem
s_fittings_theorem_on_fitting_subgroup | theorem | Fitting's theorem on Fitting subgroup
t_sylow_theory_argument | technique | Sylow theory argument
t_counting_via_group_actions | technique | Counting via group actions
t_composition_series_analysis | technique | Composition series analysis
s_ideal_of_a_ring | axiom | Ideal (of a ring)
s_quotient_ring | axiom | Quotient ring
s_nilpotent_element | axiom | Nilpotent element
s_nilradical_of_a_ring | state | Nilradical of a ring
s_jacobson_radical_of_a_ring | state | Jacobson radical of a ring
s_fraction_field | state | Fraction field
s_group_of_units | state | Group of units
s_first_isomorphism_theorem_for_rings | theorem | First isomorphism theorem for rings
s_every_maximal_ideal_is_prime | theorem | Every maximal ideal is prime
s_every_pid_is_a_ufd | theorem | Every PID is a UFD
s_euclidean_domain_is_a_pid | theorem | Euclidean domain is a PID
s_zorns_lemma_implies_maximal_ideals_exist | theorem | Zorn's lemma implies maximal ideals exist
s_nilradical_equals_intersection_of_primes | theorem | Nilradical equals intersection of primes
s_gausss_lemma_content_of_polynomials | theorem | Gauss's lemma (content of polynomials)
s_eisensteins_irreducibility_criterion | theorem | Eisenstein's irreducibility criterion
s_polynomial_ring_over_ufd_is_ufd | theorem | Polynomial ring over UFD is UFD
s_module_over_a_ring | axiom | Module over a ring
s_submodule | axiom | Submodule
s_quotient_module | axiom | Quotient module
s_projective_module | axiom | Projective module
s_exact_sequence | axiom | Exact sequence
s_short_exact_sequence | axiom | Short exact sequence
s_length_of_a_module | state | Length of a module
s_bimodule | axiom | Bimodule
s_cokernel_of_a_module_homomorphism | state | Cokernel of a module homomorphism
s_structure_theorem_for_modules_over_a_pid | theorem | Structure theorem for modules over a PID
s_characterization_of_projective_modules | theorem | Characterization of projective modules
s_baers_criterion_for_injectivity | theorem | Baer's criterion for injectivity
s_every_module_embeds_in_an_injective_module | theorem | Every module embeds in an injective module
s_schanuels_lemma | theorem | Schanuel's lemma
s_eilenberg_swindle | theorem | Eilenberg swindle
s_jordanholder_theorem_for_modules | theorem | Jordan-Hölder theorem for modules
t_localization_of_modules | technique | Localization of modules
t_diagram_chasing | technique | Diagram chasing
t_nakayamas_lemma_argument | technique | Nakayama's lemma argument
s_content_of_a_polynomial | axiom | Content of a polynomial
s_primitive_polynomial | axiom | Primitive polynomial
s_symmetric_polynomial | axiom | Symmetric polynomial
s_power_sum_symmetric_polynomials | state | Power sum symmetric polynomials
s_resultant_of_two_polynomials | state | Resultant of two polynomials
s_discriminant_of_a_polynomial | state | Discriminant of a polynomial
s_newtons_identities | theorem | Newton's identities
s_minimal_polynomial | axiom | Minimal polynomial
s_algebraic_extension | axiom | Algebraic extension
s_separable_extension | axiom | Separable extension
s_purely_inseparable_extension | axiom | Purely inseparable extension
s_separable_degree | state | Separable degree
s_inseparable_degree | state | Inseparable degree
s_finite_field | axiom | Finite field
s_existence_and_uniqueness_of_finite_fields | theorem | Existence and uniqueness of finite fields
s_frobenius_generates_galois_group_of_finite_field | theorem | Frobenius generates Galois group of finite field
s_galois_extension | axiom | Galois extension
s_fixed_field | axiom | Fixed field
s_radical_extension | axiom | Radical extension
s_krull_topology | axiom | Krull topology
s_artins_lemma_on_fixed_fields | theorem | Artin's lemma on fixed fields
s_dedekinds_independence_of_characters | theorem | Dedekind's independence of characters
s_hilberts_theorem_90_additive_form | theorem | Hilbert's Theorem 90 (additive form)
s_kummer_theory_classification | theorem | Kummer theory classification
s_solvability_by_radicals_criterion | theorem | Solvability by radicals criterion
s_infinite_galois_correspondence | theorem | Infinite Galois correspondence
t_galois_correspondence_method | technique | Galois correspondence method
t_kummer_theory_technique | technique | Kummer theory technique
t_radical_extension_analysis | technique | Radical extension analysis
s_integral_element_over_a_ring | axiom | Integral element over a ring
s_integrally_closed_domain | axiom | Integrally closed domain
s_noetherian_ring | axiom | Noetherian ring
s_jacobson_ring | axiom | Jacobson ring
s_integrality_is_transitive | theorem | Integrality is transitive
s_incomparability_theorem | theorem | Incomparability theorem
s_hilberts_nullstellensatz_weak_form | theorem | Hilbert's Nullstellensatz (weak form)
s_hilberts_nullstellensatz_strong_form | theorem | Hilbert's Nullstellensatz (strong form)
s_fractional_ideals_of_dedekind_domains_are_invertible | theorem | Fractional ideals of Dedekind domains are invertible
s_krulls_intersection_theorem | theorem | Krull's intersection theorem
t_goingupgoingdown_method | technique | Going-up/going-down method
t_noether_normalization_technique | technique | Noether normalization technique
s_algebraic_independence | axiom | Algebraic independence
s_purely_transcendental_extension | axiom | Purely transcendental extension
s_linearly_disjoint_extensions | axiom | Linearly disjoint extensions
s_separably_generated_extension | axiom | Separably generated extension
s_module_of_kahler_differentials | axiom | Module of Kähler differentials
s_derivation_on_an_algebra | axiom | Derivation on an algebra
s_transcendence_degree_is_welldefined | theorem | Transcendence degree is well-defined
s_luroths_theorem | theorem | Lüroth's theorem
s_mac_lanes_separability_criterion | theorem | Mac Lane's separability criterion
s_first_exact_sequence_of_kahler_differentials | theorem | First exact sequence of Kähler differentials
s_second_exact_sequence_of_kahler_differentials | theorem | Second exact sequence of Kähler differentials
s_affine_algebraic_set | axiom | Affine algebraic set
s_coordinate_ring_of_an_algebraic_set | state | Coordinate ring of an algebraic set
s_irreducible_algebraic_set | axiom | Irreducible algebraic set
s_function_field_of_a_variety | state | Function field of a variety
s_nullstellensatz_correspondence | theorem | Nullstellensatz correspondence
s_dimension_equals_transcendence_degree | theorem | Dimension equals transcendence degree
s_reduced_trace | state | Reduced trace
s_reduced_norm | state | Reduced norm
s_index_of_a_central_simple_algebra | state | Index of a central simple algebra
s_tensor_product_of_central_simple_algebras | theorem | Tensor product of central simple algebras
s_brauer_group_is_torsion | theorem | Brauer group is torsion
s_representation_of_a_group | axiom | Representation of a group
s_mackey_decomposition_formula | theorem | Mackey decomposition formula
s_mackey_irreducibility_criterion | theorem | Mackey irreducibility criterion
t_character_table_computation | technique | Character table computation
t_frobenius_reciprocity_method | technique | Frobenius reciprocity method
s_general_linear_group | axiom | General linear group
s_special_linear_group | axiom | Special linear group
s_orthogonal_group | axiom | Orthogonal group
s_unitary_group | axiom | Unitary group
s_parabolic_subgroup | axiom | Parabolic subgroup
s_elementary_matrix | axiom | Elementary matrix
s_sl_n_generated_by_elementary_matrices | theorem | SL_n generated by elementary matrices
s_psl_2f_q_is_simple | theorem | PSL_2(F_q) is simple
s_bilinear_form | axiom | Bilinear form
s_quadratic_form | axiom | Quadratic form
s_nondegenerate_bilinear_form | axiom | Non-degenerate bilinear form
s_hyperbolic_plane | axiom | Hyperbolic plane
s_anisotropic_form | axiom | Anisotropic form
s_witt_index | state | Witt index
s_witt_ring | state | Witt ring
s_even_clifford_algebra | state | Even Clifford algebra
s_pfaffian | state | Pfaffian
s_witts_cancellation_theorem | theorem | Witt's cancellation theorem
s_witt_decomposition_theorem | theorem | Witt decomposition theorem
s_sylvesters_law_of_inertia | theorem | Sylvester's law of inertia
s_structure_of_clifford_algebras | theorem | Structure of Clifford algebras
s_every_alternating_form_has_even_rank | theorem | Every alternating form has even rank
s_minimal_polynomial_of_a_linear_map | state | Minimal polynomial of a linear map
s_adjoint_functor_pair | axiom | Adjoint functor pair
s_representable_functor | axiom | Representable functor
s_additive_category | axiom | Additive category
s_product_of_objects_in_a_category | axiom | Product of objects in a category
s_coproduct_of_objects_in_a_category | axiom | Coproduct of objects in a category
s_direct_limit_filtered_colimit | axiom | Direct limit (filtered colimit)
s_inverse_limit | axiom | Inverse limit
s_left_exact_functor | axiom | Left exact functor
s_right_exact_functor | axiom | Right exact functor
s_exact_functor | axiom | Exact functor
s_projective_resolution | axiom | Projective resolution
s_injective_resolution | axiom | Injective resolution
s_enough_projectives | axiom | Enough projectives
s_enough_injectives | axiom | Enough injectives
s_hom_is_left_exact | theorem | Hom is left exact
s_tensor_is_right_exact | theorem | Tensor is right exact
s_comparison_theorem_for_resolutions | theorem | Comparison theorem for resolutions
s_horseshoe_lemma | theorem | Horseshoe lemma
s_long_exact_sequence_from_short_exact_sequence_of_complexes | theorem | Long exact sequence from short exact sequence of complexes
s_nine_lemma_33_lemma | theorem | Nine lemma (3×3 lemma)
s_group_extension | axiom | Group extension
s_factor_set_2cocycle | state | Factor set (2-cocycle)
s_corestriction_transfer_map | state | Corestriction (transfer) map
s_cup_product_in_group_cohomology | state | Cup product in group cohomology
s_crossed_homomorphism | axiom | Crossed homomorphism
s_h_classifies_group_extensions | theorem | H² classifies group extensions
s_herbrand_quotient_is_multiplicative | theorem | Herbrand quotient is multiplicative
s_cohomology_of_cyclic_groups_is_periodic | theorem | Cohomology of cyclic groups is periodic
s_left_derived_functor | axiom | Left derived functor
s_right_derived_functor | axiom | Right derived functor
s_universal_functor | axiom | Universal δ-functor
s_effaceable_functor | axiom | Effaceable functor
s_projective_dimension | state | Projective dimension
s_injective_dimension | state | Injective dimension
s_global_dimension_of_a_ring | state | Global dimension of a ring
s_weak_flat_dimension | state | Weak (flat) dimension
s_ext_is_independent_of_resolution | theorem | Ext is independent of resolution
s_tor_is_independent_of_resolution | theorem | Tor is independent of resolution
s_long_exact_sequence_for_ext | theorem | Long exact sequence for Ext
s_long_exact_sequence_for_tor | theorem | Long exact sequence for Tor
s_ext_classifies_extensions | theorem | Ext¹ classifies extensions
s_tor_characterizes_flatness | theorem | Tor₁ characterizes flatness
s_grothendiecks_universality_theorem | theorem | Grothendieck's universality theorem
s_dimension_shifting | theorem | Dimension shifting
s_auslanderbuchsbaumserre_theorem | theorem | Auslander-Buchsbaum-Serre theorem
t_dimension_shifting_technique | technique | Dimension shifting technique
t_resolution_comparison_technique | technique | Resolution comparison technique
t_derived_functor_computation_via_acyclic_resolutions | technique | Derived functor computation via acyclic resolutions
s_filtered_complex | axiom | Filtered complex
s_double_complex | axiom | Double complex
s_total_complex_of_a_double_complex | state | Total complex of a double complex
s_edge_homomorphism | state | Edge homomorphism
s_balancing_tor_via_double_complexes | theorem | Balancing Tor via double complexes
t_spectral_sequence_argument | technique | Spectral sequence argument
t_double_complex_filtration_technique | technique | Double complex filtration technique
s_depth_of_a_module | state | Depth of a module
s_cohenmacaulay_module | axiom | Cohen-Macaulay module
s_koszul_complex_acyclicity | theorem | Koszul complex acyclicity
s_regular_local_ring_is_ufd | theorem | Regular local ring is UFD
s_regular_local_ring_has_finite_global_dimension | theorem | Regular local ring has finite global dimension
t_koszul_complex_technique | technique | Koszul complex technique
t_long_exact_sequence_construction | technique | Long exact sequence construction
s_ordered_field | axiom | Ordered field
s_valuation_ring_of_an_absolute_value | state | Valuation ring of an absolute value
s_sturms_theorem | theorem | Sturm's theorem
s_ostrowskis_theorem | theorem | Ostrowski's theorem
s_extension_of_absolute_values | theorem | Extension of absolute values
s_dihedral_group | state | Dihedral Group D_{2n}
s_quaternion_group_q8 | state | Quaternion Group Q_8
s_klein_four_group | state | Klein Four-Group V_4
s_lattice_of_subgroups | state | Lattice of Subgroups
s_abelianization | state | Abelianization
t_classification_groups_small_order | technique | Classification of Groups of Small Order
t_holder_program | technique | Holder Program
s_chinese_remainder_theorem_rings | theorem | Chinese Remainder Theorem for Commutative Rings
s_quotient_by_prime_is_domain | theorem | Quotient by Prime Ideal is Integral Domain
s_quotient_by_maximal_is_field | theorem | Quotient by Maximal Ideal is Field
s_associate_elements | state | Associate Elements in a Ring
s_irreducible_element | state | Irreducible Element
s_prime_element | state | Prime Element in a Ring
s_polynomial_ring_over_field_is_ed | theorem | Polynomial Ring over a Field is a Euclidean Domain
s_rational_root_theorem | theorem | Rational Root Theorem
t_mod_p_irreducibility_test | technique | Mod p Irreducibility Test
s_evaluation_homomorphism | state | Evaluation Homomorphism
s_root_factor_theorem | theorem | Root-Factor Theorem
s_splitting_lemma_modules | theorem | Splitting Lemma for Modules
s_companion_matrix | state | Companion Matrix
s_jordan_block | state | Jordan Block
s_fx_module_structure_vector_space | state | F[x]-Module Structure on a Vector Space
t_galois_groups_small_degree | technique | Computing Galois Groups of Small Degree Polynomials
s_galois_closure | state | Galois Closure
s_composite_field_extension | state | Composite Field Extension (Compositum)
s_inner_product_of_class_functions | state | Inner Product of Class Functions
s_permutation_representation | state | Permutation Representation
s_symmetric_exterior_power_representations | state | Symmetric and Exterior Powers of Representations
s_frobenius_schur_indicator | state | Frobenius-Schur Indicator
t_representations_as_modules | technique | Representations as k[G]-Modules
t_character_theory_simplicity_proofs | technique | Character Theory for Simplicity Proofs
s_von_dycks_theorem | theorem | Von Dyck's theorem
s_chief_series | state | Chief series
s_chief_factors | state | Chief factors
s_indecomposable_group | axiom | Indecomposable group
s_krull_schmidt_theorem_groups | theorem | Krull-Schmidt theorem for groups
s_free_product_with_amalgamation | axiom | Free product with amalgamation
s_subnormal_subgroup | axiom | Subnormal subgroup
s_zero_divisor | axiom | Zero divisor
s_nil_ideal | axiom | Nil ideal
s_nilpotent_ideal | axiom | Nilpotent ideal
s_primitive_ring | axiom | Primitive ring
s_semiprimitive_ring | axiom | Semiprimitive ring
s_dense_ring_of_linear_transformations | state | Dense ring of linear transformations
s_group_ring | axiom | Group ring
s_semiprime_ring | axiom | Semiprime ring
s_essential_submodule | axiom | Essential submodule
s_divisible_abelian_group | axiom | Divisible abelian group
s_invariant_basis_number | state | Invariant basis number
s_comparison_theorem_resolutions | theorem | Comparison theorem for resolutions
s_torsion_module | state | Torsion module
s_cyclic_module | axiom | Cyclic module
s_progenerator | state | Progenerator
s_endomorphism_ring_module | axiom | Endomorphism ring of a module
s_maclane_separability_criterion | theorem | MacLane separability criterion
s_separating_transcendence_basis | state | Separating transcendence basis
s_separable_closure_in_extension | state | Separable closure in an extension
s_compositum_of_fields | axiom | Compositum of fields
s_artin_schreier_map | state | Artin-Schreier map
s_natural_transformation | axiom | Natural transformation
s_functor_category | axiom | Functor category
s_equalizer | axiom | Equalizer
s_coequalizer | axiom | Coequalizer
s_pullback_categorical | axiom | Pullback (categorical)
s_pushout_categorical | axiom | Pushout (categorical)
s_complete_category | axiom | Complete category
s_cocomplete_category | axiom | Cocomplete category
s_initial_object | axiom | Initial object
s_terminal_object | axiom | Terminal object
s_zero_object_category | axiom | Zero object (category)
s_cone_over_diagram | state | Cone over a diagram
s_cocone_under_diagram | state | Cocone under a diagram
s_unit_of_adjunction | state | Unit of adjunction
s_counit_of_adjunction | state | Counit of adjunction
s_additive_functor | axiom | Additive functor
s_image_in_abelian_category | state | Image in abelian category
s_triangle_identities_adjunction | theorem | Triangle identities for adjunction
s_limits_from_products_and_equalizers | theorem | Limits from products and equalizers
s_colimits_from_coproducts_and_coequalizers | theorem | Colimits from coproducts and coequalizers
s_right_adjoints_preserve_limits | theorem | Right adjoints preserve limits
s_left_adjoints_preserve_colimits | theorem | Left adjoints preserve colimits
s_characterization_equivalence_categories | theorem | Characterization of equivalence of categories
s_p_primary_component | state | p-primary component
s_matrix_equivalence | axiom | Matrix equivalence
s_matrix_similarity | axiom | Matrix similarity
s_embedded_associated_prime | state | Embedded associated prime
s_isolated_associated_prime | state | Isolated associated prime
s_minimal_primary_decomposition | state | Minimal primary decomposition
t_von_dyck_presentation_argument | technique | Von Dyck presentation argument
t_horseshoe_lemma_construction | technique | Horseshoe lemma construction
t_comparison_theorem_lifting | technique | Comparison theorem lifting
t_universal_property_argument | technique | Universal property argument
t_yoneda_argument | technique | Yoneda argument
t_adjunction_construction_universal_arrows | technique | Adjunction construction via universal arrows
t_diagram_chasing_abelian_categories | technique | Diagram chasing in abelian categories
t_cycle_decomposition_of_permutations | technique | Cycle decomposition of permutations
s_simplicity_of_alternating_group | theorem | Simplicity of A_n for n ≥ 5 ;; A_n is simple
s_ufd_polynomial_ring_theorem | theorem | R UFD implies R[x] UFD
s_fundamental_theorem_of_algebra_via_real_closed_fields | theorem | Fundamental theorem of algebra via real closed fields ;; FTA via real closed fields
s_budan_fourier_theorem | theorem | Budan–Fourier theorem
s_distributive_lattice | axiom | Distributive lattice
s_complemented_lattice | axiom | Complemented lattice
s_jordan_dedekind_chain_condition | theorem | Jordan–Dedekind chain condition
s_universal_algebra | axiom | Universal algebra (Ω-algebra) ;; Ω-algebra ;; general algebra
s_congruence_relation | axiom | Congruence relation
s_free_algebra_universal | state | Free algebra (universal algebra)
s_variety_equational_class | axiom | Variety (equational class) ;; equational class ;; variety of algebras
s_birkhoff_hsp_theorem | theorem | Birkhoff's HSP theorem ;; Birkhoff's variety theorem
s_subdirect_product | state | Subdirect product
s_birkhoff_subdirect_representation_theorem | theorem | Birkhoff's subdirect representation theorem
s_completely_reducible_module | axiom | Completely reducible module ;; semisimple module
s_composition_length | state | Composition length
s_krull_dimension_of_ring | state | Krull dimension ;; Krull dimension of a ring
s_prufer_domain | axiom | Prüfer domain ;; Prufer domain
s_prufer_domain_characterizations | theorem | Characterizations of Prüfer domains
s_homology_of_chain_complex | state | Homology of a chain complex
s_abelian_p_extensions_via_witt_vectors | theorem | Abelian p-extensions via Witt vectors
s_integrality_of_characters | theorem | Integrality of characters
t_modular_representation_theory | technique | Modular representation theory
s_division_algorithm_for_integers | theorem | Division algorithm for integers
t_euclidean_algorithm_for_integers | technique | Euclidean algorithm for integers
s_division_algorithm_for_polynomials | theorem | Division algorithm for polynomials
t_euclidean_algorithm_for_polynomials | technique | Euclidean algorithm for polynomials
s_irreducible_polynomials_over_r | theorem | Irreducible polynomials over R
s_sign_of_a_permutation | axiom | Sign of a permutation
t_row_reduction_gaussian_elimination | technique | Row reduction (Gaussian elimination)
s_matrix_algebra | axiom | Matrix algebra
s_matrix_inverse | axiom | Matrix inverse
s_one_sided_inverse_implies_two_sided | theorem | One-sided inverse implies two-sided inverse
s_vector_space | axiom | Vector space
s_subspace_of_vector_space | axiom | Subspace of a vector space
s_linear_independence | axiom | Linear independence
s_basis_extension_theorem | theorem | Basis extension theorem
s_linear_map | axiom | Linear map
s_kernel_and_image_of_linear_map | axiom | Kernel and image of a linear map
s_matrix_of_a_linear_map | axiom | Matrix of a linear map
s_quotient_vector_space | axiom | Quotient vector space V/W
s_internal_direct_sum_of_subspaces | axiom | Internal direct sum of subspaces
s_external_direct_sum_of_vector_spaces | axiom | External direct sum of vector spaces
s_dimension_formula_for_subspaces | theorem | Dimension formula for subspaces
s_multiplicativity_of_determinants | theorem | Multiplicativity of determinants
s_determinant_and_invertibility | theorem | Determinant and invertibility
s_inner_product | axiom | Inner product
s_self_adjoint_linear_map | axiom | Self-adjoint (Hermitian) linear map
s_unitary_orthogonal_linear_map | axiom | Unitary (orthogonal) linear map
s_positive_semidefinite_operator | axiom | Positive semidefinite operator
s_polar_decomposition | theorem | Polar decomposition
s_free_abelian_group | axiom | Free abelian group
s_classification_of_groups_of_order_pq | theorem | Classification of groups of order pq
s_minimal_polynomial_divides_characteristic | theorem | Minimal polynomial divides characteristic polynomial
s_diagonalizability_criterion | theorem | Diagonalizability criterion
s_projection_operator_idempotent | axiom | Projection operator (idempotent)
s_t_invariant_subspace | axiom | T-invariant subspace
s_primary_decomposition_theorem_linear_algebra | theorem | Primary decomposition theorem (linear algebra)
s_generalized_eigenspace | state | Generalized eigenspace
s_determinant_over_commutative_ring | axiom | Determinant over a commutative ring
s_gram_matrix | state | Gram matrix
s_equivalence_class_partition_theorem | theorem | Equivalence class partition theorem
s_well_ordering_principle | theorem | Well-ordering principle
s_semisimple_module_decomposition_theorem | theorem | Semisimple module decomposition theorem
s_ascending_chain_condition_modules | axiom | Ascending chain condition (Noetherian condition for modules)
s_descending_chain_condition_modules | axiom | Descending chain condition (Artinian condition for modules)
s_composition_series_existence_criterion | theorem | Composition series existence criterion
s_tensor_product_distributes_over_direct_sum | theorem | Tensor product distributes over direct sum
s_cochain_complex | axiom | Cochain complex
s_cohomology_of_cochain_complex | state | Cohomology of a cochain complex
s_chain_map | axiom | Chain map
t_left_derived_functor_construction | technique | Derived functor construction (left derived functors)
t_right_derived_functor_construction | technique | Derived functor construction (right derived functors)
s_long_exact_sequence_of_derived_functors | theorem | Long exact sequence of derived functors
t_hom_tensor_change_of_rings | technique | Hom-tensor change of rings
s_hom_exactness_failure | state | Hom functor exactness failure
s_tensor_exactness_failure | state | Tensor product exactness failure
s_diagonalization_of_symmetric_bilinear_forms | theorem | Diagonalization of symmetric bilinear forms
s_normal_form_of_alternating_bilinear_forms | theorem | Normal form of alternating bilinear forms
s_tensor_product_of_vector_spaces | axiom | Tensor product of two vector spaces
s_existence_and_uniqueness_of_tensor_product | theorem | Existence and uniqueness of tensor product
s_tensor_product_of_linear_maps | state | Tensor product of linear maps
s_tensor_product_of_multiple_vector_spaces | axiom | Tensor product of multiple vector spaces
s_associativity_of_tensor_product | theorem | Associativity of tensor product
s_universal_property_of_the_tensor_algebra | theorem | Universal property of the tensor algebra
s_universal_property_of_the_symmetric_algebra | theorem | Universal property of the symmetric algebra
s_universal_property_of_the_exterior_algebra | theorem | Universal property of the exterior algebra
s_determinant_via_exterior_algebra | state | Determinant via exterior algebra
s_basis_for_exterior_powers | theorem | Basis for exterior powers
s_dual_space_and_tensor_products | state | Dual space and tensor products
s_universal_property_of_free_groups | theorem | Universal property of free groups
s_schreier_index_formula | theorem | Schreier index formula
s_simple_extension | axiom | Simple extension
t_lagrange_resolvent | technique | Lagrange resolvent
s_field_polynomial | state | Field polynomial
s_discriminant_determines_alternating_galois_group | theorem | Discriminant determines alternating Galois group
t_reduction_modulo_p_for_galois_groups | technique | Reduction modulo p for Galois groups
s_characterization_of_semisimple_modules | theorem | Characterization of semisimple modules
s_acc_and_dcc_imply_composition_series | theorem | ACC and DCC imply composition series
s_indecomposable_module | axiom | Indecomposable module
s_direct_sum_and_direct_product_of_modules | axiom | Direct sum and direct product of modules
s_symmetric_group_as_galois_group | theorem | Symmetric group S_n as Galois group
s_multiplicativity_of_norm_and_trace | theorem | Multiplicativity of norm and trace in towers
s_primary_decomposition_for_modules_over_pids | theorem | Primary decomposition for modules over PIDs
s_localization_is_exact | theorem | Localization is exact
s_archimedean_and_nonarchimedean_places | axiom | Archimedean and non-archimedean places
s_artins_theorem_on_simple_rings | theorem | Artin's theorem on simple rings
t_buchbergers_algorithm | technique | Buchberger's algorithm
s_buchbergers_criterion | theorem | Buchberger's criterion
s_burnsides_theorem_on_matrix_algebras | theorem | Burnside's theorem on matrix algebras
s_change_of_rings_for_ext | theorem | Change of rings for Ext
s_change_of_rings_for_tor | theorem | Change of rings for Tor
s_cocycle_and_coboundary_group_cohomology | axiom | Cocycle and coboundary (group cohomology)
s_comparison_theorem_projective_resolutions | theorem | Comparison theorem (projective resolutions)
t_composition_of_binary_quadratic_forms_gauss_composition | technique | Composition of binary quadratic forms (Gauss composition)
s_correspondence_between_forms_and_ideals | theorem | Correspondence between forms and ideals
s_crossed_product_algebra | state | Crossed product algebra
s_cyclic_algebra | state | Cyclic algebra
s_universal_delta_functor | axiom | Delta-functor (universal)
s_discriminant_of_a_number_field | state | Discriminant of a number field
s_real_complex_embeddings_of_number_field | axiom | Embedding of K into ℝ^n (real and complex places)
s_euler_product | state | Euler product
s_exponent_of_a_central_simple_algebra | state | Exponent of a central simple algebra
s_ext1_classifies_extensions | theorem | Ext^1 and extensions
s_finiteness_of_form_classes | theorem | Finiteness of form classes
s_form_class_group | state | Form class group
s_frobenius_theorem_real_division_algebras | theorem | Frobenius theorem (real division algebras)
s_fundamental_identity_efg_equals_n | theorem | Fundamental identity e·f·g = n
s_gausss_lemma_quadratic_residues | theorem | Gauss's lemma (quadratic residues)
s_genus_of_a_binary_quadratic_form | axiom | Genus of a binary quadratic form
s_hilberts_theorem_on_hilbert_polynomial | theorem | Hilbert's theorem on Hilbert polynomial
s_hom_functor | axiom | Hom functor
s_hurwitz_formula | theorem | Hurwitz formula
s_induced_module | state | Induced module
s_inner_automorphism | axiom | Inner automorphism
s_brauer_group_h2_isomorphism | theorem | Isomorphism Br(K/F) ≅ H²(Gal(K/F), K×)
s_jacobian_criterion_for_nonsingularity | theorem | Jacobian criterion for nonsingularity
s_left_noetherian_ring | axiom | Left Noetherian ring
s_macaulays_theorem | theorem | Macaulay's theorem
s_minkowskis_discriminant_bound | theorem | Minkowski's discriminant bound
s_minkowskis_lattice_point_theorem | theorem | Minkowski's lattice point theorem
s_norm_map_nkf | state | Norm map N_{K/F}
s_nullstellensatz_fieldtheoretic_form | theorem | Nullstellensatz (field-theoretic form)
s_periodicity_of_cohomology_for_cyclic_groups | theorem | Periodicity of cohomology for cyclic groups
s_principal_genus | axiom | Principal genus
s_proj_construction | state | Proj construction
s_proper_equivalence_of_binary_quadratic_forms | axiom | Proper equivalence of binary quadratic forms
s_quadratic_number_field | axiom | Quadratic number field
s_quaternion_algebra | axiom | Quaternion algebra
s_ramification_of_primes | axiom | Ramification of primes
s_reduced_binary_quadratic_form | axiom | Reduced binary quadratic form
s_relative_brauer_group | state | Relative Brauer group
s_relative_brauer_group_for_cyclic_extensions | theorem | Relative Brauer group for cyclic extensions
s_restriction_and_corestriction_maps | state | Restriction and corestriction maps
s_riemann_inequality_for_curves | theorem | Riemann's inequality
s_riemann_roch_space | state | Riemann-Roch space
s_riemann_roch_theorem_for_curves | theorem | Riemann-Roch theorem for algebraic curves
s_second_cohomology_group_h2 | state | Second cohomology group H²(G,M)
s_segre_embedding | state | Segre embedding
t_semisimple_decomposition_via_idempotents | technique | Semisimple decomposition via idempotents
s_semisimplicity_and_tensor_products | theorem | Semisimplicity and tensor products
s_splitting_field_of_a_central_simple_algebra | axiom | Splitting field of a central simple algebra
t_splitting_field_of_a_polynomial_over_a_finite_field | technique | Splitting field of a polynomial over a finite field
s_splitting_of_primes | axiom | Splitting of primes
s_tensor_product_of_algebras | axiom | Tensor product of algebras
s_tor_and_flatness_criterion | theorem | Tor and flatness criterion
s_trace_map_trkf | state | Trace map Tr_{K/F}
s_units_of_quadratic_number_fields | state | Units of quadratic number fields
s_universal_coefficients_theorem_algebraic | theorem | Universal coefficients theorem (algebraic)
s_veronese_embedding | state | Veronese embedding
s_wedderburns_main_theorem | theorem | Wedderburn's main theorem
s_wedderburns_theorem_semisimple_rings | theorem | Wedderburn's theorem (semisimple rings)
s_wedderburnartin_radical | state | Wedderburn-Artin radical
s_weierstrass_equation | state | Weierstrass equation
s_weil_differential_repartition | axiom | Weil differential (repartition)
s_zariski_topology_on_specr | axiom | Zariski topology on Spec(R)
s_zariski_topology_on_affine_space | axiom | Zariski topology on affine space
s_zariski_topology_on_projective_space | axiom | Zariski topology on projective space
s_zariskis_theorem_on_nonsingular_points | theorem | Zariski's theorem on nonsingular points
s_adele_of_a_function_field | axiom | adele of a function field
s_adelic_dirichlet_unit_theorem | theorem | adelic Dirichlet unit theorem
s_adelic_finiteness_of_class_number | theorem | adelic finiteness of class number
s_affine_variety | axiom | affine variety
s_algebraic_curve | axiom | algebraic curve
s_birational_equivalence_of_smooth_projective_curves | theorem | birational equivalence of smooth projective curves
s_canonical_embedding | state | canonical embedding
s_classification_of_irreducible_curves | theorem | classification of irreducible curves
s_compactness_of_adele_quotient | theorem | compactness of adele quotient
s_compactness_of_idele_class_group_normone | theorem | compactness of idele class group (norm-one)
s_complete_linear_system_determines_morphism | theorem | complete linear system determines morphism
s_completion_of_a_valued_field | state | completion of a valued field
s_decomposition_field | state | decomposition field
s_degree_of_a_divisor | axiom | degree of a divisor
s_degree_of_a_morphism_of_curves | axiom | degree of a morphism of curves
s_degree_of_a_principal_divisor_is_zero | theorem | degree of a principal divisor is zero
s_degree_of_a_projective_variety | state | degree of a projective variety
s_different_of_an_extension | state | different of an extension
s_dimension_of_an_affine_variety | axiom | dimension of an affine variety
s_discriminant_of_an_extension | state | discriminant of an extension
s_divisor_class_group_of_a_curve | state | divisor class group of a curve
s_effective_divisor | axiom | effective divisor
s_elimination_ideal | state | elimination ideal
s_elimination_theorem | theorem | elimination theorem
s_elliptic_curve_genusone_curve_with_point | axiom | elliptic curve (genus-one curve with point)
s_equivalence_of_absolute_values | axiom | equivalence of absolute values
s_extension_theorem | theorem | extension theorem
s_function_field_of_an_algebraic_curve | axiom | function field of an algebraic curve
s_fundamental_identity_for_valuations | theorem | fundamental identity for valuations
s_fundamental_theorem_for_infinite_galois_extensions | theorem | fundamental theorem for infinite Galois extensions
s_generic_point_of_a_scheme | state | generic point of a scheme
s_genus_of_an_algebraic_curve | axiom | genus of an algebraic curve
s_geometric_dimension_noetherian_topological_space | axiom | geometric dimension (Noetherian topological space)
s_homogeneous_coordinate_ring | state | homogeneous coordinate ring
s_homogeneous_ideal | axiom | homogeneous ideal
s_homogeneous_polynomial | axiom | homogeneous polynomial
s_hyperelliptic_curve | axiom | hyperelliptic curve
s_hypersurface | axiom | hypersurface
s_ideal_of_an_algebraic_set | state | ideal of an algebraic set
s_inertia_field | state | inertia field
s_infinite_galois_group | state | infinite Galois group
s_intersection_multiplicity_two_curves | axiom | intersection multiplicity (two curves)
s_intersections_in_projective_space | theorem | intersections in projective space
s_irreducible_decomposition | theorem | irreducible decomposition
s_linear_system_of_divisors | state | linear system of divisors
s_local_ring_at_a_point_of_a_variety | state | local ring at a point of a variety
s_localization_of_a_ring_at_a_prime | state | localization of a ring at a prime
s_locally_ringed_space | axiom | locally ringed space
s_monomial_order | axiom | monomial order
s_morphism_of_affine_schemes | theorem | morphism of affine schemes
s_morphism_of_affine_varieties | axiom | morphism of affine varieties
s_morphism_of_projective_varieties | axiom | morphism of projective varieties
t_multivariate_division_algorithm | technique | multivariate division algorithm
s_nonsingular_point_of_a_variety | axiom | nonsingular point of a variety
s_normalization_of_a_curve | state | normalization of a curve
s_padic_absolute_value | axiom | p-adic absolute value
s_padic_integers | axiom | p-adic integers
s_padic_numbers | axiom | p-adic numbers
s_place_of_a_field | axiom | place of a field
s_presheaf_on_a_topological_space | axiom | presheaf on a topological space
s_prime_spectrum_specr | axiom | prime spectrum Spec(R)
s_projective_algebraic_set | axiom | projective algebraic set
s_projective_closure | state | projective closure
s_projective_plane_curve | axiom | projective plane curve
s_projective_space_pn | axiom | projective space P^n
s_projective_variety | axiom | projective variety
s_ramification_divisor | state | ramification divisor
s_rational_curve | axiom | rational curve
s_rational_function_on_a_variety | axiom | rational function on a variety
s_rational_map_of_varieties | axiom | rational map of varieties
s_regular_function_at_a_point | axiom | regular function at a point
s_residue_class_degree | axiom | residue class degree
s_separable_closure | state | separable closure
s_sheaf_on_a_topological_space | axiom | sheaf on a topological space
s_singular_point_of_a_variety | axiom | singular point of a variety
s_splitting_of_primes_in_extensions | state | splitting of primes in extensions
s_structure_sheaf_on_specr | state | structure sheaf on Spec(R)
s_tangent_space_zariski | state | tangent space (Zariski)
s_tensor_product_decomposition_of_completions | theorem | tensor product decomposition of completions
s_totally_ramified_extension | axiom | totally ramified extension
s_unramified_extension | axiom | unramified extension
s_zeta_function_of_a_variety_over_a_finite_field | state | zeta function of a variety over a finite field
s_ideal_sum | axiom | Sum of ideals
s_ideal_product | axiom | Product of ideals
s_ideal_quotient | axiom | Ideal quotient (colon ideal)
s_coprime_ideals | axiom | Coprime ideals
s_prime_avoidance_lemma | theorem | Prime avoidance lemma
s_extension_and_contraction_of_ideals | axiom | Extension and contraction of ideals
s_jacobson_radical_characterization | theorem | Jacobson radical characterization
s_characterization_of_local_rings | theorem | Characterization of local rings
s_radical_equals_intersection_of_primes_containing_ideal | theorem | Radical equals intersection of primes containing ideal
s_restriction_of_scalars | axiom | Restriction of scalars
s_algebra_over_a_ring | axiom | Algebra over a ring
s_flatness_ideal_criterion | theorem | Ideal-theoretic characterization of flatness
s_ring_of_fractions | axiom | Ring of fractions
s_module_of_fractions | axiom | Module of fractions
s_local_property | axiom | Local property of modules
s_flatness_is_a_local_property | theorem | Flatness is a local property
s_prime_correspondence_under_localization | theorem | Prime correspondence under localization
s_integral_closure_is_a_subring | theorem | Integral closure is a subring
s_integral_element_equivalences | theorem | Equivalences for integral elements
s_valuation_ring_characterization | theorem | Characterization of valuation rings
s_integral_closure_via_valuation_rings | theorem | Integral closure via valuation rings
s_noetherian_characterization_exact_sequences | theorem | Noetherian characterization via exact sequences
s_hilbert_basis_theorem_power_series | theorem | Hilbert basis theorem for power series rings
s_irreducible_ideal | axiom | Irreducible ideal
s_irreducible_ideals_are_primary_in_noetherian_rings | theorem | Irreducible ideals are primary in Noetherian rings
s_every_ideal_is_finite_intersection_of_irreducibles | theorem | Every ideal is a finite intersection of irreducible ideals
s_symbolic_power_of_a_prime | axiom | Symbolic power of a prime ideal
s_artinian_ring_has_dimension_zero | theorem | Artinian rings have Krull dimension zero
s_structure_theorem_for_artinian_rings | theorem | Structure theorem for Artinian rings
s_artinian_ring_finitely_many_maximal_ideals | theorem | Artinian ring has finitely many maximal ideals
s_dvr_characterizations | theorem | Characterizations of discrete valuation rings
s_dedekind_domain_characterizations | theorem | Characterizations of Dedekind domains
s_filtered_module | axiom | Filtered module
s_completion_is_exact_on_fg_modules | theorem | Completion is exact on finitely generated modules
s_flatness_of_completion | theorem | Flatness of completion
s_completion_of_noetherian_ring_is_noetherian | theorem | Completion of a Noetherian ring is Noetherian
s_height_of_a_prime_ideal | axiom | Height of a prime ideal
s_coheight_of_a_prime_ideal | axiom | Coheight of a prime ideal
s_regular_local_ring_is_integral_domain | theorem | Regular local ring is an integral domain
s_dimension_inequality_for_extensions | theorem | Dimension inequality for ring extensions
t_primary_decomposition_technique | technique | Primary decomposition technique
t_integral_closure_technique | technique | Integral closure technique
t_noetherian_induction | technique | Noetherian induction
t_completion_technique | technique | Completion technique
t_graded_ring_technique | technique | Graded/associated-graded technique
t_hilbert_function_method | technique | Hilbert function method
t_determinant_trick | technique | Determinant trick
s_residue_field | state | Residue field
s_idempotents_and_connected_components_of_spec | theorem | Idempotents and connected components of Spec
s_prime_filtration | state | Prime filtration of a module
s_finiteness_of_associated_primes | theorem | Finiteness of associated primes
s_zero_divisors_equal_union_of_associated_primes | theorem | Zero divisors equal union of associated primes
s_ufd_implies_integrally_closed | theorem | UFD implies integrally closed
s_zariskis_lemma | theorem | Zariski's lemma
s_filtration_of_a_module | axiom | Filtration of a module
s_stable_filtration | state | Stable filtration (I-stable)
s_tangent_cone | state | Tangent cone
s_localization_is_flat | theorem | Localization is flat
s_flat_base_change_for_tor | theorem | Flat base change for Tor
s_lazards_theorem | theorem | Lazard's theorem
s_flat_equals_locally_free_for_fp | theorem | Flat equals locally free for finitely presented modules
s_weierstrass_preparation_theorem_formal | theorem | Weierstrass preparation theorem (formal power series)
t_lifting_idempotents | technique | Lifting idempotents
s_embedding_dimension | state | Embedding dimension
s_fiber_ring | state | Fiber ring of a ring map
s_total_ring_of_fractions | state | Total ring of fractions
s_i_adic_valuation | state | I-adic valuation
s_reduced_ring | axiom | Reduced ring
s_homogeneous_nullstellensatz | theorem | Homogeneous Nullstellensatz (projective Nullstellensatz)
t_rees_algebra_technique | technique | Rees algebra technique
s_geometric_primary_decomposition | state | Geometric interpretation of primary decomposition
s_module_of_finite_length | state | Module of finite length
s_completion_is_faithfully_flat | theorem | Completion is faithfully flat
s_affine_rings_universally_catenary | theorem | Affine rings are universally catenary
s_krull_dimension_of_a_module | state | Krull dimension of a module
s_codimension_of_an_ideal | state | Codimension (height) of an ideal
s_equidimensional_ring | axiom | Equidimensional ring
s_dimension_via_m_primary_ideals | theorem | Dimension via 𝔪-primary ideals
s_existence_of_systems_of_parameters | theorem | Existence of systems of parameters
s_dimension_drop_by_one_element | theorem | Dimension drop by one element
s_parameter_ideal | state | Parameter ideal
s_regular_system_of_parameters | state | Regular system of parameters
s_dimension_of_fiber | theorem | Dimension of fibers of a ring map
s_serre_criterion_for_normality | theorem | Serre's criterion for normality (R₁ + S₂)
s_serre_condition_rk | axiom | Serre's condition Rₖ
s_serre_condition_sk | axiom | Serre's condition Sₖ
s_serre_criterion_for_reducedness | theorem | Serre's criterion for reducedness (R₀ + S₁)
s_divisor_class_group_ring | state | Divisor class group of a normal domain
s_reflexive_module | axiom | Reflexive module
s_ufd_iff_trivial_class_group | theorem | UFD iff trivial divisor class group
s_conductor_ideal | state | Conductor ideal
s_integral_closure_1dim_is_dedekind | theorem | Integral closure of a 1-dimensional Noetherian domain is Dedekind
s_multiplicity_one_implies_regularity | theorem | Multiplicity one implies regularity
s_associativity_formula_for_multiplicities | theorem | Associativity formula for multiplicities
s_reduction_of_an_ideal | state | Reduction of an ideal
s_noether_normalization_nagata | theorem | Noether normalization (Nagata version)
s_noether_normalization_graded | theorem | Noether normalization for graded rings
s_finiteness_of_integral_closure_affine | theorem | Finiteness of integral closure of affine domains
s_generic_freeness_lemma | theorem | Generic freeness lemma (Grothendieck)
s_dimension_formula_affine_domains | theorem | Dimension formula for affine domains
s_semicontinuity_of_fiber_dimension | theorem | Semicontinuity of fiber dimension
s_going_down_for_flat_extensions | theorem | Going-down for flat extensions
s_hilbert_polynomial_of_hypersurface | theorem | Hilbert polynomial of a hypersurface
t_dimension_via_parameter_chains | technique | Dimension computation via parameter chains
s_extension_theorem_elimination | theorem | Extension theorem (elimination theory)
s_fiber_dimension_theorem | theorem | Fiber dimension theorem
s_lexicographic_order | axiom | Lexicographic monomial order
s_grevlex_order | axiom | Graded reverse lexicographic order (grevlex)
s_initial_term | state | Initial term (leading term)
s_initial_ideal | state | Initial ideal
s_s_polynomial | state | S-polynomial
s_reduced_groebner_basis | state | Reduced Gröbner basis
s_flat_degeneration_to_initial_ideal | theorem | Flat degeneration to initial ideal
s_dicksons_lemma | theorem | Dickson's lemma
t_groebner_basis_syzygies | technique | Gröbner basis and syzygies technique
s_universal_derivation | state | Universal derivation
s_conormal_module | state | Conormal module I/I²
s_kahler_differentials_polynomial_ring | theorem | Kähler differentials of a polynomial ring
s_kahler_differentials_and_separability | theorem | Kähler differentials and separability
s_smoothness_via_differentials | theorem | Smoothness via Kähler differentials
s_generic_smoothness | theorem | Generic smoothness
s_jacobian_criterion_smoothness | theorem | Jacobian criterion for smoothness
s_koszul_complex_as_tensor_product | theorem | Koszul complex as tensor product
s_koszul_homology | state | Koszul homology
s_depth_via_koszul_homology | theorem | Depth via Koszul homology
s_self_duality_of_koszul_complex | theorem | Self-duality of the Koszul complex
s_koszul_and_tor | theorem | Koszul homology computes Tor
s_regular_sequence_permutation | theorem | Regular sequences are permutable in local rings
s_grade_of_an_ideal | state | Grade of an ideal
s_depth_inequality | theorem | Depth inequality (depth ≤ dim)
s_cm_polynomial_ring | theorem | Polynomial rings over Cohen-Macaulay rings are Cohen-Macaulay
s_flatness_and_depth | theorem | Flatness and depth formula
s_cm_implies_catenary | theorem | Cohen-Macaulay rings are universally catenary
s_cm_equidimensional | theorem | Cohen-Macaulay local rings are equidimensional
s_minimal_free_resolution | state | Minimal free resolution
s_uniqueness_minimal_free_resolution | theorem | Uniqueness of minimal free resolutions
s_betti_numbers | state | Betti numbers (algebraic)
s_regularity_preserved_by_localization | theorem | Regularity is preserved by localization
s_stably_free_is_free_over_local | theorem | Stably free modules are free over local rings
s_ferrand_vasconcelos | theorem | Ferrand-Vasconcelos theorem
s_nagatas_lemma | theorem | Nagata's lemma
s_fitting_ideal | state | Fitting ideal
s_fitting_ideals_and_annihilator | theorem | Fitting ideals and annihilator
s_buchsbaum_eisenbud_acyclicity | theorem | Buchsbaum-Eisenbud acyclicity criterion
s_eagon_northcott_complex | state | Eagon-Northcott complex
s_determinantal_ideal | axiom | Determinantal ideal
s_expected_codimension_determinantal | theorem | Expected codimension of determinantal ideals
s_perfect_ideal | axiom | Perfect ideal
s_perfect_ideals_unmixed | theorem | Perfect ideals are unmixed
s_syzygy_module | state | Syzygy module
s_existence_of_canonical_module | theorem | Existence of canonical module
s_gorenstein_iff_canonical_is_ring | theorem | Gorenstein iff canonical module is the ring
s_type_of_cm_ring | state | Type of a Cohen-Macaulay ring
s_complete_intersection_ring | axiom | Complete intersection ring
s_complete_intersection_is_gorenstein | theorem | Complete intersections are Gorenstein
s_duality_maximal_cm_modules | theorem | Duality for maximal Cohen-Macaulay modules
s_maximal_cohen_macaulay_module | axiom | Maximal Cohen-Macaulay module
s_bass_numbers | state | Bass numbers
s_structure_of_injective_modules_noetherian | theorem | Structure of injective modules over Noetherian rings
s_poincare_duality_gorenstein | theorem | Poincaré duality for Gorenstein rings
s_local_cohomology_via_cech | theorem | Local cohomology via Čech complex
s_vanishing_local_cohomology | theorem | Vanishing of local cohomology
s_grothendieck_vanishing | theorem | Grothendieck vanishing theorem
s_depth_via_local_cohomology | theorem | Depth via local cohomology
s_local_cohomology_sheaf_cohomology | theorem | Local cohomology and sheaf cohomology comparison
s_divided_power_algebra | axiom | Divided power algebra
s_schur_functor | axiom | Schur functor
s_buchsbaum_rim_complex | state | Buchsbaum-Rim complex
s_mapping_cone | state | Mapping cone
s_spectral_sequence_filtered_complex | state | Spectral sequence of a filtered complex
s_spectral_sequence_double_complex | theorem | Spectral sequences of a double complex
s_mittag_leffler_condition | axiom | Mittag-Leffler condition
s_p_basis | state | p-basis of a field
s_auslander_depth_formula | theorem | Auslander depth formula
s_formally_smooth_algebra | axiom | Formally smooth algebra ;; smooth algebra ;; formally smooth morphism
s_formally_unramified_algebra | axiom | Formally unramified algebra ;; unramified algebra
s_formally_etale_algebra | axiom | Formally étale algebra ;; étale algebra ;; formally étale morphism
s_g_ring | axiom | G-ring (Grothendieck ring) ;; Grothendieck ring ;; G-ring
s_quasi_excellent_ring | axiom | Quasi-excellent ring ;; quasi-excellent
s_n_1_ring | axiom | N-1 ring ;; N-1 domain
s_buchsbaum_ring | axiom | Buchsbaum ring ;; Buchsbaum local ring
s_geometrically_regular_formal_fibres | axiom | Geometrically regular formal fibres ;; regular formal fibres
s_attached_primes | axiom | Attached primes ;; attached prime ideals ;; Att(M)
s_dualizing_complex | axiom | Dualizing complex ;; residual complex
s_singular_locus | state | Singular locus ;; non-regular locus ;; Sing(R)
s_formal_fibre | state | Formal fibre ;; formal fiber
s_henselization | state | Henselization ;; Henselization of a local ring ;; R^h
s_strict_henselization | state | Strict Henselization ;; strict Henselization ;; R^{sh}
s_local_criterion_for_flatness | theorem | Local criterion for flatness ;; local flatness criterion
s_equational_criterion_for_flatness | theorem | Equational criterion for flatness ;; equational flatness criterion ;; Lazard criterion
s_generic_flatness_theorem | theorem | Generic flatness theorem ;; generic freeness ;; generic flatness lemma
s_grothendieck_non_vanishing_theorem | theorem | Grothendieck's non-vanishing theorem ;; non-vanishing of local cohomology at dimension
s_hochster_big_cm_modules | theorem | Hochster's theorem on big Cohen-Macaulay modules ;; existence of big CM modules ;; Hochster's big CM theorem
s_openness_of_regular_locus | theorem | Openness of the regular locus ;; regular locus is open
s_fibre_dimension_formula_flat | theorem | Fibre dimension formula for flat local homomorphisms ;; dimension formula for flat extensions
s_finitistic_dimension_theorem | theorem | Finitistic dimension theorem ;; finitistic projective dimension equals depth
s_cohen_structure_theorem_mixed_char | theorem | Cohen structure theorem (mixed characteristic) ;; Cohen's structure theorem for mixed characteristic
s_power_series_ring_is_local | theorem | Power series ring over a local ring is local
s_power_series_ring_faithfully_flat | theorem | Power series ring is faithfully flat
s_grothendieck_formal_functions | theorem | Grothendieck's theorem on formal functions ;; theorem on formal functions
t_cohen_macaulay_approximation | technique | Cohen-Macaulay approximation ;; Auslander-Buchweitz approximation
t_secondary_representation | technique | Secondary representation ;; secondary decomposition
s_quasi_isomorphism | axiom | Quasi-isomorphism ;; quasi-iso ;; quasi isomorphism ;; homology isomorphism
s_mapping_cylinder_chain | state | Mapping cylinder (chain complexes) ;; mapping cylinder of a chain map
s_split_exact_sequence | axiom | Split exact sequence ;; split short exact sequence ;; splitting
s_homotopy_equivalence_complexes | state | Homotopy equivalence of chain complexes ;; chain homotopy equivalence
s_erasable_functor | axiom | Erasable (effaceable/coeffaceable) functor ;; erasable functor ;; effaceable functor definition
s_flat_resolution | state | Flat resolution ;; flat resolution of a module
s_weak_global_dimension | axiom | Weak (global) dimension ;; weak dimension ;; Tor-dimension ;; w.gl.dim
s_yoneda_ext | state | Yoneda Ext ;; Yoneda extension group ;; Yoneda product
s_baer_sum | state | Baer sum of extensions ;; Baer addition
s_lim1_derived_inverse_limit | state | lim¹ (first derived functor of inverse limit) ;; lim^1 ;; R^1 lim ;; derived inverse limit
s_auslander_buchsbaum_serre_theorem | theorem | Auslander–Buchsbaum–Serre theorem ;; regularity and global dimension theorem
s_grade_of_module | axiom | Grade of a module ;; j-number ;; grade
s_derived_couple | state | Derived couple ;; derived exact couple
s_transgression_spectral_sequence | state | Transgression ;; transgression map ;; transgressive element
s_cartan_eilenberg_resolution | state | Cartan–Eilenberg resolution ;; CE resolution
s_hyperhomology | state | Hyperhomology ;; hyper-homology ;; left hyper-derived functor ;; hypertor
s_hypercohomology | state | Hypercohomology ;; hyper-cohomology ;; right hyper-derived functor ;; hyperext
s_acyclic_assembly_lemma | theorem | Acyclic assembly lemma
s_group_invariants | axiom | G-invariants M^G ;; invariants ;; fixed points of G-module ;; M^G
s_coinvariants | axiom | G-coinvariants M_G ;; coinvariants ;; module of coinvariants ;; M_G
s_normalized_bar_resolution | state | Normalized bar resolution ;; normalized standard resolution ;; reduced bar resolution
s_five_term_exact_sequence_group | theorem | Five-term exact sequence (group cohomology) ;; inflation-restriction exact sequence ;; five-term exact sequence
s_principal_crossed_homomorphism | state | Principal crossed homomorphism ;; principal derivation ;; inner derivation (group cohomology)
s_factor_set | axiom | Factor set (group extension) ;; 2-cocycle for group extension ;; factor system
s_tate_cohomology | axiom | Tate cohomology ;; Tate cohomology groups ;; Ĥ*(G,M) ;; complete cohomology
s_complete_resolution | state | Complete (Tate) resolution ;; Tate resolution ;; complete projective resolution
s_norm_map_group | state | Norm map (group cohomology) ;; norm element ;; norm homomorphism
s_cup_product_group_cohomology | state | Cup product in group cohomology ;; cup product on H*(G,M)
s_cohomological_dimension_group | axiom | Cohomological dimension of a group ;; cd(G) ;; cohomological dimension
s_group_type_fp | axiom | Group of type FP ;; type FL ;; type FP_n ;; finiteness property of groups
s_lie_algebra_homology | axiom | Lie algebra homology H_n(𝔤, M) ;; Lie algebra homology ;; H_n(g,M)
s_lie_algebra_cohomology | axiom | Lie algebra cohomology H^n(𝔤, M) ;; Lie algebra cohomology ;; H^n(g,M)
s_chevalley_eilenberg_complex | state | Chevalley–Eilenberg complex ;; CE complex ;; Koszul–Chevalley–Eilenberg complex
s_casimir_element | state | Casimir element ;; Casimir operator ;; Casimir invariant
s_relative_lie_algebra_cohomology | state | Relative Lie algebra cohomology H^n(𝔤, 𝔨; M) ;; relative cohomology of Lie pair
s_simplicial_object | axiom | Simplicial object ;; simplicial object in a category
s_cosimplicial_object | axiom | Cosimplicial object ;; cosimplicial object in a category
s_simplicial_set | axiom | Simplicial set ;; simplicial set ;; semi-simplicial set
s_face_map_simplicial | axiom | Face map (simplicial) ;; face operator ;; face morphism
s_degeneracy_map_simplicial | axiom | Degeneracy map (simplicial) ;; degeneracy operator
s_moore_complex | state | Moore complex ;; Moore complex of a simplicial group
s_normalized_chain_complex_simplicial | state | Normalized chain complex (simplicial) ;; normalized Moore complex ;; N(A)
s_dold_kan_correspondence | theorem | Dold–Kan correspondence ;; Dold–Kan theorem ;; Dold–Kan equivalence ;; Dold–Puppe theorem
s_alexander_whitney_map | state | Alexander–Whitney map ;; AW map ;; Alexander–Whitney diagonal
s_shuffle_map | state | Shuffle map (Eilenberg–Zilber) ;; Eilenberg–Zilber map ;; shuffle product ;; ∇
s_eilenberg_maclane_space | axiom | Eilenberg–MacLane space K(π, n) ;; K(π,n) ;; Eilenberg–MacLane space
s_comonad | axiom | Comonad (cotriple) ;; cotriple ;; comonad
s_cotriple_resolution | state | Cotriple resolution ;; comonadic resolution ;; comonad resolution
s_andre_quillen_homology | state | André–Quillen homology ;; AQ homology ;; cotangent homology
s_cotangent_complex | state | Cotangent complex ;; L_{A/k} ;; Illusie cotangent complex
s_bar_construction | state | Bar construction ;; bar construction B(T,A) ;; simplicial bar construction
s_kan_complex | axiom | Kan complex ;; Kan fibrant simplicial set ;; fibrant simplicial set
s_hochschild_homology | axiom | Hochschild homology HH_n(A, M) ;; Hochschild homology ;; HH_*(A,M)
s_gerstenhaber_bracket | state | Gerstenhaber bracket ;; Gerstenhaber algebra structure ;; Gerstenhaber Lie bracket
s_formal_deformation_algebra | axiom | Formal deformation of an algebra ;; infinitesimal deformation ;; deformation quantization
s_cyclic_homology | axiom | Cyclic homology HC_n(A) ;; cyclic homology ;; HC_*(A)
s_mixed_complex | axiom | Mixed complex ;; mixed complex (b, B)
s_connes_b_operator | state | Connes' B operator ;; Connes operator ;; Connes boundary B
s_sbi_exact_sequence | theorem | SBI exact sequence (Connes) ;; Connes' exact sequence ;; Connes' SBI sequence ;; periodicity exact sequence
s_cyclic_bicomplex | state | Cyclic bicomplex ;; Connes' bicomplex ;; CC(A)
s_periodic_cyclic_homology | state | Periodic cyclic homology HP_n(A) ;; periodic cyclic homology ;; HP_*(A)
s_negative_cyclic_homology | state | Negative cyclic homology HN_n(A) ;; negative cyclic homology ;; HN_*(A)
s_hkr_theorem | theorem | Hochschild–Kostant–Rosenberg theorem ;; HKR theorem ;; HKR isomorphism
s_loday_quillen_tsygan_theorem | theorem | Loday–Quillen–Tsygan theorem ;; LQT theorem
s_dennis_trace_map | state | Dennis trace map ;; Dennis trace
s_morita_invariance_hochschild | theorem | Morita invariance of Hochschild and cyclic homology ;; Morita invariance of HH and HC
s_homotopy_category_chain | axiom | Homotopy category K(A) ;; homotopy category of chain complexes ;; K(A)
s_distinguished_triangle | axiom | Distinguished triangle ;; exact triangle ;; distinguished triangle in triangulated category
s_octahedral_axiom | axiom | Octahedral axiom (TR4) ;; TR4 ;; octahedral axiom
s_verdier_quotient | state | Verdier quotient ;; Verdier localization ;; localization of triangulated category
s_ore_conditions | axiom | Ore conditions (calculus of fractions) ;; calculus of fractions ;; left Ore condition ;; right Ore condition
s_derived_category | axiom | Derived category D(A) ;; derived category ;; D(A) ;; D^b(A) ;; bounded derived category
s_bounded_derived_category | axiom | Bounded derived category D^b(A) ;; D^b ;; D^+(A) ;; D^-(A) ;; bounded above/below derived category
s_total_left_derived_functor | state | Total left derived functor LF ;; LF ;; left hyper-derived functor ;; 𝕃F
s_total_right_derived_functor | state | Total right derived functor RF ;; RF ;; right hyper-derived functor ;; ℝF
s_rhom_complex | state | RHom complex ;; RHom ;; derived Hom ;; ℝHom
s_derived_tensor_product | state | Derived tensor product ⊗^L ;; ⊗^L ;; derived tensor ;; left derived tensor product
s_way_out_functor | axiom | Way-out functor ;; way-out right ;; way-out left
s_way_out_lemma | theorem | Way-out lemma (Hartshorne) ;; Hartshorne's way-out lemma
s_t_structure | axiom | t-structure on a triangulated category ;; t-structure ;; truncation structure
s_heart_of_t_structure | state | Heart of a t-structure ;; core of a t-structure
s_truncation_functors | state | Truncation functors τ≤n and τ≥n ;; truncation ;; τ≤n ;; τ≥n ;; canonical truncation
s_grothendieck_ab_axioms | axiom | Grothendieck's AB axioms ;; AB3 ;; AB4 ;; AB5 ;; AB3* ;; AB4* ;; AB5* ;; Grothendieck category axioms
t_chevalley_eilenberg_complex_computation | technique | Chevalley–Eilenberg complex computation
t_cartan_eilenberg_resolution | technique | Cartan–Eilenberg resolution construction
t_dold_kan_correspondence | technique | Dold–Kan correspondence
t_spectral_sequence_of_double_complex | technique | Spectral sequence of a double complex
t_connes_sbi_computation | technique | SBI exact sequence computation
t_derived_category_localization | technique | Derived category construction (localization)
t_cotriple_resolution_technique | technique | Cotriple (comonadic) resolution
t_tate_cohomology_computation | technique | Tate cohomology computation
t_hyperhomology_computation | technique | Hyperhomology/hypercohomology computation
s_differential_in_chain_complex | axiom | Differential ;; boundary map ;; boundary operator
s_cycles_of_chain_complex | state | Cycles ;; n-cycles ;; cycle group
s_boundaries_of_chain_complex | state | Boundaries ;; n-boundaries ;; boundary group
s_cocycles_of_cochain_complex | state | Cocycles ;; n-cocycles
s_coboundaries_of_cochain_complex | state | Coboundaries ;; n-coboundaries
s_bounded_chain_complex | axiom | Bounded chain complex ;; bounded complex ;; bounded below complex ;; bounded above complex
s_chain_homotopy | axiom | Chain homotopy ;; homotopy of chain maps
s_chain_homotopy_implies_quasi_isomorphism | theorem | Chain homotopy equivalence implies quasi-isomorphism
s_naturality_of_long_exact_sequence | theorem | Naturality of the long exact sequence in homology
s_mapping_cylinder | axiom | Mapping cylinder
s_mapping_cone_exact_triangle | theorem | Mapping cone exact triangle
s_translation_shift_functor | axiom | Translation (shift) functor ;; shift functor ;; suspension functor ;; degree shift
s_kernel_in_abelian_category | axiom | Kernel (abelian category)
s_cokernel_in_abelian_category | axiom | Cokernel (abelian category)
s_coimage | state | Coimage
s_category_of_chain_complexes | state | Category Ch(A) of chain complexes ;; Ch(A)
s_homotopy_category_K_A | state | Homotopy category K(A) ;; K(A)
s_homology_is_functor | theorem | Homology is a functor
s_hom_complex | state | Hom complex ;; internal Hom complex
s_existence_of_projective_resolutions | theorem | Existence of projective resolutions
s_existence_of_injective_resolutions | theorem | Existence of injective resolutions
s_comparison_theorem_injective_resolutions | theorem | Comparison theorem for injective resolutions
s_cohomological_delta_functor | axiom | Cohomological delta-functor
s_coeffaceable_functor | axiom | Coeffaceable functor
s_effaceable_delta_functors_are_universal | theorem | Effaceable delta-functors are universal
s_L0_naturally_isomorphic_to_F | theorem | L_0F is naturally isomorphic to F
s_R0_naturally_isomorphic_to_G | theorem | R^0G is naturally isomorphic to G
s_left_derived_functors_form_universal_delta_functor | theorem | Left derived functors form a universal delta-functor
s_right_derived_functors_form_universal_delta_functor | theorem | Right derived functors form a universal delta-functor
s_derived_functors_vanish_on_adapted_objects | theorem | Derived functors vanish on adapted objects
s_left_adjoints_are_right_exact | theorem | Left adjoints are right exact
s_right_adjoints_are_left_exact | theorem | Right adjoints are left exact
s_balancing_ext | theorem | Balancing Ext
s_independence_of_resolution_derived_functors | theorem | Independence of resolution for derived functors
s_tor_0_is_tensor_product | theorem | Tor_0 is the tensor product
s_ext_0_is_hom | theorem | Ext^0 is Hom
t_construction_connecting_homomorphism_via_snake | technique | Construction of connecting homomorphism via Snake Lemma
s_free_module_is_projective | theorem | Free modules are projective
s_baer_criterion_for_injectivity | theorem | Baer criterion for injectivity ;; Baer's criterion
t_balancing_via_double_complexes | technique | Balancing via double complexes
s_flat_dimension | axiom | Flat dimension ;; weak dimension of a module
s_ext_characterization_projective_dimension | theorem | Ext characterization of projective dimension
s_ext_characterization_injective_dimension | theorem | Ext characterization of injective dimension
s_rings_of_global_dimension_zero | theorem | Rings of global dimension zero are semisimple
s_hereditary_ring | axiom | Hereditary ring
s_rings_of_global_dimension_at_most_one | theorem | Rings of global dimension at most one
s_change_of_rings_dimension_quotient | theorem | Change of rings theorem for dimension (quotient ring)
s_global_dimension_regular_local_rings | theorem | Global dimension of regular local rings
s_koszul_homology_definition | axiom | Koszul homology (with coefficients)
s_koszul_acyclicity_for_regular_sequences | theorem | Koszul acyclicity for regular sequences
s_global_dimension_equals_supremum_over_cyclic_modules | theorem | Global dimension equals supremum over cyclic modules
s_weak_dimension_leq_global_dimension | theorem | Weak dimension is at most global dimension
s_auslander_theorem_global_dimension | theorem | Auslander's theorem on global dimension
s_koszul_provides_resolution_of_residue_field | state | Koszul complex provides resolution of residue field
s_koszul_H1_detects_regular_sequences | theorem | Koszul H_1 detects regular sequences
t_mapping_cone_and_koszul | technique | Mapping cone and Koszul complex
t_horseshoe_lemma_application_dimension_shifting | technique | Horseshoe lemma application in dimension shifting
t_restriction_induction_of_scalars | technique | Restriction and induction of scalars ;; base change ;; extension and restriction of scalars
s_spectral_sequence | axiom | Spectral sequence
s_associated_graded_complex | state | Associated graded complex
s_bounded_filtration | axiom | Bounded filtration
s_weak_convergence_spectral_sequence | axiom | Weak convergence of a spectral sequence
s_bounded_convergence_spectral_sequence | axiom | Bounded convergence of a spectral sequence
s_first_quadrant_spectral_sequence | axiom | First quadrant spectral sequence
s_transgression | state | Transgression
s_five_term_exact_sequence_spectral | theorem | Five-term exact sequence of a spectral sequence
s_hyper_derived_functors_left | axiom | Left hyper-derived functors ;; hyper-Tor
s_hyper_derived_functors_right | axiom | Right hyper-derived functors ;; hyper-Ext
s_hyperhomology_spectral_sequences | theorem | Hyperhomology spectral sequences
s_existence_cartan_eilenberg_resolutions | theorem | Existence of Cartan-Eilenberg resolutions
s_universal_coefficients_spectral_sequence | theorem | Universal coefficients spectral sequence
s_kunneth_spectral_sequence | theorem | Kunneth spectral sequence
s_mapping_lemma_spectral_sequences | theorem | Mapping lemma for spectral sequences ;; comparison theorem for spectral sequences
s_degeneration_at_E2 | state | Degeneration at E_2 ;; collapse at E_2
t_exact_couple_construction_spectral_sequences | technique | Exact couple construction of spectral sequences
t_spectral_sequence_comparison | technique | Spectral sequence comparison technique
t_grothendieck_spectral_sequence_construction | technique | Grothendieck spectral sequence construction technique
s_augmentation_map_epsilon | axiom | Augmentation map epsilon: ZG -> Z ;; augmentation homomorphism
s_coinvariants_M_G | axiom | Coinvariants M_G ;; module of coinvariants
s_invariants_M_G | axiom | Invariants M^G ;; fixed points ;; module of invariants
s_H0_G_M_is_coinvariants | theorem | H_0(G, M) = M_G
s_H0_cohomology_is_invariants | theorem | H^0(G, M) = M^G
s_H1_classifies_crossed_homomorphisms | theorem | H^1(G, M) classifies crossed homomorphisms
s_H1_G_Z_is_abelianization | theorem | H_1(G, Z) = G^{ab}
s_hopf_formula_H2 | theorem | Hopf formula for H_2 ;; Hopf's formula
s_inhomogeneous_bar_resolution | axiom | Inhomogeneous bar resolution ;; unnormalized bar resolution
s_homogeneous_bar_resolution | axiom | Homogeneous bar resolution
s_coinduced_module | axiom | Coinduced module Coind_H^G N ;; co-induced module
s_ind_equals_coind_finite_index | theorem | Ind = Coind for finite index subgroups
s_cor_res_is_index | theorem | cor composed with res equals multiplication by [G:H]
s_lyndon_hochschild_serre_spectral_sequence | theorem | Lyndon-Hochschild-Serre spectral sequence ;; LHS spectral sequence
s_five_term_exact_sequence_group_cohomology | theorem | Five-term exact sequence in group cohomology ;; inflation-restriction exact sequence (extended)
s_five_term_exact_sequence_group_homology | theorem | Five-term exact sequence in group homology
s_cohomological_dimension_cd_G | theorem | cd(G) = proj.dim_{ZG}(Z)
s_group_of_type_FL | axiom | Group of type FL
s_group_of_type_FP | axiom | Group of type FP
s_free_groups_have_cd_1 | theorem | Free groups have cd = 1 ;; Stallings-Swan theorem
s_complete_tate_resolution | axiom | Complete (Tate) resolution ;; Tate resolution
s_tate_H0_formula | theorem | Tate cohomology H-hat^0 = M^G / N_G(M)
s_tate_H_minus1_formula | theorem | Tate cohomology H-hat^{-1} = ker(N_G) / I_G M
s_tate_long_exact_sequence | theorem | Tate cohomology long exact sequence
s_tate_induced_vanishes | theorem | Tate cohomology of induced modules vanishes
s_cup_product_tate_cohomology | axiom | Cup product in Tate cohomology
s_periodicity_via_cup_product | theorem | Periodicity in group cohomology via cup product
s_maschke_theorem_group_cohomology | theorem | Maschke's theorem for group cohomology
s_universal_coefficient_theorem_group_cohomology | theorem | Universal coefficient theorem for group cohomology
s_cohomology_of_finite_cyclic_groups | state | Cohomology of finite cyclic groups
s_cohomology_of_free_abelian_groups | state | Cohomology of free abelian groups
t_diagonal_approximation_bar_resolution | technique | Diagonal approximation on bar resolution
t_bar_resolution_technique_computing | technique | Bar resolution technique for computing group (co)homology
t_dimension_shifting_group_cohomology | technique | Dimension shifting in group cohomology
s_lie_algebra_module | axiom | Lie algebra module ;; g-module
s_augmentation_of_U_g | axiom | Augmentation of U(g)
s_H0_lie_cohomology_invariants | state | H^0(g, M) as invariants
s_H0_lie_homology_coinvariants | state | H_0(g, M) as coinvariants
s_H1_lie_cohomology_derivations | state | H^1(g, M) as derivations modulo inner derivations
s_H2_lie_classifies_extensions | theorem | H^2(g, M) classifies abelian Lie algebra extensions
s_H1_lie_homology_abelianization | state | H_1(g, k) as abelianization
s_abelian_lie_algebra_extension | axiom | Abelian Lie algebra extension
s_chevalley_eilenberg_as_free_resolution | state | Chevalley-Eilenberg complex as U(g)-free resolution
s_invariant_forms_and_lie_cohomology | state | Invariant forms and H^*(g, k)
s_cohomology_of_gl_n | state | Cohomology of gl_n
s_cohomology_of_sl_n | state | Cohomology of sl_n
s_inner_derivation_of_lie_algebra | axiom | Inner derivation of a Lie algebra
s_Hn_vanishes_semisimple_nontrivial | theorem | H^n vanishes for semisimple g and nontrivial irreducible M
s_H2_lie_homology_central_extensions | state | H_2(g, k) and central extensions
s_hopf_algebra_structure_U_g | state | Hopf algebra structure on U(g)
s_primitive_elements_of_U_g | state | Primitive elements of U(g)
s_hochschild_serre_spectral_sequence_lie | theorem | Hochschild-Serre spectral sequence for Lie algebras
t_chevalley_eilenberg_coboundary_formula | technique | Chevalley-Eilenberg coboundary formula
t_casimir_operator_vanishing | technique | Casimir operator technique for vanishing theorems
s_simplex_category | axiom | Simplex category ;; Delta ;; simplicial category
s_face_maps | axiom | Face maps
s_degeneracy_maps | axiom | Degeneracy maps
s_simplicial_identities | axiom | Simplicial identities
s_simplicial_abelian_group | axiom | Simplicial abelian group
s_standard_n_simplex | axiom | Standard n-simplex ;; representable simplicial set
s_singular_simplicial_set | state | Singular simplicial set ;; singular complex
s_geometric_realization | axiom | Geometric realization
s_simplicial_map | axiom | Simplicial map ;; simplicial morphism
s_simplicial_homotopy | axiom | Simplicial homotopy
s_simplicial_homotopy_groups | state | Simplicial homotopy groups
s_normalized_chain_complex | state | Normalized chain complex
s_degeneracy_subcomplex | state | Degeneracy subcomplex
s_dold_kan_inverse_functor_K | state | Dold-Kan inverse functor K
t_alexander_whitney_map | technique | Alexander-Whitney map ;; AW map
t_shuffle_map | technique | Shuffle map (Eilenberg-MacLane map) ;; EML map ;; Eilenberg-MacLane map
s_shuffle | axiom | Shuffle ;; (p,q)-shuffle
s_nerve_of_a_category | state | Nerve of a category
s_classifying_space_via_nerve | state | Classifying space of a group (via nerve) ;; BG
s_cotriple_comonad | axiom | Cotriple (comonad) ;; comonad
s_cotriple_homology | state | Cotriple homology
s_andre_quillen_cohomology | state | Andre-Quillen cohomology ;; AQ cohomology
s_bisimplicial_object | axiom | Bisimplicial object ;; bisimplicial set
s_diagonal_of_bisimplicial_object | state | Diagonal of a bisimplicial object
s_normalization_theorem | theorem | Normalization theorem
s_simplicial_homotopy_invariance_homology | theorem | Simplicial homotopy invariance of homology
s_acyclicity_of_cotriple_resolutions | theorem | Acyclicity of cotriple resolutions
s_transitivity_exact_sequence_AQ | theorem | Transitivity exact sequence (Andre-Quillen) ;; Jacobi-Zariski exact sequence
s_smooth_algebra_vanishing_AQ | theorem | Smooth algebra vanishing (Andre-Quillen)
t_extra_degeneracy_contracting_homotopy | technique | Extra degeneracy (contracting homotopy) ;; extra degeneracy argument
s_hochschild_complex | axiom | Hochschild complex (bar complex for algebras) ;; Hochschild bar complex
s_formal_deformation_of_algebra | axiom | Formal deformation of an algebra
s_H2_classifies_infinitesimal_deformations | theorem | HH^2 classifies infinitesimal deformations
s_connes_B_operator | axiom | Connes' B operator
s_SBI_exact_sequence | theorem | SBI exact sequence ;; Connes' exact sequence ;; periodicity exact sequence
s_connes_bicomplex | axiom | Connes' bicomplex
s_morita_invariance_hochschild_cyclic | theorem | Morita invariance of Hochschild and cyclic homology
s_hochschild_kostant_rosenberg | theorem | Hochschild-Kostant-Rosenberg theorem ;; HKR theorem
s_loday_quillen_tsygan | theorem | Loday-Quillen-Tsygan theorem
s_excision_hochschild_cyclic | theorem | Excision in Hochschild and cyclic homology
s_triangulated_category_axioms | axiom | Triangulated category axioms (TR1-TR4) ;; TR1-TR4
s_RHom_complex | axiom | RHom complex ;; derived Hom
t_calculus_of_fractions | technique | Calculus of fractions
t_replacing_spectral_sequences_with_triangles | technique | Replacing spectral sequences with triangles
s_abelian_category_axioms_AB | axiom | Abelian category axioms (AB1-AB5) ;; Grothendieck's axioms
s_grothendieck_category | axiom | Grothendieck category ;; AB5 category with generator
s_inverse_limit_functor_lim | axiom | Inverse limit functor lim ;; projective limit
s_mittag_leffler_implies_lim1_vanishes | theorem | Mittag-Leffler implies lim^1 vanishes
s_lim1_short_exact_sequences_towers | theorem | lim^1 and short exact sequences of towers
s_universal_coefficient_theorem_homology | theorem | Universal coefficient theorem for homology
s_universal_coefficient_theorem_cohomology | theorem | Universal coefficient theorem for cohomology
s_symmetry_of_tor | theorem | Symmetry of Tor
s_tor_for_abelian_groups | state | Tor for abelian groups
s_ext_for_abelian_groups | state | Ext for abelian groups
s_derivation_grothendieck_LHS_spectral | theorem | Derivation of LHS via Grothendieck spectral sequence
s_metacategory | axiom | Metacategory
s_category | axiom | Category ;; abstract category
s_small_category | axiom | Small category
s_locally_small_category | axiom | Locally small category
s_functor | axiom | Functor ;; covariant functor
s_contravariant_functor | axiom | Contravariant functor
s_natural_isomorphism | axiom | Natural isomorphism
s_isomorphism_of_categories | axiom | Isomorphism of categories
s_monomorphism | axiom | Monomorphism ;; monic ;; mono
s_epimorphism | axiom | Epimorphism ;; epic ;; epi
s_section_categorical | axiom | Section (right inverse)
s_retraction_categorical | axiom | Retraction (left inverse)
s_universal_element | axiom | Universal element
s_universal_arrow | axiom | Universal arrow
s_vertical_composition | axiom | Vertical composition of natural transformations
s_horizontal_composition | axiom | Horizontal composition (Godement product) ;; Godement product
s_interchange_law | theorem | Interchange law
s_whiskering | axiom | Whiskering
t_yoneda_embedding_technique | technique | Yoneda embedding technique
s_opposite_category | axiom | Opposite category ;; dual category ;; C^op
t_duality_principle | technique | Duality principle (categorical)
s_product_category | axiom | Product category
s_comma_category | axiom | Comma category ;; comma category (F ↓ G)
s_slice_category | axiom | Slice category ;; over category
s_coslice_category | axiom | Coslice category ;; under category
s_arrow_category | axiom | Arrow category ;; morphism category
s_free_category_on_graph | state | Free category on a graph
s_congruence_on_category | axiom | Congruence on a category
s_quotient_category | state | Quotient category
s_presheaf_categorical | axiom | Presheaf (categorical) ;; presheaf ;; set-valued presheaf
s_limit_of_functor | axiom | Limit of a functor ;; projective limit ;; inverse limit
s_colimit_of_functor | axiom | Colimit of a functor ;; inductive limit ;; direct limit
t_limit_via_products_equalizers | technique | Limit computation via products and equalizers
t_colimit_via_coproducts_coequalizers | technique | Colimit computation via coproducts and coequalizers
s_preservation_of_limits | axiom | Preservation of limits
s_reflection_of_limits | axiom | Reflection of limits
s_creation_of_limits | axiom | Creation of limits
s_subobject | axiom | Subobject
s_quotient_object | axiom | Quotient object
s_adjunction | axiom | Adjunction ;; adjoint pair ;; adjoint functors
s_special_adjoint_functor_theorem | theorem | Special adjoint functor theorem (SAFT) ;; SAFT
s_solution_set_condition | axiom | Solution set condition
s_free_forgetful_adjunction | state | Free-forgetful adjunction
s_forgetful_functor | axiom | Forgetful functor
s_free_functor | axiom | Free functor
s_faithful_functor | axiom | Faithful functor
s_full_functor | axiom | Full functor
s_fully_faithful_functor | axiom | Fully faithful functor ;; full and faithful functor
s_essentially_surjective_functor | axiom | Essentially surjective functor
s_equivalence_iff_ff_eso | theorem | Equivalence iff fully faithful and essentially surjective
s_adjunctions_give_monads | theorem | Adjunctions give rise to monads
s_filtered_category | axiom | Filtered category
s_filtered_colimits_commute_finite_limits | theorem | Filtered colimits commute with finite limits in Set
s_generator_separator | axiom | Generator (separator) ;; separator
s_cogenerator | axiom | Cogenerator (coseparator) ;; coseparator
s_well_powered_category | axiom | Well-powered category
s_ends | axiom | End (category theory) ;; end ;; wedge
s_coends | axiom | Coend (category theory) ;; coend ;; cowedge
s_nat_trans_as_end | theorem | Natural transformations as an end
s_limit_as_end | theorem | Limit as an end
s_co_yoneda_lemma | theorem | Co-Yoneda lemma (density formula) ;; density formula ;; Yoneda density
s_category_of_elements | axiom | Category of elements ;; Grothendieck construction for sets
s_repr_functors_preserve_limits | theorem | Representable functors preserve limits
s_algebra_over_monad | axiom | Algebra over a monad (T-algebra) ;; T-algebra ;; Eilenberg-Moore algebra
s_eilenberg_moore_category | axiom | Eilenberg-Moore category ;; category of algebras over T
s_kleisli_category | axiom | Kleisli category
s_comparison_functor | state | Comparison functor (Eilenberg-Moore)
s_every_monad_from_adjunction | theorem | Every monad arises from an adjunction
s_beck_monadicity_precise | theorem | Beck's monadicity theorem (precise form) ;; Beck's precise monadicity theorem ;; precise tripleability theorem
s_beck_monadicity_crude | theorem | Beck's monadicity theorem (crude form) ;; crude monadicity theorem
s_split_coequalizer | axiom | Split coequalizer
s_reflexive_coequalizer | axiom | Reflexive coequalizer
s_absolute_coequalizer | axiom | Absolute coequalizer
s_monadic_functor | axiom | Monadic functor ;; tripleable functor
s_creation_limits_monadic | theorem | Creation of limits by monadic functors
s_free_algebra_monad | state | Free algebra over a monad
t_monad_resolution | technique | Monad resolution
s_completeness_eilenberg_moore | theorem | Completeness of Eilenberg-Moore category
t_beck_conditions_verification | technique | Beck's conditions verification
s_monoid_object | axiom | Monoid object (in a monoidal category)
s_strict_monoidal_category | axiom | Strict monoidal category
s_pentagon_axiom | axiom | Pentagon axiom
s_triangle_axiom | axiom | Triangle axiom (unit coherence)
s_monoidal_functor | axiom | Monoidal functor (lax) ;; lax monoidal functor
s_strong_monoidal_functor | axiom | Strong monoidal functor
s_strict_monoidal_functor | axiom | Strict monoidal functor
s_closed_monoidal_category | axiom | Closed monoidal category
s_cartesian_closed_category | axiom | Cartesian closed category ;; CCC
s_internal_hom | axiom | Internal hom (exponential object) ;; exponential object ;; function object
s_eckmann_hilton_argument | theorem | Eckmann-Hilton argument
s_category_of_monoids | state | Category of monoids Mon(C)
s_preadditive_category | axiom | Preadditive category ;; Ab-enriched category ;; Ab-category
s_biproduct | axiom | Biproduct
s_kernel_categorical | axiom | Kernel (categorical)
s_cokernel_categorical | axiom | Cokernel (categorical)
s_image_categorical | axiom | Image (categorical)
s_freyd_mitchell_embedding | theorem | Freyd-Mitchell embedding theorem ;; Mitchell embedding theorem ;; Freyd embedding theorem
s_regular_epimorphism | axiom | Regular epimorphism
s_extremal_epimorphism | axiom | Extremal epimorphism
s_abelian_group_object | axiom | Abelian group object
s_filtered_colimit_algebraic | theorem | Filtered colimits in algebraic categories
s_interchange_limits_colimits | theorem | Interchange of limits and colimits
s_final_functor | axiom | Final functor (cofinal) ;; cofinal functor
s_final_functor_theorem | theorem | Final functor theorem
s_reflexive_coeq_existence | theorem | Reflexive coequalizer existence
s_limits_in_functor_categories | theorem | Limits in functor categories
s_creating_limits_theorem | theorem | Creating limits theorem
s_left_kan_extension | axiom | Left Kan extension
s_right_kan_extension | axiom | Right Kan extension
s_pointwise_kan_extension | axiom | Pointwise Kan extension
s_left_kan_colimit_formula | theorem | Left Kan extension as colimit formula
s_right_kan_limit_formula | theorem | Right Kan extension as limit formula
s_kan_along_yoneda | theorem | Kan extension along Yoneda embedding
s_every_functor_kan_extension | theorem | Every functor is a Kan extension of itself
s_adjunctions_as_kan_extensions | theorem | Adjunctions as Kan extensions
s_all_concepts_kan_extensions | theorem | All concepts are Kan extensions (Mac Lane)
s_dense_functor | axiom | Dense functor ;; adequate functor
s_density_theorem_categorical | theorem | Density theorem (categorical)
t_kan_extension_computation | technique | Kan extension computation
s_symmetric_monoidal_category | axiom | Symmetric monoidal category ;; SMC
s_braided_monoidal_category | axiom | Braided monoidal category ;; BMC
s_hexagon_axioms | axiom | Hexagon axioms
s_coherence_braided | theorem | Coherence for braided monoidal categories
s_coherence_symmetric | theorem | Coherence for symmetric monoidal categories
s_braid_group_categorical | state | Braid group from braided monoidal categories
s_yang_baxter_categorical | theorem | Yang-Baxter equation (categorical)
s_enriched_category | axiom | Enriched category ;; V-category ;; V-enriched category
s_2_category | axiom | 2-category ;; strict 2-category
s_elementary_topos | axiom | Elementary topos ;; topos ;; elementary topos
s_subobject_classifier | axiom | Subobject classifier
s_grothendieck_topos | axiom | Grothendieck topos ;; sheaf topos
s_grothendieck_topology | axiom | Grothendieck topology
s_sieve | axiom | Sieve
s_covering_family | axiom | Covering family
t_naturality_argument | technique | Naturality argument
t_comma_category_construction | technique | Comma category construction technique
t_representability_criterion | technique | Representability criterion
t_comparison_functor_construction | technique | Comparison functor construction
t_coherence_proof | technique | Coherence proof technique
t_transfinite_composition | technique | Transfinite composition
t_adjoint_functor_construction | technique | Adjoint functor construction technique
t_godement_calculus | technique | Godement calculus
s_dual_contragredient_representation | state | Dual (contragredient) representation ;; contragredient representation ;; dual representation
s_hom_representation | state | Hom representation ;; Hom(V,W) representation
s_space_of_invariants | state | Space of invariants V^G ;; G-invariants ;; fixed subspace
s_second_orthogonality_relation | theorem | Second orthogonality relation for characters ;; column orthogonality
s_character_determines_representation | theorem | Character determines representation
s_projection_formula_representation | state | Projection formula for representations ;; idempotent projection formula
s_representations_of_abelian_groups | theorem | Representations of abelian groups
s_standard_representation_of_S_n | state | Standard representation of S_n ;; standard representation of symmetric group
s_sign_representation | state | Sign (alternating) representation ;; alternating representation ;; signature representation ;; sgn representation
s_character_of_induced_representation | theorem | Character of induced representation ;; induced character formula
s_conjugate_partition | state | Conjugate (transpose) partition ;; transpose partition ;; dual partition
s_row_group_column_group | state | Row group and column group of a Young tableau ;; row stabilizer ;; column stabilizer
s_young_symmetrizer | state | Young symmetrizer ;; Young idempotent
s_specht_module | state | Specht module V_lambda ;; Specht representation
s_branching_rule_S_d | theorem | Branching rule for S_d ;; restriction rule for symmetric groups
s_sign_twist_specht | theorem | Sign twist of Specht modules
s_frobenius_character_formula | theorem | Frobenius character formula
s_murnaghan_nakayama_rule | theorem | Murnaghan--Nakayama rule ;; Murnaghan-Nakayama formula
s_youngs_rule | theorem | Young's rule
s_semistandard_young_tableau | axiom | Semistandard Young tableau ;; SSYT ;; column-strict tableau
s_young_subgroup | state | Young subgroup ;; parabolic subgroup of S_d
s_representation_ring_of_symmetric_groups | state | Representation ring of symmetric groups
s_center_of_group_algebra | state | Center of the group algebra ;; class sums
s_idempotents_in_group_algebra | state | Primitive idempotents in the group algebra ;; primitive central idempotents
s_schur_weyl_duality | theorem | Schur--Weyl duality ;; Schur-Weyl duality
s_plethysm | state | Plethysm
s_matrix_lie_group | axiom | Matrix Lie group ;; linear Lie group
s_one_parameter_subgroup | state | One-parameter subgroup of a Lie group
s_representation_of_lie_group | axiom | Representation of a Lie group ;; Lie group representation
s_lie_bracket_of_matrices | state | Lie bracket of matrices ;; matrix commutator bracket
s_semisimple_direct_sum_of_simples | theorem | Semisimple Lie algebra is a direct sum of simple ideals
s_preservation_of_jordan_decomposition | theorem | Preservation of Jordan decomposition under representations
s_lie_group_lie_algebra_rep_correspondence | theorem | Correspondence between Lie group and Lie algebra representations ;; Lie's third theorem for representations
s_sl2_C | state | The Lie algebra sl_2(C) ;; sl(2,C) ;; sl_2
s_standard_basis_sl2 | state | Standard basis {H,E,F} of sl_2
s_heisenberg_algebra | state | Heisenberg algebra ;; Heisenberg Lie algebra
s_irreducible_representations_of_sl2 | theorem | Classification of irreducible representations of sl_2 ;; irreps of sl(2)
s_highest_weight_vector | axiom | Highest weight vector ;; maximal vector ;; primitive vector
s_sl3_C | state | The Lie algebra sl_3(C) ;; sl(3,C) ;; sl_3
s_root_and_root_space | axiom | Root and root space ;; root of a Lie algebra
s_weight_diagram | state | Weight diagram ;; weight lattice diagram
s_partial_order_on_weights | state | Partial order on weights ;; dominance order on weights
s_weight_lattice | state | Weight lattice ;; integral weight lattice
s_weyl_group_action_on_weights | theorem | Weyl group action on weights
s_weyl_vector_rho | state | Weyl vector rho ;; half-sum of positive roots
s_rank_of_lie_algebra | state | Rank of a Lie algebra ;; Lie algebra rank
s_fundamental_representations_sl_n | state | Fundamental representations of sl_n
s_irreps_sl_n_via_young_diagrams | state | Irreducible representations of sl_n via Young diagrams
s_weyl_dimension_formula_sl_n | theorem | Weyl dimension formula for sl_n
s_fundamental_representations_so_n | state | Fundamental representations of so_n
s_fundamental_representations_sp_2n | state | Fundamental representations of sp_{2n}
s_exceptional_isomorphisms_low_rank | theorem | Exceptional isomorphisms of low-rank Lie algebras ;; accidental isomorphisms
s_spin_representation_odd | state | Spin representation of so_{2m+1} ;; spinor representation of type B
s_half_spin_representations | state | Half-spin representations of so_{2m} ;; chiral spinor representations ;; Weyl spinors
s_triality_so_8 | theorem | Triality for so_8 ;; D_4 triality
s_weyl_denominator_formula | theorem | Weyl denominator formula ;; Weyl denominator identity
s_steinbergs_formula | theorem | Steinberg's formula for tensor product decomposition ;; Steinberg tensor product multiplicity formula
s_klimyks_formula | theorem | Klimyk's formula ;; Klimyk's tensor product formula
s_cartan_involution | axiom | Cartan involution
s_maximal_compact_subgroup | state | Maximal compact subgroup
s_real_forms_classification | state | Classification of real forms of a complex semisimple Lie algebra ;; real forms of semisimple Lie algebras
s_split_real_form | state | Split real form ;; maximally split real form
s_satake_diagram | axiom | Satake diagram ;; Tits-Satake diagram
s_longest_element_w0 | state | Longest element w_0 of the Weyl group ;; w_0 ;; longest Weyl group element
s_dual_representation_via_w0 | theorem | Dual representation via the longest element
s_prv_theorem | theorem | Parthasarathy--Ranga Rao--Varadarajan (PRV) theorem ;; PRV theorem ;; PRV conjecture
s_exceptional_lie_algebra_G2 | state | Exceptional Lie algebra G_2 ;; G2
s_exceptional_lie_algebra_F4 | state | Exceptional Lie algebra F_4 ;; F4
s_exceptional_lie_algebra_E6 | state | Exceptional Lie algebra E_6 ;; E6
s_exceptional_lie_algebra_E7 | state | Exceptional Lie algebra E_7 ;; E7
s_exceptional_lie_algebra_E8 | state | Exceptional Lie algebra E_8 ;; E8
t_decomposition_via_inner_products | technique | Decomposition via character inner products
t_weight_string_analysis | technique | Weight string analysis ;; alpha-string through mu
t_clifford_construction_of_spin_reps | technique | Clifford algebra construction of spin representations
s_character_of_tensor_product | theorem | Character of tensor product
s_character_of_dual_representation | theorem | Character of dual representation
s_decomposition_multiplicity_formula | theorem | Decomposition multiplicity formula
s_regular_representation_decomposition | theorem | Regular representation decomposition
s_dimension_divides_group_order | theorem | Dimension divides group order
s_character_irreducibility_criterion | theorem | Character criterion for irreducibility
s_brauer_character_orthonormality | theorem | Orthonormality of Brauer characters
s_brauer_nesbitt_theorem | theorem | Brauer-Nesbitt theorem
s_number_simples_equals_p_regular_classes | theorem | Number of simple modules equals p-regular classes
s_cartan_matrix_symmetric_positive_definite | theorem | Cartan matrix is symmetric positive definite
s_p_regular_element | axiom | p-regular element
s_p_singular_element | axiom | p-singular element
s_brauer_character | axiom | Brauer character
s_block_of_group_algebra | axiom | Block of group algebra
s_defect_group | axiom | Defect group
s_p_modular_system | axiom | p-modular system
s_decomposition_map | state | Decomposition map
s_decomposition_matrix | state | Decomposition matrix
s_cartan_matrix_modular_representations | state | Cartan matrix for modular representations
s_projective_indecomposable_module | state | Projective indecomposable module
s_cde_triangle | state | CDE triangle
s_cartan_map | state | Cartan map
s_lifting_map_brauer | state | Lifting map (Brauer)
s_grothendieck_group_modular | state | Grothendieck group of modular representations
s_generalized_decomposition_numbers | state | Generalized decomposition numbers
s_defect_zero_block | state | Defect zero block
s_block_idempotent | state | Block idempotent
s_kernel_of_representation | state | Kernel of representation
s_inflation_of_representation | state | Inflation of representation
s_projective_cover_modular | state | Projective cover
t_semidirect_product_representations | technique | Semi-direct product representation construction
t_averaging_projection | technique | Averaging projection technique
t_reduction_mod_p_representations | technique | Reduction modulo p for representations
s_general_linear_lie_algebra | axiom | General linear Lie algebra gl(V)
s_special_linear_lie_algebra_sl_n | axiom | Special linear Lie algebra sl(n,F) ;; sl_n ;; type A_n Lie algebra
s_classical_lie_algebras_bcd | axiom | Classical Lie algebras (types B, C, D) ;; orthogonal Lie algebra ;; symplectic Lie algebra ;; so(n) ;; sp(2n)
s_toral_subalgebra | axiom | Toral subalgebra
s_maximal_toral_subalgebra | axiom | Maximal toral subalgebra
s_regular_element_lie_algebra | axiom | Regular element of a Lie algebra
s_saturated_set_of_weights | axiom | Saturated set of weights
s_admissible_lattice_lie | axiom | Admissible lattice
s_derived_series_lie | state | Derived series of a Lie algebra
s_lower_central_series_lie | state | Lower central series of a Lie algebra
s_nilradical_of_lie_algebra | state | Nilradical of a Lie algebra
s_reflection_s_alpha | state | Reflection s_α in a root system ;; simple reflection ;; Weyl reflection
s_root_string_through_beta | state | Root string ;; α-string through β
s_height_of_root | state | Height of a root
s_fundamental_weyl_chamber | state | Fundamental Weyl chamber ;; dominant chamber
s_length_function_weyl_group | state | Length function on the Weyl group
s_automorphism_group_of_root_system | state | Automorphism group of a root system ;; diagram automorphisms
s_triangular_decomposition | state | Triangular decomposition
s_filtration_of_enveloping_algebra | state | Filtration of the universal enveloping algebra
s_center_of_enveloping_algebra | state | Center of the universal enveloping algebra
s_simple_highest_weight_module | state | Simple highest weight module L(λ) ;; irreducible highest weight module
s_dot_action_shifted_weyl | state | Dot action (shifted Weyl group action) ;; shifted action ;; ρ-shifted action
s_character_of_verma_module | state | Character of a Verma module
s_formal_characters_ring | state | Ring of formal characters
s_integral_structure_constants | state | Integral structure constants of a Chevalley basis
s_kostant_z_form | state | Kostant's ℤ-form ;; Kostant integral form
s_root_subgroup | state | Root subgroup of a Chevalley group
s_chevalley_group_steinberg_relations | state | Steinberg relations in Chevalley groups ;; Chevalley commutator formula
s_dual_coxeter_number | state | Dual Coxeter number
s_highest_root_maximal | state | Highest root ;; maximal root ;; longest root
s_invariant_bilinear_form_on_module | state | Invariant bilinear form on a module
s_minuscule_weight | state | Minuscule weight
s_simultaneous_triangularization_theorem | theorem | Simultaneous triangularization (corollary of Lie's theorem)
s_semisimple_decomposition_into_simples | theorem | Decomposition of semisimple Lie algebras into simple ideals
s_all_derivations_inner_semisimple | theorem | All derivations are inner (semisimple case)
s_root_decomposition_properties | theorem | Properties of the root space decomposition
s_existence_conjugacy_of_bases | theorem | Existence and conjugacy of bases of root systems
s_simple_reflections_generate_weyl_group | theorem | Simple reflections generate the Weyl group
s_cartan_matrix_determines_root_system | theorem | Cartan matrix determines the root system
s_isomorphism_theorem_semisimple_lie_algebras | theorem | Isomorphism theorem for semisimple Lie algebras
s_conjugacy_of_borel_subalgebras | theorem | Conjugacy of Borel subalgebras
s_chevalley_invariant_polynomial_theorem | theorem | Chevalley's theorem on invariant polynomials
s_universal_property_enveloping_algebra | theorem | Universal property of the enveloping algebra
s_chevalley_integral_basis_theorem | theorem | Chevalley's theorem on integral bases
s_automorphisms_of_semisimple_lie_algebra | theorem | Structure of Aut(L) for semisimple L
s_simplicity_of_chevalley_groups | theorem | Simplicity of Chevalley groups
t_dynkin_diagram_classification_algorithm | technique | Dynkin diagram classification by elimination
t_root_system_explicit_construction | technique | Explicit construction of root systems
t_pbw_basis_construction | technique | PBW basis construction
t_verma_module_induction | technique | Verma module construction by parabolic induction
t_complete_reducibility_via_casimir | technique | Complete reducibility via Casimir element
t_freudenthal_recursive_algorithm | technique | Freudenthal's recursive multiplicity algorithm
t_killing_form_root_computation | technique | Killing form computation via roots
t_cartan_subalgebra_construction | technique | Cartan subalgebra construction from regular elements
t_chevalley_group_construction | technique | Chevalley group construction
t_sl2_triple_technique | technique | sl(2)-triple technique ;; sl(2)-copy method
s_ordered_set | axiom | Ordered set (partially ordered set)
s_field_axioms | axiom | Field axioms
s_upper_bound_and_supremum | axiom | Upper bound and supremum
s_lower_bound_and_infimum | axiom | Lower bound and infimum
s_dedekind_cut_construction | state | Dedekind cuts construction of R
s_uniqueness_of_lub | theorem | Uniqueness of least upper bound
s_existence_of_nth_roots | theorem | Existence of nth roots in R
s_limit_point_definition | axiom | Limit point of a set
s_isolated_point | axiom | Isolated point
s_perfect_set | axiom | Perfect set
s_bounded_set_metric | axiom | Bounded set in a metric space
s_dense_set | axiom | Dense set
s_open_cover | axiom | Open cover
s_compact_set_definition | axiom | Compact set
s_separated_sets | axiom | Separated sets
s_countability_of_Q | theorem | Countability of the rationals
s_countable_union_countable | theorem | Countable union of countable sets is countable
s_every_neighborhood_is_open | theorem | Every neighborhood is an open set
s_de_morgan_open_closed | theorem | De Morgan laws for open and closed sets
s_compact_subset_metric_is_closed | theorem | Compact subsets of metric spaces are closed
s_infinite_subset_compact_has_limit_point | theorem | Infinite subset of compact set has a limit point
s_k_cell_compactness | theorem | Every k-cell is compact
s_compact_metric_is_separable | theorem | Compact metric space is separable
s_connected_subsets_of_R_are_intervals | theorem | Connected subsets of R are intervals
s_cantor_set | state | Cantor set
t_cantor_diagonalization | technique | Cantor diagonalization technique
t_nested_intervals | technique | Nested intervals technique
s_convergent_sequence | axiom | Convergent sequence in a metric space
s_subsequence | axiom | Subsequence
s_subsequential_limit | state | Subsequential limit
s_series_convergence | axiom | Series convergence
s_absolute_convergence | state | Absolute convergence
s_conditional_convergence | state | Conditional convergence
s_power_series | axiom | Power series
s_radius_of_convergence | state | Radius of convergence
s_rearrangement_of_series | axiom | Rearrangement of a series
s_uniqueness_of_limits | theorem | Uniqueness of limits in metric spaces
s_convergent_sequences_bounded | theorem | Convergent sequences are bounded
s_algebraic_limit_theorem | theorem | Algebraic limit theorem for sequences
s_subsequences_converge_to_same_limit | theorem | Subsequences converge to the same limit
s_completeness_of_Rk | theorem | Completeness of R^k
s_limsup_liminf_always_exist | theorem | Limsup and liminf always exist in the extended reals
s_cauchy_criterion_for_series | theorem | Cauchy criterion for series convergence
s_comparison_test | theorem | Comparison test for series
s_cauchy_condensation_test | theorem | Cauchy condensation test
s_p_series_convergence | theorem | p-series convergence
s_convergence_of_e_series | theorem | Convergence of e = sum 1/n!
s_root_test | theorem | Root test (Cauchy)
s_ratio_test | theorem | Ratio test (d'Alembert)
s_root_test_stronger_than_ratio | theorem | Root test is at least as strong as ratio test
s_absolute_convergence_implies_convergence | theorem | Absolute convergence implies convergence
s_abel_theorem_power_series | theorem | Abel's theorem on power series
s_cauchy_hadamard_formula | theorem | Cauchy-Hadamard formula for radius of convergence
s_riemann_rearrangement_theorem | theorem | Riemann rearrangement theorem
s_dirichlet_test_series | theorem | Dirichlet's test for series convergence
t_comparison_test_technique | technique | Comparison test technique
t_root_ratio_test_technique | technique | Root and ratio test technique
t_abel_summation_technique | technique | Abel summation technique
s_limit_of_function_at_point | axiom | Limit of a function at a point
s_uniformly_continuous_function | axiom | Uniformly continuous function
s_discontinuities_classification | state | Classification of discontinuities
s_composition_continuous_is_continuous | theorem | Composition of continuous functions is continuous
s_continuous_image_compact_is_compact | theorem | Continuous image of compact set is compact
s_continuous_preserves_connectedness | theorem | Continuous image of connected set is connected
s_monotonic_jump_discontinuities | theorem | Monotonic functions have countably many discontinuities
s_uniform_continuity_on_compact | theorem | Uniform continuity theorem
s_continuous_bijection_compact_hausdorff_homeomorphism | theorem | Continuous bijection from compact to Hausdorff is homeomorphism
t_epsilon_delta_technique | technique | Epsilon-delta technique
s_continuously_differentiable | axiom | Continuously differentiable function (C^1)
s_differentiability_implies_continuity | theorem | Differentiability implies continuity
s_local_extremum_theorem | theorem | Local extremum theorem
s_differentiation_vector_valued | theorem | Differentiation of vector-valued functions
t_mvt_technique | technique | Mean value theorem technique
t_lhopital_technique | technique | L'Hopital technique
t_taylor_expansion_technique | technique | Taylor expansion technique
s_partition_of_interval | axiom | Partition of an interval
s_upper_lower_rs_sums | state | Upper and lower Riemann-Stieltjes sums
s_riemann_stieltjes_integral_definition | axiom | Riemann-Stieltjes integral
s_riemann_integral | axiom | Riemann integral
s_bounded_variation | axiom | Function of bounded variation
s_rectifiable_curve | state | Rectifiable curve and arc length
s_rs_integrability_criterion | theorem | Riemann-Stieltjes integrability criterion
s_continuous_functions_rs_integrable | theorem | Continuous functions are R-S integrable
s_monotonic_functions_rs_integrable | theorem | Monotonic functions are R-S integrable
s_rs_step_integrator | theorem | R-S integral with step function integrator
s_rs_reduces_to_riemann | theorem | R-S integral reduces to Riemann integral
s_linearity_of_rs_integral | theorem | Linearity of the Riemann-Stieltjes integral
s_additivity_over_intervals | theorem | Additivity of the integral over intervals
s_ftc_first_form | theorem | Fundamental theorem of calculus (first form)
s_ftc_second_form | theorem | Fundamental theorem of calculus (second form)
s_rectifiable_curves_finite_arc_length | theorem | Rectifiable curves characterized via bounded variation
t_rs_integration_technique | technique | Riemann-Stieltjes integration technique
s_pointwise_convergence_definition | axiom | Pointwise convergence of function sequence
s_uniform_convergence_definition | axiom | Uniform convergence of function sequence
s_pointwise_bounded_family | axiom | Pointwise bounded family of functions
s_uniformly_bounded_family | axiom | Uniformly bounded family of functions
s_algebra_of_functions | axiom | Algebra of functions and subalgebra of C(X)
s_separating_points | axiom | Algebra separating points
s_self_adjoint_algebra | axiom | Self-adjoint algebra of functions
s_cauchy_criterion_uniform_convergence | theorem | Cauchy criterion for uniform convergence
s_weierstrass_m_test | theorem | Weierstrass M-test
s_uniform_limit_continuous | theorem | Uniform limit of continuous functions is continuous
s_term_by_term_integration | theorem | Term-by-term integration of uniformly convergent series
s_uniform_limit_rs_integrable | theorem | Uniform limit of R-S integrable functions is R-S integrable
s_term_by_term_differentiation | theorem | Conditions for term-by-term differentiation
s_equicontinuity_implies_uniform_convergence | theorem | Equicontinuity with pointwise convergence implies uniform convergence
t_m_test_technique | technique | Weierstrass M-test technique
t_equicontinuous_approximation_technique | technique | Equicontinuous approximation technique
t_stone_weierstrass_technique | technique | Stone-Weierstrass approximation technique
s_power_series_continuity | theorem | Power series are continuous on their disk of convergence
s_uniqueness_of_power_series | theorem | Uniqueness theorem for power series
s_term_by_term_differentiation_power_series | theorem | Term-by-term differentiation of power series
s_composition_product_power_series | theorem | Composition and product of power series
s_exponential_functional_equation | theorem | Exponential functional equation
s_gamma_recurrence | theorem | Gamma function recurrence relation
s_gamma_analyticity | theorem | Gamma function is analytic on the right half-plane
t_power_series_manipulation | technique | Power series manipulation technique
t_fourier_analysis_technique | technique | Fourier analysis technique
t_gamma_function_technique | technique | Gamma function technique
s_total_derivative | axiom | Total (Frechet) derivative
s_partial_derivative | axiom | Partial derivative
s_jacobian_matrix | state | Jacobian matrix and determinant
s_c1_mapping | axiom | C^1 mapping
s_chain_rule_several_variables | theorem | Chain rule for functions of several variables
s_c1_implies_differentiable | theorem | C^1 implies differentiable
s_clairaut_theorem | theorem | Clairaut's theorem (symmetry of mixed partials)
s_mvt_vector_valued | theorem | Mean value estimate for vector-valued functions
s_inverse_function_theorem | theorem | Inverse function theorem
s_implicit_function_theorem | theorem | Implicit function theorem
s_rank_theorem | theorem | Rank theorem (constant rank)
s_determinant_multilinear_alternating | theorem | Determinant is the unique multilinear alternating form
t_contraction_iteration_technique | technique | Contraction iteration technique
t_inverse_implicit_technique | technique | Inverse and implicit function technique
t_multilinear_algebra_technique | technique | Multilinear algebra technique
s_k_form | axiom | k-form on R^n
s_closed_differential_form | state | Closed differential form
s_exact_differential_form | state | Exact differential form
s_pullback_of_differential_form | state | Pullback of a differential form
s_k_surface | axiom | k-surface in R^n
s_boundary_operator_surface | state | Boundary operator for k-surfaces
s_orientation_of_surface | axiom | Orientation of a k-surface
s_d_squared_zero | theorem | d^2 = 0 (exterior derivative is nilpotent)
s_pullback_commutes_with_d | theorem | Pullback commutes with exterior derivative
s_exact_forms_zero_integral | theorem | Exact forms have zero integral over cycles
s_closed_forms_exact_simply_connected | theorem | Closed forms are exact in simply connected domains
t_exterior_calculus_technique | technique | Exterior calculus technique
t_pullback_technique | technique | Pullback technique for differential forms
t_partition_of_unity_technique | technique | Partition of unity technique
t_stokes_theorem_technique | technique | Stokes' theorem technique
s_lebesgue_integral_definition | axiom | Lebesgue integral
s_l1_space | axiom | L^1 space
s_measure_zero_set | axiom | Set of measure zero
s_l2_space | axiom | L^2 space
s_regularity_of_lebesgue_measure | theorem | Regularity of Lebesgue measure
s_countable_additivity_measure | theorem | Countable additivity of measures
s_measurable_functions_form_algebra | theorem | Measurable functions form an algebra
s_pointwise_limit_measurable | theorem | Pointwise limit of measurable functions is measurable
s_every_measurable_limit_of_simple | theorem | Every measurable function is a limit of simple functions
s_riemann_vs_lebesgue_comparison | theorem | Comparison of Riemann and Lebesgue integrals
s_characterization_riemann_integrability | theorem | Characterization of Riemann integrability
s_l2_is_hilbert | theorem | L^2 is a Hilbert space
t_lebesgue_integration_technique | technique | Lebesgue integration technique
t_dominated_convergence_technique | technique | Dominated convergence technique
t_monotone_convergence_technique | technique | Monotone convergence technique
t_fatou_lemma_technique | technique | Fatou's lemma technique
s_separable_implies_countable_base | theorem | Separable metric space has a countable base
s_measurable_function | state | Measurable function
s_positive_measure | axiom | Positive measure
s_complex_measure | axiom | Complex measure
s_vitali_caratheodory_theorem | theorem | Vitali-Carathéodory theorem
s_completion_of_measure_space | state | Completion of a measure space
s_hardy_littlewood_maximal_function | state | Hardy-Littlewood maximal function
s_lebesgue_point | state | Lebesgue point
s_absolutely_continuous_function | axiom | Absolutely continuous function
s_ftc_lebesgue | theorem | Fundamental theorem of calculus for Lebesgue integration
s_rising_sun_lemma | theorem | Rising sun lemma
s_cantor_function | state | Cantor function (devil's staircase)
s_fourier_inversion_formula_Rn | theorem | Fourier inversion formula on R^n
s_fourier_transform_on_schwartz_space | theorem | Fourier transform is an automorphism of Schwartz space
s_cauchy_integral_formula_derivatives | theorem | Cauchy integral formula for derivatives
s_power_series_representation_holomorphic | theorem | Power series representation of holomorphic functions
s_isolated_zeros_theorem | theorem | Isolated zeros of holomorphic functions
s_classification_of_isolated_singularities | state | Classification of isolated singularities
t_calculus_of_residues | technique | Calculus of residues
s_cauchy_estimates | theorem | Cauchy estimates for derivatives
s_mobius_transformation | state | Möbius transformation
s_automorphisms_of_unit_disk | theorem | Automorphisms of the unit disk
s_normal_family_holomorphic | state | Normal family of holomorphic functions
s_blaschke_product | state | Blaschke product
s_blaschke_condition | theorem | Blaschke condition
s_muntz_szasz_theorem | theorem | Müntz-Szász theorem
s_hardy_space_Hp_disk | axiom | Hardy space H^p on the disk
s_fatou_theorem_radial_limits | theorem | Fatou's theorem on radial limits
s_poisson_kernel_disk | state | Poisson kernel for the unit disk
s_poisson_integral_representation | theorem | Poisson integral representation
s_f_and_m_riesz_theorem | theorem | F. and M. Riesz theorem
s_inner_function | state | Inner function
s_outer_function | state | Outer function
s_canonical_factorization_Hp | theorem | Canonical factorization of H^p functions
s_beurling_theorem_invariant_subspaces | theorem | Beurling's theorem on invariant subspaces of H^2
s_subharmonic_function | axiom | Subharmonic function
s_poisson_kernel | state | Poisson kernel
s_m_riesz_conjugate_function_theorem | theorem | M. Riesz conjugate function theorem
t_vitali_caratheodory_approximation | technique | Vitali-Carathéodory approximation technique
s_nevanlinna_class | axiom | Nevanlinna class
t_duality_technique_Lp | technique | Duality technique for L^p bounds
t_maximal_function_technique | technique | Maximal function technique
s_szego_theorem | theorem | Szegő theorem
s_logarithmic_integral_condition_Hp | theorem | Log-integrability condition for H^p
t_conformal_mapping_technique | technique | Conformal mapping technique
s_schwarz_christoffel_formula | theorem | Schwarz-Christoffel formula
s_sigma_algebra | axiom | σ-algebra
s_semifinite_measure | axiom | Semifinite measure
s_counting_measure | state | Counting measure
s_monotonicity_of_measures | theorem | Monotonicity of measures
s_subadditivity_of_measures | theorem | Subadditivity of measures
s_continuity_from_below_measures | theorem | Continuity from below for measures
s_continuity_from_above_measures | theorem | Continuity from above for measures
s_complete_measure | axiom | Complete measure
s_caratheodory_measurability | axiom | Carathéodory measurability (μ*-measurable set)
s_caratheodory_theorem_outer_measure | theorem | Carathéodory's theorem (outer measure restriction)
s_algebra_of_sets | axiom | Algebra of sets
s_outer_measure_induced_by_premeasure | state | Outer measure induced by a premeasure
s_translation_invariance_lebesgue | theorem | Translation invariance of Lebesgue measure
s_dilation_property_lebesgue | theorem | Dilation property of Lebesgue measure
s_vitali_nonmeasurable_set | state | Non-Lebesgue-measurable set (Vitali set)
s_distribution_function | axiom | Distribution function (Lebesgue-Stieltjes)
s_borel_measure_distribution_function_correspondence | theorem | Correspondence between Borel measures and distribution functions
s_cantor_measure | state | Cantor measure (Cantor-Lebesgue measure)
s_borel_measurable_function | state | Borel measurable function
s_lebesgue_measurable_function | state | Lebesgue measurable function
s_continuous_functions_are_measurable | theorem | Continuous functions are Borel measurable
s_measurability_criterion_preimages | theorem | Measurability criterion via generating sets
s_closure_measurable_functions_under_limits | theorem | Closure of measurable functions under limits
s_simple_function | axiom | Simple function
s_integral_nonneg_simple_function | axiom | Integral of a nonnegative simple function
s_integral_nonneg_measurable_function | axiom | Integral of a nonnegative measurable function
s_linearity_integration_nonneg | theorem | Linearity of integration for nonnegative functions
s_almost_everywhere | axiom | Almost everywhere (a.e.)
s_convergence_in_L1 | axiom | Convergence in L¹
s_convergence_in_measure_subsequence | theorem | Convergence in measure subsequence theorem
s_product_sigma_algebra | state | Product σ-algebra
s_measurability_of_sections | theorem | Measurability of sections
s_lebesgue_measure_Rn | state | Lebesgue measure on ℝⁿ
s_polar_coordinates_integration | theorem | Polar coordinates integration formula
s_volume_unit_ball_Rn | state | Volume of the unit ball in ℝⁿ
s_L1_convergence_implies_measure_convergence | theorem | Convergence in L¹ implies convergence in measure
s_positive_set_signed_measure | axiom | Positive set for a signed measure
s_negative_set_signed_measure | axiom | Negative set for a signed measure
s_positive_variation_measure | state | Positive variation of a signed measure
s_negative_variation_measure | state | Negative variation of a signed measure
s_absolutely_continuous_measure | axiom | Absolutely continuous measure
s_chain_rule_radon_nikodym | theorem | Chain rule for Radon-Nikodym derivatives
s_total_variation_complex_measure | state | Total variation of a complex measure
s_total_variation_complex_is_finite | theorem | Total variation of a complex measure is finite
s_polar_decomposition_complex_measure | theorem | Polar decomposition of a complex measure
s_radon_nikodym_complex_measures | theorem | Radon-Nikodym theorem for complex measures
s_lebesgue_set | state | Lebesgue set
s_differentiation_borel_measures_Rn | theorem | Differentiation of Borel measures on ℝⁿ
s_total_variation_function | state | Total variation function
s_bv_difference_of_monotone | theorem | BV functions as differences of monotone functions
s_ac_function_measure_correspondence | theorem | AC functions and absolute continuity of measures
s_bv_functions_differentiable_ae | theorem | BV functions are differentiable a.e.
s_singular_function | axiom | Singular function
s_decomposition_bv_functions | theorem | Decomposition of BV functions
s_integration_by_parts_ac | theorem | Integration by parts for absolutely continuous functions
s_lebesgue_radon_nikodym_on_R | theorem | Lebesgue-Radon-Nikodym theorem on ℝ
s_net_convergence | axiom | Net (generalized sequence)
s_subnet | axiom | Subnet
s_topology_characterized_by_nets | theorem | Characterization of topology by nets
s_compactness_via_nets | theorem | Compactness via nets
s_topological_vector_space | axiom | Topological vector space (TVS)
s_seminorm | axiom | Seminorm
s_bounded_linear_operator | axiom | Bounded linear operator
s_operator_norm | state | Operator norm
s_operator_space_banach | theorem | B(X,Y) is Banach when Y is Banach
s_separation_of_points_by_functionals | theorem | Separation of points by continuous linear functionals
s_second_category_set | axiom | Second category set (non-meager set)
s_banach_steinhaus_pointwise_convergence | theorem | Banach-Steinhaus theorem (pointwise convergence corollary)
s_bounded_inverse_theorem | theorem | Bounded inverse theorem
s_weak_topology_on_normed_space | axiom | Weak topology on a normed space
s_closest_point_convex_set | theorem | Closest point to a closed convex set in Hilbert space
s_separable_hilbert_isomorphic_l2 | theorem | Separable Hilbert spaces are isomorphic to ℓ²
s_quotient_normed_space | state | Quotient normed space
s_chebyshev_markov_Lp | theorem | Chebyshev-Markov inequality for Lp
s_lp_inclusions_finite_measure | theorem | Lp inclusions under finite measure
s_dual_of_Lp | theorem | Dual of Lp
s_minkowski_integral_inequality | theorem | Minkowski's inequality for integrals
s_weak_Lp_space | axiom | Weak Lp space
s_Cc_space | axiom | Cc(X) — continuous functions with compact support
s_C0_space | axiom | C₀(X) — continuous functions vanishing at infinity
s_lusin_theorem_radon | theorem | Lusin's theorem for Radon measures
s_dual_of_C0 | theorem | Dual of C₀(X)
s_approximate_identity_Rn | state | Approximate identity on ℝⁿ
s_fourier_transform_L2_extension | state | Fourier transform on L² (extension from L¹ ∩ L²)
s_test_function_space | axiom | Test function space D(Ω) = C_c^∞(Ω)
s_distribution_space | axiom | Distribution space D'(Ω)
s_support_of_distribution | state | Support of a distribution
s_distributions_compact_support | state | Distributions with compact support E'(Ω)
s_tempered_distribution | axiom | Tempered distribution S'(ℝⁿ)
s_convolution_of_distributions | state | Convolution of distributions
s_sobolev_space_Wkp | axiom | Sobolev space W^{k,p}(Ω)
s_order_of_distribution | state | Order of a distribution
s_singular_support | state | Singular support of a distribution
s_fourier_transform_tempered_distributions | state | Fourier transform of tempered distributions
s_borel_sigma_algebra_on_Rn | axiom | Borel σ-algebra on ℝⁿ
s_random_vector | axiom | Random vector
s_pi_system | axiom | π-system
s_lambda_system | axiom | λ-system (Dynkin system)
s_independence_of_sigma_algebras | axiom | Independence of σ-algebras
s_convergence_in_Lp | axiom | Convergence in Lᵖ
s_stopping_time | axiom | Stopping time
s_tail_sigma_algebra | axiom | Tail σ-algebra
s_backward_martingale | axiom | Backward (reversed) martingale
s_regular_conditional_distribution | axiom | Regular conditional distribution
s_sufficient_statistic | axiom | Sufficient statistic
s_moment_generating_function | axiom | Moment generating function
s_martingale_transform | axiom | Martingale transform
s_convergence_determining_class | axiom | Convergence-determining class
s_separating_class | axiom | Separating class
s_properties_of_conditional_expectation | state | Properties of conditional expectation
s_relations_between_convergence_modes | state | Relations between modes of convergence
s_properties_of_characteristic_functions | state | Properties of characteristic functions
s_dynkin_pi_lambda_theorem | theorem | Dynkin π-λ theorem
s_monotone_class_theorem | theorem | Monotone class theorem
s_kolmogorov_maximal_inequality | theorem | Kolmogorov's maximal inequality
s_levy_upward_theorem | theorem | Lévy's upward theorem
s_levy_downward_theorem | theorem | Lévy's downward theorem
s_levy_inversion_formula | theorem | Lévy inversion formula
s_uniqueness_theorem_characteristic_functions | theorem | Uniqueness theorem for characteristic functions
s_cramer_wold_device | theorem | Cramér-Wold device
s_method_of_moments | theorem | Method of moments (convergence)
s_conditional_jensen_inequality | theorem | Conditional Jensen's inequality
s_neyman_fisher_factorization_theorem | theorem | Neyman-Fisher factorization theorem
s_disintegration_of_measures | theorem | Disintegration of measures
s_scheffe_lemma | theorem | Scheffé's lemma
s_de_la_vallee_poussin_criterion | theorem | de la Vallée-Poussin criterion
t_truncation_technique | technique | Truncation technique
t_symmetrization_technique | technique | Symmetrization technique
t_characteristic_function_method | technique | Characteristic function method
t_tightness_prokhorov_method | technique | Tightness-Prokhorov method
t_monotone_class_argument | technique | Monotone class argument
t_dyadic_approximation_technique | technique | Dyadic approximation technique (measure theory)
t_upcrossing_argument | technique | Upcrossing argument (Doob)
t_martingale_convergence_technique | technique | Martingale technique for proving convergence
s_g_delta_set | axiom | G_δ set
s_f_sigma_set | axiom | F_σ set
s_positive_negative_parts | axiom | Positive and negative parts of a function
s_cauchy_in_measure | axiom | Cauchy in measure
s_non_atomic_measure | axiom | Non-atomic (diffuse) measure
s_atom_of_measure | axiom | Atom of a measure
s_baire_sigma_algebra | axiom | Baire σ-algebra
s_uniform_integrability_definition | axiom | Uniform integrability
s_weak_convergence_in_lp | axiom | Weak convergence in Lp
s_modular_function_group | state | Modular function of a locally compact group
s_density_of_rationals_in_r | theorem | Density of rationals in ℝ
s_simple_approximation_theorem | theorem | Simple approximation theorem
s_continuity_of_measure_below | theorem | Continuity of measure from below
s_continuity_of_measure_above | theorem | Continuity of measure from above
s_regularity_lebesgue_measure | theorem | Regularity of Lebesgue measure
s_clarkson_inequalities | theorem | Clarkson's inequalities
s_uniform_convexity_lp | theorem | Uniform convexity of Lp (1 < p < ∞)
s_radon_riesz_theorem | theorem | Radon–Riesz theorem
s_banach_saks_theorem | theorem | Banach–Saks theorem
s_weak_lower_semicontinuity_norm | theorem | Weak lower semicontinuity of the norm
s_weak_sequential_compactness_lp | theorem | Weak sequential compactness in Lp (1 < p < ∞)
s_separability_of_lp | theorem | Separability of Lp (1 ≤ p < ∞)
s_l_infinity_not_separable | theorem | L∞ is not separable
s_inclusion_lp_finite_measure | theorem | Inclusion of Lp spaces on finite measure sets
s_interpolation_lp_norms | theorem | Interpolation of Lp norms
s_schur_property_l1 | theorem | Schur property of ℓ¹
s_schauder_compact_adjoint | theorem | Schauder's theorem (compact adjoint)
s_compact_operators_closed_ideal | theorem | Compact operators form a closed ideal
s_ulam_tightness_theorem | theorem | Ulam's theorem (tightness on Polish spaces)
s_intermediate_value_nonatomic | theorem | Intermediate value property for non-atomic measures
s_completeness_convergence_measure | theorem | Completeness of convergence in measure
s_inversion_formula_haar | theorem | Inversion formula for Haar measure
s_condensation_of_singularities | theorem | Condensation of singularities
s_lp_approximation_by_simple | theorem | Density of simple functions in Lp
s_lp_approximation_by_continuous | theorem | Density of continuous functions in Lp(ℝ)
s_cc_dense_in_lp | theorem | Density of Cc in Lp for Radon measures
s_ac_maps_null_to_null | theorem | Absolutely continuous functions map null sets to null sets
s_lebesgue_decomposition_bv | theorem | Lebesgue decomposition of BV functions
s_daniell_stone_representation | theorem | Daniell–Stone representation theorem
s_reflexivity_of_lp | theorem | Reflexivity of Lp (1 < p < ∞)
s_l1_not_reflexive | theorem | L¹ is not reflexive
s_haar_measure_uniqueness | theorem | Uniqueness of Haar measure
s_l2_is_hilbert_space | state | L² is a Hilbert space
t_exhaustion_sigma_finite | technique | Exhaustion technique (σ-finite decomposition)
t_density_argument_lp | technique | Density argument in Lp
t_duality_method_functional_analysis | technique | Duality method (functional analysis)
t_uniform_integrability_technique | technique | Uniform integrability technique
t_weak_compactness_argument | technique | Weak compactness argument
t_egoroff_lusin_chain | technique | Egoroff–Lusin technique chain
t_baire_category_method | technique | Baire category method in analysis
t_radon_nikodym_technique | technique | Radon–Nikodym technique
t_completeness_via_rapidly_cauchy | technique | Completeness via rapidly Cauchy subsequences
t_tonelli_fubini_criterion | technique | Tonelli–Fubini criterion
s_chordal_metric | axiom | Chordal metric on the Riemann sphere
s_stereographic_projection | axiom | Stereographic projection
s_argument_of_complex_number | axiom | Argument of a complex number
s_polar_representation_complex | axiom | Polar representation of complex numbers
s_complex_conjugate | axiom | Complex conjugate
s_modulus_complex_number | axiom | Modulus of a complex number
s_complex_power | axiom | Complex power z^alpha
s_complex_trigonometric_functions | axiom | Complex trigonometric functions
s_cross_ratio | axiom | Cross-ratio
s_riemann_surface_of_sqrt_z | axiom | Riemann surface of sqrt(z)
s_riemann_surface_of_log_z | axiom | Riemann surface of log z
s_simply_periodic_function | axiom | Simply periodic function
s_period_lattice | axiom | Period lattice
s_fundamental_parallelogram | axiom | Fundamental parallelogram
s_order_of_entire_function | axiom | Order of an entire function
s_genus_of_entire_function | axiom | Genus of an entire function
s_schwarz_triangle_function | axiom | Schwarz triangle function
s_analytic_continuation_along_path | axiom | Analytic continuation along a path
s_complete_analytic_function | axiom | Complete analytic function (Weierstrass)
s_algebraic_function | axiom | Algebraic function
s_riemann_surface_of_algebraic_function | axiom | Riemann surface of an algebraic function
s_monodromy_group_linear_ode | axiom | Monodromy group of a linear ODE
s_harmonic_conjugate | axiom | Harmonic conjugate
s_residue_of_f_at_z0 | axiom | Residue of f at an isolated singularity
s_complex_line_integral | axiom | Complex line integral
s_pole_of_order_m | axiom | Pole of order m
s_symmetry_principle_mobius | theorem | Symmetry principle for Möbius transformations
s_conformality_of_analytic_functions | theorem | Conformality of analytic functions
s_automorphisms_upper_half_plane | theorem | Automorphisms of the upper half-plane
s_cauchy_theorem_multiply_connected | theorem | Cauchy's theorem for multiply connected regions
s_minimum_modulus_principle | theorem | Minimum modulus principle
s_schwarz_theorem_harmonic | theorem | Schwarz's theorem (Poisson integral solves Dirichlet problem for the disk)
s_harnack_principle | theorem | Harnack's principle
s_taylor_theorem_complex | theorem | Taylor's theorem (complex analysis)
s_laurent_theorem | theorem | Laurent's theorem
s_conformal_mapping_of_rectangles | theorem | Conformal mapping of rectangles to the upper half-plane
s_canonical_mapping_multiply_connected | theorem | Canonical conformal mapping of multiply connected regions
s_liouville_theorems_elliptic | theorem | Liouville's theorems on elliptic functions
s_differential_equation_weierstrass_p | theorem | Differential equation for the Weierstrass p-function
s_monodromy_theorem | theorem | Monodromy theorem
s_cauchy_theorem_homological | theorem | General Cauchy theorem (homological version)
t_normal_families_method | technique | Normal families method
s_wirtinger_derivatives | axiom | Wirtinger derivatives
s_primitive_of_holomorphic_function | axiom | Primitive (antiderivative) of a holomorphic function
s_ml_inequality | theorem | ML inequality (estimation lemma)
s_uniform_limit_of_holomorphic_functions | theorem | Uniform limit of holomorphic functions is holomorphic
s_zero_of_order_n | axiom | Zero of order n
s_principal_part_of_laurent_series | axiom | Principal part of a Laurent series
s_meromorphic_function | axiom | Meromorphic function
s_branch_of_logarithm | axiom | Branch of the logarithm
s_simply_connected_characterization | theorem | Characterization of simply connected domains
t_keyhole_contour | technique | Keyhole contour technique
s_riemann_sphere | axiom | Riemann sphere
s_automorphisms_of_riemann_sphere | theorem | Automorphisms of the Riemann sphere
s_cayley_transform | state | Cayley transform (complex analysis)
s_elliptic_function | axiom | Elliptic function
s_no_elliptic_function_of_order_one | theorem | No nonconstant elliptic function of order one
s_elliptic_function_rational_in_p_and_p_prime | theorem | Every elliptic function is rational in ℘ and ℘'
s_theta_weierstrass_connection | state | Connection between theta functions and Weierstrass functions
s_r2_representation_number | state | Representation number r₂(n) via theta functions
s_r4_representation_number | state | Representation number r₄(n) via theta functions
s_euler_partition_identity | theorem | Euler's partition identity
s_weierstrass_p_via_theta_null | state | Weierstrass ℘-function from theta-null values
s_two_square_theorem_via_theta | theorem | Two-square theorem via theta functions
t_generating_function_theta_method | technique | Generating function method via theta functions
s_triangle_inequality_complex | theorem | Triangle inequality (complex)
s_branch_of_multivalued_function | axiom | Branch of a multi-valued function
s_fixed_point_classification_mobius | state | Fixed-point classification of Möbius transformations
s_homotopy_of_curves | axiom | Homotopy of curves
s_cycle_complex_analysis | axiom | Cycle (complex analysis)
s_homology_null_homologous_cycle | axiom | Null-homologous cycle
s_maximum_modulus_theorem | theorem | Maximum modulus theorem
s_space_C_G_Omega | axiom | Space C(G, Ω)
s_germ_of_analytic_function | axiom | Germ of an analytic function
s_sheaf_of_germs_holomorphic | state | Sheaf of germs of holomorphic functions
s_riemann_surface_abstract | axiom | Riemann surface (abstract)
s_converse_mean_value_property | theorem | Converse of the mean value property
s_maximum_principle_harmonic | theorem | Maximum principle for harmonic functions
s_dirichlet_problem_disk | state | Dirichlet problem for the disk
s_superharmonic_function | axiom | Superharmonic function
s_maximum_principle_subharmonic | theorem | Maximum principle for subharmonic functions
s_barrier_boundary_point | axiom | Barrier at a boundary point
s_regular_boundary_point | state | Regular boundary point
s_poisson_jensen_formula | theorem | Poisson–Jensen formula
s_type_of_entire_function | state | Type of an entire function
s_canonical_product | state | Canonical product
s_exponent_of_convergence | state | Exponent of convergence
s_landau_theorem | theorem | Landau's theorem (complex analysis)
s_schottky_theorem | theorem | Schottky's theorem
s_montel_three_omitted | theorem | Montel's theorem (omitting two values)
s_simple_connectivity_characterization | theorem | Characterization of simple connectivity
s_conformal_equivalence | axiom | Conformal equivalence
s_hyperbolic_distance | state | Hyperbolic distance on the disk
s_caratheodory_boundary_correspondence | theorem | Carathéodory boundary correspondence theorem
s_covering_space | axiom | Covering space
s_analytic_covering_map | axiom | Analytic covering map
s_deck_transformation | state | Deck transformation
s_universal_covering_surface | state | Universal covering surface
s_schlicht_class_S | axiom | Schlicht class S
s_koebe_function | state | Koebe function
s_area_theorem_gronwall | theorem | Area theorem (Gronwall)
s_bieberbach_inequality | theorem | Bieberbach's inequality
s_distortion_theorem | theorem | Distortion theorem (Koebe)
s_growth_theorem | theorem | Growth theorem (univalent functions)
t_loewner_method | technique | Loewner's method
s_loewner_differential_equation | state | Loewner's differential equation (radial)
s_loewner_chain | state | Loewner chain
s_robertson_conjecture | state | Robertson conjecture
s_milin_conjecture | state | Milin conjecture
s_boundary_values_Hp | theorem | Boundary values of H^p functions
s_Hp_norm_completeness | theorem | H^p norm and completeness
s_singular_inner_function | state | Singular inner function
s_Hp_duality | theorem | H^p duality
s_green_function_domain | state | Green's function of a domain
s_symmetry_green_function | theorem | Symmetry of Green's function
s_logarithmic_potential | state | Logarithmic potential
s_energy_of_measure | state | Energy of a measure
s_equilibrium_measure | state | Equilibrium measure
s_logarithmic_capacity | state | Logarithmic capacity
s_robin_constant | state | Robin constant
s_polar_set | axiom | Polar set (potential theory)
s_riesz_decomposition_subharmonic | theorem | Riesz decomposition theorem (subharmonic functions)
s_evans_theorem | theorem | Evans's theorem
s_weierstrass_product_domains | theorem | Weierstrass product theorem for domains
s_lindelof_boundary_theorem | theorem | Lindelöf's theorem (boundary behavior)
s_corona_theorem | theorem | Corona theorem
s_separation_properties_tvs | theorem | Separation properties of topological vector spaces
s_continuity_boundedness_equivalence_linear_maps | theorem | Continuity–boundedness equivalence for linear maps
s_seminorm_characterization_local_convexity | theorem | Seminorm characterization of local convexity
s_bilinear_map_continuity_theorem | theorem | Bilinear map continuity theorem (separate implies joint)
s_balanced_convex_hull | axiom | Balanced convex hull
s_extreme_point | axiom | Extreme point
s_milman_converse_theorem | theorem | Milman's converse theorem
s_weak_star_convergence | axiom | Weak* convergence
s_schauder_basis | axiom | Schauder basis
s_day_lp_trivial_dual | theorem | Trivial dual of Lp for 0 < p < 1
s_lyapunov_vector_measure_theorem | theorem | Lyapunov's vector measure theorem
s_space_E_omega | axiom | Space E(Ω) = C^∞(Ω)
t_regularization_by_convolution | technique | Regularization of distributions by convolution
s_completeness_of_distribution_space | theorem | Completeness of distribution spaces
s_sobolev_lemma | theorem | Sobolev's lemma (regularity from L² derivatives)
s_local_structure_theorem_distributions | theorem | Local structure theorem for distributions
s_elliptic_regularity_theorem | theorem | Elliptic regularity theorem
s_hypoelliptic_operator | axiom | Hypoelliptic operator
s_characterization_hypoelliptic_constant_coefficients | theorem | Characterization of hypoelliptic operators with constant coefficients
s_spanning_translates_L1 | theorem | Spanning of translates in L¹(ℝⁿ)
s_spectral_synthesis_problem | state | Spectral synthesis problem
s_failure_of_spectral_synthesis | theorem | Failure of spectral synthesis (Malliavin)
s_wiener_general_tauberian_theorem | theorem | Wiener's general Tauberian theorem
s_ideals_quotient_banach_algebra | axiom | Ideals and quotient algebras in Banach algebras
s_shilov_boundary | axiom | Shilov boundary
s_wiener_lemma | theorem | Wiener's lemma
t_symbolic_calculus_banach_algebra | technique | Symbolic calculus in commutative Banach algebras
s_toeplitz_operator | axiom | Toeplitz operator
s_shift_operator | axiom | Shift operator (unilateral and bilateral)
s_positive_square_root | theorem | Positive square root of a positive operator
s_unbounded_operator | axiom | Unbounded operator
s_closable_operator | axiom | Closable operator
s_symmetric_operator | axiom | Symmetric operator
s_adjoint_unbounded_operator | axiom | Adjoint of an unbounded operator
s_essentially_self_adjoint_operator | axiom | Essentially self-adjoint operator
s_deficiency_indices | axiom | Deficiency indices
s_von_neumann_extension_theory | theorem | Von Neumann's extension theory for symmetric operators
s_nelson_analytic_vectors_theorem | theorem | Nelson's theorem on analytic vectors
s_semigroup_generation_by_laplacian | theorem | Semigroup generation by the Laplacian
t_neumann_series | technique | Neumann series
s_trace_class_operator | axiom | Trace class operator
s_fredholm_index | state | Fredholm index
s_spectral_multiplicity_theory | state | Spectral multiplicity theory
s_tensor_product_of_hilbert_spaces | axiom | Tensor product of Hilbert spaces
s_relative_bound | axiom | Relative bound (Kato bound)
s_klmn_theorem | theorem | KLMN theorem
s_form_representation_theorem | theorem | Form representation theorem (Kato)
t_friedrichs_extension | technique | Friedrichs extension
s_analytic_vector | axiom | Analytic vector
s_kato_inequality | theorem | Kato's inequality
s_hypercontractive_semigroup | state | Hypercontractive semigroup
s_nelson_best_hypercontractive_estimate | theorem | Nelson's best hypercontractive estimate
s_diamagnetic_inequality | theorem | Diamagnetic inequality
s_nelson_commutator_theorem | theorem | Nelson's commutator theorem
s_kato_class | axiom | Kato class
t_cook_method | technique | Cook's method
t_enss_method | technique | Enss method
s_kato_smoothness | axiom | Kato smoothness (H-smoothness)
s_kato_smoothness_absolute_continuity | theorem | Kato smoothness implies absolute continuity
s_lippmann_schwinger_equation | state | Lippmann–Schwinger equation
t_stationary_scattering_theory | technique | Stationary scattering theory
s_multichannel_scattering | state | Multichannel scattering
s_asymptotic_completeness_n_body | theorem | Asymptotic completeness for N-body systems
s_dollard_modified_wave_operators | state | Dollard modified wave operators
s_hvz_theorem | theorem | HVZ theorem (Hunziker–van Winter–Zhislin)
s_zhislin_theorem | theorem | Zhislin's theorem
s_analytic_family_type_A | axiom | Analytic family of type (A)
s_analytic_family_type_B | axiom | Analytic family of type (B)
t_asymptotic_perturbation_theory | technique | Asymptotic perturbation theory
t_semiclassical_approximation | technique | Semiclassical approximation
s_perturbation_determinant | state | Perturbation determinant
s_resonance_schrodinger | state | Resonance (Schrödinger operator)
s_absence_of_singular_continuous_spectrum | theorem | Absence of singular continuous spectrum
s_virial_theorem | theorem | Virial theorem
s_temple_inequality | theorem | Temple's inequality
s_kato_birman_invariance_principle | theorem | Kato–Birman invariance principle
t_ims_localization_formula | technique | IMS localization formula
s_rollnik_class | axiom | Rollnik class
s_simon_absence_positive_eigenvalues | theorem | Simon's theorem on absence of positive eigenvalues
t_decoupling_inequalities | technique | Decoupling inequalities
s_hahn_banach_first_geometric_form | theorem | Hahn-Banach First Geometric Form
s_hahn_banach_second_geometric_form | theorem | Hahn-Banach Second Geometric Form
s_conjugate_convex_function | axiom | Conjugate Convex Function (Legendre-Fenchel Transform)
s_fenchel_moreau_theorem | theorem | Fenchel-Moreau Theorem
s_young_fenchel_inequality | theorem | Young-Fenchel Inequality
s_subdifferential | axiom | Subdifferential
s_complemented_subspace | axiom | Complemented Subspace
s_weak_star_topology | axiom | Weak* Topology σ(E*, E)
s_finite_rank_operator | axiom | Finite-Rank Operator
s_courant_fischer_min_max | theorem | Courant-Fischer Min-Max Principle
s_maximal_monotone_operator | axiom | Maximal Monotone Operator
s_yosida_approximation | axiom | Yosida Approximation
s_resolvent_of_monotone_operator | axiom | Resolvent of a Monotone Operator
s_m_dissipative_operator | axiom | m-Dissipative Operator
s_c0_semigroup | axiom | Strongly Continuous (C₀) Semigroup
s_infinitesimal_generator | axiom | Infinitesimal Generator of a Semigroup
s_exponential_formula_semigroups | theorem | Exponential Formula for Semigroups
s_crandall_liggett_theorem | theorem | Crandall-Liggett Theorem
s_sobolev_W1p_1d_characterization | theorem | Characterization of W^{1,p} in One Dimension
s_compact_embedding_W1p_C_1d | theorem | Compact Embedding W^{1,p} into C in One Dimension
s_eigenvalues_second_derivative_1d | state | Eigenvalues of -d²/dx² on (0,L)
s_sobolev_extension_operator | theorem | Sobolev Extension Operator
s_morrey_inequality | theorem | Morrey's Inequality
s_poincare_wirtinger_inequality | theorem | Poincaré-Wirtinger Inequality
s_green_formula_sobolev | theorem | Green's Formula in Sobolev Spaces
s_negative_sobolev_space_H_minus_1 | axiom | Negative Sobolev Space H^{-1}(Ω)
s_sobolev_space_W0_1_p | axiom | Sobolev Space W₀^{1,p}(Ω)
s_trace_space_fractional_sobolev | axiom | Trace Space W^{1-1/p,p}(∂Ω)
s_eigenvalues_dirichlet_laplacian | theorem | Eigenvalues of -Δ on H¹₀(Ω)
s_variational_characterization_lambda_1 | theorem | Variational Characterization of First Eigenvalue
s_domain_monotonicity_eigenvalues | theorem | Domain Monotonicity of Dirichlet Eigenvalues
s_faber_krahn_inequality | theorem | Faber-Krahn Inequality
s_bv_space | axiom | Space of Functions of Bounded Variation BV(Ω)
s_polarization_identity | theorem | Polarization Identity
s_orthogonal_decomposition_hilbert | theorem | Orthogonal Decomposition Theorem (Hilbert Space)
s_duality_lp_spaces | theorem | Duality of Lp Spaces
s_lp_reflexivity | theorem | Reflexivity of Lp for 1 < p < ∞
s_density_Cc_infinity_in_Lp | theorem | Density of C_c^∞ in Lp
s_continuity_of_translation_in_lp | theorem | Continuity of Translation in Lp
s_interpolation_inequality_lp | theorem | Interpolation Inequality for Lp Norms
s_obstacle_problem_variational | state | Obstacle Problem (Variational Inequality Formulation)
s_gagliardo_nirenberg_interpolation_general | theorem | Gagliardo-Nirenberg Interpolation Inequality (General)
s_necas_inequality | theorem | Nečas Inequality
t_minkowski_gauge | technique | Minkowski Gauge Technique
t_difference_quotient_regularity | technique | Difference Quotient Technique for Regularity
t_symmetric_rearrangement_technique | technique | Symmetric Decreasing Rearrangement Technique
t_weak_convergence_methods_sobolev | technique | Weak Convergence Methods in Sobolev Spaces
s_hardy_littlewood_maximal_theorem | theorem | Hardy–Littlewood maximal theorem
s_dyadic_cubes | axiom | Dyadic cubes in ℝⁿ
s_weak_type_p_p_estimate | state | Weak-type (p,p) estimate
s_maximal_function_characterization_of_Lp | theorem | Maximal function characterization of L^p
s_hilbert_transform | axiom | Hilbert transform
s_truncated_singular_integral | state | Truncated singular integral
s_pointwise_convergence_truncated_singular_integrals | theorem | Pointwise convergence of truncated singular integrals
s_maximal_singular_integral_operator | state | Maximal singular integral operator
s_cotlar_inequality | theorem | Cotlar's inequality
t_calderon_zygmund_rotation_method | technique | Calderón–Zygmund rotation method
s_hormander_kernel_condition | axiom | Hörmander kernel condition
s_fourier_multiplier_representation_singular_integrals | theorem | Fourier multiplier representation of singular integrals
s_Lp_boundedness_of_riesz_transforms | theorem | L^p boundedness of Riesz transforms
s_riesz_transforms_and_laplacian | theorem | Riesz transforms and the Laplacian
s_conjugate_poisson_integral | state | Conjugate Poisson integral
s_spherical_harmonic_decomposition | theorem | Spherical harmonic decomposition of L²(S^{n−1})
s_hecke_identity_spherical_harmonics | theorem | Hecke's identity for spherical harmonics
s_boundary_behavior_poisson_integrals | theorem | Boundary behavior of Poisson integrals in ℝ^{n+1}_+
s_multiplier_formula_spherical_harmonics | theorem | Multiplier formula via spherical harmonics
s_littlewood_paley_g_function | axiom | Littlewood–Paley g-function
s_littlewood_paley_g_lambda_star_function | axiom | Littlewood–Paley g*_λ function
s_lusin_area_integral | axiom | Lusin area integral (S-function)
s_S_function_Lp_equivalence | theorem | S-function L^p norm equivalence
s_marcinkiewicz_multiplier_theorem | theorem | Marcinkiewicz multiplier theorem
s_partial_sums_operator_fourier | state | Partial sums operator for Fourier integrals
s_Lp_boundedness_partial_sums_1d | theorem | L^p boundedness of Fourier partial sums in one dimension
s_bessel_potential_space | axiom | Bessel potential space L^p_α
s_equivalence_sobolev_bessel_potential_spaces | theorem | Equivalence of Sobolev and Bessel potential spaces
s_lipschitz_space_Lambda_alpha | axiom | Lipschitz space Λ_α
s_zygmund_class | axiom | Zygmund class Λ_*
s_characterization_Lambda_alpha_via_poisson | theorem | Characterization of Λ_α via Poisson integral
s_characterization_Lp_alpha_via_littlewood_paley | theorem | Characterization of L^p_α via Littlewood–Paley theory
s_homogeneous_sobolev_space | axiom | Homogeneous Sobolev space Ẇ^{s,p}
s_riesz_potential_characterization_of_sobolev_spaces | theorem | Riesz potential characterization of Sobolev spaces
s_difference_characterization_bessel_potential_spaces | theorem | Difference characterization of Bessel potential spaces
s_extension_theorem_sobolev_lipschitz_domains | theorem | Extension theorem for Sobolev spaces on Lipschitz domains
s_total_extension_operator | state | Stein's total extension operator
s_restriction_theorem_riesz_potentials | theorem | Restriction theorem for Riesz potentials
t_reflection_method | technique | Reflection method for Sobolev extension
s_nontangential_maximal_function | axiom | Nontangential maximal function
s_nontangential_convergence | state | Nontangential convergence
s_fatou_theorem_upper_half_space | theorem | Fatou's theorem for the upper half-space
s_harmonic_function_characterization_area_integral | theorem | Area integral characterization of harmonic functions
s_Hp_Lp_equivalence | theorem | H^p–L^p equivalence for p > 1
s_singular_integrals_Linfty_to_BMO | theorem | Singular integrals map L^∞ to BMO
s_area_integral_characterization_Hp | theorem | Area integral characterization of H^p
s_nontangential_maximal_characterization_Hp | theorem | Nontangential maximal function characterization of H^p
s_conjugate_function_characterization_Hp | theorem | Conjugate function characterization of H^p
s_sharp_function | axiom | Fefferman–Stein sharp function
s_fefferman_stein_sharp_function_inequality | theorem | Fefferman–Stein inequality for the sharp function
s_Lp_differentiability | state | L^p differentiability
s_calderon_theorem_Lp_differentiability | theorem | Calderón's theorem on L^p differentiability
s_CZ_theorem_ae_differentiability | theorem | Calderón–Zygmund theorem on a.e. differentiability
s_maximal_operator_sobolev_functions | state | Maximal operator for Sobolev functions
s_lebesgue_point_theorem_sobolev_functions | theorem | Lebesgue point theorem for Sobolev functions
s_p_capacity | axiom | p-capacity
t_real_interpolation_method | technique | Real interpolation method (K-functional)
t_complex_interpolation_method | technique | Complex interpolation method
t_good_lambda_inequality | technique | Good-λ inequality technique
t_CZ_L2_boundedness_via_fourier | technique | L² boundedness of CZ operators via Fourier transform
t_duality_argument_weak_type | technique | Duality argument for weak-type estimates
t_subordination_formula | technique | Subordination formula for Poisson semigroup
s_multiplier_problem_for_the_ball | state | Multiplier problem for the ball
s_bochner_riesz_conjecture | state | Bochner–Riesz conjecture
s_CZ_lemma_dyadic | theorem | Calderón–Zygmund lemma (dyadic version)
s_fractional_laplacian | axiom | Fractional Laplacian (−Δ)^s
s_singular_integral_on_domain | state | Singular integral operator restricted to a domain
s_elliptic_regularity_in_Lp | theorem | L^p elliptic regularity via singular integrals
s_newtonian_potential | axiom | Newtonian potential
s_calderon_commutator_theorem | theorem | Calderón commutator theorem
s_homogeneous_distribution | axiom | Homogeneous distribution
s_laplace_beltrami_operator_sphere | axiom | Laplace–Beltrami operator on S^{n−1}
s_stein_maximal_principle | theorem | Stein's maximal principle
s_semigroup_characterization_bessel_potentials | theorem | Semigroup characterization of Bessel potentials
s_heat_kernel_Rn | axiom | Heat kernel on ℝⁿ
s_dual_characterization_H1 | theorem | Dual characterization of H¹
s_interpolation_H1_Linfty | theorem | Interpolation between H¹ and L^∞
s_calderon_formula_sobolev_norms | theorem | Calderón's formula for Sobolev norms
s_circle_group_T | axiom | Circle group 𝕋
s_fourier_coefficients_on_T | axiom | Fourier coefficients on 𝕋
s_trigonometric_polynomial | axiom | Trigonometric polynomial
s_convolution_on_T | state | Convolution on 𝕋
s_approximate_identity_on_T | axiom | Approximate identity on 𝕋
s_dirichlet_kernel_katznelson | state | Dirichlet kernel on 𝕋
s_fejer_kernel_katznelson | state | Fejér kernel on 𝕋
s_fejer_theorem_katznelson | theorem | Fejér's theorem (uniform Cesàro convergence)
s_poisson_kernel_on_T | state | Poisson kernel on 𝕋
s_abel_summability_fourier | theorem | Abel summability of Fourier series
s_uniqueness_theorem_fourier_series | theorem | Uniqueness theorem for Fourier series
s_riemann_lebesgue_on_T | theorem | Riemann–Lebesgue lemma on 𝕋
s_parseval_theorem_on_T | theorem | Parseval's theorem on 𝕋
s_bessel_inequality_on_T | theorem | Bessel's inequality on 𝕋
s_weierstrass_trigonometric_approximation | theorem | Weierstrass trigonometric approximation theorem
s_formal_multiplication_fourier_series | state | Formal multiplication of Fourier series
s_partial_sum_operator_on_T | state | Partial sum operator on 𝕋
s_dini_test | theorem | Dini's test for pointwise convergence
s_jordan_test | theorem | Jordan's test for Fourier convergence
s_dirichlet_jordan_theorem | theorem | Dirichlet–Jordan theorem
s_du_bois_reymond_theorem | theorem | du Bois-Reymond theorem (divergent Fourier series)
s_localization_principle | theorem | Localization principle for Fourier series
s_kolmogorov_divergence_example | theorem | Kolmogorov's divergence example
s_uniform_boundedness_and_divergence | theorem | Banach–Steinhaus theorem applied to Fourier divergence
s_lebesgue_constants | state | Lebesgue constants for Fourier partial sums
s_best_approximation_fourier | theorem | Best approximation by trigonometric polynomials
s_conjugate_fourier_series | state | Conjugate Fourier series
s_conjugate_function_on_T | state | Conjugate function on 𝕋
s_hilbert_transform_on_T | axiom | Hilbert transform on 𝕋
s_m_riesz_theorem_conjugate_function | theorem | M. Riesz theorem (L^p boundedness of conjugate function)
s_kolmogorov_weak_type_conjugate | theorem | Kolmogorov's weak-type (1,1) inequality for conjugate function
s_zygmund_L_log_L_theorem | theorem | Zygmund's L log L theorem for conjugate function
s_hardy_space_Hp_T | axiom | Hardy space H^p on the disk (Katznelson)
s_conjugate_poisson_kernel | state | Conjugate Poisson kernel on 𝕋
s_analytic_function_factorization_Hp | theorem | Inner-outer factorization in H^p
s_helson_lowdenslager_prediction | theorem | Helson–Lowdenslager prediction theorem
s_riesz_thorin_katznelson | theorem | Riesz–Thorin interpolation theorem (Katznelson)
s_marcinkiewicz_interpolation_katznelson | theorem | Marcinkiewicz interpolation theorem (Katznelson)
s_hausdorff_young_katznelson | theorem | Hausdorff–Young inequality (Katznelson)
s_young_convolution_katznelson | theorem | Young's convolution inequality on 𝕋 (Katznelson)
t_three_lines_lemma | technique | Three-lines lemma (Hadamard)
s_lacunary_series_hadamard | axiom | Lacunary series (Hadamard gaps)
s_lacunary_convergence_theorem | theorem | Lacunary convergence theorem
s_sidon_set | axiom | Sidon set
s_sidon_set_characterization | theorem | Characterization of Sidon sets
s_riesz_product | state | Riesz product
s_quasi_analytic_class | axiom | Quasi-analytic class C{M_n}
s_denjoy_carleman_theorem | theorem | Denjoy–Carleman theorem
s_rudin_shapiro_polynomials | state | Rudin–Shapiro polynomials
s_lacunary_Lp_equivalence | theorem | L^p equivalence for lacunary series
s_fourier_transform_L1_R | axiom | Fourier transform on L¹(ℝ)
s_fourier_inversion_formula_R | theorem | Fourier inversion formula on ℝ
s_plancherel_theorem_R | theorem | Plancherel theorem on ℝ
s_schwartz_space_katznelson | axiom | Schwartz space 𝒮(ℝ) (Katznelson)
s_tempered_distributions_katznelson | axiom | Tempered distributions 𝒮′(ℝ) (Katznelson)
s_poisson_summation_formula_katznelson | theorem | Poisson summation formula (Katznelson)
s_heisenberg_uncertainty_katznelson | theorem | Heisenberg uncertainty principle (Fourier analysis formulation)
s_wiener_tauberian_theorem_katznelson | theorem | Wiener's Tauberian theorem (Katznelson)
s_fourier_transform_of_measures_R | state | Fourier–Stieltjes transform on ℝ
s_wiener_algebra_A_R | state | Wiener algebra A(ℝ)
s_bernstein_inequality_fourier | theorem | Bernstein's inequality for trigonometric polynomials
s_lca_group_katznelson | axiom | Locally compact abelian group (Katznelson)
s_haar_measure_katznelson | theorem | Existence and uniqueness of Haar measure (Katznelson)
s_dual_group | axiom | Dual group (Pontryagin dual)
s_pontryagin_duality_katznelson | theorem | Pontryagin duality theorem (Katznelson)
s_fourier_transform_lca | state | Fourier transform on LCA groups
s_plancherel_theorem_lca | theorem | Plancherel theorem for LCA groups
s_fourier_inversion_lca | theorem | Fourier inversion formula on LCA groups
s_bochner_theorem_katznelson | theorem | Bochner's theorem on LCA groups (Katznelson)
s_structure_theorem_lca | theorem | Structure theorem for LCA groups
s_dual_group_examples | state | Standard dual group identifications
s_annihilator_subgroup | state | Annihilator of a subgroup
s_positive_definite_function_lca | axiom | Positive definite function on LCA group
s_commutative_banach_algebra_katznelson | axiom | Commutative Banach algebra (Katznelson)
s_maximal_ideal_space | state | Maximal ideal space (spectrum) of a Banach algebra
s_gelfand_transform_katznelson | state | Gelfand transform (Katznelson)
s_gelfand_representation_theorem_katznelson | theorem | Gelfand representation theorem (Katznelson)
s_wiener_lemma_katznelson | theorem | Wiener's lemma (absolutely convergent Fourier series)
s_wiener_algebra_A_T | axiom | Wiener algebra A(𝕋)
s_gelfand_mazur_katznelson | theorem | Gelfand–Mazur theorem (Katznelson)
s_spectral_synthesis | axiom | Spectral synthesis
s_spectral_analysis_katznelson | theorem | Spectral analysis holds in L¹(G)
s_malliavin_theorem | theorem | Malliavin's theorem (failure of spectral synthesis)
s_regular_banach_algebra | axiom | Regular Banach algebra
s_hull_kernel_topology | state | Hull-kernel topology
s_ideal_theory_group_algebra | state | Ideal theory of L¹(G)
t_gelfand_theory_technique | technique | Gelfand theory technique
t_approximate_identity_method | technique | Approximate identity method in harmonic analysis
t_summability_method | technique | Summability method for Fourier series
s_de_la_vallee_poussin_kernel | state | de la Vallée-Poussin kernel
s_L1_algebra_characterization_via_gelfand | theorem | Gelfand space of L¹(G) is Ĝ
s_vibrating_string_equation | axiom | Vibrating string equation
s_dalembert_solution_wave | theorem | D’Alembert’s solution to the wave equation
s_standing_wave_solution | state | Standing wave solution
t_superposition_principle_linear_PDE | technique | Superposition principle for linear PDEs
t_fourier_program | technique | Fourier’s program for solving PDEs
s_formal_solution_dirichlet_problem_disk | state | Formal solution to Dirichlet problem on the disk
s_riemann_integrable_on_circle | axiom | Riemann integrable function on the circle
s_convolution_identity_fourier_coefficients | theorem | Convolution identity for Fourier coefficients
s_approximation_theorem_good_kernels | theorem | Approximation theorem for good kernels
s_cesaro_mean_definition | axiom | Cesàro mean (Nth)
s_dirichlet_kernel_not_good | theorem | Dirichlet kernel is not a good kernel
s_fejer_kernel_is_good | theorem | Fejér kernel is a good kernel
s_poisson_kernel_is_good | theorem | Poisson kernel is a good kernel
s_L2_inner_product_circle | axiom | L² inner product on the circle
s_orthonormality_of_exponentials | theorem | Orthonormality of exponentials on 𝕋
s_mean_square_convergence_fourier | theorem | Mean-square convergence of Fourier series
s_completeness_trigonometric_system | theorem | Completeness of the trigonometric system
s_holder_pointwise_convergence_fourier | theorem | Pointwise convergence of Fourier series under Hölder condition
t_fourier_proof_isoperimetric | technique | Fourier analytic proof of the isoperimetric inequality
s_weyl_criterion_equidistribution | theorem | Weyl’s criterion for equidistribution
s_continuous_nowhere_differentiable | state | Continuous nowhere-differentiable function
t_lacunary_fourier_construction | technique | Construction via lacunary Fourier series
s_heat_equation_circle | axiom | Heat equation on the circle
s_heat_equation_circle_fourier_solution | theorem | Solution of heat equation on circle via Fourier series
s_heat_kernel_circle | state | Heat kernel on the circle
s_heat_kernel_circle_good_kernel | theorem | Heat kernel on circle is a good kernel
s_smoothing_property_heat | theorem | Smoothing property of the heat equation
s_uniqueness_heat_equation_circle | theorem | Uniqueness of solution to heat equation on circle
s_steady_state_heat_solution | state | Steady-state solution of the heat equation
s_basic_properties_fourier_transform | theorem | Basic properties of the Fourier transform on ℝ
s_fourier_transform_of_gaussian | theorem | Fourier transform of the Gaussian
s_fourier_maps_schwartz_to_schwartz | theorem | Fourier transform maps 𝒮(ℝ) to 𝒮(ℝ)
s_multiplication_formula_fourier | theorem | Multiplication formula for the Fourier transform
s_convolution_theorem_R | theorem | Convolution theorem on ℝ
s_equality_case_uncertainty | theorem | Equality case in the Heisenberg uncertainty principle
s_heat_equation_solution_R | theorem | Solution of the heat equation on ℝ via Fourier transform
s_heat_kernel_R_good_kernel | theorem | Heat kernel on ℝ is a good kernel
s_fourier_differentiation_correspondence | theorem | Fourier transform and differentiation correspondence
s_moderate_decrease_functions | axiom | Functions of moderate decrease
s_schwartz_space_Rd | axiom | Schwartz space 𝒮(ℝ^d)
t_wave_equation_solution_via_FT | technique | Solution of wave equation via Fourier transform
s_spherical_means | axiom | Spherical means
s_kirchhoff_formula_wave_R3 | theorem | Kirchhoff’s formula for the wave equation in ℝ³
s_huygens_principle | theorem | Huygens’ principle
s_failure_huygens_even_dim | theorem | Failure of Huygens’ principle in even dimensions
t_method_of_descent | technique | Hadamard’s method of descent
s_wave_solution_R2_poisson | theorem | Wave equation solution in ℝ² (Poisson’s formula)
s_radon_transform | axiom | Radon transform
s_fourier_slice_theorem | theorem | Fourier slice theorem (projection-slice theorem)
s_radon_inversion_formula | theorem | Radon inversion formula
s_radial_functions_FT_bessel | theorem | Fourier transform of radial functions via Bessel functions
s_FT_indicator_ball | state | Fourier transform of the indicator of a ball
s_discrete_fourier_transform | axiom | Discrete Fourier transform (DFT)
s_inverse_DFT | theorem | Inverse discrete Fourier transform
s_parseval_identity_DFT | theorem | Parseval’s identity for the DFT
s_convolution_on_ZNZ | axiom | Convolution on ℤ/Nℤ
s_convolution_theorem_DFT | theorem | Convolution theorem for the DFT
t_fast_fourier_transform | technique | Fast Fourier transform (FFT)
s_FFT_complexity | theorem | FFT complexity
t_polynomial_multiplication_FFT | technique | Polynomial multiplication via FFT
t_fast_integer_multiplication | technique | Fast integer multiplication via FFT
s_non_vanishing_L1_complex_characters | theorem | Non-vanishing of L(1,χ) for complex Dirichlet characters
s_non_vanishing_L1_real_characters | theorem | Non-vanishing of L(1,χ) for real Dirichlet characters
s_analytic_continuation_L_function_s_gt_0 | theorem | Analytic continuation of L(s,χ) to Re(s) > 0
s_natural_density_primes_AP | theorem | Natural density of primes in arithmetic progressions
s_gibbs_phenomenon | theorem | Gibbs phenomenon
s_absolute_convergence_fourier_smoothness | theorem | Absolute convergence of Fourier series under smoothness
s_decay_fourier_coefficients_smoothness | theorem | Decay of Fourier coefficients and smoothness
s_wirtinger_inequality | theorem | Wirtinger’s inequality
s_gronwall_inequality_integral_form | theorem | Gronwall inequality (integral form)
s_continuous_dependence_on_initial_conditions | theorem | Continuous dependence on initial conditions
s_continuous_dependence_on_parameters | theorem | Continuous dependence on parameters
s_differentiability_wrt_initial_conditions | theorem | Differentiability with respect to initial conditions
s_variational_equation | axiom | Variational equation
s_characteristic_multipliers | state | Characteristic multipliers (Floquet multipliers)
s_floquet_normal_form | state | Floquet normal form
s_stability_zones_periodic_ode | state | Stability zones for periodic equations
s_self_adjointness_sturm_liouville | theorem | Self-adjointness of Sturm–Liouville operators
s_sturm_separation_theorem | theorem | Sturm separation theorem
s_sturm_oscillation_theorem | theorem | Sturm oscillation theorem
s_greens_function_bvp | state | Green’s function for ODE boundary value problem
s_eigenfunction_expansion_theorem | theorem | Eigenfunction expansion theorem (Sturm–Liouville)
s_min_max_principle_eigenvalues | theorem | Min-max principle for eigenvalues (Sturm–Liouville)
s_asymptotic_stability | axiom | Asymptotic stability
s_lyapunov_instability_theorem | theorem | Lyapunov instability theorem
s_lyapunov_equation_linear_systems | theorem | Lyapunov equation for linear systems
s_exponential_stability | axiom | Exponential stability
s_uniform_stability | axiom | Uniform stability
s_phase_plane | axiom | Phase plane
s_omega_limit_set | axiom | ω-limit set
s_alpha_limit_set | axiom | α-limit set
s_limit_cycle | state | Limit cycle
s_bendixson_criterion | theorem | Bendixson criterion (negative criterion)
s_orbital_stability | axiom | Orbital stability
s_index_of_closed_curve | state | Index of a closed curve (Poincaré index)
s_index_theorem_equilibria | theorem | Index theorem for equilibria
s_poincare_expansion_theorem | theorem | Poincaré expansion theorem
s_singular_perturbation | axiom | Singular perturbation
s_boundary_layer | state | Boundary layer
s_classification_equilibria_r2 | state | Classification of equilibria in ℝ²
s_separatrices | state | Separatrices
s_topological_equivalence_phase_portraits | axiom | Topological equivalence of phase portraits
s_weyl_limit_point_limit_circle | theorem | Weyl limit-point/limit-circle classification
s_weyl_m_function | state | Weyl m-function
s_titchmarsh_weyl_theory | theorem | Titchmarsh–Weyl theory
s_singular_sturm_liouville_problem | axiom | Singular Sturm–Liouville problem
s_eigenfunction_expansion_half_line | theorem | Eigenfunction expansion on half-line
s_regular_singular_point | axiom | Regular singular point of an ODE
s_irregular_singular_point | axiom | Irregular singular point of an ODE
s_logarithmic_solutions_regular_singular | state | Logarithmic solutions at regular singular points
s_fuchs_theorem | theorem | Fuchs’s theorem
s_asymptotic_solutions_irregular_singular | theorem | Asymptotic solutions at irregular singular points
s_ordinary_point_ode_complex | axiom | Ordinary point of an ODE in ℂ
s_monodromy_representation | state | Monodromy representation
s_riemann_hilbert_problem_ode | theorem | Riemann–Hilbert problem (for linear ODE)
s_perron_asymptotic_integration | theorem | Perron’s theorem on asymptotic integration
s_levinson_asymptotic_integration | theorem | Levinson’s theorem (asymptotic integration)
s_global_existence_linear_ode | theorem | Global existence for linear ODE systems
t_euler_polygonal_approximation | technique | Euler polygonal approximation method
t_greens_function_construction_bvp | technique | Green’s function construction for BVP
t_prufer_transformation | technique | Prüfer transformation
t_regular_perturbation | technique | Regular perturbation method
t_matched_asymptotic_expansions | technique | Matched asymptotic expansions
t_averaging_method | technique | Averaging method
t_analytic_continuation_of_solutions | technique | Analytic continuation of ODE solutions
t_reduction_of_order | technique | Reduction of order
t_differential_inequality | technique | Differential inequality technique
t_higher_order_ode_reduction | technique | Higher-order linear ODE reduction to first-order system
s_phase_space_ode | axiom | Phase space of a dynamical system
s_phase_curve | axiom | Phase curve (integral curve of an ODE)
s_phase_portrait | axiom | Phase portrait
s_phase_flow | axiom | Phase flow
s_phase_velocity_vector_field | axiom | Phase velocity vector field
s_non_autonomous_ode | axiom | Non-autonomous ODE
s_direction_field | axiom | Direction field (line element field)
t_quadrature_barrow | technique | Solution by quadrature (Barrow's formula)
t_separation_of_variables_ode | technique | Separation of variables for ODEs
s_rectification_theorem | theorem | Rectification theorem (flow box theorem)
s_extension_theorem_ode | theorem | Extension theorem for ODE solutions
s_differentiable_dependence_on_initial_conditions | theorem | Differentiable dependence on initial conditions
s_superposition_principle_linear_ode | theorem | Superposition principle for linear ODE systems
s_wronskian | state | Wronskian
s_quasi_polynomial | axiom | Quasi-polynomial
t_complexification_linear_operator | technique | Complexification of a real linear operator
t_decomplexification | technique | Decomplexification (realification)
s_stable_node | axiom | Stable node
s_unstable_node | axiom | Unstable node
s_saddle_point_ode | axiom | Saddle point (planar linear system)
s_stable_focus | axiom | Stable focus (spiral sink)
s_unstable_focus | axiom | Unstable focus (spiral source)
s_center_equilibrium | axiom | Center (elliptic equilibrium)
s_topological_classification_planar_linear_systems | theorem | Topological classification of planar linear systems
s_stability_equilibrium_linear_system | theorem | Stability of equilibrium of a linear system
t_proof_differentiable_dependence_gronwall | technique | Proof of differentiable dependence via Gronwall
s_tangent_vector | axiom | Tangent vector
s_tangent_space | axiom | Tangent space
s_one_parameter_group_of_diffeomorphisms | axiom | One-parameter group of diffeomorphisms
s_completeness_of_vector_field | axiom | Completeness of a vector field
s_compactness_implies_completeness_vector_field | theorem | Compactness implies completeness of vector fields
s_lie_bracket_vector_fields | axiom | Lie bracket of vector fields
s_lie_algebra_of_vector_fields | state | Lie algebra of vector fields
s_commutativity_of_flows_vanishing_bracket | theorem | Commutativity of flows iff vanishing Lie bracket
s_lie_derivative_vector_field | state | Lie derivative of a vector field
s_pushforward_of_vector_field | axiom | Pushforward of a vector field by a diffeomorphism
t_linearization_at_equilibrium | technique | Linearization at an equilibrium
s_lyapunov_indirect_method | theorem | Lyapunov's indirect method (stability by linearization)
s_periodic_orbit | axiom | Periodic orbit
s_poincare_index | axiom | Poincaré index of a singular point
s_sturm_liouville_equation | axiom | Sturm-Liouville equation
s_boundary_value_problem_ode | axiom | Boundary value problem for second-order ODEs
s_homoclinic_orbit | axiom | Homoclinic orbit
s_heteroclinic_orbit | axiom | Heteroclinic orbit (saddle connection)
s_first_integral | axiom | First integral (constant of motion)
s_invariance_ode_under_coordinate_changes | theorem | Invariance of ODE systems under coordinate changes
s_transport_equation_PDE | axiom | Transport equation
s_classification_second_order_PDE | state | Classification of second-order linear PDEs
s_fundamental_solution_laplace | state | Fundamental solution of the Laplace equation
s_poisson_formula_ball | theorem | Poisson's formula for the ball
s_smoothness_of_harmonic_functions | theorem | Smoothness of harmonic functions
s_derivative_estimates_harmonic | theorem | Derivative estimates for harmonic functions
s_analyticity_of_harmonic_functions | theorem | Analyticity of harmonic functions
s_infinite_speed_of_propagation_heat | state | Infinite speed of propagation for the heat equation
s_mean_value_property_heat | theorem | Mean value property for the heat equation
s_strong_maximum_principle_heat | theorem | Strong maximum principle for the heat equation
s_euler_poisson_darboux_equation | state | Euler-Poisson-Darboux equation
s_finite_speed_of_propagation_wave | theorem | Finite speed of propagation for the wave equation
s_domain_of_dependence_wave | state | Domain of dependence for the wave equation
s_greens_identities | state | Green's identities
s_characteristic_equations | state | Characteristic equations for first-order PDEs
s_hamilton_jacobi_equation_PDE | axiom | Hamilton-Jacobi equation (PDE)
s_hopf_lax_formula | theorem | Hopf-Lax formula
s_scalar_conservation_law | axiom | Scalar conservation law
s_weak_solution_conservation_law | state | Weak solution of a conservation law
s_lax_entropy_condition | state | Lax entropy condition
s_entropy_entropy_flux_pair | axiom | Entropy/entropy-flux pair
s_lax_oleinik_formula | theorem | Lax-Oleinik formula
s_rarefaction_wave | state | Rarefaction wave
s_formation_of_singularities_first_order | theorem | Formation of singularities for first-order PDEs
t_hopf_cole_transformation | technique | Hopf-Cole transformation
t_similarity_solutions | technique | Similarity solutions (self-similar reduction)
s_sobolev_conjugate_exponent | state | Sobolev conjugate exponent
s_trace_zero_characterization | theorem | Trace zero characterization of W^{1,p}_0
s_difference_quotient_characterization_sobolev | theorem | Difference quotient characterization of Sobolev spaces
s_general_sobolev_inequalities | theorem | General Sobolev inequalities (higher order)
s_negative_sobolev_space_W_minus_k_p | axiom | Negative Sobolev space W^{−k,p'}(Ω)
s_interpolation_inequalities_sobolev | theorem | Interpolation inequalities for Sobolev spaces
s_campanato_space | axiom | Campanato space L^{p,λ}(Ω)
s_campanato_characterization_holder | theorem | Campanato's characterization of Hölder spaces
s_uniform_ellipticity | axiom | Uniform ellipticity condition
s_bilinear_form_elliptic_operator | state | Bilinear form B[u,v] for elliptic operator
s_interior_H2_regularity_elliptic | theorem | Interior H^2 regularity for elliptic equations
s_boundary_H2_regularity_elliptic | theorem | Boundary H^2 regularity for elliptic equations
t_elliptic_regularity_bootstrap | technique | Elliptic regularity bootstrapping
s_weak_maximum_principle_elliptic_divergence | theorem | Weak maximum principle for elliptic equations in divergence form
s_strong_maximum_principle_elliptic_divergence | theorem | Strong maximum principle for elliptic equations in divergence form
s_eigenvalues_symmetric_elliptic_operator | theorem | Eigenvalue theory for symmetric elliptic operators
s_principal_eigenvalue_elliptic | state | Principal eigenvalue of an elliptic operator
s_energy_estimates_elliptic | theorem | Energy estimates for elliptic PDEs
s_parabolic_cylinder | axiom | Parabolic cylinder
s_weak_solution_parabolic_PDE | state | Weak solution of a parabolic PDE
s_existence_weak_solutions_parabolic | theorem | Existence of weak solutions for parabolic equations
s_energy_estimates_parabolic | theorem | Energy estimates for parabolic equations
s_parabolic_regularity | theorem | Parabolic regularity
s_weak_maximum_principle_parabolic | theorem | Weak maximum principle for parabolic equations
s_strong_maximum_principle_parabolic | theorem | Strong maximum principle for parabolic equations
s_lions_magenes_lemma | theorem | Lions-Magenes lemma
s_weak_solution_hyperbolic_PDE | state | Weak solution of a second-order hyperbolic PDE
s_existence_weak_solutions_hyperbolic | theorem | Existence of weak solutions for hyperbolic equations
s_energy_estimates_hyperbolic | theorem | Energy estimates for hyperbolic equations
s_finite_speed_propagation_hyperbolic_general | theorem | Finite speed of propagation for general hyperbolic equations
s_quasiconvexity_morrey | axiom | Quasiconvexity (Morrey)
s_polyconvexity_ball | axiom | Polyconvexity (Ball)
s_weak_lower_semicontinuity_integral_functionals | theorem | Weak lower semicontinuity of integral functionals
s_null_lagrangian | state | Null Lagrangian
s_regularity_of_minimizers | theorem | Regularity of minimizers in the calculus of variations
t_minimax_methods_critical_points | technique | Minimax methods for critical points
s_deformation_lemma | theorem | Deformation lemma
s_convexity_conditions_lagrangian | state | Convexity conditions on the Lagrangian
t_direct_method_calculus_of_variations | technique | Direct method of the calculus of variations
t_method_of_sub_supersolutions | technique | Method of sub- and supersolutions
t_moser_iteration | technique | Moser iteration technique
t_stampacchia_truncation | technique | Stampacchia truncation method
s_leray_schauder_fixed_point | theorem | Leray-Schauder fixed point theorem
s_comparison_principle_viscosity | theorem | Comparison principle for viscosity solutions
t_doubling_of_variables | technique | Doubling of variables technique
s_dynamic_programming_principle | theorem | Dynamic programming principle (Bellman)
s_stability_viscosity_solutions | theorem | Stability of viscosity solutions
s_hopf_lax_as_viscosity_solution | theorem | Hopf-Lax formula gives the viscosity solution
t_vanishing_viscosity_method | technique | Vanishing viscosity method
s_system_of_conservation_laws | axiom | System of conservation laws
s_strict_hyperbolicity | axiom | Strict hyperbolicity
s_genuine_nonlinearity | axiom | Genuine nonlinearity
s_linear_degeneracy | axiom | Linear degeneracy
s_contact_discontinuity | state | Contact discontinuity
s_lax_theorem_riemann_problem | theorem | Lax's theorem on the Riemann problem
t_glimm_scheme | technique | Glimm's random choice method
s_glimm_functional | state | Glimm interaction functional
s_existence_global_entropy_solutions_glimm | theorem | Existence of global entropy solutions (Glimm's theorem)
s_rankine_hugoniot_systems | state | Rankine-Hugoniot condition for systems
s_lax_entropy_condition_systems | state | Lax entropy condition for systems
s_simple_waves_systems | state | Simple waves (rarefaction waves) for systems
s_young_measures | axiom | Young measures
t_tartar_compensated_compactness_framework | technique | Tartar's compensated compactness framework
s_semilinear_wave_local_existence | theorem | Local existence for semilinear wave equations
s_global_existence_small_data_wave | theorem | Global existence for small data (semilinear wave)
s_blow_up_finite_time_semilinear_wave | theorem | Blow-up in finite time for semilinear wave equations
t_levine_concavity_method | technique | Levine's concavity method for blow-up
s_strauss_conjecture | state | Strauss conjecture (critical exponent for semilinear wave)
s_bochner_space | axiom | Bochner space L^p(0,T; X)
s_evolution_triple | axiom | Evolution triple (Gelfand triple)
s_holder_space_Ck_gamma | axiom | Hölder space C^{k,γ}(Ω̄)
t_duhamel_principle | technique | Duhamel's principle
s_poisson_equation_newtonian_potential | theorem | Solution of Poisson's equation via Newtonian potential
t_energy_wave_equation | technique | Energy methods for the wave equation
s_open_subset_Rn | axiom | Open subset Ω ⊂ ℝⁿ
s_multi_index_notation | axiom | Multi-index notation
s_smooth_functions_C_infinity | axiom | Smooth functions C^∞(Ω)
s_test_function_space_Cc_infinity | axiom | Test function space C^∞_c(Ω) = 𝒟(Ω)
s_LF_space_topology_on_D | axiom | Strict inductive limit topology on 𝒟(Ω)
s_partition_of_unity_smooth | state | Smooth partition of unity
s_cutoff_function | state | Smooth cutoff function
s_mollifier_standard | state | Standard mollifier (Friedrich's mollifier)
t_mollification | technique | Mollification (Friedrich's smoothing)
s_distribution_on_omega | axiom | Distribution u ∈ 𝒟'(Ω)
s_locally_integrable_as_distribution | state | Regular distribution (L¹_loc ↪ 𝒟')
s_distributions_compact_support_E_prime | axiom | Distributions of compact support ℰ'(Ω)
s_sheaf_property_distributions | theorem | Sheaf property of distributions
s_fundamental_solution | axiom | Fundamental solution of a differential operator
s_leibniz_rule_distributions | theorem | Leibniz rule for distributions
s_heaviside_derivative_is_delta | state | Derivative of the Heaviside function
s_division_of_distributions | state | Division of distributions by smooth functions
s_principal_value_distribution | state | Principal value distribution p.v.(1/x)
s_structure_theorem_distributions | theorem | Structure theorem for distributions
s_structure_theorem_compact_support | theorem | Structure theorem for ℰ'(ℝⁿ)
s_distributions_supported_at_point | theorem | Distributions supported at a point
s_convolution_algebra_E_prime | state | Convolution algebra ℰ'(ℝⁿ)
s_convolution_support_rule | theorem | Convolution support theorem
s_singular_support_convolution | theorem | Singular support of a convolution
s_regularization_by_convolution | theorem | Regularization of distributions by convolution
s_approximation_by_smooth_functions | theorem | Density of C^∞_c in 𝒟'
t_regularization_convolution | technique | Regularization by convolution with mollifier
s_tensor_product_distributions | state | Tensor product of distributions
s_schwartz_kernel_theorem_hormander | theorem | Schwartz kernel theorem
t_kernel_method | technique | Schwartz kernel method
s_pullback_distribution | state | Pullback of a distribution by a diffeomorphism
s_pullback_submersion | state | Pullback of a distribution by a submersion
s_distribution_on_submanifold | state | Distribution on a submanifold (restriction/trace)
s_push_forward_distribution | state | Push-forward of a distribution
t_change_of_variables_distributions | technique | Change of variables for distributions
s_schwartz_space_S | axiom | Schwartz space 𝒮(ℝⁿ)
s_tempered_distributions_S_prime | axiom | Tempered distributions 𝒮'(ℝⁿ)
s_fourier_transform_on_S | state | Fourier transform on 𝒮(ℝⁿ)
s_fourier_transform_on_S_prime | state | Fourier transform on 𝒮'(ℝⁿ)
s_fourier_transform_exchange_rules | theorem | Fourier exchange rules (derivative ↔ multiplication)
s_plancherel_theorem_hormander | theorem | Plancherel theorem (L² isometry)
s_fourier_transform_of_E_prime | theorem | Fourier transform of ℰ'(ℝⁿ) (entire analytic functions)
s_paley_wiener_schwartz_theorem | theorem | Paley–Wiener–Schwartz theorem
s_paley_wiener_smooth | theorem | Paley–Wiener theorem for C^∞_c
s_sobolev_space_Hs | axiom | Sobolev space H^s(ℝⁿ) via Fourier transform
s_sobolev_embedding_via_fourier | theorem | Sobolev embedding via Fourier transform
s_fourier_inversion_formula | theorem | Fourier inversion formula
s_convolution_fourier_exchange | theorem | Convolution theorem (Fourier)
t_fourier_analysis_of_PDE | technique | Fourier transform method for constant-coefficient PDE
s_wavefront_set_definition | axiom | Wavefront set WF(u)
s_wavefront_set_projection_is_sing_supp | theorem | Projection of WF(u) is sing supp u
s_wavefront_set_of_derivative | theorem | Wavefront set under differentiation
s_wavefront_set_of_product | theorem | Wavefront set of a product of distributions
s_hormander_condition_product | theorem | Hörmander's condition for multiplication of distributions
s_wavefront_set_pullback | theorem | Wavefront set under pullback
s_microlocal_regularity_elliptic | theorem | Elliptic regularity (microlocal version)
s_characteristic_set | state | Characteristic set of a differential operator
s_cone_and_conic_set | axiom | Conic set in T*Ω \ 0
s_microlocal_smoothness | axiom | Microlocal smoothness (absence from WF)
s_wavefront_set_convolution | theorem | Wavefront set of a convolution
t_microlocal_analysis | technique | Microlocal analysis (wavefront set calculus)
s_oscillatory_integral | state | Oscillatory integral
t_stationary_phase | technique | Stationary phase method
s_distribution_of_finite_order_global | state | Distribution of finite order
s_positive_distribution | theorem | Positive distributions are measures
s_fourier_laplace_transform | state | Fourier–Laplace transform
s_hormander_propagation_of_singularities_distribution | theorem | Propagation of singularities (distribution-theoretic version)
s_fbi_transform | axiom | FBI transform
s_analytic_wave_front_set | axiom | Analytic wave front set WF_A(u)
s_fbi_characterization_analytic_regularity | theorem | FBI characterization of analytic regularity
t_schwartz_kernel_method | technique | Schwartz kernel method
s_p_convexity_for_supports | axiom | P-convexity for supports
s_p_convexity_for_singular_supports | axiom | P-convexity for singular supports
s_malgrange_exponential_polynomial_approximation | theorem | Exponential-polynomial solution approximation (Malgrange)
s_geometric_characterization_p_convexity | theorem | Geometric characterization of P-convexity
s_existence_theorem_p_convex_domains | theorem | Existence theorem for PDE on P-convex domains
s_convexity_implies_p_convexity | theorem | Convexity implies P-convexity
s_regularity_index_hypoelliptic | state | Regularity index of a hypoelliptic operator
s_analytically_hypoelliptic_operator | axiom | Analytically hypoelliptic operator
s_characterization_analytically_hypoelliptic | theorem | Characterization of analytically hypoelliptic operators
s_well_posedness_garding_petrowsky | theorem | Well-posedness of Cauchy problem (Gårding–Petrowsky)
s_hyperbolic_polynomial | axiom | Hyperbolic polynomial
s_petrowsky_lacuna | axiom | Petrowsky lacuna
s_herglotz_petrovsky_leray_formula | theorem | Herglotz–Petrovsky–Leray formula
s_atiyah_bott_garding_lacuna_criterion | theorem | Atiyah–Bott–Gårding lacuna criterion
s_operator_of_constant_strength | axiom | Operator of constant strength
s_existence_theorem_constant_strength | theorem | Existence theorem for operators of constant strength
s_hypoellipticity_constant_strength | theorem | Hypoellipticity for operators of constant strength
s_abstract_cauchy_kovalevsky_theorem | theorem | Abstract Cauchy–Kovalevsky theorem (Nirenberg–Nishida)
s_holmgren_john_global_uniqueness | theorem | Holmgren–John global uniqueness
s_ehrenpreis_fundamental_principle | theorem | Ehrenpreis fundamental principle
s_mean_periodic_function | axiom | Mean-periodic function
s_spectral_synthesis_mean_periodic | theorem | Spectral synthesis for mean-periodic functions
s_noetherian_operator_pde | state | Noetherian operator (PDE)
s_division_theorem_ehrenpreis_malgrange | theorem | Division theorem for distributions (Ehrenpreis–Malgrange)
s_slowly_decreasing_condition | axiom | Slowly decreasing condition
s_surjectivity_convolution_operators_smooth | theorem | Surjectivity of convolution operators on C^∞
s_supports_theorem_convolution | theorem | Supports theorem for convolution (Titchmarsh generalization)
s_subelliptic_estimate | state | Subelliptic estimate
t_kohn_subelliptic_multiplier_method | technique | Kohn's subelliptic multiplier method
s_rothschild_stein_lifting_theorem | theorem | Rothschild–Stein lifting theorem
s_classical_symbol_class | axiom | Classical symbol class S^m_cl
s_pseudodifferential_operator_definition | state | Pseudodifferential operator Op(a)
s_asymptotic_expansion_of_symbols | state | Asymptotic expansion of symbols
s_composition_formula_pseudodifferential | theorem | Composition formula for pseudodifferential operators
s_adjoint_formula_pseudodifferential | theorem | Adjoint formula for pseudodifferential operators
s_kernel_characterization_pseudodifferential | theorem | Kernel characterization of pseudodifferential operators
s_elliptic_symbol | axiom | Elliptic symbol
s_elliptic_parametrix_construction | theorem | Elliptic parametrix construction
s_elliptic_regularity_pseudodifferential | theorem | Elliptic regularity for pseudodifferential operators
s_calderon_vaillancourt_theorem | theorem | Calderón–Vaillancourt theorem
s_beals_characterization_pseudodifferential | theorem | Beals characterization of pseudodifferential operators
s_weyl_quantization | state | Weyl quantization
s_moyal_product | state | Moyal product
s_sharp_garding_inequality | theorem | Sharp Gårding inequality
s_fefferman_phong_inequality | theorem | Fefferman–Phong inequality
t_symbol_smoothing | technique | Symbol smoothing (Friedrichs symmetrization)
s_pseudodifferential_operators_on_manifolds | axiom | Pseudodifferential operators on manifolds Ψ^m(M)
s_principal_symbol_on_manifold | state | Principal symbol on a manifold
s_fredholm_property_elliptic_compact_manifold | theorem | Fredholm property of elliptic operators on compact manifolds
s_seeley_complex_powers | state | Seeley's complex powers
s_weyl_law_sharp_remainder | theorem | Weyl's law with sharp remainder (Hörmander)
s_calderon_projector | state | Calderón projector
s_shapiro_lopatinskii_condition | axiom | Shapiro–Lopatinskii condition
s_boutet_de_monvel_calculus | axiom | Boutet de Monvel calculus
s_poisson_operator_boutet | state | Poisson operator (Boutet de Monvel)
s_singular_green_operator | state | Singular Green operator
s_fredholm_property_elliptic_bvp | theorem | Fredholm property for elliptic BVPs
s_hodge_decomposition_elliptic_complexes | theorem | Hodge decomposition for elliptic complexes
s_symplectic_classification_submanifolds | state | Symplectic classification of submanifolds
s_bicharacteristic_strip | state | Bicharacteristic strip
s_operator_of_principal_type | axiom | Operator of principal type
s_condition_psi_nirenberg_treves | axiom | Condition (Ψ) (Nirenberg–Treves)
s_subelliptic_operators_loss_delta | state | Subelliptic operators (loss of δ derivatives)
s_microlocal_hypoellipticity | state | Microlocal hypoellipticity
s_strictly_hyperbolic_operator | axiom | Strictly hyperbolic operator (PDE)
s_well_posedness_strictly_hyperbolic_cauchy | theorem | Well-posedness of strictly hyperbolic Cauchy problem
s_parametrix_strictly_hyperbolic_cauchy | state | Parametrix for strictly hyperbolic Cauchy problem
s_propagation_singularities_hyperbolic | theorem | Propagation of singularities for hyperbolic equations
t_lax_parametrix_construction | technique | Lax parametrix construction
t_hadamard_fundamental_solution_construction | technique | Hadamard's construction of fundamental solutions
s_glancing_rays | state | Glancing rays
s_diffractive_rays | state | Diffractive rays
s_melrose_sjostrand_propagation_theorem | theorem | Melrose–Sjöstrand propagation theorem
s_gliding_rays | state | Gliding rays
s_non_degenerate_phase_function | axiom | Non-degenerate phase function
s_canonical_relation | axiom | Canonical relation
s_lagrangian_distribution | axiom | Lagrangian distribution I^μ(X, Λ)
s_principal_symbol_lagrangian_distribution | state | Principal symbol of a Lagrangian distribution
s_maslov_index | axiom | Maslov index
s_maslov_bundle | state | Maslov bundle
s_clean_intersection_composition_fio | theorem | Clean intersection composition of FIOs
s_transversal_composition_fio | theorem | Transversal composition of FIOs
s_l2_continuity_fio | theorem | L² continuity of FIOs
s_hormander_sato_lemma | theorem | Hörmander–Sato lemma on WF of FIOs
t_global_calculus_oscillatory_integrals | technique | Global calculus of oscillatory integrals
t_refined_symbol_calculus_type_1_0 | technique | Refined symbol calculus for type (1,0)
s_carleman_estimate | theorem | Carleman estimate (general form)
s_strong_pseudo_convexity_condition | axiom | Strong pseudo-convexity condition
s_holmgren_uniqueness_microlocal | theorem | Holmgren's uniqueness (microlocal version)
s_calderon_uniqueness_theorem | theorem | Calderón uniqueness theorem
s_strong_unique_continuation_property | theorem | Strong unique continuation property
s_unique_continuation_across_non_characteristic | theorem | Unique continuation across non-characteristic surfaces
t_tauberian_theorem_spectral_function | technique | Tauberian theorem for spectral function
s_spectral_function_asymptotic | theorem | Spectral function asymptotic
s_wave_trace_formula | theorem | Wave trace formula
s_sharpness_weyl_remainder | theorem | Sharpness of remainder in Weyl's law
s_local_weyl_law | theorem | Local Weyl law
s_asymptotic_completeness_long_range | theorem | Asymptotic completeness for long-range potentials
s_limiting_absorption_principle_long_range | theorem | Limiting absorption principle (long-range)
s_scattering_matrix_long_range | state | Scattering matrix for long-range potentials
t_phase_space_scattering_long_range | technique | Phase space characterization of long-range scattering
s_sommerfeld_radiation_condition | axiom | Sommerfeld radiation condition (generalized)
s_differentiable_manifold | axiom | Differentiable manifold
s_submanifold | axiom | Submanifold
s_orientation_of_manifold | axiom | Orientation of a manifold
s_vector_field | axiom | Vector field
s_riemannian_manifold | axiom | Riemannian manifold
s_length_of_curve | state | Length of a curve
s_riemannian_distance | state | Riemannian distance
s_isometry | axiom | Isometry (Riemannian)
s_local_isometry | axiom | Local isometry
s_pullback_metric | state | Pullback metric (induced metric)
s_product_metric | state | Product metric
s_affine_connection | axiom | Affine connection
s_covariant_derivative_along_curve | axiom | Covariant derivative along a curve
s_geodesic_equation_local_coords | state | Geodesic equation in local coordinates
s_normal_coordinates | state | Normal coordinates
s_normal_neighborhood | state | Normal neighborhood
s_geodesic_ball_and_sphere | state | Geodesic ball and geodesic sphere
s_gauss_lemma_riemannian | theorem | Gauss lemma (Riemannian geometry)
s_minimizing_property_of_geodesics | theorem | Minimizing property of geodesics
s_convex_neighborhood | axiom | Convex neighborhood
s_whitehead_convex_neighborhood_theorem | theorem | Whitehead's theorem on convex neighborhoods
s_symmetries_of_curvature_tensor | state | Symmetries of the curvature tensor
s_first_bianchi_identity | theorem | First Bianchi identity
s_sectional_curvature | axiom | Sectional curvature
s_ricci_curvature | axiom | Ricci curvature
s_scalar_curvature | axiom | Scalar curvature
s_curvature_determined_by_sectional | theorem | Curvature tensor determined by sectional curvature
s_ricci_determines_curvature_dim_3 | state | Ricci determines curvature in dimension 3
s_jacobi_fields_as_geodesic_variations | state | Jacobi fields as geodesic variations
s_conjugate_point | axiom | Conjugate point
s_multiplicity_of_conjugate_point | state | Multiplicity of a conjugate point
s_conjugate_points_exp_singularity | theorem | Conjugate points as singularities of exp
s_geodesics_cease_minimizing_past_conjugate | theorem | Geodesics cease to minimize past conjugate points
s_isometric_immersion | axiom | Isometric immersion
s_second_fundamental_form | axiom | Second fundamental form
s_normal_connection | state | Normal connection
s_shape_operator | state | Shape operator (Weingarten map)
s_gauss_equation | theorem | Gauss equation
s_codazzi_equation | theorem | Codazzi equation (Codazzi-Mainardi)
s_ricci_equation_submanifold | theorem | Ricci equation (normal curvature equation)
s_totally_geodesic_submanifold | state | Totally geodesic submanifold
s_minimal_submanifold | state | Minimal submanifold
s_geodesic_completeness | axiom | Geodesic completeness
s_cut_point | axiom | Cut point
s_model_spaces | state | Model spaces S^n, R^n, H^n
s_hyperbolic_space | state | Hyperbolic space H^n
s_cartan_theorem_space_forms | theorem | Cartan's theorem on space forms
s_classification_of_space_forms | state | Classification of space forms
s_energy_functional | axiom | Energy functional
s_first_variation_of_energy | theorem | First variation of energy
s_first_variation_of_arc_length | theorem | First variation of arc length
s_second_variation_of_energy | theorem | Second variation of energy
s_index_form | axiom | Index form
s_synge_lemma | theorem | Synge's lemma
s_berger_lemma | theorem | Berger's lemma
s_klingenberg_injectivity_radius_estimate | theorem | Klingenberg's injectivity radius estimate
s_injectivity_radius | state | Injectivity radius
s_toponogov_hinge_version | state | Toponogov comparison (hinge version)
s_index_of_geodesic | axiom | Index of a geodesic
s_nullity_of_geodesic | state | Nullity of a geodesic
s_morse_index_theorem | theorem | Morse index theorem
s_byers_theorem | theorem | Byers' theorem
s_closed_geodesic_in_free_homotopy_class | theorem | Closed geodesic in free homotopy class
s_geodesic_variation | state | Geodesic variation
s_energy_vs_length_minimizers | state | Energy vs length minimizers
s_laplacian_comparison | state | Laplacian comparison theorem
s_geodesic_convexity | state | Geodesic convexity
t_normal_coordinate_computation | technique | Normal coordinate computation
t_sturm_liouville_comparison | technique | Sturm-Liouville comparison for Jacobi fields
t_covering_space_negative_curvature | technique | Covering space argument for negative curvature
t_comparison_of_jacobi_fields | technique | Comparison of Jacobi fields
s_smoothly_compatible_charts | axiom | Smoothly compatible charts
s_smooth_atlas | axiom | Smooth atlas
s_maximal_smooth_atlas | axiom | Maximal smooth atlas (smooth structure)
s_interior_and_boundary_of_manifold_with_boundary | state | Interior and boundary of a manifold with boundary
s_product_manifold | state | Product manifold
s_smooth_manifold_chart_lemma | theorem | Smooth manifold chart lemma
s_invariance_of_the_boundary | theorem | Invariance of the boundary
s_smooth_function_ring | state | Smooth function ring C^∞(M)
s_smooth_extension_lemma | theorem | Smooth extension lemma
s_exhaustion_function | state | Exhaustion function
s_derivation_at_a_point | axiom | Derivation at a point (tangent vector)
s_coordinate_basis_for_tangent_space | state | Coordinate basis for tangent space
s_differential_of_a_smooth_map | axiom | Differential of a smooth map
s_chain_rule_for_differentials | theorem | Chain rule for differentials
s_natural_coordinates_on_TM | state | Natural coordinates on TM
s_velocity_of_a_curve | state | Velocity of a curve
s_smooth_map_of_constant_rank | axiom | Smooth map of constant rank
s_inverse_function_theorem_manifolds | theorem | Inverse function theorem for manifolds
s_global_rank_theorem | theorem | Global rank theorem
s_local_immersion_theorem | theorem | Local immersion theorem
s_local_submersion_theorem | theorem | Local submersion theorem
s_local_section_of_submersion | theorem | Local section of a submersion
s_smooth_covering_map | axiom | Smooth covering map
s_embedded_submanifold | axiom | Embedded submanifold (regular submanifold)
s_slice_chart | axiom | Slice chart (adapted chart)
s_local_slice_criterion | theorem | Local slice criterion for embedded submanifolds
s_immersed_submanifold | axiom | Immersed submanifold
s_level_set | state | Level set
s_regular_value | axiom | Regular value
s_tangent_space_to_level_set | theorem | Tangent space to a level set
s_set_of_measure_zero_in_manifold | axiom | Set of measure zero in a manifold
s_critical_point_manifold | axiom | Critical point (manifold context)
s_parametric_transversality_theorem | theorem | Parametric transversality theorem
s_lie_group_isomorphism | axiom | Lie group isomorphism
s_embedded_lie_subgroup | state | Embedded Lie subgroup
s_left_right_translation | state | Left and right translation
s_smooth_group_action | axiom | Smooth group action
s_orbit_and_stabilizer | state | Orbit and stabilizer (isotropy group)
s_free_action | axiom | Free action
s_proper_action | axiom | Proper action
s_quotient_manifold_theorem | theorem | Quotient manifold theorem (for group actions)
s_equivariant_map | axiom | Equivariant map
s_covering_group_theorem | theorem | Covering group theorem
s_f_related_vector_fields | axiom | F-related vector fields
s_naturality_of_lie_bracket | theorem | Naturality of the Lie bracket
s_lie_algebra_homomorphism_from_lie_group | theorem | Lie algebra homomorphism induced by Lie group homomorphism
s_integral_curve | axiom | Integral curve
s_existence_uniqueness_integral_curves | theorem | Existence and uniqueness of integral curves
s_flow_domain | state | Flow domain
s_complete_vector_field | axiom | Complete vector field
s_flowout_theorem | theorem | Flowout theorem
s_lie_derivative_equals_lie_bracket | theorem | Lie derivative equals Lie bracket
s_commuting_vector_fields_and_flows | theorem | Commuting vector fields and flows
s_canonical_form_nonvanishing_vector_field | theorem | Canonical form for nonvanishing vector fields
s_local_trivialization | axiom | Local trivialization
s_transition_function_vector_bundle | state | Transition function of a vector bundle
s_section_of_vector_bundle | axiom | Section of a vector bundle
s_local_frame_vector_bundle | state | Local frame for a vector bundle
s_vector_bundle_construction_theorem | theorem | Vector bundle construction theorem
s_bundle_map | axiom | Bundle map (bundle homomorphism)
s_subbundle | axiom | Subbundle
s_covector | axiom | Covector (cotangent vector)
s_coordinate_covector_fields | state | Coordinate covector fields (dx^i)
s_covector_field | axiom | Covector field (1-form)
s_line_integral_of_covector_field | state | Line integral of a covector field
s_conservative_covector_field_characterization | theorem | Conservative covector field characterization
s_poincare_lemma_covector_fields | theorem | Poincare lemma for covector fields
s_multilinear_map_k_tensor | axiom | Multilinear map (k-tensor)
s_mixed_tensor | axiom | Mixed tensor
s_symmetric_tensor | axiom | Symmetric tensor
s_alternating_tensor | axiom | Alternating tensor
s_symmetrization_alternation_operators | state | Symmetrization and alternation operators
s_pullback_of_tensor_fields | state | Pullback of tensor fields
s_tensor_characterization_lemma | theorem | Tensor characterization lemma
s_existence_of_riemannian_metrics | theorem | Existence of Riemannian metrics
s_musical_isomorphisms | state | Musical isomorphisms (flat and sharp)
s_pseudo_riemannian_metric | axiom | Pseudo-Riemannian metric
s_pseudo_riemannian_manifold | state | Pseudo-Riemannian manifold
s_interior_multiplication | axiom | Interior multiplication (contraction)
s_invariant_formula_exterior_derivative | theorem | Invariant formula for the exterior derivative
s_orientation_of_a_vector_space | axiom | Orientation of a vector space
s_oriented_double_cover | state | Oriented double cover
s_riemannian_volume_form | state | Riemannian volume form
s_integration_of_forms_on_manifolds | axiom | Integration of differential forms on oriented manifolds
s_induced_boundary_orientation | state | Induced boundary orientation
s_manifold_with_corners | axiom | Manifold with corners
s_stokes_for_manifolds_with_corners | theorem | Stokes's theorem for manifolds with corners
s_induced_map_on_cohomology | state | Induced map on de Rham cohomology
s_homotopy_invariance_de_rham | theorem | Homotopy invariance of de Rham cohomology
t_homotopy_operator | technique | Homotopy operator (cochain homotopy)
s_poincare_lemma | theorem | Poincare lemma
s_cohomology_of_spheres | theorem | Cohomology of spheres
s_smooth_singular_homology | state | Smooth singular homology
s_de_rham_homomorphism | state | De Rham homomorphism
s_distribution_tangent_subbundle | axiom | Distribution (tangent subbundle)
s_integral_manifold | axiom | Integral manifold of a distribution
s_integrable_distribution | axiom | Integrable distribution
s_differential_form_criterion_involutivity | theorem | Differential form criterion for involutivity
s_lie_subalgebra_subgroup_correspondence | theorem | Lie subalgebra-Lie subgroup correspondence
s_exponential_map_lie_group | state | Exponential map of a Lie group
s_continuous_homomorphisms_are_smooth | theorem | Continuous homomorphisms between Lie groups are smooth
s_infinitesimal_generator_group_action | state | Infinitesimal generator of a group action
s_adjoint_representation_lie_algebra | state | Adjoint representation of the Lie algebra
s_homogeneous_space | state | Homogeneous space
s_covering_manifold | state | Covering manifold
s_symplectic_form_on_vector_space | axiom | Symplectic form on a vector space
s_symplectomorphism | axiom | Symplectomorphism
s_contact_structure | axiom | Contact structure
s_contact_manifold | axiom | Contact manifold
s_cotangent_bundle_symplectic | state | Cotangent bundle as symplectic manifold
t_chart_computation | technique | Chart computation technique
t_rank_theorem_technique | technique | Rank theorem technique
t_flow_box_straightening | technique | Flow box / straightening technique
t_mayer_vietoris_induction | technique | Mayer-Vietoris induction
t_exhaustion_by_compact_sets | technique | Exhaustion by compact sets technique
t_gluing_via_partition_of_unity | technique | Smooth maps into Euclidean space (gluing lemma)
s_space_form | axiom | Space form
s_principal_curvatures | state | Principal curvatures
s_mean_curvature_surface | state | Mean curvature of a surface
s_riemannian_submanifold | axiom | Riemannian submanifold
s_riemannian_distance_induces_topology | theorem | Riemannian distance induces the manifold topology
s_riemannian_density | state | Riemannian density
s_euclidean_space_riemannian | axiom | Euclidean space (Riemannian)
s_round_metric_sphere | axiom | Round metric on Sⁿ
s_poincare_ball_model | state | Poincaré ball model
s_poincare_half_space_model | state | Poincaré half-space model
s_homogeneous_riemannian_manifold | axiom | Homogeneous Riemannian manifold
s_isotropic_riemannian_manifold | axiom | Isotropic Riemannian manifold
s_frame_homogeneous_riemannian_manifold | axiom | Frame-homogeneous Riemannian manifold
s_classification_frame_homogeneous | theorem | Classification of frame-homogeneous spaces
s_riemannian_covering | state | Riemannian covering
s_connection_in_local_frame | state | Connection in a local frame
s_parallel_transport_determines_connection | theorem | Parallel transport characterizes connection
s_metric_connection | axiom | Metric connection
s_symmetric_connection | axiom | Symmetric (torsion-free) connection
s_christoffel_symbols_levi_civita | state | Christoffel symbols of the Levi-Civita connection
s_geodesics_constant_speed | theorem | Geodesics have constant speed
s_exp_local_diffeomorphism | theorem | Exponential map is a local diffeomorphism
s_fermi_coordinates | state | Fermi coordinates
s_riemannian_geodesics_minimize_length | theorem | Riemannian geodesics as locally length-minimizing curves
s_variation_of_curve | axiom | Variation of a curve
s_uniformly_normal_neighborhood | state | Uniformly normal neighborhood
s_distance_function_from_point | state | Distance function from a point
s_cut_locus_measure_zero | theorem | Cut locus has measure zero
s_riemann_curvature_endomorphism | state | Riemann curvature endomorphism
s_second_bianchi_identity | theorem | Second Bianchi identity
s_schouten_tensor | state | Schouten tensor
s_curvature_tensor_decomposition | theorem | Decomposition of the Riemann curvature tensor
s_curvature_of_constant_curvature_spaces | theorem | Curvature of constant-curvature spaces
s_contracted_second_bianchi_identity | theorem | Contracted second Bianchi identity
s_curvature_of_product_manifolds | theorem | Curvature of product manifolds
s_gauss_formula_submanifold | theorem | Gauss formula
s_weingarten_equation | theorem | Weingarten equation
s_curvature_of_hypersurfaces | theorem | Curvature of hypersurfaces
s_fundamental_theorem_hypersurface_theory | theorem | Fundamental theorem of hypersurface theory
s_gauss_bonnet_local | theorem | Local Gauss-Bonnet theorem
s_gauss_bonnet_with_boundary | theorem | Gauss-Bonnet theorem with boundary
s_euler_characteristic | axiom | Euler characteristic
s_jacobi_fields_existence_uniqueness | theorem | Existence and uniqueness of Jacobi fields
s_jacobi_fields_constant_curvature | theorem | Jacobi fields on constant-curvature spaces
s_no_conjugate_before_cut | theorem | No conjugate points before cut point
s_second_variation_of_arc_length | theorem | Second variation of arc length
s_index_form_conjugate_points | theorem | Index form characterization of conjugate points
s_hessian_comparison_theorem | theorem | Hessian comparison theorem
s_relative_volume_comparison | theorem | Relative volume comparison
s_milnor_theorem_fundamental_group | theorem | Milnor's theorem on fundamental groups
s_weinstein_theorem | theorem | Weinstein's theorem (Riemannian)
s_berger_quarter_pinched_sphere_theorem | theorem | Berger's quarter-pinched sphere theorem
s_killing_vector_field | axiom | Killing vector field
s_killing_fields_lie_algebra | theorem | Killing fields form a Lie algebra
s_max_dimension_isometry_group | theorem | Maximum dimension of the isometry group
s_cartan_ambrose_hicks_theorem | theorem | Cartan-Ambrose-Hicks theorem
s_local_to_global_isometry_extension | theorem | Local-to-global isometry extension
s_flat_riemannian_manifold | axiom | Flat Riemannian manifold
s_geodesic_curvature | state | Geodesic curvature
s_normal_jacobi_field | state | Normal Jacobi field
s_tangential_jacobi_field | state | Tangential Jacobi field
s_parallel_transport_preserves_inner_products | theorem | Parallel transport preserves inner products
t_variational_methods_geodesics | technique | Variational methods for geodesics
t_symmetry_group_action_methods | technique | Symmetry and group action methods
t_index_form_methods | technique | Index form methods
t_geodesic_deviation_analysis | technique | Geodesic deviation analysis
s_busemann_function | state | Busemann function
t_busemann_function_technique | technique | Busemann function technique
t_maximum_principle_riemannian | technique | Maximum principle (Riemannian)
s_product_riemannian_manifold | state | Product Riemannian manifold
s_warped_product_metric | state | Warped product metric
s_curvature_warped_products | theorem | Curvature of warped products
s_riemannian_submersion | axiom | Riemannian submersion
s_oneill_formula | theorem | O'Neill's formula
s_ambrose_singer_holonomy_theorem | theorem | Ambrose-Singer holonomy theorem
s_volume_of_geodesic_balls | state | Volume of geodesic balls
s_riemannian_laplacian | state | Laplacian (Riemannian)
s_riemannian_hessian | state | Hessian (Riemannian)
s_riemannian_gradient | state | Gradient (Riemannian)
s_riemannian_divergence | state | Divergence (Riemannian)
s_conformally_flat_manifold | axiom | Conformally flat manifold
s_weyl_vanishing_conformal_flatness | theorem | Weyl tensor vanishing and conformal flatness
s_cotton_tensor | state | Cotton tensor
s_conformal_flatness_dim_3 | theorem | Conformal flatness in dimension 3
s_normal_bundle | state | Normal bundle
s_comparison_triangle | state | Comparison triangle
s_alexandrov_angle_comparison | theorem | Alexandrov angle comparison
s_ricci_identity | theorem | Ricci identity (commutation of covariant derivatives)
t_parallel_transport_along_geodesics | technique | Parallel transport along geodesics
t_curvature_estimation_via_comparison | technique | Curvature estimation via comparison
s_principal_fibre_bundle | axiom | Principal fibre bundle
s_frame_bundle | state | Frame bundle L(M)
s_orthonormal_frame_bundle | state | Orthonormal frame bundle O(M)
s_connection_on_principal_bundle | axiom | Connection on a principal bundle
s_horizontal_lift_of_curve | state | Horizontal lift of a curve
s_restricted_holonomy_group | state | Restricted holonomy group
s_exterior_covariant_derivative | state | Exterior covariant derivative
s_cartan_structure_equation | theorem | Cartan structure equation
s_bianchi_identity_principal_bundle | theorem | Bianchi identity on principal bundle
s_reduction_theorem_connections | theorem | Reduction theorem for connections
s_existence_of_connections | theorem | Existence of connections on principal bundles
s_induced_connection_associated_bundle | state | Induced connection on associated bundle
s_gauge_transformation | axiom | Gauge transformation
s_linear_connection | axiom | Linear connection
s_torsion_tensor | state | Torsion tensor
s_canonical_form_solder_form | state | Canonical form (solder form) on L(M)
s_torsion_form_on_frame_bundle | state | Torsion form on L(M)
s_structure_equations_linear_connection | theorem | Structure equations of a linear connection
t_development_of_curves | technique | Development of curves
s_parallel_section | state | Parallel section
s_fundamental_theorem_riemannian_geometry | theorem | Fundamental theorem of Riemannian geometry
s_koszul_formula | theorem | Koszul formula
s_interior_product | state | Interior product
s_cartans_magic_formula | theorem | Cartan's magic formula
s_affine_transformation | axiom | Affine transformation
s_infinitesimal_affine_transformation | state | Infinitesimal affine transformation
s_killing_equation | theorem | Killing equation
s_maximum_dimension_isometry_group | theorem | Maximum dimension of isometry group
s_de_rham_decomposition_theorem | theorem | De Rham decomposition theorem
s_holonomy_of_flat_connections | theorem | Holonomy of flat connections
s_symmetric_space | axiom | Symmetric space
s_locally_symmetric_space | axiom | Locally symmetric space
s_locally_symmetric_iff_parallel_curvature | theorem | Locally symmetric iff parallel curvature
s_cartan_local_to_global_symmetric | theorem | Cartan's local-to-global symmetric space theorem
s_totally_umbilic_submanifold | axiom | Totally umbilic submanifold
s_gauss_map | state | Gauss map
s_flat_normal_connection | axiom | Flat normal connection
s_reduction_of_codimension_theorem | theorem | Reduction of codimension theorem
s_hermitian_metric | axiom | Hermitian metric
s_kahler_form_fundamental_2_form | state | Kähler form (fundamental 2-form)
s_kahler_potential | state | Kähler potential
s_holomorphic_sectional_curvature | state | Holomorphic sectional curvature
s_constant_holomorphic_sectional_curvature | theorem | Constant holomorphic sectional curvature characterization
s_complex_space_form | axiom | Complex space form
s_complex_projective_space | state | Complex projective space ℂPⁿ
s_fubini_study_metric | axiom | Fubini–Study metric
s_type_decomposition_differential_forms | state | Type decomposition of differential forms
s_holomorphic_vector_bundle | axiom | Holomorphic vector bundle
s_ricci_form_kahler | state | Ricci form of a Kähler manifold
s_bochner_curvature_tensor | state | Bochner curvature tensor
s_kahler_submanifold | state | Kähler submanifold
s_reductive_homogeneous_space | axiom | Reductive homogeneous space
s_naturally_reductive_homogeneous_space | axiom | Naturally reductive homogeneous space
s_invariant_connection_homogeneous | state | Invariant connection on G/H
s_canonical_connection_reductive | state | Canonical connection on reductive homogeneous space
s_curvature_naturally_reductive_spaces | theorem | Curvature of naturally reductive spaces
s_isotropy_representation | state | Isotropy representation
s_invariant_metric_homogeneous | state | Invariant metric on G/H
s_invariant_almost_complex_structure_homogeneous | state | G-invariant almost complex structure on G/H
s_geodesic_symmetry | state | Geodesic symmetry
s_riemannian_symmetric_pair | axiom | Riemannian symmetric pair
s_rank_of_symmetric_space | state | Rank of a symmetric space
s_symmetric_space_compact_type | state | Symmetric space of compact type
s_symmetric_space_noncompact_type | state | Symmetric space of noncompact type
s_duality_of_symmetric_spaces | theorem | Duality of symmetric spaces
s_curvature_of_symmetric_spaces | theorem | Curvature of symmetric spaces
s_de_rham_decomposition_symmetric_spaces | theorem | De Rham decomposition for symmetric spaces
s_maximal_flat_submanifold | state | Maximal flat submanifold
s_compact_symmetric_nonneg_curvature | theorem | Compact symmetric space has non-negative curvature
s_symmetric_space_is_complete | theorem | Symmetric space is complete
s_isotropy_irreducible_symmetric_space | state | Isotropy-irreducible symmetric space
t_weil_homomorphism | technique | Weil homomorphism
t_chern_weil_construction | technique | Chern–Weil construction
s_total_chern_class | state | Total Chern class
s_whitney_product_formula_chern | theorem | Whitney product formula for Chern classes
s_first_chern_class | state | First Chern class
s_total_pontryagin_class | state | Total Pontryagin class
s_euler_class | axiom | Euler class
s_chern_number | state | Chern number
s_pontryagin_number | state | Pontryagin number
t_transgression | technique | Transgression
t_splitting_principle | technique | Splitting principle technique
s_chern_class_of_line_bundle | theorem | Chern class of a line bundle
s_characteristic_classes_flat_bundles_vanish | theorem | Characteristic classes of flat bundles vanish
s_morse_function | axiom | Morse function
s_critical_point_of_smooth_function | axiom | Critical point of a smooth function
s_index_of_critical_point | axiom | Index of a critical point
s_hessian_at_critical_point | state | Hessian of a smooth function at a critical point
s_sublevel_set | axiom | Sublevel set
s_handle_attachment_theorem | theorem | Handle attachment theorem
s_handle_of_index_lambda | axiom | Handle of index λ
s_cw_structure_from_morse_function | theorem | CW complex structure from a Morse function
s_morse_number | axiom | Morse number Cₖ
s_weak_morse_inequalities | theorem | Weak Morse inequalities
s_strong_morse_inequalities | theorem | Strong Morse inequalities
s_poincare_polynomial_morse | state | Poincaré polynomial (Morse context)
s_lacunary_principle_morse | theorem | Lacunary principle
s_euler_characteristic_via_morse | theorem | Euler characteristic via Morse theory
s_gradient_flow_morse | state | Gradient flow of a Morse function
s_gradient_like_vector_field | axiom | Gradient-like vector field
s_stable_manifold_critical_point | state | Stable manifold of a critical point
s_unstable_manifold_critical_point | state | Unstable manifold of a critical point
s_cobordism_via_morse | state | Cobordism via Morse theory
s_path_space | axiom | Path space Ω(M; p, q)
s_energy_sublevel_set | state | Energy sublevel set Ωᶜ
s_finite_dimensional_approximation_path_space | state | Finite-dimensional approximation of path space
s_morse_index_of_geodesic | state | Morse index of a geodesic
s_focal_point | axiom | Focal point
s_index_theorem_with_focal_points | theorem | Index theorem with focal points
s_homotopy_groups_classical_lie_groups | state | Homotopy groups of classical Lie groups
s_loop_space_of_symmetric_space | state | Loop space of a symmetric space
s_bott_samelson_theorem | theorem | Bott–Samelson theorem
s_bott_iteration_formula | theorem | Bott's iteration formula
s_perfect_morse_function | axiom | Perfect Morse function
s_geodesic_as_critical_point_of_energy | state | Geodesic as critical point of energy
s_existence_of_morse_functions | theorem | Existence of Morse functions
s_compact_lie_group_as_symmetric_space | state | Compact Lie group as symmetric space
s_bi_invariant_metric | axiom | Bi-invariant metric on a Lie group
s_geodesics_in_symmetric_space | state | Geodesics in a symmetric space
s_periodicity_of_geodesics_compact_symmetric | theorem | Periodicity of geodesics in compact symmetric spaces
s_morse_theory_on_loop_space_symmetric | state | Morse theory on the loop space of a symmetric space
s_non_degenerate_critical_submanifold | state | Non-degenerate critical submanifold
s_morse_bott_theory | theorem | Morse–Bott theory
s_minimax_principle_morse | theorem | Minimax principle (Morse theory)
s_normal_bundle_of_critical_submanifold | state | Normal bundle of a critical submanifold (negative bundle)
s_palais_smale_condition | axiom | Condition (C) (Palais–Smale condition)
s_energy_satisfies_condition_c | theorem | Energy functional satisfies Condition (C)
s_homotopy_type_path_space_sublevel | theorem | Homotopy type of path space sublevel sets
s_existence_of_geodesics_via_morse | theorem | Existence of geodesics via Morse theory
s_theorem_of_serre | theorem | Theorem of Serre on closed geodesics
s_curvature_bounds_and_conjugate_points | theorem | Curvature bounds and conjugate points
s_homology_of_loop_space_via_morse | state | Homology of the loop space via Morse theory
t_gradient_flow_deformation | technique | Gradient flow deformation
t_handle_attachment_technique | technique | Handle attachment technique
t_second_variation_analysis | technique | Second variation analysis
t_jacobi_field_index_computation | technique | Jacobi field method for index computation
t_finite_dimensional_approximation_paths | technique | Finite-dimensional approximation of path spaces
t_symmetric_space_morse_theory | technique | Symmetric space Morse theory
t_morse_sublevel_deformation | technique | Morse theory technique (deforming sublevel sets)
s_normal_bundle_along_geodesic | state | Normal bundle along a geodesic
s_broken_geodesic | state | Broken geodesic (piecewise geodesic path)
s_topology_of_spheres_via_morse | state | Topology of Sⁿ via Morse theory
s_topology_of_projective_spaces_via_morse | state | Topology of projective spaces via Morse theory
s_topology_of_grassmannians_via_morse | state | Topology of Grassmannians via Morse theory
s_stable_unitary_group | axiom | Stable unitary group U
s_stable_orthogonal_group | axiom | Stable orthogonal group O
s_stable_symplectic_group | axiom | Stable symplectic group Sp
s_bott_periodicity_chain | state | Bott periodicity chain of symmetric spaces
s_index_form_positive_definite | theorem | Index form is positive definite before first conjugate point
s_isotropy_representation_symmetric_space | state | Isotropy representation of a symmetric space
s_quasi_projective_variety | axiom | Quasi-projective variety
s_ringed_space | axiom | Ringed space
s_separated_morphism | axiom | Separated morphism
s_proper_morphism | axiom | Proper morphism
s_projective_morphism | axiom | Projective morphism
s_quasi_coherent_sheaf | axiom | Quasi-coherent sheaf
s_coherent_sheaf | axiom | Coherent sheaf
s_invertible_sheaf | axiom | Invertible sheaf
s_ideal_sheaf | state | Ideal sheaf
s_very_ample_divisor | axiom | Very ample divisor
s_flat_morphism | axiom | Flat morphism
s_regular_scheme | axiom | Regular scheme
s_normal_scheme | axiom | Normal scheme
s_arithmetic_genus | axiom | Arithmetic genus
s_geometric_genus | axiom | Geometric genus
s_ruled_surface | axiom | Ruled surface
s_rational_surface | axiom | Rational surface
s_minimal_model_surface | axiom | Minimal model (surface)
s_exceptional_curve | axiom | Exceptional curve of the first kind
s_closed_immersion | axiom | Closed immersion
s_open_immersion | axiom | Open immersion
s_finite_morphism | axiom | Finite morphism
s_unramified_morphism | axiom | Unramified morphism
s_formal_scheme | axiom | Formal scheme
s_elliptic_curve | axiom | Elliptic curve
s_canonical_sheaf | state | Canonical sheaf
s_dualizing_sheaf | state | Dualizing sheaf
s_conormal_sheaf | state | Conormal sheaf
s_euler_sequence | state | Euler sequence
s_twisted_sheaf | state | Twisting sheaf O(n)
s_sheaf_of_differentials | state | Sheaf of relative differentials
s_strict_transform | state | Strict transform
s_higher_direct_image | state | Higher direct image sheaf
s_irregularity | state | Irregularity
s_albanese_variety | state | Albanese variety
s_cubic_surface | state | Cubic surface
s_serre_vanishing_theorem | theorem | Serre vanishing theorem
s_adjunction_formula | theorem | Adjunction formula
s_noether_formula | theorem | Noether's formula
s_semicontinuity_theorem | theorem | Semicontinuity theorem (sheaf cohomology)
s_flat_base_change_theorem | theorem | Flat base change theorem
s_projection_formula_sheaves | theorem | Projection formula (sheaves)
s_enriques_classification | theorem | Enriques classification of surfaces
s_castelnuovo_de_franchis_theorem | theorem | Castelnuovo–de Franchis theorem
s_degree_genus_formula | theorem | Degree-genus formula
s_valuative_criterion_properness | theorem | Valuative criterion of properness
s_valuative_criterion_separatedness | theorem | Valuative criterion of separatedness
s_serre_criterion_affineness | theorem | Serre's criterion for affineness
s_cohomology_of_projective_space | theorem | Cohomology of projective space
s_cartier_theorem_smoothness | theorem | Cartier's theorem on smoothness
s_stein_factorization | theorem | Stein factorization
t_spec_construction | technique | Spec construction
t_sheaf_hom | technique | Sheaf Hom construction
t_tensor_product_of_sheaves | technique | Tensor product of sheaves
t_veronese_embedding_technique | technique | Veronese embedding technique
t_segre_embedding_technique | technique | Segre embedding technique
t_linear_system_embedding | technique | Linear system embedding
t_adjunction_technique_ag | technique | Adjunction technique (algebraic geometry)
t_flattening_stratification | technique | Flattening stratification
t_semicontinuity_method | technique | Semicontinuity method
t_formal_completion | technique | Formal completion
t_castelnuovo_contraction | technique | Castelnuovo contraction
t_minimal_model_program_step | technique | Minimal model program step
t_taking_stalks | technique | Taking stalks
t_cech_cohomology_computation | technique | Čech cohomology computation
t_canonical_embedding_technique | technique | Canonical embedding technique
s_holomorphic_map | axiom | Holomorphic map
s_dolbeault_complex | axiom | Dolbeault complex
s_theta_divisor | axiom | Theta divisor
s_schubert_cycle | axiom | Schubert cycle
s_grothendieck_residue | axiom | Grothendieck residue
s_hodge_filtration | axiom | Hodge filtration
s_variation_of_hodge_structure | axiom | Variation of Hodge structure
s_dolbeault_cohomology | state | Dolbeault cohomology
s_abel_jacobi_map | state | Abel–Jacobi map
s_lefschetz_pencil | state | Lefschetz pencil
s_pluricanonical_system | state | Pluricanonical system
s_period_mapping | state | Period mapping
s_dolbeault_theorem | theorem | Dolbeault theorem
s_hard_lefschetz_theorem | theorem | Hard Lefschetz theorem
s_lefschetz_1_1_theorem | theorem | Lefschetz (1,1) theorem
s_riemann_roch_for_surfaces | theorem | Riemann–Roch for surfaces
s_abel_theorem | theorem | Abel's theorem
s_jacobi_inversion_theorem | theorem | Jacobi inversion theorem
s_torelli_theorem | theorem | Torelli theorem
s_bertini_theorem | theorem | Bertini's theorem
s_gaga_principle | theorem | GAGA principle
s_riemann_bilinear_relations | theorem | Riemann bilinear relations
s_wirtinger_theorem | theorem | Wirtinger's theorem
s_newlander_nirenberg_theorem | theorem | Newlander–Nirenberg theorem
s_castelnuovo_rationality_criterion | theorem | Castelnuovo's rationality criterion
s_enriques_kodaira_classification | theorem | Enriques–Kodaira classification
s_exponential_sheaf_sequence | theorem | Exponential sheaf sequence
s_grothendieck_residue_theorem | theorem | Grothendieck residue theorem
s_kodaira_spencer_deformation_theory | theorem | Kodaira–Spencer deformation theory
t_dolbeault_cohomology_computation | technique | Dolbeault cohomology computation
t_hodge_theoretic_methods | technique | Hodge-theoretic methods
t_residue_calculus_on_manifolds | technique | Residue calculus on manifolds
t_schubert_calculus | technique | Schubert calculus
t_blow_up_technique | technique | Blow-up technique
t_adjunction_formula_technique | technique | Adjunction formula technique
t_lefschetz_pencil_method | technique | Lefschetz pencil method
t_period_mapping_torelli_arguments | technique | Period mapping and Torelli-type arguments
t_deformation_theory_methods | technique | Deformation theory methods
s_smooth_variety | axiom | Smooth variety
s_complete_variety | axiom | Complete variety
s_unirational_variety | axiom | Unirational variety
s_rational_variety | axiom | Rational variety
s_algebraic_cycle | axiom | Algebraic cycle
s_pure_hodge_structure | axiom | Pure Hodge structure
s_plurigenera | state | Plurigenera
s_formal_neighborhood | state | Formal neighborhood
s_multiplicity_of_a_singularity | state | Multiplicity of a singularity
s_intersection_number_global | state | Intersection number (global)
s_degree_of_a_finite_morphism | state | Degree of a finite morphism
s_genus_formula_plane_curves | theorem | Genus formula for plane curves
s_chevalley_theorem_constructible_sets | theorem | Chevalley's theorem on constructible sets
s_moving_lemma | theorem | Moving lemma
s_noether_theorem_rational_curves_surfaces | theorem | Noether's theorem on rational curves on surfaces
t_adjunction_formula | technique | Adjunction formula
t_linear_system_technique | technique | Linear system technique
t_projection_from_point | technique | Projection from a point
t_normalization_technique | technique | Normalization of a variety
t_base_change_technique | technique | Base change technique
t_descent_technique | technique | Descent technique
t_specialization_argument | technique | Specialization argument
t_generic_point_argument | technique | Generic point argument
s_smooth_homotopy | axiom | Smooth homotopy
s_degree_mod_2 | state | Degree modulo 2
s_sign_of_preimage_point | axiom | Sign of a regular preimage point
s_homotopy_invariance_of_degree | theorem | Homotopy invariance of Brouwer degree
s_boundary_theorem_degree | theorem | Boundary theorem for degree
s_degree_of_composition | theorem | Degree of a composition
s_degree_determines_surjectivity | theorem | Nonzero degree implies surjectivity
s_classification_of_smooth_compact_1_manifolds | theorem | Classification of compact 1-manifolds
s_smooth_approximation_theorem | theorem | Smooth approximation theorem
s_index_of_isolated_zero_of_vector_field | state | Index of an isolated zero of a vector field
s_linking_number | state | Linking number
s_hopf_fibration | state | Hopf fibration
s_smooth_retraction | axiom | Smooth retraction
s_framed_submanifold | axiom | Framed submanifold
s_framed_cobordism | axiom | Framed cobordism
s_pontryagin_manifold | state | Pontryagin manifold
s_smooth_isotopy | axiom | Smooth isotopy
s_density_of_regular_values | theorem | Density of regular values
s_well_definedness_of_mod_2_degree | theorem | Well-definedness of mod 2 degree
s_boundary_theorem_mod_2 | theorem | Boundary theorem (mod 2)
s_index_nondegenerate_zero_equals_sign_det | theorem | Index of a non-degenerate zero
s_nonvanishing_vector_fields_on_odd_spheres | theorem | Nonvanishing vector fields on odd-dimensional spheres
s_linking_number_homotopy_invariance | theorem | Linking number is a homotopy invariant
s_existence_of_maps_with_arbitrary_degree | theorem | Existence of maps of any given degree
s_browns_refinement_of_sard | theorem | Brown's refinement of Sard's theorem
t_induction_on_dimension_sard | technique | Induction on dimension (Sard proof)
t_proof_of_brouwer_via_sard | technique | Proof of Brouwer fixed point theorem via Sard
t_one_manifold_classification_technique | technique | One-manifold classification argument
t_diagonal_self_intersection | technique | Diagonal map and self-intersection number
t_degree_based_surjectivity | technique | Degree-based surjectivity argument
s_local_parametrization | axiom | Local parametrization
s_derivative_of_smooth_map_between_manifolds | axiom | Derivative of a smooth map between manifolds
s_tangent_space_of_preimage | state | Tangent space of a preimage
s_transversal_preimage_theorem | theorem | Transversal preimage theorem
s_transverse_intersection_dimension_formula | theorem | Transverse intersection dimension formula
s_stability_of_immersions | theorem | Stability of immersions
s_stability_of_submersions | theorem | Stability of submersions
s_stability_of_embeddings | theorem | Stability of embeddings
s_stability_of_diffeomorphisms | theorem | Stability of diffeomorphisms
s_stability_of_transversality | theorem | Stability of transversality
s_nondegenerate_critical_points_are_isolated | theorem | Nondegenerate critical points are isolated
s_compact_morse_finite_critical_points | theorem | Compact manifolds have finitely many Morse critical points
s_boundary_is_manifold | theorem | Boundary of a manifold is a manifold
s_epsilon_neighborhood_theorem | theorem | Epsilon-neighborhood theorem
s_transversality_homotopy_theorem | theorem | Transversality homotopy theorem
s_extension_theorem_transversality | theorem | Extension theorem for transversality
s_intersection_number_mod_2 | axiom | Intersection number mod 2
s_intersection_number_mod_2_homotopy_invariance | theorem | Homotopy invariance of intersection number mod 2
s_mod_2_intersection_number_of_submanifolds | state | Mod 2 intersection number of submanifolds
s_degree_mod_2_homotopy_invariance | theorem | Homotopy invariance of degree mod 2
s_no_retraction_theorem | theorem | No retraction theorem
s_brouwer_fixed_point_mod_2 | theorem | Brouwer fixed-point theorem (mod 2 proof)
s_oriented_intersection_number | state | Oriented intersection number
s_oriented_intersection_number_homotopy_invariance | theorem | Homotopy invariance of oriented intersection number
s_oriented_intersection_number_of_submanifolds | state | Oriented intersection number of submanifolds
s_oriented_boundary_theorem | theorem | Oriented boundary theorem
s_brouwer_degree | state | Brouwer degree
s_hopf_degree_theorem | theorem | Hopf degree theorem
s_local_lefschetz_number | state | Local Lefschetz number
s_euler_characteristic_via_diagonal_intersection | theorem | Euler characteristic as self-intersection of diagonal
s_euler_equals_lefschetz_of_identity | theorem | Euler characteristic equals Lefschetz number of identity
s_self_intersection_number | state | Self-intersection number
s_euler_characteristic_of_spheres | theorem | Euler characteristic of spheres
s_brouwer_fixed_point_oriented | theorem | Brouwer fixed-point theorem (oriented proof)
s_intersection_number_symmetry | theorem | Intersection number symmetry
s_pullback_commutes_with_wedge | theorem | Pullback commutes with wedge product
s_change_of_variables_for_forms | theorem | Change of variables for differential forms
s_degree_formula_via_integration | theorem | Degree formula via integration
t_dimension_counting_transversality | technique | Dimension counting for transversality
t_parametric_transversality_method | technique | Parametric transversality method (Thom's method)
t_cobordism_argument | technique | Cobordism argument
t_isotopy_deformation_argument | technique | Isotopy and deformation argument
t_diagonal_intersection_method | technique | Diagonal intersection method
t_degree_counting_preimages | technique | Degree computation by counting preimages
t_integration_pairing_topological_invariants | technique | Integration pairing for topological invariants
s_compactly_supported_de_rham_cohomology | axiom | Compactly supported de Rham cohomology
s_compact_vertical_support_cohomology | axiom | Cohomology with compact vertical support
s_bump_form | axiom | Bump form
s_relative_de_rham_cohomology | axiom | Relative de Rham cohomology
s_acyclic_cover | axiom | Acyclic cover (Leray cover)
s_tautological_bundle_grassmannian | axiom | Tautological bundle over the Grassmannian
s_integration_along_fiber | state | Integration along the fiber
s_global_angular_form | state | Global angular form
s_diagonal_class | state | Diagonal class
s_spectral_sequence_collapse | state | Collapsing of a spectral sequence
s_cohomology_of_SO_n | state | Cohomology of SO(n)
s_flag_bundle | state | Flag bundle
s_poincare_lemma_compact_support | theorem | Poincaré lemma for compactly supported cohomology
s_mayer_vietoris_compact_support | theorem | Mayer-Vietoris sequence for compact supports
s_finite_dimensionality_de_rham | theorem | Finite-dimensionality of de Rham cohomology
s_generalized_mayer_vietoris_principle | theorem | Generalized Mayer-Vietoris principle
s_partition_of_unity_presheaf_acyclicity | theorem | Partition of unity argument for presheaves of forms
s_long_exact_sequence_compact_support | theorem | Long exact sequence of a pair for compact supports
s_infinite_grassmannian_classifying_space | theorem | Infinite Grassmannian as classifying space
s_cohomology_of_BU_k | theorem | Cohomology of BU(k)
s_cohomology_of_BO_k | theorem | Cohomology of BO(k)
s_naturality_of_characteristic_classes | theorem | Naturality of characteristic classes
s_homotopy_axiom_cech_cohomology | theorem | Homotopy axiom for Čech cohomology
t_zigzag_double_complex | technique | Zigzag argument in a double complex
t_cochain_homotopy | technique | Cochain homotopy construction
t_compact_exhaustion_direct_limit | technique | Compact exhaustion and direct limit for H*_c
s_symplectic_subspace | axiom | Symplectic subspace
s_isotropic_subspace | axiom | Isotropic subspace
s_coisotropic_subspace | axiom | Coisotropic subspace
s_lagrangian_subspace | axiom | Lagrangian subspace
s_symplectic_complement | state | Symplectic complement
s_lagrangian_grassmannian | state | Lagrangian Grassmannian Λ(n)
s_compatible_complex_structure_symplectic | axiom | Compatible complex structure on symplectic vector space
s_positive_compatible_triple | state | Positive compatible triple (ω, J, g)
s_space_of_compatible_complex_structures | state | Space of compatible complex structures J(V,ω)
s_polar_decomposition_sp2n | theorem | Polar decomposition in Sp(2n)
s_maslov_index_for_paths | state | Maslov index for paths of Lagrangian subspaces
s_maslov_cycle | state | Maslov cycle
s_conley_zehnder_index | state | Conley-Zehnder index
s_moser_isotopy_lemma_relative | theorem | Moser isotopy lemma (relative version)
s_weinstein_isotropic_neighbourhood | theorem | Weinstein isotropic neighborhood theorem
s_isotropic_submanifold | axiom | Isotropic submanifold
s_coisotropic_submanifold | axiom | Coisotropic submanifold
s_symplectic_fiber_bundle | state | Symplectic fiber bundle
s_symplectic_connection_on_fibration | state | Symplectic connection on a fibration
s_hamiltonian_isotopy | axiom | Hamiltonian isotopy
s_poisson_bracket | state | Poisson bracket
s_existence_uniqueness_moment_maps | theorem | Existence and uniqueness of moment maps
s_symplectic_reduction_at_general_level | theorem | Symplectic reduction at a general level
s_coadjoint_orbit | state | Coadjoint orbit
s_coadjoint_orbits_symplectic | theorem | Coadjoint orbits as symplectic manifolds
s_flux_homomorphism | state | Flux homomorphism
s_flux_conjecture_ono | theorem | Flux conjecture (Ono)
s_calabi_homomorphism | state | Calabi homomorphism
s_hofer_metric | axiom | Hofer metric on Ham(M,ω)
s_hofer_nondegeneracy_theorem | theorem | Hofer's nondegeneracy theorem
s_hofer_zehnder_capacity | state | Hofer-Zehnder capacity
s_displacement_energy | state | Displacement energy
s_tame_almost_complex_structure | axiom | Tame almost complex structure
s_compatible_almost_complex_structure_manifold | axiom | Compatible almost complex structure on symplectic manifold
s_contractibility_of_J_M_omega | theorem | Contractibility of J(M,ω)
s_symplectic_capacity_axiomatic | axiom | Symplectic capacity (axiomatic)
s_gromov_width | state | Gromov width
s_ekeland_hofer_capacities | state | Ekeland-Hofer capacities
s_j_holomorphic_curve | axiom | J-holomorphic curve
s_energy_j_holomorphic_curve | state | Energy of a J-holomorphic curve
s_energy_identity_j_holomorphic | theorem | Energy identity for J-holomorphic curves
s_moduli_space_j_holomorphic | state | Moduli space of J-holomorphic curves
s_fredholm_theory_dbar_J | state | Fredholm theory for the ∂̄_J operator
s_virtual_dimension_moduli | state | Virtual dimension of moduli space of J-holomorphic curves
s_simple_j_holomorphic_curve | axiom | Simple (somewhere injective) J-holomorphic curve
s_transversality_simple_curves | theorem | Transversality for simple J-holomorphic curves
s_bubbling_phenomenon | state | Bubbling phenomenon for J-holomorphic curves
s_stable_map_gw | axiom | Stable map
s_evaluation_map_gw | state | Evaluation map in Gromov-Witten theory
s_removal_of_singularities_j_holo | theorem | Removal of singularities for J-holomorphic curves
s_monotonicity_lemma_j_holo | theorem | Monotonicity lemma for J-holomorphic curves
s_symplectic_packing | state | Symplectic packing problem
s_symplectic_form_on_surfaces | state | Symplectic form on surfaces
s_eliashberg_rigidity | theorem | Eliashberg rigidity theorem
s_mcduff_rational_ruled | theorem | McDuff's theorem on rational and ruled symplectic 4-manifolds
s_taubes_sw_equals_gr | theorem | Taubes' theorem (SW = Gr)
s_existence_j_holo_curves_taubes | theorem | Existence of J-holomorphic curves via Taubes
s_symplectic_isotopy_problem | state | Symplectic isotopy problem
s_positivity_of_intersections | theorem | Positivity of intersections of J-holomorphic curves
s_adjunction_formula_j_holomorphic | theorem | Adjunction formula for J-holomorphic curves
s_recurrence_hamiltonian | theorem | Recurrence theorem for Hamiltonian diffeomorphisms
s_liouville_measure_symplectic | state | Liouville measure on a symplectic manifold
s_contact_hypersurface | axiom | Contact-type hypersurface in a symplectic manifold
s_liouville_vector_field | state | Liouville vector field
s_symplectization | state | Symplectization of a contact manifold
s_symplectic_vector_bundle | axiom | Symplectic vector bundle
s_characteristic_foliation_coisotropic | state | Characteristic foliation on a coisotropic submanifold
s_exact_lagrangian_submanifold | axiom | Exact Lagrangian submanifold
s_exact_symplectic_manifold | axiom | Exact symplectic manifold
s_spectral_invariants_floer | state | Spectral invariants in Hamiltonian Floer theory
s_floer_homology_hamiltonian | state | Hamiltonian Floer homology
s_energy_capacity_inequality | theorem | Energy-capacity inequality
t_moser_method | technique | Moser's method
t_thurston_symplectic_fibration | technique | Thurston's construction for symplectic fiber bundles
t_symplectic_cutting | technique | Symplectic cutting
t_generating_function_lagrangian | technique | Generating function for Lagrangians
t_neck_stretching | technique | Neck-stretching technique
t_gromov_compactification | technique | Gromov compactification of moduli spaces
t_symplectic_sum_construction | technique | Symplectic sum construction
t_symplectic_blowup | technique | Symplectic blowup
t_hamiltonian_perturbation | technique | Hamiltonian perturbation technique
s_uncountable_set | axiom | Uncountable set
s_subbasis_for_a_topology | axiom | Subbasis for a topology
s_box_topology | axiom | Box topology
s_metric_topology | axiom | Metric topology
s_metrizable_space | axiom | Metrizable space
s_hausdorff_unique_limits | theorem | Hausdorff spaces have unique limits
s_separation_of_a_topological_space | axiom | Separation of a topological space
s_connected_subspace | axiom | Connected subspace
s_path_connected_implies_connected | theorem | Path connectedness implies connectedness
s_path_component | axiom | Path component
s_locally_path_connected_space | axiom | Locally path-connected space
s_compact_space | axiom | Compact space
s_compact_subspace | axiom | Compact subspace
s_tube_lemma | theorem | Tube lemma
s_limit_point_compactness | axiom | Limit point compactness
s_metrizable_implies_normal | theorem | Every metrizable space is normal
s_regular_second_countable_implies_normal | theorem | Every regular second-countable space is normal
s_paracompact_hausdorff_implies_normal | theorem | Every paracompact Hausdorff space is normal
s_smirnov_metrization_theorem | theorem | Smirnov metrization theorem
s_baire_space | axiom | Baire space
s_baire_category_theorem_lch | theorem | Baire category theorem (locally compact Hausdorff version)
s_homotopy_of_paths | axiom | Homotopy of paths
s_homotopy_equivalence | axiom | Homotopy equivalence
s_path_homotopy_class | axiom | Path homotopy class
s_simply_connected_space | axiom | Simply connected space
s_fundamental_group_of_circle | theorem | Fundamental group of the circle
s_path_lifting_property | theorem | Path lifting property
s_homotopy_lifting_property | theorem | Homotopy lifting property
s_lifting_criterion | theorem | Lifting criterion (general lifting lemma)
s_deformation_retract | axiom | Deformation retract
s_deformation_retract_preserves_fundamental_group | theorem | Deformation retract preserves fundamental group
s_induced_homomorphism_fundamental_group | state | Induced homomorphism on fundamental groups
s_fundamental_group_of_product | theorem | Fundamental group of a product space
s_jordan_separation_theorem | theorem | Jordan separation theorem
s_nonseparation_theorem_for_arcs | theorem | Nonseparation theorem for arcs
s_fundamental_group_of_wedge_sum | theorem | Fundamental group of a wedge sum
t_van_kampen_surface_computation | technique | Computing pi_1 of surfaces via van Kampen
s_fundamental_group_of_torus | theorem | Fundamental group of the torus
s_fundamental_group_of_rp2 | theorem | Fundamental group of the real projective plane
s_surface_2_manifold | axiom | Surface (2-manifold)
s_fundamental_polygon | axiom | Fundamental polygon
s_orientable_surface | axiom | Orientable surface
s_equivalence_of_covering_spaces | axiom | Equivalence of covering spaces
s_lifting_correspondence | state | Lifting correspondence (monodromy action)
s_classification_of_covering_spaces | theorem | Classification of covering spaces
s_semilocally_simply_connected | axiom | Semilocally simply connected space
s_existence_of_universal_covering | theorem | Existence of universal covering space
s_normal_covering_space | axiom | Normal (regular) covering space
s_covering_index_formula | theorem | Index formula for coverings
t_urysohn_function_construction | technique | Urysohn function construction (dyadic rational technique)
t_lebesgue_number_argument | technique | Lebesgue number argument
t_tube_lemma_argument | technique | Tube lemma argument
s_sequence_lemma | theorem | Sequence lemma
s_metric_second_countable_iff_separable | theorem | Metric space is second-countable iff separable
s_long_line | axiom | Long line
s_topologists_sine_curve | axiom | Topologist's sine curve
s_t0_space | axiom | T0-space (Kolmogorov)
s_pseudometric_space | axiom | Pseudometric space
s_quotient_map | axiom | Quotient map (identification map)
s_locally_compact_space | axiom | Locally compact space
s_equicontinuous_family | axiom | Equicontinuous family
s_totally_bounded_uniform_space | axiom | Totally bounded (precompact) uniform space
s_cauchy_net | axiom | Cauchy net in a uniform space
s_base_for_uniformity | axiom | Base for a uniformity
s_even_continuity | axiom | Even continuity
s_convergence_class | axiom | Convergence class
s_realcompact_space | axiom | Realcompact space (Hewitt-Nachbin space)
s_morse_kelley_set_theory | axiom | Morse-Kelley set theory
s_mk_classification_axiom_scheme | axiom | Classification axiom scheme (MK comprehension)
s_axiom_of_choice | axiom | Axiom of choice
s_topology_generated_by_subbase | state | Topology generated by a subbase
s_eventually_in | state | Eventually in
s_frequently_in | state | Frequently in
s_cluster_point_of_net | state | Cluster point of a net
s_universal_net | state | Universal net (ultranet)
s_eventuality_filter | state | Eventuality filter (filter generated by a net)
s_evaluation_map_diagonal | state | Evaluation map (diagonal map)
s_partition_of_unity | state | Partition of unity
s_hewitt_realcompactification | state | Hewitt realcompactification
s_gauge_family_pseudometrics | state | Gauge (family of pseudometrics)
s_metrizable_uniformity | state | Metrizable uniformity
s_existence_of_universal_subnets | theorem | Existence of universal subnets
s_kelleys_convergence_class_theorem | theorem | Kelley's convergence class theorem
s_existence_of_ultrafilters | theorem | Existence of ultrafilters (ultrafilter lemma)
s_equivalence_of_nets_and_filters | theorem | Equivalence of nets and filters
s_universal_property_product_topology | theorem | Universal property of product topology
s_universal_property_quotient_topology | theorem | Universal property of quotient topology
s_product_of_continuous_maps_continuous | theorem | Product of continuous functions is continuous
s_closed_surjection_is_quotient | theorem | Closed continuous surjection is a quotient map
s_open_surjection_is_quotient | theorem | Open continuous surjection is a quotient map
s_completely_regular_separates_points_closed | theorem | Completely regular spaces separate points from closed sets
s_tychonoff_embedding_theorem | theorem | Tychonoff embedding theorem
s_normal_implies_completely_regular | theorem | Normal spaces are completely regular
s_filter_characterization_compactness | theorem | Filter characterization of compactness
s_tychonoff_equivalent_to_ac | theorem | Tychonoff theorem equivalent to axiom of choice
s_alexandroff_hausdorff_iff_locally_compact | theorem | Alexandroff compactification Hausdorff iff locally compact Hausdorff
s_stone_cech_universal_property | theorem | Universal property of Stone-Čech compactification
s_compact_implies_paracompact | theorem | Compact implies paracompact
s_compact_hausdorff_unique_uniformity | theorem | Unique uniformity on compact Hausdorff space
s_baire_category_complete_metric | theorem | Baire category theorem (complete metric space)
s_uniform_space_completely_regular | theorem | Hausdorff uniform space is completely regular
s_uniform_continuity_implies_continuity | theorem | Uniform continuity implies continuity
s_existence_uniqueness_completion | theorem | Existence and uniqueness of completion
s_extension_uniformly_continuous | theorem | Extension of uniformly continuous functions
s_compact_open_refines_pointwise | theorem | Compact-open topology refines pointwise convergence
s_evaluation_map_continuity_theorem | theorem | Evaluation map continuity
s_compact_open_equals_uniform_convergence | theorem | Compact-open equals uniform convergence for compact domain
s_equicontinuity_topology_agreement | theorem | Equicontinuity and topology agreement
s_even_continuity_equicontinuity_relation | theorem | Even continuity and equicontinuity relationship
s_ascoli_via_even_continuity | theorem | Ascoli theorem via even continuity (Kelley)
t_net_argument_compactness | technique | Net argument for compactness
t_filter_argument_compactness | technique | Filter argument for compactness
t_diagonal_argument_metrization | technique | Diagonal argument for metrization
t_zorn_in_topology | technique | Zorn's lemma in topology
t_cofinal_subnet_extraction | technique | Cofinal subnet extraction
t_cauchy_filter_completion_construction | technique | Cauchy filter completion construction
t_cofinite_topology_trick | technique | Cofinite topology trick (Kelley's AC proof)
t_evaluation_map_embedding | technique | Evaluation map embedding
s_cw_complex | axiom | CW complex
s_n_skeleton | state | n-skeleton
t_cell_attachment | technique | Cell attachment
s_contractible_space | axiom | Contractible space
s_homotopy_extension_property | axiom | Homotopy extension property
s_change_of_basepoint_isomorphism | theorem | Change of basepoint isomorphism
s_universal_covering_space | axiom | Universal covering space
s_fundamental_group_of_graph | theorem | Fundamental group of a graph
s_fundamental_group_of_surfaces | theorem | Fundamental group of closed surfaces
s_singular_n_simplex | axiom | Singular n-simplex
s_singular_chain_group | axiom | Singular chain group Cₙ(X)
s_boundary_operator | axiom | Boundary operator ∂ₙ
s_singular_homology_group | axiom | Singular homology group Hₙ(X)
s_reduced_homology | state | Reduced homology
s_homotopy_invariance_of_homology | theorem | Homotopy invariance of homology
t_chain_homotopy_technique | technique | Chain homotopy
s_relative_homology_group | axiom | Relative homology group Hₙ(X, A)
t_barycentric_subdivision | technique | Barycentric subdivision
s_good_pair | axiom | Good pair (X, A)
s_homology_of_spheres | theorem | Homology of spheres
s_local_homology_group | state | Local homology group
s_homology_with_coefficients | axiom | Homology with coefficients
s_exact_sequence_of_triple | theorem | Long exact sequence of a triple
s_homology_of_real_projective_space | state | Homology of ℝPⁿ
s_homology_of_complex_projective_space | state | Homology of ℂPⁿ
t_transfer_homomorphism_covering | technique | Transfer homomorphism (covering spaces)
s_coboundary_operator | axiom | Coboundary operator δⁿ
s_cross_product_cohomology | state | Cross product (cohomology)
s_cap_product | axiom | Cap product
s_fundamental_class_manifold | axiom | Fundamental class of a manifold
s_cohomology_with_compact_support | axiom | Cohomology with compact support
t_bockstein_homomorphism | technique | Bockstein homomorphism
s_orientation_double_cover | state | Orientation double cover
s_higher_homotopy_group | axiom | Higher homotopy group πₙ(X)
s_pi_n_abelian_for_n_geq_2 | theorem | πₙ is abelian for n ≥ 2
s_relative_homotopy_group | axiom | Relative homotopy group πₙ(X, A)
s_long_exact_sequence_of_pair_homotopy | theorem | Long exact sequence of a pair (homotopy)
s_long_exact_sequence_of_fibration | theorem | Long exact sequence of a fibration
s_cw_approximation_theorem | theorem | CW approximation theorem
s_relative_hurewicz_theorem | theorem | Relative Hurewicz theorem
s_n_connected_space | axiom | n-connected space
s_n_connected_pair | axiom | n-connected pair
s_postnikov_tower | state | Postnikov tower
s_k_invariant | state | k-invariants
t_obstruction_theory | technique | Obstruction theory
s_loopspace | state | Loop space ΩX
s_path_loop_fibration | state | Path-loop fibration
s_pi_3_S2_isomorphic_Z | theorem | π₃(S²) ≅ ℤ
s_fiber_sequence_puppe | state | Fiber sequence (Puppe sequence)
s_cofiber_sequence_puppe | state | Cofiber sequence (Puppe sequence)
s_pi_1_action_on_pi_n | state | π₁ action on πₙ
s_compression_lemma | theorem | Compression lemma
s_delta_complex | axiom | Δ-complex
t_acyclic_models | technique | Acyclic models
s_naturality_of_les | theorem | Naturality of long exact sequences
s_suspension_space | state | Suspension ΣX
s_suspension_isomorphism_cohomology | theorem | Suspension isomorphism (cohomology)
s_homology_of_wedge | state | Homology of a wedge sum
s_attaching_cells_and_homology | theorem | Effect of cell attachment on homology
s_cellular_equals_singular | theorem | Cellular homology equals singular homology
s_degree_and_local_degree | theorem | Degree as sum of local degrees
s_antipodal_map_degree | theorem | Degree of the antipodal map
s_mapping_degree_formula | theorem | Mapping degree formula
s_cohomology_ring_of_real_projective_space | theorem | Cohomology ring of ℝPⁿ
s_cohomology_ring_of_complex_projective_space | theorem | Cohomology ring of ℂPⁿ
s_graded_commutativity_of_cup_product | theorem | Graded commutativity of cup product
s_cohomological_mayer_vietoris | theorem | Cohomological Mayer-Vietoris sequence
s_cup_product_distinguishes_spaces | theorem | Cup product distinguishes spaces
s_fiber_bundle_is_serre_fibration | theorem | Fiber bundle is a Serre fibration
s_intersection_form | state | Intersection form
s_homology_manifold | axiom | Homology manifold
t_smith_theory | technique | Smith theory
s_homotopy_type | axiom | Homotopy type
s_retraction | axiom | Retraction (topological)
s_fundamental_groupoid | axiom | Fundamental groupoid
s_h_space | axiom | H-space
s_co_h_space | axiom | Co-H-space (H-cogroup)
s_hurewicz_fibration | axiom | Hurewicz fibration
s_mapping_path_space | state | Mapping path space (path-space fibration)
s_abstract_simplicial_complex | axiom | Abstract simplicial complex
s_star_and_link | axiom | Star and link of a simplex
s_cellular_map | axiom | Cellular map
s_relative_cw_complex | axiom | Relative CW complex
s_relative_singular_homology | state | Relative singular homology
s_short_five_lemma | theorem | Short five lemma
s_slant_product | state | Slant product
s_kronecker_pairing | state | Kronecker pairing
s_n_connected_map | axiom | n-connected map
s_suspension_loop_adjunction | theorem | Suspension-loop space adjunction (Σ ⊣ Ω)
s_representability_of_cohomology | theorem | Representability of cohomology
s_obstruction_cocycle | state | Obstruction cocycle
s_difference_cochain | state | Difference cochain
s_moore_postnikov_decomposition | state | Moore-Postnikov decomposition
s_cohomology_operation | axiom | Cohomology operation
s_steenrod_squares | state | Steenrod squares
s_steenrod_reduced_powers | state | Steenrod reduced powers
s_steenrod_algebra | state | Steenrod algebra
s_thom_space | state | Thom space
s_cone_topological | state | Cone CX
s_join_of_spaces | state | Join of spaces
s_homotopy_pullback | state | Homotopy pullback
s_homotopy_fiber | state | Homotopy fiber
s_induced_fibration | state | Induced fibration (pullback bundle)
s_hopf_algebra_h_space_homology | state | Hopf algebra structure on H-space homology
s_tensor_product_chain_complexes | axiom | Tensor product of chain complexes
s_smash_product | state | Smash product
s_wedge_sum | axiom | Wedge sum
s_homotopy_pushout | state | Homotopy pushout
t_spectral_sequence_computation | technique | Spectral sequence computation
s_ndr_pair | axiom | NDR pair (neighborhood deformation retract pair)
s_ndr_pair_characterization_of_cofibrations | theorem | NDR pair characterization of cofibrations
s_loop_eilenberg_maclane_delooping | theorem | Loop space of Eilenberg-MacLane space delooping
s_spectrum_stable_homotopy | axiom | Spectrum (stable homotopy theory)
s_omega_spectrum | axiom | Omega spectrum (Ω-spectrum)
s_stiefel_whitney_numbers | state | Stiefel-Whitney numbers
s_wu_formula | theorem | Wu formula
s_localization_of_spaces | state | Localization of spaces at a set of primes
s_rationalization_of_space | state | Rationalization of a space
s_p_localization_of_space | state | p-localization of a space
s_p_completion_of_space | state | p-completion of a space
s_sullivan_arithmetic_square | theorem | Sullivan arithmetic square (fracture theorem)
s_formal_space_rational_homotopy | axiom | Formal space (rational homotopy theory)
s_hopf_algebra_structure_on_loop_space_homology | state | Hopf algebra structure on loop space homology
s_cellular_chain_complex | state | Cellular chain complex
s_graph_as_1_dim_cw_complex | axiom | Graph as 1-dimensional CW complex
t_based_vs_free_homotopy | technique | Based versus free homotopy comparison
t_help_lemma | technique | HELP lemma (homotopy extension and lifting property)
t_zabrodsky_mixing | technique | Zabrodsky mixing
t_serre_ss_computation_eilenberg_maclane | technique | Computing H*(K(ℤ,n)) via Serre spectral sequence
s_boolean_lattice | axiom | Boolean lattice B_n
s_partition_lattice_pi_n | state | Partition lattice Pi_n
s_youngs_lattice | state | Young's lattice
s_zeta_function_of_a_poset | state | Zeta function of a poset
s_philip_halls_theorem_posets | theorem | Philip Hall's theorem (Möbius function)
s_crosscut_theorem_rota | theorem | Crosscut theorem (Rota)
s_euler_characteristic_mobius_function | theorem | Euler characteristic and Möbius function
s_order_complex | state | Order complex Δ(P)
s_order_polynomial | state | Order polynomial Ω(P, m)
s_strict_order_polynomial | state | Strict order polynomial Ω̄(P, m)
s_reciprocity_theorem_order_polynomials | theorem | Reciprocity theorem for order polynomials (Stanley)
s_exponential_generating_function | axiom | Exponential generating function
s_rational_generating_function | state | Rational generating function
s_d_finite_generating_function | state | D-finite (holonomic) generating function
t_transfer_matrix_method | technique | Transfer matrix method
t_sieve_inclusion_exclusion | technique | Sieve methods (inclusion-exclusion)
s_principle_of_inclusion_exclusion | theorem | Principle of inclusion-exclusion
s_composition_of_an_integer | axiom | Composition of an integer
s_descent_of_a_permutation | state | Descent of a permutation
s_descent_number_and_major_index | state | Descent number and major index
s_inversion_of_a_permutation | state | Inversion of a permutation
s_eulerian_polynomial | state | Eulerian polynomial A_n(t)
s_q_factorial | state | q-analog of n!
s_exponential_formula_combinatorics | theorem | Exponential formula (combinatorics)
s_compositional_formula | theorem | Compositional formula (exponential)
s_p_partition_generating_function | state | P-partition generating function
s_fundamental_theorem_of_p_partitions | theorem | Fundamental theorem of P-partitions (Stanley)
s_linear_extension_of_a_poset | state | Linear extension of a poset
s_graded_poset | axiom | Rank function and graded poset
s_rank_generating_function | state | Rank generating function
s_characteristic_polynomial_poset | state | Characteristic polynomial of a lattice
s_chromatic_polynomial | state | Chromatic polynomial χ_G(k)
s_stirling_numbers_second_kind | state | Stirling numbers of the second kind S(n,k)
s_stirling_numbers_first_kind | state | Stirling numbers of the first kind c(n,k)
s_binomial_poset | axiom | Binomial poset
s_eulerian_poset | axiom | Eulerian poset
s_differential_poset | axiom | Differential poset
s_monomial_symmetric_function | state | Monomial symmetric function m_λ
s_homogeneous_symmetric_function | state | Homogeneous symmetric function h_k
s_hall_inner_product | axiom | Hall inner product
s_knuth_equivalence | state | Knuth equivalence
t_jeu_de_taquin | technique | Jeu de taquin (Schützenberger)
t_promotion_and_evacuation | technique | Promotion and evacuation (Schützenberger)
s_littlewood_richardson_tableau | state | Littlewood-Richardson tableau
s_border_strip | state | Border strip (rim hook)
s_dual_cauchy_identity | theorem | Dual Cauchy identity
t_omega_involution | technique | Omega involution ω
s_skew_schur_function | state | Skew Schur function s_{λ/μ}
s_hall_littlewood_polynomial | state | Hall-Littlewood polynomial P_λ(x;t)
s_macdonald_polynomial | state | Macdonald polynomial P_λ(x;q,t)
s_macmahon_plane_partition_gf | theorem | MacMahon's plane partition generating function
s_lattice_path | axiom | Lattice path
t_reflection_principle_andre | technique | Reflection principle (André)
s_chromatic_symmetric_function | state | Chromatic symmetric function X_G (Stanley)
s_stanley_symmetric_function | state | Stanley symmetric function F_w
s_fundamental_quasisymmetric_function | state | Fundamental quasisymmetric function F_{n,S}
s_quasisymmetric_function | axiom | Quasisymmetric function
s_gessels_formula_p_partitions | theorem | Gessel's formula for P-partitions
s_schur_positivity | state | Schur positivity
s_hook_content_formula | theorem | Hook content formula
t_involution_principle | technique | Involution principle (Garsia-Milne)
s_schensted_theorem | theorem | Schensted's theorem
s_greenes_theorem | theorem | Greene's theorem
t_hillman_grassl_correspondence | technique | Hillman-Grassl correspondence
s_stanley_hook_content_formula_plane_partitions | theorem | Stanley's hook content formula for column-strict plane partitions
t_combinatorial_species | technique | Theory of combinatorial species (Joyal)
t_gaussian_elimination_q_counting | technique | Gaussian elimination and q-counting
s_f_vector_h_vector_simplicial_complex | state | f-vector and h-vector of a simplicial complex
s_shellability | axiom | Shellability
t_el_shellability | technique | EL-shellability and CL-shellability
t_polyhedral_geometry_of_partitions | technique | Polyhedral geometry of partitions
s_multigraph_definition | axiom | Multigraph
s_subgraph | axiom | Subgraph
s_induced_subgraph | axiom | Induced subgraph
s_spanning_subgraph | axiom | Spanning subgraph
s_graph_path | axiom | Path (graph theory)
s_graph_cycle | axiom | Cycle (graph theory)
s_connected_graph | axiom | Connected graph
s_tree_graph | axiom | Tree (graph theory)
s_forest_graph | axiom | Forest
s_spanning_tree | axiom | Spanning tree
s_vertex_degree | axiom | Vertex degree
s_minimum_degree_delta | axiom | Minimum degree δ(G)
s_euler_tour | axiom | Euler tour
s_euler_tour_theorem | theorem | Euler's theorem on Euler tours
s_bipartite_characterization | theorem | Bipartite characterization
s_handshaking_lemma | theorem | Handshaking lemma
s_graph_complement | axiom | Complement graph
s_complete_graph_kn | axiom | Complete graph K_n
s_complete_bipartite_graph | axiom | Complete bipartite graph K_{r,s}
s_graph_contraction | axiom | Edge contraction
s_graph_minor_definition | axiom | Graph minor
s_topological_minor_subdivision | axiom | Topological minor (subdivision)
s_vertex_cover | axiom | Vertex cover
s_independent_set_graph | axiom | Independent set
s_edge_cover | axiom | Edge cover
s_gallai_identity | theorem | Gallai's identities
s_hall_theorem_diestel | theorem | Hall's marriage theorem
s_konig_theorem_diestel | theorem | König's theorem (matching-cover duality)
s_tutte_theorem_diestel | theorem | Tutte's 1-factor theorem
s_deficiency_version_hall | theorem | Deficiency version of Hall's theorem
s_berge_theorem_matching | theorem | Berge's theorem (maximum matching)
s_k_connected_graph | axiom | k-connected graph
s_vertex_connectivity | axiom | Vertex connectivity κ(G)
s_edge_connectivity | axiom | Edge connectivity λ(G)
s_whitney_inequality | theorem | Whitney's inequality
s_menger_theorem_diestel | theorem | Menger's theorem
s_menger_edge_version | theorem | Menger's theorem (edge version)
s_menger_global_version | theorem | Menger's theorem (global version)
s_block_graph | axiom | Block (2-connected component)
s_block_cut_tree | state | Block-cut tree
s_ear_decomposition | state | Ear decomposition
s_ear_decomposition_theorem | theorem | Ear decomposition theorem (Whitney)
s_tutte_3connected_structure | theorem | Tutte's wheel theorem
t_ear_decomposition_technique | technique | Ear decomposition technique
s_face_planar_graph | axiom | Face of a plane graph
s_euler_formula_planar | theorem | Euler's formula for planar graphs
s_planar_edge_bound | theorem | Planar graph edge bound
s_k5_k33_nonplanar | theorem | Non-planarity of K₅ and K₃,₃
s_kuratowski_theorem_diestel | theorem | Kuratowski's theorem
s_wagner_theorem_diestel | theorem | Wagner's theorem
s_dual_graph | axiom | Dual graph (planar)
s_mac_lane_planarity | theorem | Mac Lane's planarity criterion
s_planar_graph_six_color | theorem | Six-color theorem for planar graphs
t_kempe_chain_argument | technique | Kempe chain argument
s_outerplanar_graph | axiom | Outerplanar graph
s_graph_genus | axiom | Genus of a graph
s_vertex_coloring_proper | axiom | Proper vertex coloring
s_chromatic_number | axiom | Chromatic number χ(G)
s_clique_number | axiom | Clique number ω(G)
s_brooks_theorem_diestel | theorem | Brooks' theorem
s_greedy_coloring_bound | theorem | Greedy coloring bound
s_chromatic_polynomial_diestel | state | Chromatic polynomial P(G, k)
s_edge_coloring | axiom | Proper edge coloring
s_chromatic_index | axiom | Chromatic index χ'(G)
s_vizing_theorem_diestel | theorem | Vizing's theorem
s_list_coloring | axiom | List coloring
s_choosability | axiom | Choosability (list chromatic number)
s_perfect_graph_theorem_lovasz | theorem | Perfect graph theorem (Lovász)
s_strong_perfect_graph_diestel | theorem | Strong perfect graph theorem (Chudnovsky–Robertson–Seymour–Thomas)
s_mycielski_construction | state | Mycielski construction
t_deletion_contraction | technique | Deletion-contraction
s_degeneracy_coloring | axiom | Degeneracy (k-degenerate graph)
s_nowhere_zero_flow | axiom | Nowhere-zero flow
s_nowhere_zero_k_flow | axiom | Nowhere-zero k-flow
s_flow_coloring_duality | theorem | Flow-coloring duality (planar graphs)
s_tutte_flow_conjectures | state | Tutte's flow conjectures
s_six_flow_theorem | theorem | Six-flow theorem (Seymour)
s_eight_flow_theorem | theorem | Eight-flow theorem (Jaeger)
s_four_color_flow_equivalence | theorem | Four-color theorem via flows
s_group_valued_flow | axiom | Group-valued flow
s_tutte_group_flow_theorem | theorem | Tutte's theorem on group-valued flows
s_turan_theorem_diestel | theorem | Turán's theorem
s_turan_graph_definition | axiom | Turán graph T(n,r)
s_extremal_number | axiom | Extremal number ex(n, H)
s_kovari_sos_turan_diestel | theorem | Kővári–Sós–Turán theorem
s_szemeredi_regularity_diestel | theorem | Szemerédi regularity lemma
s_epsilon_regular_pair | axiom | ε-regular pair
s_graph_removal_lemma_diestel | theorem | Graph removal lemma
s_ramsey_theorem_diestel | theorem | Ramsey's theorem (finite, graph version)
s_ramsey_number | axiom | Ramsey number R(r, s)
s_erdos_ramsey_lower | theorem | Erdős probabilistic lower bound for Ramsey numbers
s_zarankiewicz_problem | axiom | Zarankiewicz problem
t_regularity_method_diestel | technique | Regularity method
t_double_counting_technique | technique | Double counting (combinatorics)
t_probabilistic_method_graphs | technique | Probabilistic method (graph theory)
s_konig_infinity_lemma | theorem | König's infinity lemma
s_ray_graph | axiom | Ray (one-way infinite path)
s_end_of_graph | axiom | End of a graph
s_locally_finite_graph | axiom | Locally finite graph
s_halin_theorem_ends | theorem | Halin's theorem (grid minor from thick ends)
s_star_comb_lemma | theorem | Star-comb lemma
s_compactness_graph_coloring | theorem | De Bruijn–Erdős compactness theorem for coloring
s_ramsey_theorem_infinite_diestel | theorem | Infinite Ramsey theorem
s_canonical_ramsey_theorem | theorem | Canonical Ramsey theorem (Erdős–Rado)
s_graph_ramsey_number | axiom | Graph Ramsey number R(G, H)
s_ramsey_multiplicity | state | Ramsey multiplicity
s_hamilton_cycle | axiom | Hamilton cycle
s_hamilton_path | axiom | Hamilton path
s_dirac_theorem_hamilton | theorem | Dirac's theorem (Hamilton cycles)
s_chvatal_theorem_hamilton | theorem | Chvátal's theorem (degree sequence condition)
s_chvatal_erdos_theorem_diestel | theorem | Chvátal–Erdős theorem
s_tutte_hamilton_planar | theorem | Tutte's theorem (4-connected planar graphs are Hamiltonian)
s_toughness | axiom | Toughness of a graph
s_chvatal_toughness_conjecture | state | Chvátal's toughness conjecture
s_bondy_chvatal_closure | theorem | Bondy–Chvátal closure theorem
s_erdos_renyi_model | axiom | Erdős–Rényi random graph model G(n, p)
s_threshold_function | axiom | Threshold function
s_connectivity_threshold | theorem | Connectivity threshold for G(n, p)
s_giant_component_threshold | theorem | Giant component threshold (Erdős–Rényi)
s_evolution_random_graphs | state | Evolution of random graphs
s_zero_one_law | theorem | Zero-one law for random graphs
s_chromatic_number_random | theorem | Chromatic number of random graphs
s_hamilton_threshold | theorem | Hamilton cycle threshold
t_first_moment_method | technique | First moment method (Markov bound)
t_second_moment_method | technique | Second moment method (Chebyshev)
s_tree_width | axiom | Tree-width
s_tree_decomposition | axiom | Tree-decomposition
s_bramble | axiom | Bramble (screen)
s_tree_width_duality | theorem | Tree-width duality theorem (Seymour–Thomas)
s_robertson_seymour_theorem | theorem | Robertson–Seymour theorem (graph minor theorem)
s_minor_closed_characterization | theorem | Finite forbidden minor characterization
s_well_quasi_ordering | axiom | Well-quasi-ordering (WQO)
s_graph_structure_theorem | theorem | Graph structure theorem (Robertson–Seymour)
s_grid_minor_theorem | theorem | Grid minor theorem (Robertson–Seymour)
s_excluded_minor_planar | theorem | Excluded minor characterization of planar graphs
s_path_width | axiom | Path-width
s_clique_sum | axiom | Clique-sum
s_haven | axiom | Haven
s_cops_and_robber | state | Cops and robbers characterization of tree-width
t_well_quasi_ordering_technique | technique | Well-quasi-ordering technique
t_graph_minor_algorithm | technique | Graph minor testing algorithm
s_petersen_graph | axiom | Petersen graph
s_line_graph | axiom | Line graph
s_tutte_polynomial_diestel | state | Tutte polynomial T(G; x, y)
s_girth | axiom | Girth of a graph
s_erdos_girth_chromatic | theorem | Erdős theorem on girth and chromatic number
s_kirchhoff_matrix_tree | theorem | Kirchhoff's matrix tree theorem
s_kruskal_tree_theorem | theorem | Kruskal's tree theorem
s_higman_lemma | theorem | Higman's lemma
s_gallai_milgram_theorem | theorem | Gallai–Milgram theorem
s_alon_tarsi_theorem | theorem | Alon–Tarsi theorem
s_hadwiger_conjecture | state | Hadwiger's conjecture
t_augmenting_path_method | technique | Augmenting path method
t_discharging_method_diestel | technique | Discharging method
t_extremal_graph_counting | technique | Extremal counting (Zykov symmetrization)
s_network_flow_theory | axiom | Network flow
s_max_flow_min_cut_diestel | theorem | Max-flow min-cut theorem (Ford–Fulkerson)
s_mader_theorem_average_degree | theorem | Mader's theorem (average degree forces topological minors)
s_cycle_space | axiom | Cycle space
s_cut_space | axiom | Cut space
t_tree_decomposition_technique | technique | Tree-decomposition dynamic programming
s_graph_product_types | axiom | Graph products (Cartesian, tensor, strong)
s_heawood_map_coloring | theorem | Heawood's map coloring theorem
s_edge_connectivity_submodularity | theorem | Submodularity of the edge boundary
s_degree_of_vertex | axiom | Degree of a vertex
s_walk_graph | axiom | Walk (graph theory)
s_trail_graph | axiom | Trail (graph theory)
s_prufer_sequence | axiom | Prüfer sequence
s_konig_edge_coloring_theorem | theorem | König's edge-coloring theorem
s_clique | axiom | Clique
s_electrical_network_on_graph | axiom | Electrical network on a graph
s_kirchhoff_current_law_graph | axiom | Kirchhoff's current law (graph)
s_kirchhoff_voltage_law_graph | axiom | Kirchhoff's voltage law (graph)
s_effective_resistance | state | Effective resistance
s_rayleigh_monotonicity_law | theorem | Rayleigh's monotonicity law
t_series_parallel_reduction | technique | Series-parallel reduction
s_random_walk_on_graph | axiom | Random walk on a graph
s_hitting_time | state | Hitting time
s_commute_time | state | Commute time
s_random_walk_electrical_correspondence | theorem | Random walk–electrical network correspondence
s_nash_williams_criterion | theorem | Nash-Williams recurrence criterion
t_ford_fulkerson_algorithm | technique | Ford–Fulkerson algorithm
s_f_factor | axiom | f-factor
s_tutte_f_factor_theorem | theorem | Tutte's f-factor theorem
s_erdos_szekeres_ramsey_bound | theorem | Erdős–Szekeres bound on Ramsey numbers
s_thomassen_5_list_coloring | theorem | Thomassen's theorem (5-list-coloring of planar graphs)
s_voigt_non_4_choosable_planar | state | Voigt's example (non-4-choosable planar graph)
s_chordal_graph | axiom | Chordal graph
s_shannon_multigraph_bound | theorem | Shannon's multigraph edge-coloring bound
s_vizing_adjacency_lemma | theorem | Vizing's adjacency lemma
s_class_1_class_2_graph | axiom | Class 1 and Class 2 graphs
t_vizing_fan_argument | technique | Vizing fan argument
s_ramsey_theorem_multicolor | theorem | Ramsey's theorem (multicolor version)
s_diagonal_ramsey_bounds | state | Diagonal Ramsey number bounds
s_off_diagonal_ramsey_bounds | state | Off-diagonal Ramsey bounds
s_schur_number | axiom | Schur number S(r)
s_combinatorial_line | axiom | Combinatorial line
s_stepping_up_lemma | theorem | Erdős–Rado stepping-up lemma
s_compactness_principle_ramsey | theorem | Compactness principle for Ramsey theory
s_gnm_random_graph_model | axiom | Uniform random graph model G(n, M)
s_subgraph_appearance_threshold | theorem | Subgraph appearance threshold
s_double_jump_random_graph | state | Double jump in random graph evolution
s_clique_number_random_graph | state | Clique number of random graphs
s_janson_inequality | theorem | Janson's inequality
s_graph_automorphism_group | axiom | Automorphism group of a graph
s_vertex_transitive_graph | axiom | Vertex-transitive graph
s_frucht_theorem | theorem | Frucht's theorem
s_adjacency_matrix_graph | axiom | Adjacency matrix of a graph
s_graph_spectrum | axiom | Spectrum of a graph
s_spectral_radius_graph | state | Spectral radius of a graph
s_algebraic_connectivity | state | Algebraic connectivity (Fiedler value)
s_expander_graph | axiom | Expander graph
s_spectral_gap_graph | state | Spectral gap of a graph
s_expander_mixing_lemma | theorem | Expander mixing lemma
s_discrete_cheeger_inequality | theorem | Discrete Cheeger inequality
s_cheeger_constant_graph | axiom | Cheeger constant of a graph
s_ramanujan_graph | axiom | Ramanujan graph
s_alon_boppana_bound | theorem | Alon–Boppana bound
s_lps_construction | state | Lubotzky–Phillips–Sarnak (LPS) construction
s_strongly_regular_graph | axiom | Strongly regular graph srg(n, k, λ, μ)
s_eigenvalue_interlacing | theorem | Eigenvalue interlacing
s_cover_time | state | Cover time of a graph
s_blanket_time | state | Blanket time
s_mixing_time | state | Mixing time of a Markov chain
s_total_variation_distance_markov | axiom | Total variation distance (Markov chains)
s_cutoff_phenomenon | state | Cutoff phenomenon
s_rapid_mixing | state | Rapid mixing
s_whitney_rank_polynomial | state | Whitney rank polynomial (rank-nullity generating function)
s_flow_polynomial | state | Flow polynomial
s_reliability_polynomial | state | Reliability polynomial
s_cographic_matroid | axiom | Cographic matroid
s_matroid_minor | axiom | Matroid minor
s_potts_model | axiom | Potts model
s_tutte_polynomial_complexity | state | Computational complexity of the Tutte polynomial
s_partition_function_graph | state | Partition function of a graph (statistical mechanics)
s_dichromatic_polynomial | state | Dichromatic polynomial (Whitney)
t_uniform_random_coloring | technique | Uniform random coloring
s_property_b_existence_uniform | theorem | Property B existence for uniform hypergraphs
s_max_cut_probabilistic_lower_bound | theorem | Max-cut probabilistic lower bound
s_sum_free_set_probabilistic_bound | theorem | Sum-free subset probabilistic bound
a_property_b_hypergraph | axiom | Property B of a hypergraph
t_linearity_of_expectation | technique | Linearity of expectation technique
s_tournament_domination_bound | theorem | Tournament dominating set bound
s_balancing_vectors_bound | theorem | Balancing vectors bound
s_unbalancing_lights | theorem | Unbalancing lights
s_independence_number_greedy_expectation | theorem | Independence number greedy-expectation bound
s_crossing_number_inequality | theorem | Crossing number inequality
s_ramsey_lower_bound_alteration | theorem | Ramsey lower bound via alteration
s_shearer_independence_triangle_free | theorem | Shearer's bound for triangle-free graphs
s_independent_transversal_alteration | theorem | Independent transversal via alteration
s_paley_zygmund_inequality | theorem | Paley-Zygmund inequality
s_paley_graph_properties | state | Paley graph properties
a_variance | axiom | Variance
t_lovasz_local_lemma_technique | technique | Lovasz Local Lemma technique
s_lovasz_local_lemma_symmetric | theorem | Lovasz Local Lemma (symmetric form)
s_lopsided_lovasz_local_lemma | theorem | Lopsided Lovasz Local Lemma
s_moser_tardos_algorithmic_lll | theorem | Moser-Tardos algorithmic LLL
s_k_sat_lll_bound | theorem | k-SAT LLL bound
s_latin_transversal_lll | theorem | Latin transversal via LLL
a_lopsidependency_graph | axiom | Lopsidependency graph
s_fkg_inequality | theorem | FKG inequality
s_four_functions_theorem | theorem | Four functions theorem (Ahlswede-Daykin)
s_reimer_inequality | theorem | Reimer inequality (BK inequality)
a_increasing_event | axiom | Increasing event
a_disjoint_occurrence | axiom | Disjoint occurrence (box product)
t_azuma_hoeffding_technique | technique | Azuma-Hoeffding technique
t_vertex_exposure_martingale | technique | Vertex exposure martingale
t_edge_exposure_martingale | technique | Edge exposure martingale
s_chromatic_number_concentration | theorem | Chromatic number concentration
s_kim_vu_polynomial_concentration | theorem | Kim-Vu polynomial concentration
a_doob_martingale_combinatorial | axiom | Doob martingale (combinatorial)
a_lipschitz_function_product_space | axiom | Lipschitz function on product space
t_stein_chen_poisson_approximation | technique | Stein-Chen Poisson approximation
s_triangle_threshold_gnp | theorem | Triangle threshold in G(n,p)
s_bonferroni_inequalities | theorem | Bonferroni inequalities
t_expander_mixing_technique | technique | Expander mixing technique
s_paley_graph_pseudo_random | theorem | Paley graph pseudo-randomness
a_ndlambda_graph | axiom | (n,d,lambda)-graph
a_erdos_renyi_gnm | axiom | Erdos-Renyi G(n,m) model
s_katona_cyclic_permutation_proof | state | Katona's cyclic permutation proof of EKR
s_lovasz_theta_sandwich | theorem | Lovasz theta sandwich theorem
s_shannon_capacity_c5 | theorem | Shannon capacity of C_5
s_lovasz_theta_ramsey_bound | theorem | Ramsey bound via Lovasz theta function
a_shannon_capacity | axiom | Shannon capacity of a graph
a_lovasz_theta_function | axiom | Lovasz theta function
t_partial_coloring_method | technique | Partial coloring method
t_beck_fiala_technique | technique | Beck-Fiala technique
s_partial_coloring_lemma | theorem | Partial coloring lemma
s_discrepancy_arithmetic_progressions | theorem | Discrepancy of arithmetic progressions
a_combinatorial_discrepancy | axiom | Combinatorial discrepancy
t_conditional_expectations_method | technique | Method of conditional expectations
t_pairwise_independence_derandomization | technique | Pairwise independence derandomization
t_k_wise_independence_derandomization | technique | k-wise independence derandomization
s_derandomized_ramsey_bound | theorem | Derandomized Ramsey lower bound
s_deterministic_max_cut | theorem | Deterministic max-cut algorithm
a_k_wise_independent_rv | axiom | k-wise independent random variables
a_pairwise_independent_rv | axiom | Pairwise independent random variables
t_graph_property_testing | technique | Graph property testing
s_graph_property_testability | theorem | Testability of hereditary graph properties
s_triangle_removal_lemma | theorem | Triangle removal lemma
s_ruzsa_szemeredi_theorem | theorem | Ruzsa-Szemeredi (6,3)-theorem
t_chernoff_bound_technique | technique | Chernoff bound technique
t_janson_inequality_technique | technique | Janson inequality technique
s_janson_extended_inequality | theorem | Janson extended inequality
s_subgraph_count_concentration | theorem | Subgraph count concentration
s_weierstrass_product_inequality | theorem | Weierstrass product inequality
t_entropy_method_combinatorics | technique | Entropy method in combinatorics
s_shearer_entropy_lemma | theorem | Shearer's entropy lemma
s_bregman_theorem | theorem | Bregman's theorem (permanent bound)
s_kahn_zhao_independent_sets | theorem | Kahn-Zhao theorem on independent sets
s_shearer_independent_set_bound | theorem | Shearer's independent set bound via entropy
t_container_method | technique | Hypergraph container method
s_saxton_thomason_container_theorem | theorem | Saxton-Thomason container theorem
s_balogh_morris_samotij_container_theorem | theorem | Balogh-Morris-Samotij container theorem
s_sum_free_sets_count_container | theorem | Sum-free subsets count via containers
s_triangle_free_graph_count_container | theorem | Triangle-free graph count via containers
t_sparse_regularity_method | technique | Sparse regularity method
s_sparse_regularity_lemma | theorem | Kohayakawa-Rodl sparse regularity lemma
s_sparse_counting_lemma | theorem | Sparse counting lemma
s_sparse_ramsey_theorem | theorem | Rodl-Rucinski sparse Ramsey theorem
s_shamir_spencer_chromatic_concentration | theorem | Shamir-Spencer chromatic concentration
s_ramsey_upper_bound_lll | state | Off-diagonal Ramsey bound via LLL
s_chromatic_lower_bound_theta | theorem | Chromatic number lower bound via theta
a_2_density | axiom | 2-density of a graph
a_upper_uniformity_condition | axiom | Upper uniformity condition
s_k_uniform_set_system | axiom | k-uniform set system (k-graph)
s_chain_in_poset | axiom | Chain in a poset
s_shattering_trace | axiom | Shattering
s_trace_of_set_system | axiom | Trace of a set system
s_covering_number_tau | axiom | Covering number (τ)
s_packing_number_matching | axiom | Packing number (matching number)
s_epsilon_net_def | axiom | Epsilon-net
s_boolean_function_def | axiom | Boolean function
s_sensitivity_boolean | axiom | Sensitivity of a Boolean function
s_block_sensitivity | axiom | Block sensitivity
s_certificate_complexity | axiom | Certificate complexity
s_decision_tree_complexity | axiom | Decision tree complexity
s_influence_of_variable | axiom | Influence of a variable
s_total_influence | axiom | Total influence
s_error_correcting_code | axiom | Error-correcting code
s_partition_regularity | axiom | Partition regularity
s_noise_sensitivity | axiom | Noise sensitivity
s_monotone_boolean_function | axiom | Monotone Boolean function
s_turan_density | state | Turán density π(F)
s_extremal_number_set_system | state | Extremal number for set systems
s_hypergraph_ramsey_number | state | Hypergraph Ramsey number
s_van_der_waerden_number | state | Van der Waerden number W(k,r)
s_hales_jewett_number | state | Hales-Jewett number HJ(k,r)
s_dual_set_system | state | Dual set system
s_growth_function_shatter | state | Growth function (shatter function)
s_nisan_wigderson | state | Nisan-Wigderson design
s_symmetric_chain_decomposition | state | Symmetric chain decomposition of 2^[n]
s_normalized_matching_property | state | Normalized matching property
s_turan_forbidden_trace | state | Turán problem for forbidden traces
s_hilton_milner | theorem | Hilton-Milner theorem
s_frankl_t_intersecting | theorem | Frankl's theorem on t-intersecting families
s_lovasz_kruskal_katona | theorem | Lovász continuous Kruskal-Katona theorem
s_ramsey_theorem_hypergraph | theorem | Ramsey's theorem (hypergraph version)
s_rado_partition_regularity | theorem | Rado's theorem (partition regularity)
s_pajor_lemma | theorem | Pajor's lemma
s_haussler_packing | theorem | Haussler packing lemma
s_epsilon_net_theorem | theorem | Epsilon-net theorem (Haussler-Welzl)
s_fractional_helly | theorem | Fractional Helly theorem
s_colorful_helly | theorem | Colorful Helly theorem (Bárány)
s_p_q_theorem | theorem | (p,q)-theorem (Alon-Kleitman)
s_ray_chaudhuri_wilson | theorem | Ray-Chaudhuri-Wilson theorem
s_oddtown_theorem | theorem | Oddtown theorem
s_two_distance_sets | theorem | Two-distance sets theorem
s_combinatorial_nullstellensatz | theorem | Combinatorial Nullstellensatz (Alon)
s_johnson_bound | theorem | Johnson bound
s_kkl_theorem | theorem | KKL theorem (Kahn-Kalai-Linial)
s_friedgut_sharp_thresholds | theorem | Friedgut's theorem on sharp thresholds
s_bollobas_two_families_general | theorem | Bollobás two-families theorem (generalized)
s_katona_intersection | theorem | Katona's intersection theorem
s_bondy_theorem_traces | theorem | Bondy's theorem on traces
s_alon_babai_suzuki | theorem | Alon-Babai-Suzuki theorem
s_elias_bassalygo | theorem | Elias-Bassalygo bound
s_threshold_phenomenon | theorem | Threshold phenomenon for monotone properties
s_sunflower_conjecture | theorem | Improved sunflower lemma (Alweiss-Lovett-Wu-Zhang)
t_compression_shifting | technique | Compression / shifting
t_sunflower_extraction | technique | Sunflower extraction (greedy peeling)
t_dimension_argument | technique | Dimension argument (linear algebra bound)
t_lexicographic_cascading | technique | Cascading representation
t_weight_function_method | technique | Weight function method
t_product_tensor_argument | technique | Product / tensor argument
t_stepping_up_ramsey | technique | Stepping-up technique
t_derandomization | technique | Derandomization via conditional expectations
t_fourier_analysis_boolean | technique | Fourier analysis on the Boolean cube
t_chain_partition | technique | Chain partition technique
t_multilinear_extension | technique | Multilinear polynomial representation
t_algebraic_rank_argument | technique | Algebraic rank argument
t_turan_extremal_method | technique | Turán-type extremal method
t_amalgamation | technique | Amalgamation technique
t_inclusion_exclusion | technique | Inclusion-exclusion as counting technique
s_difference_set | axiom | Difference set A−B
s_iterated_sumset | axiom | Iterated sumset kA
s_doubling_constant | axiom | Doubling constant σ(A)
s_ruzsa_distance | axiom | Ruzsa distance d(A,B)
s_additive_energy | axiom | Additive energy E(A,B)
s_freiman_homomorphism | axiom | Freiman homomorphism of order s
s_freiman_isomorphism | axiom | Freiman isomorphism of order s
s_generalized_arithmetic_progression | axiom | Generalized arithmetic progression (GAP)
s_proper_gap | axiom | Proper generalized arithmetic progression
s_bohr_set | axiom | Bohr set B(S,ε)
s_dissociated_set | axiom | Dissociated set
s_k_approximate_group | axiom | K-approximate group
s_combinatorial_dimension | axiom | Combinatorial dimension of a Bohr set
s_progression_free_set | axiom | Progression-free set (3-AP-free set)
s_sumset_growth_function | axiom | Sumset growth function
s_plunnecke_graph | axiom | Plünnecke graph
s_magnification_ratio | axiom | Magnification ratio
s_additive_energy_normalized | axiom | Normalized additive energy
s_regularity_pair | axiom | ε-regular pair (Szemerédi)
s_popular_difference | axiom | Popular difference
s_additive_quadruple | axiom | Additive quadruple
s_coset_progression | axiom | Coset progression
s_convolution_counting_function | axiom | Convolution (representation) function r_{A+B}
s_spectrum_of_set | axiom | Spectrum (large Fourier coefficients) of a set
s_pseudorandom_measure | axiom | Pseudorandom measure (Green–Tao)
s_w_trick | axiom | W-trick (Green–Tao)
s_linear_forms_condition | axiom | Linear forms condition
s_correlation_condition | axiom | Correlation condition (Green–Tao)
s_gowers_uniformity_norm | axiom | Gowers uniformity norm U^k
s_gowers_anti_uniformity | state | Gowers anti-uniform function
s_plunnecke_inequality | theorem | Plünnecke's inequality
s_ruzsa_triangle_inequality | theorem | Ruzsa triangle inequality
s_ruzsa_covering_lemma | theorem | Ruzsa covering lemma
s_ruzsa_sum_triangle_inequality | theorem | Ruzsa sum-triangle inequality
s_ruzsa_distance_controls_energy | theorem | Ruzsa distance–energy duality
s_freiman_ruzsa_f2n | theorem | Freiman–Ruzsa theorem in 𝔽₂ⁿ
s_balog_szemeredi_gowers | theorem | Balog–Szemerédi–Gowers theorem
s_erdos_heilbronn_conjecture | theorem | Erdős–Heilbronn conjecture (Dias da Silva–Hamidoune)
s_sum_product_estimate_reals | theorem | Elekes sum-product estimate
s_erdos_szemeredi_conjecture | state | Erdős–Szemerédi sum-product conjecture
s_bergelson_leibman | theorem | Bergelson–Leibman theorem (polynomial Szemerédi)
s_inverse_littlewood_offord | theorem | Inverse Littlewood–Offord theorem (Tao–Vu)
s_littlewood_offord_problem | state | Littlewood–Offord problem
s_chang_theorem | theorem | Chang's theorem (large Fourier spectrum)
s_bogolyubov_lemma | theorem | Bogolyubov's lemma
s_sanders_quasi_polynomial_freiman_ruzsa | theorem | Sanders' quasi-polynomial Freiman–Ruzsa theorem
s_polynomial_freiman_ruzsa_conjecture | state | Polynomial Freiman–Ruzsa conjecture
s_gowers_inverse_u3 | theorem | Gowers inverse theorem for U³
s_gowers_inverse_uk | theorem | Gowers inverse conjecture for U^k (Green–Tao–Ziegler)
s_generalized_von_neumann | theorem | Generalized von Neumann theorem
s_solymosi_sum_product | theorem | Solymosi's sum-product estimate
s_sum_product_finite_fields | theorem | Sum-product estimate in finite fields (Bourgain–Katz–Tao)
s_ruzsa_modeling_lemma | theorem | Ruzsa modeling lemma
s_ruzsa_analog_freiman | theorem | Ruzsa's proof of Freiman's theorem (via covering lemma)
s_gowers_bound_szemeredi | theorem | Gowers quantitative bound for Szemerédi's theorem
s_regularity_counting_lemma | theorem | Regularity counting lemma (embedding lemma)
s_koopman_von_neumann_structure | theorem | Koopman–von Neumann structure theorem (in additive combinatorics)
s_arithmetic_regularity_lemma | theorem | Arithmetic regularity lemma (Green)
s_goldston_yildirim_majorant | state | Goldston–Yıldırım pseudorandom majorant for primes
s_hypergraph_regularity | theorem | Hypergraph regularity lemma (Gowers, Nagle–Rödl–Schacht–Skokan)
s_hypergraph_removal | theorem | Hypergraph removal lemma
s_multidimensional_szemeredi | theorem | Multidimensional Szemerédi theorem (Furstenberg–Katznelson)
s_long_aps_in_sumsets | theorem | Long arithmetic progressions in sumsets
s_bourgain_3ap_bound | theorem | Bourgain's bound for 3-AP-free sets
s_meshulam_roth_f3n | theorem | Meshulam's theorem (Roth in 𝔽₃ⁿ)
s_bateman_katz_f3n | theorem | Bateman–Katz bound for cap sets
s_rankin_construction | state | Rankin's construction (k-AP-free sets)
s_green_tao_linear_equations_primes | theorem | Green–Tao linear equations in primes
t_plunnecke_graph_method | technique | Plünnecke graph method
t_ruzsa_calculus | technique | Ruzsa calculus
t_fourier_analysis_finite_abelian_groups | technique | Fourier analysis on finite abelian groups
t_density_increment_argument | technique | Density increment argument
t_energy_increment_argument | technique | Energy increment argument
t_regularity_method_arithmetic | technique | Regularity method for arithmetic combinatorics
t_graph_removal_technique | technique | Graph removal technique
t_probabilistic_method_additive_comb | technique | Probabilistic method in additive combinatorics
t_polynomial_method_additive | technique | Polynomial method (Combinatorial Nullstellensatz applications)
t_sum_product_technique | technique | Sum-product technique
t_incidence_geometry_method | technique | Incidence geometry method
t_tensor_product_trick | technique | Tensor product trick (Ruzsa)
t_ergodic_theory_approach | technique | Ergodic theory approach (Furstenberg)
t_hypergraph_regularity_method | technique | Hypergraph regularity method
t_cell_decomposition_technique | technique | Cell decomposition (for incidence bounds)
t_dyadic_pigeonholing | technique | Dyadic pigeonholing
t_compression_symmetrization | technique | Compression/symmetrization
t_additive_energy_amplification | technique | Additive energy amplification
t_bohr_set_method | technique | Bohr set method
t_weyl_differencing | technique | Weyl differencing
t_van_der_corput_trick | technique | Van der Corput trick/inequality
t_dependent_random_choice | technique | Dependent random choice
t_stepanov_method_addcomb | technique | Stepanov method
t_character_sum_method | technique | Character sum method
t_geometry_of_numbers_in_freiman | technique | Geometry of numbers in Freiman's theorem
t_structure_randomness_dichotomy | technique | Structure–randomness dichotomy
t_epsilon_net_argument | technique | ε-net/covering argument for Bohr sets
s_nilsequence | axiom | Nilsequence
s_nilmanifold | axiom | Nilmanifold G/Γ
s_stabilizer_of_sumset | axiom | Stabilizer (period) of a sumset
s_davenport_constant | axiom | Davenport constant D(G)
s_sum_free_set | axiom | Sum-free set
s_covering_number_sumset | axiom | Covering number (Ruzsa)
s_szemeredi_regularity_graph_arithmetic | theorem | Szemerédi's theorem via graph regularity
s_roth_via_fourier | theorem | Roth's theorem via Fourier analysis
s_roth_via_triangle_removal | theorem | Roth's theorem via triangle removal
s_green_tao_structure | state | Structure of the Green–Tao proof
s_additive_combinatorics_in_fp | axiom | Additive combinatorics in 𝔽_p (finite field model)
s_inverse_sumset_problem | state | Inverse sumset problem
s_cap_set_problem | state | Cap set problem
s_gowers_norm_controls_ap_count | theorem | Gowers norms control AP counts
t_balog_szemeredi_gowers_graph | technique | Balog–Szemerédi–Gowers graph argument
s_ap_count_formula_fourier | theorem | AP count formula via Fourier analysis
s_sumset_lower_bound_cauchy_davenport | state | Sumset lower bound problem
s_vosper_theorem | theorem | Vosper's theorem
s_freiman_inverse_2_3k_4 | theorem | Freiman's 3k−4 theorem
s_erdos_littlewood_offord | theorem | Erdős–Littlewood–Offord inequality
s_halasz_inequality | theorem | Halász-type inequality (additive combinatorics)
s_small_ball_probability | state | Small ball probability (Lévy concentration function)
s_probability_space | axiom | Probability space ;; probability triple
s_probability_measure | axiom | Probability measure
s_random_variable | axiom | Random variable ;; measurable function to ℝ
s_law_of_random_variable | axiom | Distribution (law) of a random variable ;; law of X ;; push-forward measure
s_independence_of_events | axiom | Independence of events ;; mutual independence of events
s_independence_of_random_variables | axiom | Independence of random variables
s_almost_sure_convergence | axiom | Almost sure convergence ;; convergence with probability one ;; a.s. convergence
s_convergence_in_probability | axiom | Convergence in probability
s_convergence_in_distribution | axiom | Convergence in distribution ;; weak convergence of random variables ;; convergence in law
s_characteristic_function | axiom | Characteristic function ;; Fourier transform of a distribution
s_stable_distribution | axiom | Stable distribution ;; α-stable law ;; Lévy stable distribution
s_domain_of_attraction | axiom | Domain of attraction ;; domain of normal attraction
s_tightness_of_measures | axiom | Tightness of a family of probability measures ;; uniform tightness
s_poisson_limit_theorem | theorem | Poisson limit theorem ;; law of rare events ;; Poisson approximation theorem
s_local_central_limit_theorem | theorem | Local central limit theorem ;; local CLT ;; local limit theorem
s_strong_markov_property_rw | theorem | Strong Markov property for random walks ;; strong Markov property (discrete)
s_arc_sine_law | theorem | Arc sine law ;; arc sine law for random walks ;; Lévy's arc sine law
s_gambler_ruin | theorem | Gambler's ruin ;; gambler's ruin problem
s_ladder_variables | axiom | Ladder variables ;; ladder heights ;; ladder epochs
s_convergence_of_random_series | state | Convergence of random series
s_kronecker_lemma | theorem | Kronecker's lemma
s_tower_property | theorem | Tower property of conditional expectation ;; law of total expectation ;; iterated expectations
s_ui_and_l1_convergence | theorem | Uniform integrability and L¹ convergence ;; UI characterization of L¹ convergence
s_doob_decomposition | theorem | Doob's decomposition theorem (discrete) ;; Doob decomposition
s_backward_martingale_convergence | theorem | Backward martingale convergence theorem ;; reversed martingale convergence
s_levy_zero_one_law | theorem | Lévy's 0-1 law
s_branching_process | axiom | Branching process (Galton-Watson) ;; Galton-Watson process
s_branching_process_extinction | theorem | Extinction criterion for branching processes ;; Galton-Watson extinction theorem
s_martingale_clt | theorem | Martingale central limit theorem ;; CLT for martingale differences
s_exponential_martingale | state | Exponential martingale ;; Wald's exponential martingale
s_markov_chain | axiom | Markov chain ;; discrete-time Markov chain ;; DTMC
s_transition_matrix | axiom | Transition matrix ;; stochastic matrix ;; transition probability matrix
s_chapman_kolmogorov_equations | theorem | Chapman-Kolmogorov equations
s_markov_property | theorem | Markov property ;; memoryless property
s_strong_markov_property_mc | theorem | Strong Markov property for Markov chains
s_communication_classes | axiom | Communication classes
s_irreducibility_mc | axiom | Irreducibility of a Markov chain ;; irreducible chain
s_recurrence_mc | axiom | Recurrence for Markov chains ;; recurrent state
s_transience_mc | axiom | Transience for Markov chains ;; transient state
s_recurrence_is_class_property | theorem | Recurrence is a class property
s_positive_recurrence | axiom | Positive recurrence ;; positive recurrent state ;; ergodic state
s_null_recurrence | axiom | Null recurrence ;; null recurrent state
s_period_of_state | axiom | Period of a state ;; periodicity of a Markov chain state
s_existence_uniqueness_stationary | theorem | Existence and uniqueness of stationary distribution
s_convergence_theorem_mc | theorem | Convergence theorem for Markov chains ;; basic limit theorem for Markov chains
s_detailed_balance | axiom | Detailed balance ;; reversibility condition ;; time-reversibility
s_ratio_limit_theorem | theorem | Ratio limit theorem
s_mean_return_time | state | Mean return time ;; expected return time
s_absorbing_state | axiom | Absorbing state ;; absorbing barrier
s_ergodicity_definition | axiom | Ergodicity ;; metric transitivity ;; ergodic transformation
s_maximal_ergodic_lemma | theorem | Maximal ergodic lemma ;; Hopf maximal ergodic lemma
s_strong_mixing | axiom | Strong mixing ;; mixing
s_stationary_process | axiom | Stationary process ;; strictly stationary process
s_shannon_mcmillan_breiman | theorem | Shannon-McMillan-Breiman theorem ;; asymptotic equipartition property ;; AEP
s_kac_lemma | theorem | Kac's lemma ;; Kac's recurrence theorem
s_kolmogorov_centsov_continuity | theorem | Kolmogorov-Čentsov continuity theorem ;; Kolmogorov continuity criterion
s_markov_property_bm | theorem | Markov property of Brownian motion
s_strong_markov_property_bm | theorem | Strong Markov property of Brownian motion
s_blumenthal_zero_one_law | theorem | Blumenthal's 0-1 law
s_reflection_principle_bm | theorem | Reflection principle for Brownian motion
s_distribution_of_max_bm | theorem | Distribution of the maximum of Brownian motion
s_arc_sine_law_bm | theorem | Arc sine law for Brownian motion ;; Lévy's arc sine laws
s_nowhere_differentiability_bm | theorem | Nowhere differentiability of Brownian motion ;; Paley-Wiener-Zygmund theorem
s_quadratic_variation_bm | theorem | Quadratic variation of Brownian motion
s_bm_harmonic_functions | theorem | Brownian motion and harmonic functions ;; Kakutani's theorem ;; probabilistic solution of Dirichlet problem
s_recurrence_transience_bm | theorem | Recurrence and transience of Brownian motion
s_bm_time_inversion | theorem | Time inversion of Brownian motion ;; BM inversion
s_bm_scaling_property | theorem | Scaling property of Brownian motion ;; Brownian scaling ;; self-similarity of BM
s_bm_as_martingale | state | Brownian motion as a martingale
s_construction_of_bm | state | Construction of Brownian motion ;; Lévy-Ciesielski construction ;; Wiener's construction
s_hitting_times_bm | state | Hitting times of Brownian motion ;; first passage time for BM
t_subsequence_argument | technique | Subsequence argument
t_borel_cantelli_technique | technique | Borel-Cantelli technique
t_moment_method | technique | Moment method (probabilistic)
t_skorokhod_representation_technique | technique | Skorokhod representation technique
t_maximal_inequality_technique | technique | Maximal inequality technique
t_stopping_time_technique | technique | Stopping time technique
t_diagonalization_technique | technique | Diagonalization technique (probability)
s_sample_space | axiom | Sample space
s_classical_probability | axiom | Classical probability (Laplace)
s_multinomial_coefficient | axiom | Multinomial coefficient
s_multinomial_distribution | axiom | Multinomial distribution
s_hypergeometric_distribution | axiom | Hypergeometric distribution
s_birthday_problem | state | Birthday problem
s_bose_einstein_statistics | axiom | Bose–Einstein statistics
s_fermi_dirac_statistics | axiom | Fermi–Dirac statistics
s_first_passage_time | state | First passage time
s_matching_problem | state | Matching problem (problème des rencontres)
s_law_of_total_probability | theorem | Law of total probability
s_polya_urn_model | state | Pólya urn model
s_bernoulli_trial | axiom | Bernoulli trial
s_negative_binomial_distribution | axiom | Negative binomial distribution
s_covariance | axiom | Covariance
s_standard_deviation | axiom | Standard deviation
s_probability_generating_function | axiom | Probability generating function
s_pgf_convolution_theorem | theorem | Convolution theorem for PGFs
s_continuity_theorem_pgf | theorem | Continuity theorem for probability generating functions
s_random_sum | state | Random sum (compound distribution)
s_compound_poisson_distribution | state | Compound Poisson distribution
s_branching_process_criticality | axiom | Criticality classification of branching processes
s_recurrent_event_feller | axiom | Recurrent event (Feller)
s_persistent_recurrent_event | axiom | Persistent (certain) recurrent event
s_transient_recurrent_event | axiom | Transient (uncertain) recurrent event
s_periodic_recurrent_event | axiom | Periodic and aperiodic recurrent event
s_renewal_equation | theorem | Renewal equation
s_elementary_renewal_theorem | theorem | Elementary renewal theorem
s_excess_life | state | Excess life (residual waiting time)
s_delayed_renewal_process | state | Delayed (modified) renewal process
s_duration_gamblers_ruin | theorem | Duration of the gambler's ruin
t_duality_principle_random_walk | technique | Duality principle for random walks
s_aperiodic_markov_chain | axiom | Aperiodic Markov chain
s_convergence_periodic_markov_chain | theorem | Convergence theorem for periodic Markov chains
s_absorption_probability | state | Absorption probability
s_mean_absorption_time | state | Mean absorption time
t_canonical_form_absorbing_chain | technique | Canonical form for absorbing Markov chains
s_doubly_stochastic_matrix | axiom | Doubly stochastic matrix
s_exponential_distribution | axiom | Exponential distribution
s_memoryless_property | theorem | Memoryless property
s_superposition_poisson_processes | theorem | Superposition of Poisson processes
s_thinning_poisson_process | theorem | Thinning of a Poisson process
s_birth_process | axiom | Pure birth process
s_yule_process | state | Yule process (pure birth with linear rate)
s_birth_death_process | axiom | Birth-death process
s_kolmogorov_forward_equation | theorem | Kolmogorov forward equations
s_generator_matrix | axiom | Generator matrix (Q-matrix)
s_inspection_paradox | state | Inspection paradox (waiting time paradox)
t_method_of_generating_functions | technique | Method of generating functions
t_first_step_analysis | technique | First-step analysis
s_inversion_formula_characteristic_function | theorem | Inversion formula for characteristic functions
s_uniqueness_theorem_distributions | theorem | Uniqueness theorem for characteristic functions
s_gamma_distribution | axiom | Gamma distribution
s_beta_distribution | axiom | Beta distribution
s_cauchy_distribution | axiom | Cauchy distribution
s_uniform_distribution_continuous | axiom | Continuous uniform distribution
s_exponential_family | axiom | Exponential family of distributions
s_f_distribution | state | F-distribution (Fisher–Snedecor)
s_lognormal_distribution | state | Lognormal distribution
s_pareto_distribution | state | Pareto distribution
s_weibull_distribution | state | Weibull distribution
s_logistic_distribution | state | Logistic distribution
s_order_statistics | axiom | Order statistics
s_marginal_distribution | axiom | Marginal distribution
s_edgeworth_expansion | state | Edgeworth expansion
t_saddlepoint_approximation | technique | Saddlepoint approximation
s_cramer_decomposition_theorem | theorem | Cramér's decomposition theorem
s_raikov_theorem | theorem | Raikov's theorem
s_key_renewal_theorem | theorem | Key renewal theorem
s_age_renewal_theory | state | Age (backward recurrence time)
s_equilibrium_renewal_distribution | state | Equilibrium (stationary) renewal distribution
s_renewal_reward_theorem | theorem | Renewal reward theorem
s_abelian_theorem_laplace | theorem | Abelian theorem for Laplace transforms
s_regularly_varying_function | axiom | Regularly varying function
s_feller_semigroup | axiom | Feller semigroup
s_diffusion_process | axiom | Diffusion process
t_esscher_transform | technique | Esscher transform (exponential tilting)
s_levy_inequality | theorem | Lévy's inequality
s_occupation_time | state | Occupation time
s_local_time_random_walk | state | Local time for random walk
s_resolvent_markov_process | state | Resolvent of a Markov process
s_levy_khintchine_triplet | state | Lévy–Khintchine triplet
s_compound_poisson_representation | theorem | Compound Poisson representation of infinitely divisible laws
s_maximal_coupling | state | Maximal coupling
s_symmetrization_inequality | theorem | Symmetrization inequality
s_record_values | state | Record values
s_cf_inversion_lattice | theorem | Characteristic function inversion formula on a lattice
s_type_convergence_of_types | axiom | Type (convergence of types)
s_stable_laws_classification | theorem | Classification of stable laws (Lévy)
s_operator_stable_law | state | Operator stable distribution
s_subexponential_distribution | axiom | Subexponential distribution
s_self_decomposable_distribution | axiom | Self-decomposable distribution
s_ehrenfest_urn_model | state | Ehrenfest urn model (diffusion model)
s_ruin_probability_half_line | theorem | Ruin probability on the half-line
s_fourier_analysis_on_groups | axiom | Fourier analysis on locally compact abelian groups
s_domain_of_normal_attraction | state | Domain of normal attraction
s_n_step_transition_probability | state | n-step transition probability
s_initial_distribution_mc | axiom | Initial distribution of a Markov chain
t_canonical_construction_mc | technique | Canonical construction of a Markov chain
s_accessibility_mc | axiom | Accessibility (leads to) in a Markov chain
s_closed_communicating_set | axiom | Closed (absorbing) set in a Markov chain
s_solidarity_of_period | theorem | Solidarity of period
s_cyclic_decomposition_mc | theorem | Cyclic decomposition of a periodic chain
s_essential_inessential_states | axiom | Essential and inessential states
s_first_return_time_decomposition | state | First return time decomposition
s_recurrence_criterion_green_function | theorem | Recurrence criterion via Green function
s_solidarity_positive_null_recurrence | theorem | Solidarity of positive/null recurrence
s_chung_fuchs_theorem | theorem | Chung-Fuchs theorem
s_number_of_visits_geometric | state | Number of visits to a transient state
s_invariant_measure_mc | axiom | Invariant measure for a Markov chain
s_existence_uniqueness_invariant_measure_mc | theorem | Existence and uniqueness of invariant measure for irreducible recurrent chains
s_ratio_limit_theorem_chung_erdos_orey | theorem | Ratio limit theorem (Chung-Erdős-Orey form)
s_cesaro_convergence_mc | theorem | Cesàro convergence for Markov chains
s_orey_convergence_theorem | theorem | Orey's convergence theorem
s_doeblin_condition | axiom | Doeblin's condition
t_doeblin_decomposition | technique | Doeblin decomposition
s_geometric_ergodicity_mc | state | Geometric ergodicity
s_taboo_probability | axiom | Taboo probability
s_taboo_green_function | state | Taboo Green function
s_last_exit_decomposition | theorem | Last exit decomposition
s_harmonic_function_mc | axiom | Harmonic function for a Markov chain
s_liouville_property_recurrent_mc | theorem | Liouville property for recurrent Markov chains
t_discrete_dirichlet_problem | technique | Discrete Dirichlet problem
s_superharmonic_optional_stopping | theorem | Superharmonic functions and optional stopping
s_potential_kernel_recurrent_mc | state | Potential kernel (deviation matrix) for recurrent chains
s_potential_of_charge_mc | state | Potential of a charge (recurrent chain)
s_green_kernel_transient_mc | state | Green kernel for transient Markov chains
s_riesz_decomposition_discrete_mc | theorem | Riesz decomposition for Markov chains
s_excessive_function_mc | axiom | Excessive function (α-excessive)
s_excessive_function_characterization | theorem | Characterization of excessive functions
s_minimum_principle_excessive | theorem | Minimum principle for excessive functions
s_domination_principle_mc | theorem | Domination principle for potentials
s_cone_of_excessive_functions | state | Cone of excessive functions
s_energy_of_measure_mc | axiom | Energy of a measure (discrete potential theory)
s_dirichlet_form_discrete | axiom | Dirichlet form (discrete Markov chain)
s_capacity_discrete_mc | state | Capacity of a set (discrete Markov chain)
s_equilibrium_measure_mc | state | Equilibrium measure (discrete potential theory)
s_equilibrium_potential_mc | state | Equilibrium potential (discrete Markov chain)
s_capacity_transience_criterion | theorem | Capacity and transience criterion
s_energy_principle_mc | theorem | Energy principle (discrete potential theory)
t_balayage_mc | technique | Balayage (sweeping) for Markov chains
s_reduite_mc | state | Réduite (reduced function) for Markov chains
s_balayage_of_measure_mc | state | Balayage of a measure (discrete potential theory)
s_balayage_principle_mc | theorem | Balayage principle for Markov chains
s_hunt_balayage_theorem | theorem | Hunt's balayage theorem
s_ctmc_definition | axiom | Continuous-time Markov chain (CTMC)
s_transition_semigroup_ctmc | axiom | Transition semigroup of a CTMC
s_holding_time_ctmc | state | Holding time of a CTMC
s_jump_chain_ctmc | state | Jump chain (embedded chain) of a CTMC
s_minimal_solution_ctmc | state | Minimal solution of the Kolmogorov equations
s_explosion_ctmc | state | Explosion of a CTMC
s_non_explosion_criterion_ctmc | theorem | Non-explosion criterion for CTMCs
t_uniformization_ctmc | technique | Uniformization of a CTMC
s_resolvent_as_lambda_potential | state | Resolvent as λ-potential kernel
s_lambda_excessive_function | axiom | λ-excessive function
s_resolvent_characterization_excessive | theorem | Resolvent characterization of λ-excessive functions
s_potential_operator_continuous | state | Potential operator for continuous-time processes
s_recovery_semigroup_from_resolvent | theorem | Recovery of semigroup from resolvent
s_recurrence_transience_ctmc | axiom | Recurrence and transience for CTMCs
s_invariant_measure_ctmc | axiom | Invariant measure for a CTMC
s_ergodic_theorem_ctmc | theorem | Ergodic theorem for CTMCs
s_continuous_time_green_function | state | Green function for continuous-time processes
s_riesz_decomposition_continuous_mc | theorem | Riesz decomposition for continuous-time processes
s_capacity_continuous_process | state | Capacity for continuous-time processes
s_hunt_process | axiom | Hunt process
s_quasi_left_continuity | axiom | Quasi-left-continuity
s_additive_functional | axiom | Additive functional
s_revuz_measure | state | Revuz measure
s_revuz_correspondence | theorem | Revuz correspondence
s_fine_topology | axiom | Fine topology
t_time_change_markov | technique | Time change of a Markov process
t_killing_subprocess | technique | Killing and subprocess construction
t_renewal_argument_mc | technique | Renewal argument for Markov chains
t_cycle_trick_mc | technique | Cycle trick for Markov chains
t_doeblin_coupling_mc | technique | Doeblin's coupling method
t_lyapunov_foster_criterion | technique | Lyapunov function method (Foster's criterion)
s_foster_lyapunov_geometric_ergodicity | theorem | Foster-Lyapunov criterion for geometric ergodicity
t_taboo_probability_technique | technique | Taboo probability technique
s_harris_recurrence | axiom | Harris recurrence
t_doob_h_transform | technique | Doob's h-transform
s_martin_boundary | state | Martin boundary
s_martin_representation_theorem | theorem | Martin representation theorem
s_convergence_parameter | state | Convergence parameter
s_R_recurrence_transience | axiom | R-recurrence and R-transience
s_solidarity_theorem_general | theorem | Solidarity theorem (general form)
s_birth_death_chain_discrete | axiom | Birth-death chain (discrete time)
t_subadditive_argument_period | technique | Subadditive argument for period
s_wiener_test_discrete | theorem | Wiener test (discrete potential theory)
s_markov_kernel | axiom | Markov kernel (transition kernel on measurable space)
s_phi_irreducibility | axiom | φ-irreducibility (general state space)
s_maximal_irreducibility_measure | theorem | Existence and uniqueness of maximal irreducibility measure
s_small_set | axiom | Small set (Meyn-Tweedie)
s_petite_set | axiom | Petite set
s_sampled_chain | axiom | Sampled chain (sampling distribution)
s_atom_general_state_space | axiom | Atom (general state space Markov chain)
s_minorization_condition | axiom | Minorization condition
s_aperiodicity_general_state_space | axiom | Aperiodicity (φ-irreducible chain)
s_t_chain | axiom | T-chain
s_e_chain | axiom | e-chain (equicontinuous chain)
s_feller_property_markov_chain | axiom | Feller property (Markov chain on topological space)
s_strong_feller_property | axiom | Strong Feller property
s_skeleton_chain | state | Skeleton chain
s_occupation_measure_mc | state | Occupation measure (Markov chain)
s_recurrence_transience_dichotomy_general | theorem | Recurrence-transience dichotomy (general state space)
s_positive_harris_recurrence | axiom | Positive Harris recurrence
s_kac_theorem_general | theorem | Kac's theorem (general state space)
s_drift_criterion_recurrence | theorem | Foster's drift criterion for recurrence (general state space)
s_drift_criterion_transience | theorem | Drift criterion for transience
s_drift_criterion_positive_recurrence | theorem | Foster's drift criterion for positive recurrence (general state space)
s_ergodicity_general_state_space | theorem | Ergodicity (Meyn-Tweedie sense)
s_uniform_ergodicity | state | Uniform ergodicity
s_geometric_ergodicity_general | theorem | Geometric ergodicity (general state space, V-norm)
s_v_norm_ergodicity | axiom | V-norm (f-norm) for measures
s_subgeometric_ergodicity | state | Subgeometric ergodicity
s_polynomial_ergodicity | state | Polynomial ergodicity
s_drift_criterion_geometric_ergodicity_general | theorem | Drift criterion for geometric ergodicity (general state space)
s_drift_criterion_subgeometric | theorem | Drift criterion for subgeometric ergodicity
s_lln_harris_chain | theorem | Law of large numbers for Harris recurrent chains
s_clt_harris_chain | theorem | Central limit theorem for Harris recurrent chains
s_lil_harris_chain | theorem | Law of the iterated logarithm for Harris chains
s_asymptotic_variance_mc | state | Asymptotic variance (Markov chain CLT)
s_kendall_theorem | theorem | Kendall's theorem (uniform ergodicity characterization)
s_harris_ergodic_theorem | theorem | Harris' ergodic theorem
s_comparison_theorem_drift | theorem | Comparison theorem for drift functions
s_multiplicative_mean_drift_criterion | theorem | Multiplicative mean drift criterion (geometric case)
s_doeblin_minorization_equivalence | theorem | Doeblin-minorization equivalence for uniform ergodicity
s_moment_bound_ergodic_chain | theorem | Moment bounds from drift conditions
s_stochastic_monotonicity_mc | axiom | Stochastic monotonicity (Markov chain)
s_iterated_function_system | axiom | Iterated function system (IFS) as Markov chain
s_state_space_collapse_petite | theorem | Compact sets are petite for T-chains
s_v_uniform_ergodicity | state | V-uniform ergodicity
s_irreducibility_plus_strong_feller_implies_harris | theorem | Irreducibility plus strong Feller implies Harris recurrence
s_ergodic_decomposition_harris | theorem | Ergodic decomposition for general Markov chains
s_number_of_visits_general | state | Expected number of visits (general state space)
s_nonlinear_state_space_model | axiom | Nonlinear state space model as Markov chain
s_stability_criterion_nonlinear_sde | theorem | Stability criterion for Markov chains from SDEs
s_lyapunov_criterion_continuous_time | theorem | Lyapunov criterion for continuous-time chains (generator form)
s_basic_adjoint_relationship | theorem | Basic adjoint relationship (Revuz measure)
s_resolvent_characterization_general | theorem | Resolvent characterization of recurrence (general state space)
s_two_series_theorem_chain | theorem | Convergence parameter characterization of R-transience
s_mcmc_ergodicity | theorem | MCMC ergodicity via Meyn-Tweedie theory
s_f_modulated_drift | theorem | f-modulated drift criterion
s_total_variation_convergence_bound | theorem | Explicit total variation convergence bound via drift and minorization
s_splitting_construction_continuous_time | state | Splitting construction for continuous-time chains
s_accessible_set_general | axiom | Accessible set (general state space)
s_absorbing_set_general | axiom | Absorbing set (general state space)
s_taboo_kernel_general | state | Taboo kernel (general state space)
t_nummelin_splitting | technique | Nummelin splitting technique
t_regeneration_technique | technique | Regeneration technique (for general state space chains)
t_coupling_general_state_space | technique | Coupling construction for general state space chains
t_monotone_coupling | technique | Monotone coupling (stochastically ordered chains)
s_conditional_expectation_as_projection | state | Conditional expectation as L² projection
s_stochastic_process_cinlar | axiom | Stochastic process
s_natural_filtration | state | Natural filtration
s_predictable_process | axiom | Predictable process
s_predictable_sigma_algebra | axiom | Predictable σ-algebra
s_optional_sigma_algebra | axiom | Optional σ-algebra
s_stopped_sigma_algebra | state | Stopped σ-algebra
s_stopped_process | state | Stopped process
s_submartingale | axiom | Submartingale
s_right_continuous_martingale | axiom | Right-continuous martingale
s_usual_conditions | axiom | Usual conditions (conditions habituelles)
s_cadlag_modification | theorem | Existence of càdlàg modification
s_class_D_process | axiom | Class (D) process
s_class_DL_process | axiom | Class (DL) process
s_predictable_quadratic_variation | state | Predictable quadratic variation (angle bracket process)
s_semimartingale | axiom | Semimartingale
s_markov_process_cinlar | axiom | Markov process (general state space)
s_feller_process | axiom | Feller process
s_resolvent_operator | state | Resolvent operator (of a semigroup/generator)
s_resolvent_equation | theorem | Resolvent equation
s_positive_maximum_principle | axiom | Positive maximum principle
s_martingale_problem | axiom | Martingale problem (Stroock–Varadhan)
s_poisson_process_as_markov | state | Poisson process as a Markov/Lévy process
s_compound_poisson_process | state | Compound Poisson process
s_bm_construction_kolmogorov | state | Construction of Brownian motion via Kolmogorov extension
s_bm_generator | state | Generator of Brownian motion
s_bm_path_properties | state | Path properties of Brownian motion
s_right_continuous_filtration | state | Right-continuous filtration
s_debut_theorem | theorem | Début theorem
s_section_theorem | theorem | Section theorem
s_optional_projection | state | Optional projection
s_predictable_projection | state | Predictable projection
s_compensator_process | state | Compensator (dual predictable projection)
s_levy_ito_decomposition | theorem | Lévy–Itô decomposition
s_poisson_random_measure | axiom | Poisson random measure
s_markov_semigroup_on_banach | state | Markov semigroup on a Banach space
s_generator_core | axiom | Core of a generator
s_diffusion_generator | state | Generator of a diffusion process
s_ito_diffusion | state | Itô diffusion
s_stochastic_exponential | state | Stochastic exponential (Doléans-Dade exponential)
s_novikov_condition | theorem | Novikov's condition
s_multiplicative_functional | axiom | Multiplicative functional
s_killed_process | state | Killed process
t_coupling_technique | technique | Coupling technique
t_optional_stopping_technique_cinlar | technique | Optional stopping technique
t_martingale_method_cinlar | technique | Martingale method
t_semigroup_method | technique | Semigroup method
t_localization_technique | technique | Localization via stopping times
t_generator_characterization | technique | Generator characterization technique
t_doob_meyer_technique | technique | Doob–Meyer technique
t_change_of_measure_technique | technique | Change of measure (Girsanov) technique
t_lyapunov_function_technique | technique | Lyapunov function technique (for Markov processes)
t_potential_theory_technique | technique | Potential-theoretic technique for Markov processes
s_conditional_dominated_convergence | theorem | Conditional dominated convergence theorem
s_conditional_fatou | theorem | Conditional Fatou's lemma
s_martingale_Lp_convergence | theorem | Martingale Lᵖ convergence theorem
s_dambis_dubins_schwarz | theorem | Dambis–Dubins–Schwarz theorem
s_occupation_times_formula | theorem | Occupation times formula
s_local_time | state | Local time of a continuous semimartingale
s_tanaka_formula | theorem | Tanaka's formula
s_entrance_boundary | axiom | Entrance boundary (Feller classification)
s_exit_boundary | axiom | Exit boundary (Feller classification)
s_speed_measure | state | Speed measure of a one-dimensional diffusion
s_scale_function | state | Scale function of a one-dimensional diffusion
s_feller_boundary_classification | state | Feller boundary classification for one-dimensional diffusions
s_column_space | axiom | Column Space (Range)
s_row_space | axiom | Row Space
s_null_space | axiom | Null Space (Kernel)
s_matrix_rank | axiom | Rank of a Matrix
s_unitary_matrix | axiom | Unitary Matrix
s_conjugate_transpose | axiom | Conjugate Transpose (Adjoint)
s_frobenius_norm | axiom | Frobenius Norm
s_matrix_operator_norm | axiom | Matrix Operator Norm (Induced Norm)
s_submultiplicativity_of_matrix_norms | theorem | Submultiplicativity of Matrix Norms
s_singular_values | axiom | Singular Values
s_reduced_svd | state | Reduced (Thin) SVD
s_eckart_young_theorem | theorem | Eckart–Young–Mirsky Theorem
s_moore_penrose_pseudoinverse | axiom | Moore–Penrose Pseudoinverse
s_orthogonal_projector | axiom | Orthogonal Projector
s_qr_factorization | state | QR Factorization
s_upper_triangular_matrix | axiom | Upper Triangular Matrix
s_lower_triangular_matrix | axiom | Lower Triangular Matrix
s_householder_reflector | axiom | Householder Reflector
s_least_squares_problem | axiom | Least Squares Problem
s_normal_equations | state | Normal Equations
s_condition_number_of_linear_system | theorem | Conditioning of Ax = b
s_machine_epsilon | axiom | Machine Epsilon
s_fundamental_axiom_of_floating_point | axiom | Fundamental Axiom of Floating-Point Arithmetic
s_backward_stability | axiom | Backward Stability
s_backward_stability_implies_accuracy | theorem | Backward Stability Implies Forward Accuracy
s_accuracy_equals_condition_times_stability | theorem | Accuracy = Condition × Stability
s_backward_stability_of_householder_qr | theorem | Backward Stability of Householder QR
s_lu_factorization | state | LU Factorization
s_partial_pivoting | axiom | Partial Pivoting
s_permutation_matrix | axiom | Permutation Matrix
s_growth_factor | state | Growth Factor
s_backward_stability_of_gepp | theorem | Backward Stability of GEPP
s_hermitian_positive_definite_matrix | axiom | Hermitian Positive Definite Matrix
s_cholesky_factorization | state | Cholesky Factorization
s_hessenberg_matrix | axiom | Hessenberg Matrix
s_tridiagonal_matrix | axiom | Tridiagonal Matrix
s_givens_rotation | axiom | Givens Rotation
s_bidiagonal_matrix | axiom | Bidiagonal Matrix
s_sparse_matrix | axiom | Sparse Matrix
s_preconditioner | axiom | Preconditioner
s_wilkinson_shift | state | Wilkinson Shift
s_ritz_values | state | Ritz Values and Vectors
s_ilu_preconditioner | state | Incomplete LU (ILU) Preconditioner
s_incomplete_cholesky_preconditioner | state | Incomplete Cholesky Preconditioner
s_domain_decomposition_preconditioner | state | Domain Decomposition Preconditioner
s_convergence_via_polynomial_approximation | theorem | Convergence of Krylov Methods via Polynomial Approximation
s_gmres_convergence | theorem | GMRES Convergence Bound
s_lanczos_gauss_quadrature_connection | theorem | Lanczos–Gauss Quadrature Connection
s_cg_lanczos_equivalence | theorem | CG–Lanczos Equivalence
s_cg_convergence_bound | theorem | CG Convergence Bound
s_faber_manteuffel_theorem | theorem | Faber–Manteuffel Theorem
s_bauer_fike_theorem | theorem | Bauer–Fike Theorem
s_cauchy_interlacing_theorem | theorem | Cauchy Interlacing Theorem
t_classical_gram_schmidt | technique | Classical Gram–Schmidt (CGS)
t_modified_gram_schmidt | technique | Modified Gram–Schmidt (MGS)
t_householder_triangularization | technique | Householder Triangularization
t_least_squares_via_qr | technique | Least Squares via QR Factorization
t_least_squares_via_svd | technique | Least Squares via SVD
t_back_substitution | technique | Back Substitution
t_forward_substitution | technique | Forward Substitution
t_gaussian_elimination_with_partial_pivoting | technique | Gaussian Elimination with Partial Pivoting (GEPP)
t_cholesky_algorithm | technique | Cholesky Algorithm
t_hessenberg_reduction | technique | Hessenberg Reduction
t_power_iteration | technique | Power Iteration
t_inverse_iteration | technique | Inverse Iteration
t_rayleigh_quotient_iteration | technique | Rayleigh Quotient Iteration
t_qr_algorithm | technique | QR Algorithm (Unshifted)
t_shifted_qr_algorithm | technique | Shifted QR Algorithm
t_simultaneous_iteration | technique | Simultaneous Iteration (Subspace Iteration)
t_implicit_qr_step_francis | technique | Implicit QR Step (Francis Algorithm)
t_deflation | technique | Deflation in Eigenvalue Computation
t_jacobi_eigenvalue_algorithm | technique | Jacobi Eigenvalue Algorithm
t_divide_and_conquer_eigenvalue | technique | Divide-and-Conquer Eigenvalue Algorithm
t_bisection_for_eigenvalues | technique | Bisection for Symmetric Tridiagonal Eigenvalues
t_mrrr_algorithm | technique | MRRR Algorithm (Multiple Relatively Robust Representations)
t_svd_algorithm | technique | SVD Algorithm (Golub–Kahan)
t_bidiagonalization | technique | Bidiagonalization (Golub–Kahan)
t_arnoldi_iteration | technique | Arnoldi Iteration
t_implicitly_restarted_arnoldi | technique | Implicitly Restarted Arnoldi (IRAM / ARPACK)
t_gmres | technique | GMRES (Generalized Minimal Residual)
t_lanczos_iteration | technique | Lanczos Iteration
t_bicg | technique | BiCG (Biconjugate Gradient)
t_cgs_method | technique | CGS (Conjugate Gradient Squared)
t_bicgstab | technique | BiCGSTAB (Bi-Conjugate Gradient Stabilized)
t_qmr | technique | QMR (Quasi-Minimal Residual)
t_preconditioned_cg | technique | Preconditioned Conjugate Gradient
s_chebyshev_points_second_kind | axiom | Chebyshev points of the second kind ;; Chebyshev-Lobatto points ;; Chebyshev extreme points ;; Chebyshev grid
s_chebyshev_three_term_recurrence | theorem | Chebyshev three-term recurrence ;; Recurrence for Chebyshev polynomials
s_chebyshev_orthogonality | theorem | Chebyshev orthogonality ;; Orthogonality of Chebyshev polynomials
s_chebyshev_series | state | Chebyshev series ;; Chebyshev expansion (series object)
s_equivalence_chebyshev_fourier_cosine | theorem | Equivalence of Chebyshev and Fourier cosine series ;; Chebyshev-Fourier correspondence ;; cos-substitution equivalence
s_chebyshev_projection | state | Chebyshev projection (truncated Chebyshev series) ;; Truncated Chebyshev series ;; Chebyshev L2 projection
s_chebyshev_aliasing | theorem | Aliasing of Chebyshev coefficients ;; Chebyshev aliasing ;; Aliasing relation interpolant vs projection ;; Trigonometric aliasing on Chebyshev grid
t_barycentric_interpolation_formula | technique | Barycentric interpolation formula ;; Second barycentric formula ;; Barycentric Lagrange interpolation
s_chebyshev_barycentric_weights | state | Barycentric weights for Chebyshev points ;; Chebyshev barycentric weights
s_numerical_stability_barycentric | theorem | Numerical stability of the barycentric formula ;; Backward stability of barycentric interpolation
s_bernstein_ellipse | axiom | Bernstein ellipse ;; Bernstein regularity ellipse ;; ρ-ellipse
t_joukowski_map | technique | Joukowski map ;; Joukowsky map ;; Zhukovsky transform
s_chebyshev_coeff_decay_bv | theorem | Chebyshev coefficient decay for bounded-variation functions ;; BV Chebyshev coefficient bound
s_chebyshev_coeff_decay_differentiable | theorem | Chebyshev coefficient decay for differentiable functions ;; Algebraic Chebyshev coefficient decay
s_chebyshev_coeff_decay_analytic | theorem | Chebyshev coefficient decay for analytic functions ;; Geometric Chebyshev coefficient decay
s_geometric_convergence_analytic_chebyshev | theorem | Geometric convergence of Chebyshev approximation for analytic functions ;; Spectral/geometric convergence of Chebyshev interpolation
s_convergence_rate_differentiable_chebyshev | theorem | Algebraic convergence of Chebyshev approximation for differentiable functions ;; Finite-smoothness Chebyshev convergence rate
s_optimal_chebyshev_rate_rho | theorem | Estimate of optimal Chebyshev convergence rate ρ ;; Root test for Chebyshev convergence rate
s_lebesgue_constant_interpolation | state | Lebesgue constant (interpolation operator norm) ;; Lebesgue constant for interpolation
s_lebesgue_constant_chebyshev_points | theorem | Lebesgue constant for Chebyshev points ;; Logarithmic growth of Chebyshev Lebesgue constant
s_lebesgue_constant_equispaced_points | theorem | Lebesgue constant for equispaced points ;; Exponential growth of equispaced Lebesgue constant
s_lebesgue_constant_bound | theorem | Lebesgue constant interpolation error bound ;; Near-best interpolation bound ;; Operator-norm inequality for interpolation
s_conditioning_polynomial_interpolation | theorem | Conditioning of polynomial interpolation ;; Stability of polynomial interpolation
s_existence_uniqueness_best_approximation | theorem | Existence and uniqueness of best polynomial approximation ;; Uniqueness of minimax polynomial
s_de_la_vallee_poussin_alternation_lemma | theorem | de la Vallée Poussin lower bound (alternation lemma) ;; Vallée-Poussin alternation lemma ;; de la Vallée Poussin lower bound
s_haar_condition | axiom | Haar condition (Haar/Chebyshev system) ;; Chebyshev system ;; Haar system
s_linear_approximation_scheme | axiom | Linear approximation scheme ;; Linear approximation
s_nonlinear_approximation | axiom | Nonlinear approximation ;; Nonlinear approximation theory
s_node_polynomial | state | Node polynomial ;; Nodal polynomial
s_hermite_integral_formula | theorem | Hermite integral formula (Cauchy interpolation error formula) ;; Cauchy interpolation formula ;; Hermite-Cauchy error formula
t_potential_theory_of_approximation | technique | Potential theory of polynomial approximation ;; Potential-theoretic convergence analysis
s_green_function_complement_log_singularity | state | Green's function of the complement with logarithmic pole ;; Green's function with logarithmic singularity ;; complement Green's function
s_convergence_rate_from_green_function | theorem | Convergence rate from Green's function ;; Geometric convergence region (potential theory)
s_equispaced_equilibrium_potential | state | Equispaced equilibrium potential ;; Uniform-density logarithmic potential
s_impossibility_stable_equispaced_approximation | theorem | Impossibility of stable high-order equispaced approximation ;; No stable geometric equispaced reconstruction ;; Platte-Trefethen-Kuijlaars theorem
s_interlacing_orthogonal_polynomial_roots | theorem | Interlacing of orthogonal polynomial roots ;; Separation/interlacing of orthogonal polynomial zeros
s_colleague_matrix | state | Colleague matrix ;; Chebyshev companion matrix
t_rootfinding_via_eigenvalues | technique | Rootfinding via eigenvalues (colleague/companion method) ;; Eigenvalue rootfinding ;; companion matrix rootfinding
s_max_degree_exactness_gauss_optimality | theorem | Maximal algebraic degree of exactness (Gauss optimality) ;; Gauss quadrature optimality
t_golub_welsch_algorithm | technique | Golub-Welsch algorithm ;; Golub-Welsch ;; Jacobi matrix quadrature
s_comparable_accuracy_cc_gauss | theorem | Comparable accuracy of Clenshaw-Curtis and Gauss quadrature ;; Trefethen's observation on Clenshaw-Curtis vs Gauss
t_caratheodory_fejer_approximation | technique | Carathéodory-Fejér (CF) approximation ;; CF approximation ;; Caratheodory-Fejer method
s_cf_hankel_singular_value_identity | theorem | CF singular value / Hankel eigenvalue identity ;; Hankel singular value = best approximation error
s_aak_theory | theorem | AAK (Adamyan-Arov-Krein) theory ;; Adamyan-Arov-Krein theorem ;; AAK theorem
s_rational_best_approximation_type_mn | state | Rational best approximation of type (m,n) ;; Best rational minimax approximant ;; R_{mn} best approximation
s_rational_equioscillation_characterization | theorem | Rational equioscillation characterization theorem ;; Rational Chebyshev characterization
s_defect_of_rational_approximant | axiom | Defect of a rational approximant ;; Rational approximation defect
s_existence_uniqueness_rational_best_approximation | theorem | Existence and uniqueness of rational best approximation ;; Rational best approximation existence/uniqueness
s_root_exponential_convergence_rational | theorem | Root-exponential convergence of rational approximation ;; exp(−C√n) rational convergence
t_pole_clustering | technique | Pole clustering near singularities ;; Exponential pole clustering ;; lightning/AAA pole placement
s_newman_theorem_abs_x | theorem | Newman's theorem (rational approximation of |x|) ;; Newman rational approximation of |x|
s_newman_construction_points | state | Newman's construction (Newman points) ;; Newman points ;; Newman rational
s_stahl_theorem_sharp_rates | theorem | Stahl's theorem (sharp |x| and exp rational rates) ;; Stahl asymptotics ;; Halphen constant rate
s_zolotarev_first_problem | theorem | Zolotarev's first problem (best rational approximation of sign) ;; Zolotarev sign approximation
s_zolotarev_problems | theorem | Zolotarev problems (elliptic extremal rational functions) ;; Four Zolotarev problems
s_chebyshev_pade_approximation | state | Chebyshev-Padé (Clenshaw-Lord) approximation ;; Clenshaw-Lord approximation ;; Chebyshev-Pade
s_convergence_pade_pole_detection | theorem | Convergence of Padé approximants and pole detection ;; Montessus de Ballore / Stahl convergence ;; Padé pole detection
t_analytic_continuation_via_rational_approximation | technique | Analytic continuation via rational approximation ;; Rational analytic continuation ;; Padé continuation
s_froissart_doublets | axiom | Froissart doublets (spurious poles) ;; Spurious poles ;; Froissart doublets
s_quasimatrix | axiom | Quasimatrix ;; Continuous-analogue matrix
s_chebfun | state | Chebfun (numerical function representation) ;; Chebfun representation
t_chebfun_adaptive_chopping | technique | Chebfun adaptive degree selection (chopping) ;; Chopping ;; standardChop ;; resolution test
s_spectral_differentiation_matrix | state | Spectral differentiation matrix ;; Chebyshev differentiation matrix ;; spectral derivative matrix
s_chebyshev_differentiation_matrix_entries | theorem | Chebyshev differentiation matrix entries formula ;; Negative sum trick ;; explicit Chebyshev D matrix
s_spectral_accuracy | theorem | Spectral accuracy (spectral convergence) ;; Spectral convergence ;; exponential accuracy of spectral methods
t_spectral_collocation_method | technique | Spectral collocation method for differential equations ;; Chebyshev collocation ;; pseudospectral method
s_affine_set | axiom | Affine set
s_affine_hull | axiom | Affine hull
s_convex_set | axiom | Convex set
s_convex_hull | state | Convex hull
s_convex_cone | axiom | Convex cone
s_conic_hull | state | Conic hull
s_hyperplane | axiom | Hyperplane
s_halfspace | axiom | Halfspace
s_euclidean_ball | state | Euclidean ball
s_ellipsoid_set | state | Ellipsoid (set)
s_second_order_cone | state | Second-order (norm) cone
s_polyhedron_convex_opt | state | Polyhedron
s_simplex_convex | state | Simplex
s_positive_semidefinite_cone | state | Positive semidefinite cone
s_intersection_preserves_convexity | theorem | Intersection preserves convexity
s_affine_map_preserves_convexity | theorem | Affine image and preimage preserve convexity
t_perspective_function | technique | Perspective function
t_linear_fractional_function | technique | Linear-fractional function
s_separating_hyperplane_theorem | theorem | Separating hyperplane theorem
s_dual_cone | state | Dual cone
s_self_dual_cone | theorem | Self-dual cones
s_proper_cone | axiom | Proper cone
s_generalized_inequality | axiom | Generalized inequality
s_minimal_element_generalized_inequality | state | Minimum and minimal elements
s_dual_characterization_minimal_elements | theorem | Dual characterization of minimal elements
s_convex_function | axiom | Convex function
s_concave_function | axiom | Concave function
t_extended_value_extension | technique | Extended-value extension
s_first_order_condition_convexity | theorem | First-order condition for convexity
s_second_order_condition_convexity | theorem | Second-order condition for convexity
s_epigraph | state | Epigraph
s_nonneg_weighted_sum_convexity | theorem | Nonnegative weighted sum and integral preserve convexity
s_affine_composition_convexity | theorem | Composition with an affine map preserves convexity
s_pointwise_supremum_convexity | theorem | Pointwise maximum and supremum preserve convexity
s_scalar_composition_convexity | theorem | Scalar composition rules
s_vector_composition_convexity | theorem | Vector composition rules
s_partial_minimization_convexity | theorem | Partial minimization preserves convexity
t_perspective_of_function | technique | Perspective of a function
s_quasiconvex_function | axiom | Quasiconvex function
s_quasiconvex_first_order_condition | theorem | First-order condition for quasiconvexity
s_log_concave_log_convex_function | axiom | Log-concave and log-convex functions
s_k_convex_function | axiom | K-convex function
s_optimization_standard_form | axiom | Standard-form optimization problem
s_convex_optimization_standard_form | axiom | Convex optimization problem (standard form)
s_optimality_condition_convex | theorem | Optimality condition for convex problems
t_equivalence_transformations | technique | Equivalence transformations
t_quasiconvex_bisection | technique | Quasiconvex optimization via bisection
s_linear_fractional_program | state | Linear-fractional program
s_quadratic_program | state | Quadratic program (QP)
s_qcqp | state | Quadratically constrained quadratic program (QCQP)
s_second_order_cone_program | state | Second-order cone program (SOCP)
s_robust_linear_program | state | Robust linear program
s_geometric_program | state | Geometric program (GP)
s_semidefinite_program | state | Semidefinite program (SDP)
s_cone_program | state | Conic (cone) program
s_vector_optimization | axiom | Vector (multicriterion) optimization
s_pareto_optimal_point | state | Pareto optimal point
t_scalarization | technique | Scalarization
s_lagrangian | state | Lagrangian
s_lagrange_dual_function | state | Lagrange dual function
s_lagrange_dual_problem | state | Lagrange dual problem
s_weak_duality | theorem | Weak duality
s_duality_gap | state | Duality gap
s_strong_duality | theorem | Strong duality
s_slater_condition | theorem | Slater's condition
t_epigraph_duality_interpretation | technique | Geometric/epigraph interpretation of duality
s_complementary_slackness | theorem | Complementary slackness
s_kkt_conditions | theorem | Karush-Kuhn-Tucker (KKT) conditions
s_saddle_point_duality | theorem | Max-min (saddle-point) interpretation of duality
s_strong_max_min_sion | theorem | Strong max-min equality (Sion's minimax theorem)
t_alternatives_via_duality | technique | Theorems of alternatives via duality
s_sensitivity_shadow_price | theorem | Sensitivity (shadow price) interpretation
t_global_sensitivity_analysis | technique | Perturbation and global sensitivity analysis
s_duality_generalized_inequalities | theorem | Duality for generalized inequalities
s_sdp_cone_duality | theorem | SDP and cone program duality
s_norm_approximation_problem | state | Norm approximation problem
s_chebyshev_minimax_approximation | state | Chebyshev (minimax) approximation
s_penalty_function_approximation | state | Penalty function approximation
s_regularized_approximation | state | Regularized (Tikhonov) approximation
s_robust_least_squares | state | Robust least squares
s_mle_convex_problem | state | Maximum likelihood estimation as a convex problem
s_map_estimation | state | MAP estimation as a convex problem
s_optimal_experiment_design | state | Optimal experiment design
s_chebyshev_center | state | Chebyshev center
s_max_volume_inscribed_ellipsoid | state | Maximum volume inscribed ellipsoid
s_min_volume_enclosing_ellipsoid | state | Minimum volume enclosing (Löwner-John) ellipsoid
s_linear_discrimination | state | Linear discrimination
s_support_vector_machine | state | Support vector machine
s_convex_projection_problem | state | Projection onto a convex set
s_floor_planning_convex | state | Floor planning (placement)
s_strong_convexity_assumption | axiom | Strong convexity
t_line_search | technique | Line search (exact and backtracking)
t_gradient_descent_method | technique | Gradient descent method
t_steepest_descent_method | technique | Steepest descent method
s_newton_step_decrement | state | Newton step and Newton decrement
t_newton_method_optimization | technique | Newton's method (damped and pure)
s_self_concordant_function | axiom | Self-concordant function
s_self_concordant_convergence | theorem | Convergence analysis for self-concordant functions
t_newton_system_factorization | technique | Newton system via factorization and structure
s_kkt_system_equality | state | KKT system (Newton system for equality constraints)
t_newton_method_equality_constrained | technique | Newton's method with equality constraints
t_infeasible_start_newton | technique | Infeasible-start Newton method
t_newton_step_elimination | technique | Newton step via elimination
s_logarithmic_barrier_function | state | Logarithmic barrier function
s_central_path | state | Central path
s_central_path_suboptimality_bound | theorem | Central path suboptimality bound
t_barrier_method_interior_point | technique | Barrier method (SUMT)
s_barrier_method_convergence | theorem | Barrier method convergence and complexity
t_phase_one_feasibility | technique | Phase I (finding a strictly feasible point)
t_primal_dual_interior_point | technique | Primal-dual interior-point method
s_surrogate_duality_gap | state | Surrogate duality gap
t_generalized_inequality_interior_point | technique | Generalized-inequality interior-point methods
t_schur_complement | technique | Schur complement
s_schur_complement_psd | theorem | Schur complement characterization of positive semidefiniteness
s_matrix_fractional_convexity | theorem | Convexity of the matrix-fractional function
s_log_det_concavity | theorem | Concavity of log-determinant
s_entropy | axiom | Entropy H(X)
s_joint_entropy | axiom | Joint entropy H(X,Y)
s_conditional_entropy | axiom | Conditional entropy H(Y|X)
s_mutual_information | axiom | Mutual information I(X;Y)
s_conditional_mutual_information | state | Conditional mutual information I(X;Y|Z)
s_mi_entropy_relations | theorem | Mutual information–entropy relations
s_chain_rule_entropy | theorem | Chain rule for entropy
s_chain_rule_mutual_information | theorem | Chain rule for mutual information
s_chain_rule_relative_entropy | theorem | Chain rule for relative entropy
s_information_inequality | theorem | Information inequality (Gibbs' inequality)
s_nonnegativity_mutual_information | theorem | Nonnegativity of mutual information
s_conditioning_reduces_entropy | theorem | Conditioning reduces entropy
s_uniform_maximizes_entropy | theorem | Uniform distribution maximizes entropy
s_independence_bound_entropy | theorem | Independence bound on entropy
s_log_sum_inequality | theorem | Log-sum inequality
s_convexity_relative_entropy | theorem | Convexity of relative entropy
s_concavity_of_entropy | theorem | Concavity of entropy
s_mi_concave_convex | theorem | Concavity/convexity of mutual information
s_data_processing_inequality | theorem | Data-processing inequality
s_fanos_inequality | theorem | Fano's inequality
s_le_cam_l1_entropy | theorem | L1 bound on entropy difference
s_aep | theorem | Asymptotic Equipartition Property (AEP)
s_typical_set | state | Typical set A_ε^{(n)}
s_typical_set_properties | theorem | Properties of the typical set
s_aep_source_coding | theorem | AEP-based source coding bound
s_high_probability_set | state | Smallest high-probability set B_δ^{(n)}
s_entropy_rate | axiom | Entropy rate H(𝒳)
s_entropy_rate_conditional_form | theorem | Equivalence of entropy-rate definitions
s_markov_entropy_rate | theorem | Entropy rate of a Markov chain
s_hidden_markov_entropy_rate | theorem | Entropy rate of a hidden Markov / functions of Markov chains
s_second_law_thermodynamics_markov | theorem | Markov chains and the second law
s_prefix_code | axiom | Prefix (instantaneous) code
s_entropy_bound_codeword_length | theorem | Entropy bound on expected codeword length
s_shannon_code | state | Shannon code lengths
s_shannon_code_optimality_bound | theorem | Optimal code length bound (H ≤ L < H+1)
s_wrong_code_penalty | theorem | Penalty for using the wrong code
t_huffman_code | technique | Huffman coding
s_huffman_optimality | theorem | Optimality of Huffman codes
t_shannon_fano_elias_code | technique | Shannon–Fano–Elias coding
t_arithmetic_coding | technique | Arithmetic coding
s_competitive_optimality_shannon | theorem | Competitive optimality of Shannon code
s_horse_race | axiom | Horse race / gambling model
s_doubling_rate | state | Doubling rate W(b,p)
s_kelly_proportional_betting | theorem | Kelly (proportional) gambling optimality
s_conservation_theorem_gambling | theorem | Conservation theorem (gambling)
s_gambling_side_information | theorem | Value of side information in gambling
s_dependent_horse_races | theorem | Doubling rate of dependent horse races
s_entropy_of_english | state | Entropy of English
s_discrete_channel | axiom | Discrete memoryless channel (DMC)
s_channel_capacity | state | Channel capacity C
s_bsc | state | Binary symmetric channel (BSC)
s_bec | state | Binary erasure channel (BEC)
s_symmetric_channel | state | Symmetric and weakly symmetric channels
s_jointly_typical_set | state | Jointly typical set A_ε^{(n)}
s_joint_aep | theorem | Joint AEP
s_channel_code | axiom | Channel code (M, n)
s_channel_coding_theorem | theorem | Channel coding theorem (achievability)
s_channel_coding_converse | theorem | Converse to the channel coding theorem
t_random_coding | technique | Random coding argument
t_jointly_typical_decoding | technique | Jointly typical decoding
s_feedback_capacity | theorem | Feedback capacity theorem
s_source_channel_separation | theorem | Source–channel separation theorem
s_zero_error_capacity | state | Zero-error capacity
s_differential_entropy | axiom | Differential entropy h(X)
s_differential_entropy_uniform_gaussian | state | Differential entropy of uniform and Gaussian
s_aep_differential | theorem | AEP for differential entropy
s_relation_discrete_differential | theorem | Relation of differential to discrete entropy
s_joint_conditional_differential_entropy | state | Joint and conditional differential entropy
s_relative_entropy_continuous | state | Continuous relative entropy and mutual information
s_multivariate_gaussian_entropy | theorem | Entropy of the multivariate Gaussian
s_gaussian_maximizes_differential_entropy | theorem | Gaussian maximizes differential entropy
s_hadamard_information_inequality | theorem | Hadamard's inequality (via entropy)
s_gaussian_channel | axiom | Gaussian channel
s_gaussian_channel_capacity | theorem | Capacity of the Gaussian channel
t_sphere_packing_gaussian | technique | Sphere-packing argument
s_bandlimited_channel_capacity | theorem | Capacity of the band-limited Gaussian channel
s_parallel_gaussian_channels | theorem | Parallel Gaussian channels capacity
t_water_filling | technique | Water-filling power allocation
s_colored_gaussian_noise | theorem | Capacity with colored Gaussian noise
s_gaussian_feedback_capacity | theorem | Gaussian channel with feedback
s_rate_distortion_function | state | Rate-distortion function R(D)
s_distortion_measure | axiom | Distortion measure d(x,x̂)
s_rate_distortion_theorem | theorem | Rate-distortion theorem
s_bernoulli_rate_distortion | theorem | Rate-distortion of a Bernoulli source
s_gaussian_rate_distortion | theorem | Rate-distortion of a Gaussian source
s_parallel_gaussian_rate_distortion | theorem | Rate-distortion for parallel Gaussian sources (reverse water-filling)
s_rate_distortion_convexity | theorem | Convexity and monotonicity of R(D)
t_blahut_arimoto | technique | Blahut–Arimoto algorithm
s_channel_capacity_double_minimization | theorem | Capacity / R(D) as double optimization
s_multivariate_aep_general | theorem | Strong typicality / method of types (preview)
s_binary_entropy_function | state | Binary entropy function H(p)
s_type_of_sequence | axiom | Type (empirical distribution) of a sequence
s_number_of_types_bound | theorem | Polynomial bound on the number of types
t_method_of_types | technique | Method of types
s_probability_of_sequence_by_type | theorem | Probability of a sequence under Q in terms of type
s_type_class_size_bound | theorem | Size of a type class
s_type_class_probability_bound | theorem | Probability of a type class under Q
s_universal_source_code_via_types | state | Universal source code via types
s_i_projection | state | I-projection (relative-entropy projection)
s_conditional_limit_theorem | theorem | Conditional limit theorem
s_pythagorean_relative_entropy | theorem | Pythagorean theorem for relative entropy
t_large_deviation_theory | technique | Large deviation theory
s_hypothesis_testing_problem | axiom | Binary hypothesis testing problem
s_likelihood_ratio_test | state | Likelihood ratio test
s_chernoff_stein_lemma | theorem | Chernoff–Stein lemma
s_chernoff_information | state | Chernoff information
s_bayesian_error_exponent | theorem | Bayesian probability of error exponent
s_relative_entropy_fisher_information | theorem | Relative entropy and Fisher information (local approximation)
s_score_function | state | Score function
s_maximum_entropy_principle | axiom | Maximum entropy principle
s_maximum_entropy_distribution | state | Maximum entropy distribution
s_max_entropy_theorem | theorem | Maximum entropy theorem
s_burg_maximum_entropy_theorem | theorem | Burg's maximum entropy theorem
s_maximum_entropy_spectral_estimation | state | Maximum entropy spectral estimation
s_anomalous_max_entropy_no_solution | theorem | Existence of maximum entropy distribution
s_universal_source_coding | axiom | Universal source coding
s_minimax_redundancy | state | Minimax redundancy
s_redundancy_of_code | state | Redundancy of a code
t_lz78_coding | technique | Lempel–Ziv (LZ78) coding
t_lz77_sliding_window | technique | Lempel–Ziv (LZ77) sliding-window coding
s_tree_structured_lz_parsing | state | LZ78 phrase / dictionary tree
s_optimality_of_lz78 | theorem | Optimality of LZ78
s_optimality_of_lz77 | theorem | Optimality of sliding-window (LZ77) coding
s_ziv_lempel_distinct_phrases_bound | theorem | Bound on number of distinct LZ phrases
s_kolmogorov_complexity | axiom | Kolmogorov complexity
s_conditional_kolmogorov_complexity | state | Conditional Kolmogorov complexity
s_universality_kolmogorov_complexity | theorem | Universality (invariance) of Kolmogorov complexity
s_kolmogorov_complexity_bounds | theorem | Upper bound on Kolmogorov complexity
s_kolmogorov_complexity_and_entropy | theorem | Kolmogorov complexity and entropy
s_universal_probability | state | Universal probability
s_universal_probability_complexity_relation | theorem | Universal probability ≈ 2^{-K(x)}
s_incompressible_sequence | axiom | Incompressible sequence
s_algorithmically_random_sequence | axiom | Algorithmically random (infinite) sequence
s_chaitin_omega | state | Chaitin's Omega
s_undecidability_of_kolmogorov_complexity | theorem | Noncomputability of Kolmogorov complexity
s_kolmogorov_sufficient_statistic | state | Kolmogorov sufficient statistic
s_kolmogorov_structure_function | state | Kolmogorov structure function
s_occams_razor | theorem | Occam's razor (algorithmic form)
t_universal_gambling | technique | Universal gambling scheme
s_network_information_theory | axiom | Network information theory
s_multiple_access_channel | axiom | Multiple-access channel (MAC)
s_mac_capacity_region | theorem | Multiple-access channel capacity region
s_gaussian_mac_capacity | theorem | Gaussian multiple-access channel capacity
t_successive_cancellation_decoding | technique | Successive cancellation (onion-peeling) decoding
s_broadcast_channel | axiom | Broadcast channel
s_degraded_broadcast_channel | axiom | Degraded broadcast channel
s_degraded_broadcast_capacity_region | theorem | Capacity region of the degraded broadcast channel
t_superposition_coding | technique | Superposition coding
s_gaussian_broadcast_channel | theorem | Gaussian broadcast channel capacity
s_slepian_wolf_rate_region | state | Slepian–Wolf rate region
s_correlated_sources_over_mac | theorem | Transmission of correlated sources over a MAC
s_relay_channel | axiom | Relay channel
s_relay_channel_capacity_degraded | theorem | Capacity of the degraded relay channel
s_interference_channel | axiom | Interference channel
s_gaussian_interference_strong | theorem | Capacity of the strong-interference Gaussian interference channel
s_gelfand_pinsker_channel | theorem | Gelfand–Pinsker theorem (channel with side information)
s_writing_on_dirty_paper | theorem | Writing on dirty paper (Costa)
t_binning_for_side_information | technique | Gelfand–Pinsker binning
s_source_channel_separation_networks | theorem | Failure of source–channel separation in networks
s_gaussian_multiuser_channels | axiom | Gaussian multiuser channels
s_stock_market_model | axiom | Stock market (price relative vector)
s_portfolio | axiom | Portfolio
s_doubling_rate_portfolio | state | Doubling rate
s_log_optimal_portfolio | state | Log-optimal portfolio
s_kuhn_tucker_log_optimal | theorem | Kuhn–Tucker characterization of the log-optimal portfolio
s_competitive_optimality_log_optimal | theorem | Competitive optimality of the log-optimal portfolio
s_expected_ratio_bound | theorem | Expected-ratio bound for log-optimal wealth
s_side_information_growth_rate | theorem | Side information increases growth rate by mutual information
s_growth_rate_stationary_market | theorem | Asymptotic optimality of log-optimal investment (AEP for investment)
t_universal_portfolio | technique | Universal portfolio
s_constant_rebalanced_portfolio | state | Constant rebalanced portfolio
s_universal_portfolio_optimality | theorem | Asymptotic optimality of the universal portfolio
s_entropy_power | state | Entropy power
s_entropy_power_inequality | theorem | Entropy power inequality (EPI)
s_fisher_information_matrix_vector | state | Fisher information (of a random vector under translation)
s_fisher_information_inequality | theorem | Fisher information inequality
s_cramer_rao_fisher_form | theorem | Cramér–Rao (estimation form) via Fisher information
s_de_bruijn_identity | theorem | de Bruijn's identity
s_minkowski_determinant_inequality | theorem | Minkowski determinant inequality
s_szasz_inequality | theorem | Szász's inequality
s_han_inequality | theorem | Han's inequality (subset entropy inequality)
s_han_inequality_relative_entropy | theorem | Han's inequality for relative entropies
s_subset_inequalities_determinant | theorem | Subset inequalities for determinants
s_cauchy_schwarz_inequality_it | theorem | Cauchy–Schwarz inequality (covariance form)
s_invariant_sigma_algebra | state | Invariant σ-algebra of a measure-preserving transformation
s_sigma_algebra_generated_by_rv | state | σ-algebra generated by a random variable
s_stieltjes_measure_function | axiom | Stieltjes measure function
s_feller_clt_converse | theorem | Feller's converse to the central limit theorem
s_esseen_smoothing_inequality | theorem | Esseen's smoothing inequality
t_lindeberg_replacement | technique | Lindeberg replacement (swapping) method
t_standard_machine | technique | Standard machine (measure-theoretic bootstrap)
s_mixing_implies_ergodicity | theorem | Mixing implies ergodicity
s_kolmogorov_cycle_condition | theorem | Kolmogorov cycle condition
t_multiplication_counting_principle | technique | Multiplication (fundamental counting) principle
s_ordered_samples_with_replacement | state | Ordered samples with replacement
s_ordered_samples_without_replacement | state | Ordered samples without replacement (permutations of r from n)
t_combinations_with_repetition | technique | Combinations with repetition (stars and bars)
s_occupancy_model | axiom | Occupancy (balls in cells) model
s_maxwell_boltzmann_statistics | state | Maxwell–Boltzmann statistics
s_boole_inequality | theorem | Boole's inequality (subadditivity)
s_multinomial_theorem | theorem | Multinomial theorem
s_probability_exactly_m_events | theorem | Probability of realizing exactly m of N events
s_returns_to_origin_random_walk | state | Returns to the origin (u_{2n})
s_iterated_generating_functions_branching | theorem | Iterated generating functions for branching processes
s_sparre_andersen_fluctuation | theorem | Sparre Andersen fluctuation theorem
s_independent_increments_process | axiom | Independent increments process
s_stationary_increments_process | axiom | Stationary increments
s_bessel_function_density | state | Bessel function densities
s_kolmogorov_canonical_representation | theorem | Kolmogorov canonical representation
s_index_of_stability | axiom | Index of stability α
s_types_of_distributions | axiom | Types of distributions (affine equivalence)
t_duality_lemma_random_walk | technique | Duality lemma for random walks
s_transitive_closure | state | Transitive closure TC(x) ;; TC(x)
s_burali_forti_paradox | theorem | Burali-Forti paradox ;; Burali-Forti
s_ordinal_arithmetic | state | Ordinal arithmetic ;; ordinal addition ;; ordinal multiplication
s_cantor_normal_form | theorem | Cantor normal form ;; CNF
s_epsilon_numbers | state | Epsilon numbers (ε-numbers) ;; ε-numbers ;; ε₀
s_order_type | state | Order type ;; otp
s_rank_function | state | Rank function ;; set-theoretic rank
t_epsilon_induction | technique | ∈-induction (Foundation induction)
s_well_founded_relation | axiom | Well-founded relation ;; well-founded
t_well_founded_recursion | technique | Well-founded recursion
s_hereditarily_small_sets_H_kappa | state | Hereditarily small sets H(κ) ;; H(κ) ;; H_κ
s_aleph_function | state | Aleph function ℵ_α ;; ℵ_α
s_initial_ordinal | state | Initial ordinal
s_cardinal_arithmetic | state | Cardinal arithmetic ;; κ+λ ;; κ·λ ;; κ^λ
s_hessenberg_kappa_squared | theorem | Hessenberg's theorem (κ²=κ) ;; κ²=κ
s_godel_pairing_function | state | Gödel pairing function ;; canonical well-ordering of pairs
s_cofinality | state | Cofinality cf(α) ;; cf(α)
s_regular_cardinal | axiom | Regular cardinal ;; regular
s_singular_cardinal | axiom | Singular cardinal ;; singular
s_successor_cardinals_regular | theorem | Successor cardinals are regular
s_continuum_function | state | Continuum function ;; 2^κ
s_hausdorff_formula | theorem | Hausdorff formula
s_gimel_function | state | Gimel function ℷ(κ) ;; ℷ ;; gimel
s_beth_function | state | Beth function ℶ_α ;; ℶ_α ;; beth numbers
s_generalized_continuum_hypothesis | state | Generalized Continuum Hypothesis (GCH) ;; GCH
s_tukey_lemma | theorem | Tukey's lemma
s_trichotomy_of_cardinals | theorem | Trichotomy of cardinals ;; cardinal comparability
s_ideal_set_theory | axiom | Ideal (set-theoretic) ;; dual ideal
s_kappa_complete_filter | axiom | κ-complete filter ;; κ-complete
s_principal_filter | state | Principal vs non-principal ultrafilter ;; principal ultrafilter ;; free ultrafilter
s_boolean_prime_ideal_theorem | theorem | Boolean Prime Ideal Theorem ;; BPIT ;; ultrafilter theorem
s_complete_boolean_algebra | axiom | Complete Boolean algebra ;; cBa
s_regular_open_algebra | state | Regular open algebra RO(P) ;; RO(P)
s_club_set | state | Closed unbounded (club) set ;; club ;; closed unbounded set
s_club_filter | state | Club filter
s_normal_filter | state | Normal filter ;; normal
s_club_intersection_theorem | theorem | Intersection of clubs is club
s_tree_set_theoretic | axiom | Tree (set-theoretic) ;; set-theoretic tree
s_tree_branch_level_height | state | Branch, level, and height of a tree ;; branch ;; level ;; height
s_aronszajn_tree | state | Aronszajn tree ;; κ-Aronszajn tree
s_aronszajn_tree_existence | theorem | Existence of an Aronszajn tree
s_suslin_tree | state | Suslin tree
s_suslin_problem | state | Suslin's problem (Suslin Hypothesis) ;; Suslin's Hypothesis ;; SH ;; Suslin line
s_suslin_tree_line_equivalence | theorem | Equivalence of Suslin tree and Suslin line
s_kurepa_tree | state | Kurepa tree / Kurepa's hypothesis ;; Kurepa's hypothesis ;; KH
s_special_aronszajn_tree | state | Special Aronszajn tree
s_partition_calculus | state | Partition calculus (arrow notation) ;; arrow notation ;; κ→(λ)^n_m
s_square_principle | axiom | Square principle □_κ (Jensen) ;; □_κ ;; Jensen square
s_club_principle | state | Club principle ♣ ;; ♣ ;; clubsuit
s_diamond_implies_ch | theorem | ◊ implies CH
s_singular_cardinal_hypothesis | state | Singular Cardinal Hypothesis (SCH) ;; SCH
s_levy_hierarchy | state | Lévy hierarchy (Σ_n/Π_n) ;; Σ_n ;; Π_n
t_absoluteness | technique | Absoluteness
t_elementary_submodel_method | technique | Elementary submodel method
t_skolem_hull | technique | Skolem functions / Skolem hull
t_godel_operations | technique | Gödel operations 𝒢₁–𝒢₁₀
s_constructible_wellordering | state | Constructible well-ordering <_L ;; <_L
s_ac_in_L | theorem | AC holds in L
s_relative_constructibility_L_A | state | Relative constructibility L[A] ;; L[A]
s_L_of_A | state | Constructible closure L(A) ;; L(A)
s_diamond_holds_in_L | theorem | ◊ holds in L (Jensen)
s_square_holds_in_L | theorem | □_κ holds in L (Jensen)
s_kurepa_tree_in_L | theorem | Kurepa tree exists in L
t_rudimentary_functions | technique | Rudimentary functions
s_jensen_J_hierarchy | state | Jensen J-hierarchy J_α ;; J_α ;; J-hierarchy
s_sigma_n_projectum | state | Σ_n-projectum ρ_n ;; ρ_n ;; projectum
t_fine_structural_condensation | technique | Fine-structural condensation
s_silver_indiscernibles | state | Silver indiscernibles
s_zero_sharp_implies_V_not_L | theorem | 0# exists implies V≠L
s_covering_implies_SCH | theorem | Covering lemma implies SCH
s_notion_of_forcing | state | Notion of forcing ;; forcing poset ;; forcing notion
s_dense_predense | state | Dense and predense sets ;; dense set ;; predense
s_compatible_conditions | state | Compatible and incompatible conditions ;; compatibility ;; forcing antichain
s_rasiowa_sikorski_lemma | theorem | Rasiowa–Sikorski lemma
t_p_names | technique | P-names
t_canonical_names | technique | Canonical names (x̌, Ġ)
s_forcing_relation | state | Forcing relation p⊩φ ;; p⊩φ
s_definability_lemma_forcing | theorem | Definability lemma (forcing)
s_truth_lemma | theorem | Truth lemma
s_forcing_theorem | theorem | Forcing theorem (fundamental)
s_ground_model_definability | theorem | Ground model definability
t_separative_quotient | technique | Separativity / separative quotient
t_cohen_forcing | technique | Cohen forcing
s_ccc_forcing | state | Countable chain condition (ccc) ;; ccc
s_ccc_preserves_cardinals | theorem | ccc forcing preserves cardinals
s_cohen_forcing_is_ccc | theorem | Cohen forcing is ccc
t_nice_names | technique | Nice names
s_con_zfc_not_ch_cohen | theorem | Con(ZFC)⟹Con(ZFC+¬CH) (Cohen)
s_boolean_valued_universe | state | Boolean-valued universe V^B ;; V^B
t_boolean_truth_value | technique | Boolean truth value ‖φ‖
s_mixing_lemma_forcing | theorem | Mixing lemma
s_maximum_principle_forcing | theorem | Maximum principle (forcing)
t_levy_collapse | technique | Lévy collapse Coll(ω,κ)
s_kappa_closed_forcing | state | <κ-closed forcing ;; κ-closed ;; <κ-closed
s_closed_forcing_adds_no_sequences | theorem | <κ-closed forcing adds no new <κ-sequences
s_kappa_cc_knaster | state | κ-cc / κ-Knaster ;; κ-cc ;; Knaster
s_kappa_cc_preserves_cardinals | theorem | κ-cc forcing preserves cardinals ≥κ
s_product_forcing_lemma | theorem | Product forcing lemma
t_two_step_iteration | technique | Two-step iteration P∗Q̇
s_martins_axiom_MA_kappa | axiom | Martin's Axiom MA_κ ;; MA_κ
s_martins_axiom | axiom | Martin's Axiom (MA) ;; MA
s_MA_aleph0_theorem | theorem | MA_{ℵ₀} is a theorem of ZFC
s_con_MA_not_CH | theorem | Con(MA+¬CH) (Solovay–Tennenbaum)
t_finite_support_iteration | technique | Finite-support iteration
s_fs_iteration_ccc | theorem | FS iteration of ccc is ccc
s_MA_aleph1_negates_CH | theorem | MA_{ℵ₁} implies ¬CH
s_con_SH | theorem | Con(SH) (Solovay–Tennenbaum)
s_MA_measure_category | theorem | MA and additivity of measure/category
t_countable_support_iteration | technique | Countable-support iteration
s_properness_preserved_CS_iteration | theorem | Properness preserved under CS iteration
s_mahlo_cardinal | state | Mahlo cardinal ;; Mahlo
s_weakly_compact_cardinal | state | Weakly compact cardinal ;; weakly compact
s_weakly_compact_tree_property | theorem | Tree property characterization of weak compactness
s_normal_measure | state | Normal measure ;; normal ultrafilter
t_ultrapower_Ult_V_U | technique | Ultrapower Ult(V,U)
s_measurable_embedding_characterization | theorem | Measurable cardinals and elementary embeddings ;; critical point characterization
t_iterated_ultrapowers | technique | Iterated ultrapowers
s_ramsey_cardinal | state | Ramsey cardinal ;; Ramsey
s_erdos_cardinal | state | Erdős cardinal κ→(α)^{<ω} ;; Erdős cardinal ;; κ(α)
t_ehrenfeucht_mostowski_models | technique | Ehrenfeucht–Mostowski models
s_erdos_implies_zero_sharp | theorem | κ→(ω₁)^{<ω} implies 0# exists
s_strongly_compact_cardinal | state | Strongly compact cardinal ;; strongly compact
s_huge_cardinal | state | Huge cardinal ;; huge
t_extenders | technique | Extenders
s_vopenka_principle | axiom | Vopěnka's principle ;; Vopěnka
s_proper_forcing | state | Proper forcing ;; proper
s_cantor_space | state | Cantor space 2^ω ;; 2^ω
s_baire_space_irrationals | theorem | Baire space homeomorphic to the irrationals
s_universality_of_baire_space | theorem | Universality of Baire space
s_tree_on_a_set_body | state | Tree on a set and its body [T] ;; [T] ;; tree on ω
s_well_founded_tree_rank | state | Well-founded tree and its rank ;; tree rank
s_borel_hierarchy | state | Borel hierarchy (Σ⁰_α,Π⁰_α,Δ⁰_α) ;; Σ⁰_α ;; Π⁰_α
t_borel_codes | technique | Borel codes
s_property_of_baire | state | Property of Baire ;; Baire property ;; BP
t_suslin_operation | technique | Suslin operation 𝒜
s_coanalytic_set | state | Coanalytic set Π¹₁ ;; Π¹₁
s_cantor_bendixson_rank | technique | Cantor–Bendixson derivative and rank ;; CB derivative ;; CB rank
s_perfect_set_property | state | Perfect set property (PSP) ;; PSP
s_analytic_psp | theorem | Analytic sets have the perfect set property
s_analytic_measurable | theorem | Analytic sets are Lebesgue measurable
s_analytic_baire_property | theorem | Analytic sets have the Baire property
s_universal_analytic_set | state | Universal analytic set
t_pi11_norm | technique | Π¹₁ norm / rank
s_boundedness_theorem | theorem | Σ¹₁-boundedness theorem ;; Σ¹₁-bounding
s_projective_hierarchy | state | Projective hierarchy (Σ¹_n,Π¹_n,Δ¹_n) ;; Σ¹_n ;; Π¹_n ;; projective sets
s_projective_hierarchy_proper | theorem | Projective hierarchy is proper
s_uniformization | state | Uniformization
s_sigma2_uniformization | theorem | Σ¹₂-uniformization (Novikov–Kondô–Addison)
s_prewellordering_property | state | Prewellordering property ;; PWO
s_scale_property | state | Scale property ;; scale
s_periodicity_theorems | theorem | Periodicity theorems (Moschovakis) ;; First and Second Periodicity
t_shoenfield_tree | technique | Shoenfield tree
s_sigma2_wellordering_in_L | theorem | Σ¹₂ well-ordering under V=L
s_constructible_reals | state | Constructible reals ℝ∩L ;; ℝ∩L
s_infinite_game | state | Infinite game G(A) and strategy ;; G(A) ;; winning strategy
s_determinacy_of_a_set | state | Determinacy of a set Det(A) ;; Det(A)
s_gale_stewart_theorem | theorem | Gale–Stewart theorem ;; open determinacy ;; closed determinacy
t_unraveling | technique | Unraveling / covering technique
s_AD_implies_measurable | theorem | AD implies all sets Lebesgue measurable
s_AD_implies_baire_property | theorem | AD implies the Baire property for all sets
s_AD_implies_PSP | theorem | AD implies perfect set property for all sets
t_banach_mazur_game | technique | Banach–Mazur game
s_analytic_determinacy | theorem | Analytic determinacy ;; Σ¹₁-determinacy
s_sharps_analytic_determinacy_equivalence | state | Sharps and analytic determinacy equivalence ;; x#
t_homogeneous_tree | technique | Homogeneously Suslin set / homogeneous tree
s_martin_steel_pd | theorem | Martin–Steel theorem (projective determinacy) ;; Martin–Steel
s_PD_implies_regularity | theorem | PD implies regularity for all projective sets
s_AD_equiconsistent_woodins | theorem | AD equiconsistent with infinitely many Woodins
s_wadge_reducibility | state | Wadge reducibility / Wadge hierarchy ;; ≤_W ;; Wadge degrees
t_solovay_model_construction | technique | Solovay's model construction via Lévy collapse
s_shelah_inaccessible_necessary | theorem | Inaccessible necessary for all sets Lebesgue measurable (Shelah)
s_transitive_set_def | state | Transitive set ;; transitive set
s_delta_0_formula | state | Δ₀ (bounded) formula ;; Σ₀ formula ;; Π₀ formula ;; bounded formula ;; Delta-0 formula
t_relativization | technique | Relativization φᴹ
s_diamond_plus | axiom | ◊⁺ (diamond-plus) ;; diamond-plus ;; ◊+
s_diamond_plus_kurepa | theorem | ◊⁺ implies a Kurepa tree (Jensen) ;; diamond-plus implies Kurepa tree
s_first_order_signature | axiom | First-order signature (language)
s_l_structure | axiom | L-structure
s_l_term | axiom | L-term
s_l_formula | axiom | L-formula
s_tarski_satisfaction | axiom | Tarski satisfaction relation
s_l_embedding | axiom | L-embedding
s_substructure | axiom | Substructure / extension
s_l_theory | axiom | L-theory
s_logical_consequence | axiom | Logical consequence (semantic entailment)
s_axiomatizable_theory | axiom | Axiomatizable theory
s_definable_function | axiom | Definable function and definable closure
t_henkin_construction | technique | Henkin construction
s_tarski_vaught_test | theorem | Tarski–Vaught test
s_elementary_chain_theorem | theorem | Tarski's elementary chain theorem
s_diagram_lemma | theorem | Diagram lemma
s_elementary_diagram | axiom | Elementary diagram
s_upward_lowenheim_skolem | theorem | Upward Löwenheim–Skolem theorem
s_lowenheim_skolem_tarski | theorem | Löwenheim–Skolem–Tarski theorem
s_dlo | axiom | Dense linear order without endpoints (DLO)
s_robinson_model_completeness_test | theorem | Robinson's test for model completeness
s_qe_test | theorem | Quantifier elimination test
s_completeness_decidability_acf | theorem | Completeness and decidability of ACF_p
s_lefschetz_principle | theorem | Lefschetz principle
s_ax_injective_surjective | theorem | Ax's theorem (injective endomorphisms are surjective)
s_completeness_decidability_rcf | theorem | Completeness and decidability of RCF
s_hilbert_17th_model_theoretic | theorem | Hilbert's 17th problem (model-theoretic form)
s_presburger_qe | theorem | Quantifier elimination for Presburger arithmetic
s_omitting_types_theorem_real | theorem | Omitting Types Theorem
s_isolated_principal_type | axiom | Isolated (principal) type
s_prime_model | axiom | Prime model
s_atomic_model | axiom | Atomic model
s_prime_iff_atomic | theorem | Prime ⟺ countable atomic
s_existence_uniqueness_prime_model | theorem | Existence and uniqueness of prime/atomic models
s_uniqueness_saturated_models | theorem | Uniqueness of saturated models
s_uniqueness_homogeneous_models | theorem | Uniqueness of homogeneous models
s_universal_model | axiom | Universal model
s_monster_model | axiom | Monster model
s_em_type | axiom | Ehrenfeucht–Mostowski type (EM-type)
s_existence_of_indiscernibles | theorem | Existence of indiscernibles
s_few_types_over_em_models | theorem | Few types over EM models
s_vaughtian_pair | axiom | Vaughtian pair
s_no_vaughtian_pairs_categorical | theorem | Uncountably categorical theories have no Vaughtian pairs
s_morley_degree | state | Morley degree
s_totally_transcendental | axiom | Totally transcendental theory
s_omega_stable_implies_tt | theorem | ω-stable ⟹ totally transcendental
s_definability_of_types | theorem | Definability of types (and heirs)
s_uncountably_categorical_implies_omega_stable | theorem | Uncountably categorical ⟹ ω-stable
s_omega_stable_group | axiom | ω-stable group / group of finite Morley rank
s_dcc_definable_subgroups | theorem | Descending chain condition on definable subgroups
s_connected_component_group | state | Connected component G⁰
s_generic_type_group | axiom | Generic type of a group
s_stabilizer_finite_morley | state | Stabilizer of a type
s_zilber_indecomposability | theorem | Zilber indecomposability theorem
s_macintyre_theorem | theorem | Macintyre's theorem
s_omega_stable_division_ring_commutative | theorem | ω-stable division rings are commutative
s_acl_pregeometry | theorem | Algebraic closure gives a pregeometry
s_exchange_property_acl | theorem | Exchange property for acl (Steinitz exchange)
s_pregeometry | axiom | Pregeometry (combinatorial geometry)
s_dimension_in_pregeometry | state | Dimension and independence in a pregeometry
s_dimension_determines_model | theorem | Dimension determines the model
s_trivial_geometry | axiom | Trivial (degenerate) geometry
s_locally_modular_geometry | axiom | Locally modular geometry
s_locally_modular_iff_no_field | theorem | Locally modular ⟺ no definable field
s_zilber_trichotomy_conjecture | state | Zilber's trichotomy conjecture
s_locally_modular_implies_group_based | theorem | Locally modular ⟹ group-based geometry
s_robinson_model_completeness_acf | theorem | Model completeness of ACF
s_completeness_robinson_test_application | theorem | Robinson's test application to ACF/RCF
s_signature | axiom | Signature ;; language ;; vocabulary
s_domain_universe | axiom | Domain (universe) ;; universe ;; carrier set
s_cardinality_of_structure | state | Cardinality of a structure
s_term | axiom | Term
s_term_function | state | Term function (value of a term) ;; term value
s_homomorphism | axiom | Homomorphism
s_isomorphism | axiom | Isomorphism
s_substructure_generated | state | Substructure generated by a set
s_reduct_expansion | state | Reduct and expansion
s_atomic_diagram | state | Diagram (atomic/literal) ;; literal diagram
s_canonical_term_structure | state | Canonical (term) structure ;; Herbrand structure
s_first_order_formula | axiom | First-order formula
s_atomic_formula | axiom | Atomic formula
s_free_bound_variables | axiom | Free and bound variables
s_sentence | axiom | Sentence ;; closed formula
s_parametrically_definable_set | state | Parametrically definable set ;; definable with parameters
s_theory_of_structure | state | Theory of a structure Th(A)
s_elementary_equivalence | state | Elementary equivalence
s_elementary_embedding | axiom | Elementary embedding
s_elementary_diagram_lemma | theorem | Elementary Diagram Lemma
s_back_and_forth_equivalence | state | Back-and-forth equivalence (potential isomorphism) ;; potential isomorphism
s_cantor_dlo_theorem | theorem | Cantor's theorem on DLO
s_ehrenfeucht_fraisse_game | state | Ehrenfeucht–Fraïssé game ;; EF game
s_ef_quantifier_rank_theorem | theorem | Ehrenfeucht–Fraïssé theorem ;; Fraïssé's theorem on games
s_quantifier_rank | state | Quantifier rank ;; quantifier depth
s_hintikka_formulas | state | Hintikka formulas ;; Hintikka normal form
s_age_of_structure | state | Age of a structure
s_hereditary_property | axiom | Hereditary property (HP) ;; HP
s_joint_embedding_property | axiom | Joint embedding property (JEP) ;; JEP
s_amalgamation_property | axiom | Amalgamation property (AP) ;; AP
s_fraisse_limit | state | Fraïssé limit
s_ultrahomogeneous_structure | axiom | Ultrahomogeneous structure ;; homogeneous structure
s_fraisse_theorem | theorem | Fraïssé's theorem
s_weak_homogeneity | axiom | Weak homogeneity
s_random_graph | state | The random graph (Rado graph) ;; Rado graph
s_extension_property_graphs | axiom | Extension property (graphs) ;; one-point extension property
s_random_graph_aleph0_categorical | theorem | ℵ₀-categoricity of the random graph
s_orbit_stabiliser | state | Orbit and stabiliser
s_permutation_group_topology | axiom | Permutation group topology ;; topology of pointwise convergence
s_closed_subgroup_aut_theorem | theorem | Closed-subgroup characterization of automorphism groups
s_k_transitivity | state | k-homogeneity / k-transitivity ;; k-homogeneity
s_imaginary_elements | state | Imaginary elements ;; definable quotients
s_small_index_property | axiom | Small index property
s_interpretation | state | Interpretation Γ
s_coordinate_map | state | Coordinate map
t_interpretation_reduction | technique | Reduction of theories via interpretation
s_aeq_construction | state | A^eq / M^eq construction ;; Meq
s_elimination_of_imaginaries | axiom | Elimination of imaginaries
s_pseudo_elementary_class | axiom | Pseudo-elementary class (PC) ;; PC class
t_word_construction_interpretation | technique | Word construction interpreting groups/fields
s_completeness_theorem | theorem | Gödel completeness theorem ;; Gödel's completeness theorem
s_direct_reduced_product | state | Direct and reduced product ;; reduced product
t_compactness_via_ultraproducts | technique | Compactness via ultraproducts
s_diagonal_embedding_ultrapower | state | Diagonal embedding into an ultrapower
s_los_tarski_preservation | theorem | Łoś–Tarski preservation theorem ;; Tarski–Łoś theorem
s_chang_los_suszko_theorem | theorem | Chang–Łoś–Suszko theorem
s_lyndon_preservation_theorem | theorem | Lyndon's preservation theorem
s_craig_interpolation_theorem | theorem | Craig interpolation theorem
s_beth_definability_theorem | theorem | Beth definability theorem
s_realizing_omitting_types | state | Realizing and omitting types
s_countably_saturated_model | state | Countably saturated model ;; ℵ₀-saturated model
s_aleph0_categorical_theory | axiom | ℵ₀-categorical theory ;; omega-categorical theory
s_oligomorphic_permutation_group | axiom | Oligomorphic permutation group
s_existentially_closed_structure | axiom | Existentially closed structure ;; e.c. structure
s_inductive_theory | axiom | Inductive theory ;; inductive class
s_existence_ec_models | theorem | Existence of existentially closed models
s_model_companion | state | Model companion / model completion ;; model completion
s_existence_uniqueness_model_companion | theorem | Existence and uniqueness of the model companion
s_quantifier_elimination | axiom | Quantifier elimination ;; QE
s_model_complete_prime_implies_qe | theorem | Model-complete with prime model implies QE
t_nullstellensatz_via_model_completeness | technique | Nullstellensatz via model completeness
s_kaiser_hull | state | Kaiser hull
s_stable_theory_type_counting | axiom | Stability via type-counting ;; stable theory
s_stable_superstable_omega_stable | state | Stable / superstable / ω-stable hierarchy ;; superstable theory
s_order_property | axiom | Order property ;; no order property
s_indiscernible_sequence | axiom | Indiscernible sequence ;; order-indiscernibles
s_set_indiscernibles | axiom | Set indiscernibles
s_ramsey_erdos_rado | theorem | Ramsey / Erdős–Rado theorem ;; Ramsey's theorem
s_em_model_functor | state | EM model / functor ;; EM functor
t_em_construction | technique | EM construction technique
s_forking | state | Forking ;; non-forking
s_heir_coheir | state | Heir and coheir (non-forking extensions)
s_prime_model_over_set | state | Prime model over a set (t.t.)
s_partial_computable_function | axiom | Partial computable function ;; partial recursive function ;; φ_e
s_total_computable_function | axiom | Total computable function ;; total recursive function ;; computable function
s_primitive_recursion_schema | axiom | Primitive recursion schema ;; primitive recursion
s_mu_operator_minimization | axiom | μ-operator (unbounded minimization) ;; minimization ;; unbounded search ;; μ
s_partial_recursive_functions | axiom | Partial recursive functions ;; μ-recursive functions
s_register_machine | axiom | Register machine ;; unlimited register machine ;; URM ;; counter machine
s_church_turing_thesis | axiom | Church–Turing Thesis ;; Church's thesis ;; Turing's thesis
s_computability_thesis_soare | axiom | Computability Thesis (Soare) ;; Soare's thesis
s_oracle_turing_machine | axiom | Oracle Turing machine ;; o-machine ;; relativized Turing machine
s_use_function | axiom | Use function ;; use ;; u(A;e,x)
s_godel_index | axiom | Gödel number / index ;; index ;; program number ;; e
s_turing_jump_A_prime | axiom | Turing jump A′ ;; jump ;; A'
s_arithmetical_hierarchy | axiom | Arithmetical hierarchy ;; Kleene arithmetical hierarchy ;; Σ0n/Π0n/Δ0n
s_many_one_reducibility | axiom | Many-one (m) reducibility ;; m-reducibility ;; ≤m
s_one_one_reducibility | axiom | One-one (1) reducibility ;; 1-reducibility ;; ≤1
s_truth_table_reducibility | axiom | Truth-table (tt) reducibility ;; tt-reducibility ;; ≤tt
s_weak_truth_table_reducibility | axiom | Weak truth-table (wtt) reducibility ;; wtt-reducibility ;; bounded Turing reducibility ;; ≤wtt
s_simple_set | axiom | Simple set
s_immune_set | axiom | Immune set
s_productive_set | axiom | Productive set
s_creative_set | axiom | Creative set
s_diagonal_halting_set_K0 | state | Diagonal halting set K0 ;; K_0 ;; general halting set
s_hypersimple_set | state | Hypersimple set
s_hyperimmune_set | state | Hyperimmune set ;; h-immune set
s_hyperhypersimple_set | state | Hyperhypersimple set ;; hhsimple set
s_maximal_set | state | Maximal set
s_stage_approximation_Phi | state | Stage approximation Φ_{e,s}^A(x) ;; finite-stage approximation ;; Φ_{e,s}
s_enumeration_theorem | theorem | Enumeration Theorem ;; Universal function theorem
s_kleene_normal_form_theorem | theorem | Kleene Normal Form Theorem ;; Normal form theorem ;; Kleene T-predicate
s_ce_equals_sigma01 | theorem | c.e. = Σ0_1 characterization ;; projection characterization of c.e. sets ;; domain/range characterization ;; Σ01 normal form for c.e. sets
s_post_complementation_theorem | theorem | Post's Complementation Theorem ;; Complementation theorem ;; Post's theorem (computable iff c.e. and co-c.e.)
s_recursion_theorem_with_parameters | theorem | Recursion Theorem with parameters ;; uniform recursion theorem ;; parametrized fixed-point theorem
s_myhill_isomorphism_theorem | theorem | Myhill Isomorphism Theorem ;; Myhill's theorem
s_myhill_creative_characterization | theorem | Myhill characterization of creative sets ;; creative = m-complete
s_m_completeness_of_K | theorem | m-completeness of K ;; K is m-complete
s_jump_theorem | theorem | Jump Theorem ;; properties of the jump ;; jump monotonicity
s_posts_theorem | theorem | Post's Theorem ;; Post's theorem (hierarchy/jump)
s_arithmetical_hierarchy_theorem | theorem | Hierarchy Theorem (arithmetical hierarchy is proper) ;; arithmetical hierarchy proper ;; strictness of the arithmetical hierarchy
s_limit_lemma | theorem | Shoenfield Limit Lemma ;; Limit lemma ;; Shoenfield's lemma ;; Δ02 = computable in 0'
s_modulus_lemma | theorem | Modulus Lemma ;; modulus of convergence lemma
s_post_simple_set_existence | theorem | Existence of a simple set (Post) ;; Post's simple set
s_productive_not_ce | theorem | Productive sets are not c.e.
s_relativization_principle | theorem | Relativization principle ;; relativization
t_dovetailing | technique | Dovetailing ;; interleaving computations
t_pairing_function | technique | Pairing function / coding of tuples ;; Cantor pairing ;; sequence coding ;; tupling
t_construction_by_stages_markers | technique | Construction by stages with movable markers ;; movable markers ;; stage construction
t_permitting | technique | Permitting method ;; permitting argument ;; C-permitting
t_reduction_to_K | technique | Reduction to K (proving undecidability) ;; reduction from the halting problem
t_computable_approximation_delta02 | technique | Computable approximation (Δ0_2 method) ;; finite approximation method ;; stage approximation method
s_turing_equivalence | axiom | Turing equivalence (≡_T) ;; ≡T
s_turing_degree | axiom | Turing degree ;; degree of unsolvability ;; T-degree
s_degree_structure_D | axiom | Degree structure D ;; the degrees ;; D ;; global degree structure
s_join_of_degrees | state | Join of degrees (a∨b) ;; least upper bound of degrees ;; a∨b ;; A⊕B
s_degree_0 | state | Degree 0 ;; zero degree ;; the computable degree
s_degree_0_prime | state | Degree 0′ (zero jump) ;; 0' ;; the halting degree ;; zero jump
s_jump_hierarchy | state | Jump hierarchy (0,0′,0″,…) ;; iterated jumps ;; 0,0',0''
s_ce_degrees_R | state | c.e. degrees R ;; R ;; the c.e. degrees ;; r.e. degrees
s_low_set | state | Low set / degree ;; low degree ;; low
s_lown_degree | state | Low_n degree ;; low_n ;; Low_n
s_high_set | state | High set / degree ;; high degree ;; high
s_highn_degree | state | High_n degree ;; high_n ;; High_n
s_high_low_hierarchy | state | High/Low hierarchy ;; high/low hierarchy
s_superlow_degree | state | Superlow degree ;; superlow
s_hyperimmune_free_degree | state | Hyperimmune-free degree ;; computably dominated degree ;; hyperimmune-free
s_hyperimmune_degree | state | Hyperimmune degree ;; hyperimmune degree
s_GL1_degree | state | GL1 degree (generalized low 1) ;; generalized low ;; GL_1 ;; GL1
s_GL2_degree | state | GL2 degree (generalized low 2) ;; generalized low 2 ;; GL_2 ;; GL2
s_minimal_degree | state | Minimal degree ;; degree minimal over 0
s_1_generic_set | state | 1-generic set / degree ;; 1-generic ;; weakly 1-generic
s_n_generic_degree | state | n-generic set / degree ;; n-generic
s_ershov_hierarchy | state | Ershov hierarchy ;; difference hierarchy ;; Boolean hierarchy over c.e. sets
s_n_ce_set | state | n-c.e. set ;; n-r.e. set ;; n-c.e.
s_dce_set | state | d.c.e. set / degree ;; difference c.e. ;; 2-c.e. set ;; d-c.e.
s_e_state | state | e-state ;; e-state of an element
s_posts_problem | state | Post's Problem ;; Post problem
s_kleene_post_theorem | theorem | Kleene–Post Theorem ;; Kleene–Post incomparable degrees below 0'
s_incomparable_ce_degrees | theorem | Existence of incomparable c.e. degrees ;; incomparable c.e. degrees
s_low_basis_theorem | theorem | Low Basis Theorem (Jockusch–Soare) ;; Jockusch–Soare low basis theorem
s_hyperimmune_free_basis_theorem | theorem | Hyperimmune-free Basis Theorem ;; Jockusch–Soare hyperimmune-free basis theorem
s_minimal_pair_theorem | theorem | Minimal Pair Theorem (Lachlan–Yates) ;; Lachlan–Yates minimal pair
s_spector_minimal_degree | theorem | Existence of a minimal degree (Spector) ;; Spector minimal degree theorem
s_exact_pair_theorem | theorem | Exact Pair Theorem ;; exact pair (Kleene–Post / Spector)
s_lattice_embedding_into_ce_degrees | theorem | Embedding lattices into the c.e. degrees ;; distributive lattice embedding theorem
s_nondistributivity_of_ce_degrees | theorem | Nondistributivity of the c.e. degrees (Lachlan) ;; Lachlan nondistributivity
s_nondiamond_theorem | theorem | Nondiamond Theorem (Lachlan) ;; Lachlan nondiamond theorem
s_robinson_low_splitting | theorem | Robinson Low Splitting / interpolation ;; Robinson splitting theorem ;; Robinson interpolation
s_friedberg_jump_inversion | theorem | Friedberg Jump Inversion Theorem ;; Friedberg completeness criterion ;; jump inversion (Friedberg)
t_finite_injury_priority_method | technique | Finite-injury priority method ;; finite injury method ;; Friedberg–Muchnik method
t_priority_requirements_and_injury | technique | Priority requirements and injury ;; injury ;; priority ordering
t_kleene_post_finite_extension | technique | Kleene–Post finite extension method ;; finite extension method ;; forcing the jump
t_length_of_agreement_function | technique | Length-of-agreement function ;; agreement length ;; ℓ(e,s)
t_restraint_function | technique | Restraint function ;; restraint ;; r(e,s)
t_hat_trick | technique | Hat trick ;; hatting ;; Â_s
t_infinite_injury_priority_method | technique | Infinite-injury priority method ;; infinite injury method ;; 0''-priority argument
t_tree_of_strategies | technique | Tree of strategies ;; strategy tree ;; tree method
s_true_path | state | True path ;; true path of the construction
t_thickness_lemma | technique | Thickness Lemma ;; thickness requirement
t_pinball_machine | technique | Pinball machine ;; pinball construction
t_0_triple_prime_priority_argument | technique | 0‴-priority argument ;; 0''' argument ;; triple-jump priority method
t_minimal_pair_construction | technique | Minimal pair construction ;; minimal pair method ;; gap-cogap argument
t_spector_tree_forcing | technique | Spector's construction (tree forcing) ;; perfect tree forcing ;; Spector forcing
t_cohen_forcing_in_arithmetic | technique | Cohen forcing in arithmetic ;; arithmetic forcing ;; finite-condition forcing
t_induction_on_wffs | technique | Induction principle on wffs
s_truth_table | technique | Truth table method
t_substitution_of_term | technique | Substitution of a term for a variable
s_generalization_on_constants | technique | Generalization on constants
t_alphabetic_variant | technique | Alphabetic variant (bound-variable renaming)
t_many_sorted_to_one_sorted | technique | Reduction many-sorted to one-sorted
s_sentence_symbol | axiom | Sentence symbol
s_sentential_wff | state | Sentential well-formed formula
s_unique_readability_sentential | theorem | Unique readability (sentential)
s_truth_assignment | state | Truth assignment
s_extended_truth_assignment | theorem | Extended truth assignment
s_tautology | state | Tautology
s_tautological_implication | state | Tautological implication
s_tautological_equivalence | state | Tautological equivalence
s_functional_completeness | theorem | Functional completeness
s_normal_forms_sentential | theorem | Normal forms (DNF/CNF)
s_compactness_sentential | theorem | Compactness (sentential)
s_decidability_of_tautologies | theorem | Decidability of tautologies
s_switching_circuits | state | Switching circuits
s_variable_assignment | state | Variable assignment
s_logical_implication_validity | state | Logical implication and validity
s_substitution_lemma | theorem | Substitution lemma
s_homomorphism_theorem_logic | theorem | Homomorphism theorem
s_logical_axiom_groups | axiom | Logical axiom groups
s_deduction_relation | state | Deducibility relation
s_generalization_theorem | theorem | Generalization theorem
s_soundness_theorem | theorem | Soundness theorem
s_consistency | state | Consistency
s_lindenbaum_lemma | theorem | Lindenbaum lemma
s_enumerability_of_theorems | theorem | Enumerability of theorems
s_prenex_normal_form | theorem | Prenex normal form
s_skolem_normal_form | state | Skolem normal form
s_herbrand_normal_form | state | Herbrand normal form
s_language_of_number_theory | axiom | Language of number theory
s_standard_structure_N | axiom | Standard structure N
s_robinson_arithmetic_Q | axiom | Robinson arithmetic Q
s_numeral | state | Numeral
s_definable_relation_in_N | state | Definable relation in N
s_representability_theorem | theorem | Representability theorem
s_godel_beta_function | state | Godel beta function
s_recursiveness_of_syntax | theorem | Recursiveness of syntax
s_fixed_point_lemma | theorem | Fixed-point (diagonal) lemma
s_undecidability_of_arithmetic | theorem | Undecidability of arithmetic
s_church_theorem_validity | theorem | Church's theorem
s_essential_undecidability | state | Essential undecidability
s_decidable_undecidable_theories | state | Decidable and undecidable theories
s_derivability_conditions | axiom | Hilbert-Bernays derivability conditions
s_godel_second_incompleteness | theorem | Godel second incompleteness theorem
s_incompleteness_applications_set_theory | state | Incompleteness applications to set theory
s_second_order_language | axiom | Second-order language
s_standard_full_semantics_sol | axiom | Standard (full) second-order semantics
s_categoricity_second_order | theorem | Categoricity in second-order logic
s_failure_compactness_completeness_sol | theorem | Failure of compactness/completeness (SOL)
s_general_henkin_structures | axiom | General (Henkin) structures
s_completeness_general_semantics | theorem | Completeness for general semantics
s_many_sorted_logic | axiom | Many-sorted logic
s_topological_dynamical_system | axiom | Topological dynamical system (cascade)
s_topological_transitivity | axiom | Topological transitivity
s_topological_mixing | axiom | Topological mixing
s_minimality_topological | axiom | Minimality (topological)
s_nonwandering_set | state | Nonwandering set Ω(f)
s_chain_recurrent_set | state | Chain recurrent set / ε-chains
s_birkhoff_recurrence_theorem | theorem | Birkhoff recurrence theorem
s_topological_entropy_spanning_separated | axiom | Topological entropy via spanning/separated sets (Bowen–Dinaburg)
s_topological_entropy_open_covers | axiom | Topological entropy via open covers (Adler–Konheim–McAndrew)
s_periodic_orbit_growth_pn | state | Periodic orbit growth rate p_n(f)
s_artin_mazur_zeta_function | state | Artin–Mazur zeta function
s_zeta_rationality_sft | theorem | Rationality of the zeta function for subshifts of finite type
s_li_yorke_theorem | theorem | Li–Yorke theorem (period three implies chaos)
s_denjoy_inequality | theorem | Denjoy inequality
s_denjoy_counterexample | state | Denjoy counterexample
s_poincare_classification_circle | theorem | Poincaré classification theorem for circle homeomorphisms
s_rotation_number_rational_iff_periodic | theorem | Rationality of rotation number iff periodic orbit
s_morse_smale_system | state | Morse–Smale system
s_expanding_map_of_circle_Em | state | Expanding map of the circle E_m
s_doubling_map_dynamics | state | Doubling map
s_linear_toral_automorphism_cat_map | state | Linear toral automorphism (cat map)
s_circle_rotation_R_alpha | state | Circle rotation R_α
s_full_shift_sigma_N | state | Full shift σ on Σ_N
s_topological_markov_chain | state | Topological Markov chain / subshift of finite type Σ_A
s_spectral_decomposition_topological_markov_chain | theorem | Spectral decomposition for topological Markov chains
s_cylinder_set_symbolic | state | Cylinder set (symbolic dynamics)
s_inclination_lemma | theorem | Inclination lemma (λ-lemma)
s_anosov_flow | state | Anosov flow
s_hyperbolic_linear_map | state | Hyperbolic linear map
s_unstable_subspace_Eu | state | Unstable subspace E^u
s_local_product_structure | state | Local product structure
s_expansiveness | axiom | Expansiveness
t_cone_criterion_hyperbolicity | technique | Cone criterion (cone field) for hyperbolicity
t_adapted_lyapunov_norm | technique | Adapted (Lyapunov) norm
t_graph_transform_method | technique | Graph transform method
s_expanding_map_structural_stability_shub | theorem | Structural stability of expanding maps (Shub)
s_anosov_structural_stability | theorem | Anosov structural stability theorem
s_holder_continuity_hyperbolic_splitting | theorem | Hölder continuity of the hyperbolic splitting
s_no_cycle_condition | axiom | No-cycle condition
s_omega_stability_theorem | theorem | Ω-stability theorem
s_strong_transversality_condition | axiom | Strong transversality condition
s_structural_stability_theorem_RRM | theorem | C¹ structural stability theorem (Robbin–Robinson–Mañé)
t_filtration_dynamics | technique | Filtration (dynamical)
t_markov_partition_hyperbolic | technique | Markov partition for hyperbolic sets
s_markov_partition_existence_sinai_bowen | theorem | Existence of Markov partitions (Sinai–Bowen)
s_smale_solenoid | state | Smale solenoid attractor
s_da_attractor | state | DA (derived from Anosov) attractor
s_plykin_attractor | state | Plykin attractor
s_absolute_continuity_of_foliations | theorem | Absolute continuity of stable/unstable foliations
t_hopf_argument | technique | Hopf argument
s_geodesic_flow_anosov_negative_curvature | theorem | Geodesic flow on negatively curved manifold is Anosov
s_volume_preserving_anosov_ergodic | theorem | Ergodicity of volume-preserving Anosov systems
s_livsic_theorem | theorem | Livšic theorem
s_cohomological_equation_coboundary | state | Cohomological equation / coboundary (dynamics)
s_zeta_rationality_axiom_a | theorem | Rationality/meromorphy of the zeta function for Axiom A systems
t_smale_program_synthesis | technique | Smale's differentiable dynamics program
s_unique_ergodicity | state | Unique ergodicity
s_unique_ergodicity_criterion | theorem | Unique ergodicity criterion
t_koopman_operator | technique | Koopman operator
s_spectral_characterization_mixing | theorem | Spectral characterization of weak and strong mixing
s_discrete_spectrum_system | state | Discrete (pure point) spectrum
s_halmos_von_neumann_theorem | theorem | Halmos–von Neumann theorem
s_rank_one_system | state | Rank-one system
s_rigid_transformation | state | Rigidity (measurable)
s_entropy_of_partition | state | Entropy of a partition H(ξ)
s_conditional_entropy_partitions | state | Conditional entropy of partitions H(ξ|η)
s_kolmogorov_sinai_entropy | state | Metric (Kolmogorov–Sinai) entropy h_μ(f)
s_sinai_generator_theorem | theorem | Kolmogorov–Sinai (Sinai) generator theorem
s_abramov_formula | theorem | Abramov formula
s_suspension_flow | state | Suspension (special) flow
s_kolmogorov_automorphism_k_system | state | K-system (Kolmogorov automorphism)
s_bernoulli_automorphism | state | Bernoulli automorphism
s_measure_of_maximal_entropy | state | Measure of maximal entropy
s_variational_principle_pressure | theorem | Variational principle for topological pressure
s_equilibrium_state | state | Equilibrium state
s_gibbs_measure_dynamics | state | Gibbs measure (for a Hölder potential)
t_measurable_partition_rokhlin | technique | Measurable partition (Rokhlin theory)
t_conditional_measures_disintegration | technique | Conditional measures (disintegration)
s_lebesgue_space_mod_zero | axiom | Lebesgue space (Lebesgue probability space mod 0)
s_gauss_measure | axiom | Gauss measure
s_induced_transformation | state | Induced (first-return) transformation T_A
s_first_return_time_function | state | First return-time function n_A
t_kakutani_tower_construction | technique | Kakutani–Rokhlin tower (skyscraper) construction
s_bernoulli_shift_object | state | Bernoulli shift
s_markov_shift_object | state | Markov shift
s_shift_map_sigma | axiom | Two-sided shift map σ
s_compact_group_endomorphism_mpt | state | Endomorphism/automorphism of a compact group (as m.p.t.)
s_toral_automorphism_mpt | state | Automorphism/endomorphism of the torus (matrix m.p.t.)
s_isomorphism_mod_zero | axiom | Isomorphism mod 0 (measure-theoretic isomorphism)
s_measure_algebra_isomorphism | axiom | Measure-algebra isomorphism (conjugacy of measure algebras)
s_spectral_isomorphism_mpt | axiom | Spectral isomorphism (unitary equivalence of Koopman operators)
s_conjugacy_implies_spectral_isomorphism | theorem | Conjugacy implies spectral isomorphism
s_point_spectrum_eigenvalues_UT | state | Point spectrum (eigenvalues and eigenfunctions of U_T)
s_eigenvalues_modulus_one_constant_modulus | theorem | Eigenvalues of modulus one with constant-modulus eigenfunctions
s_weak_mixing_product_characterization | theorem | Weak-mixing product characterization (T×T ergodic)
s_weak_mixing_continuous_spectrum | theorem | Weak-mixing ⇔ continuous spectrum (no nonconstant eigenfunctions)
s_weak_mixing_all_powers_ergodic | theorem | Weak-mixing equivalent to all powers ergodic
s_koopman_von_neumann_lemma | theorem | Koopman–von Neumann lemma
s_ergodicity_of_group_rotations | theorem | Ergodicity criterion for group rotations
s_ergodicity_of_toral_automorphisms | theorem | Ergodicity criterion for toral automorphisms
s_ergodicity_mixing_of_shifts | theorem | Ergodicity and mixing of Bernoulli and Markov shifts
s_fekete_subadditive_lemma | theorem | Fekete subadditive sequence lemma
s_entropy_of_transformation_wrt_partition | state | Entropy of a transformation with respect to a partition h(T,ξ)
s_entropy_of_transformation_hT | state | Measure-theoretic entropy of a transformation h(T)
s_entropy_powers_inverse_properties | theorem | Entropy behaviour under powers and inverses
s_generator_partition | axiom | Generator (one-sided/two-sided)
s_join_of_partitions | state | Join (refinement) of partitions ξ∨η
s_basic_entropy_inequalities | theorem | Basic entropy (in)equalities for partitions
s_pinsker_sigma_algebra | state | Pinsker σ-algebra
s_space_M_X_probability_measures | state | Space M(X) of Borel probability measures
s_space_M_X_T_invariant_measures | state | Space M(X,T) of T-invariant probability measures
s_ergodic_measures_extreme_points | theorem | Ergodic measures are the extreme points of M(X,T)
s_bowen_metric_dn | state | Bowen (dynamical) metric d_n
s_Lp_mean_ergodic_theorem | theorem | L^p mean ergodic theorem
s_upper_semicontinuity_entropy_map | theorem | Upper semicontinuity of the entropy map and existence of equilibrium states
s_levy_khinchin_continued_fraction_constants | theorem | Khinchin and Lévy constants for continued fractions
s_eigenvalue_group_complete_invariant | theorem | Eigenvalue group as complete invariant for discrete spectrum
s_galilean_spacetime | axiom | Galilean space-time structure
s_galilean_group | axiom | Galilean group
s_galilean_relativity_principle | axiom | Galileo's principle of relativity
s_newton_equation_of_motion | axiom | Newton's equation of motion
s_newton_determinacy_principle | axiom | Newton's principle of determinacy
s_galilean_invariance_of_forces | theorem | Galilean invariance constraints on forces
s_conservative_force_field | axiom | Conservative (potential) force field
s_conservation_of_momentum | theorem | Conservation of momentum
s_conservation_of_angular_momentum | theorem | Conservation of angular momentum
s_period_1d_motion_integral | state | Period of one-dimensional motion (energy integral)
s_reduced_mass_two_body | theorem | Reduction of the two-body problem (reduced mass)
t_reduction_central_field_angular_momentum | technique | Reduction of central motion via angular momentum
s_effective_potential | state | Effective potential
s_central_orbit_quadrature | state | Orbit equation of central motion (quadrature)
s_kepler_problem | state | Kepler's problem
s_laplace_runge_lenz_vector | state | Laplace–Runge–Lenz vector
t_fall_to_center | technique | Analysis of fall to the center
s_canonical_momentum | state | Generalized (canonical) momentum
s_routh_reduction | theorem | Cyclic coordinates and Routh reduction
s_holonomic_constraint | axiom | Holonomic constraint
s_dalembert_lagrange_principle | axiom | D'Alembert–Lagrange principle
s_lagrangian_mechanics_on_manifolds | state | Lagrangian mechanics on a manifold
t_legendre_transformation_mechanics | technique | Legendre transformation (Lagrangian to Hamiltonian)
t_small_oscillations_linearization | technique | Small oscillations / linearization about equilibrium
s_normal_modes | theorem | Normal modes and characteristic frequencies
t_simultaneous_diagonalization_quadratic_forms | technique | Simultaneous diagonalization of two quadratic forms
s_parametric_resonance | theorem | Parametric resonance
s_rigid_body_configuration_so3 | axiom | Rigid-body configuration space SO(3)
s_inertia_tensor | state | Inertia tensor
s_principal_axes_of_inertia | state | Principal axes of inertia
s_angular_velocity_body_space | state | Angular velocity in body and space frames
s_euler_rigid_body_equations | theorem | Euler's equations for a rigid body
s_euler_angles | state | Euler angles
s_poinsot_construction | theorem | Poinsot's construction (inertia ellipsoid)
s_lagrange_top | state | Lagrange top (heavy symmetric top)
s_stability_rotation_principal_axes | theorem | Stability of rotation about principal axes
s_poincare_cartan_integral_invariant | theorem | Poincaré–Cartan integral invariant
s_relative_absolute_integral_invariant | theorem | Relative and absolute integral invariants
t_separation_of_variables_hamilton_jacobi | technique | Separation of variables in the Hamilton–Jacobi equation
s_euler_lie_algebra_equations | theorem | Euler equations on a Lie algebra
s_geodesics_on_lie_groups | theorem | Geodesics on Lie groups (hydrodynamics analogy)
s_lie_poisson_structure | state | Lie–Poisson structure on the dual of a Lie algebra
s_small_denominators_problem | state | Small denominators problem
t_adiabatic_invariant | technique | Adiabatic invariant
t_canonical_perturbation_theory | technique | Canonical perturbation theory
s_caustic | state | Caustic (Lagrangian singularity)
t_wkb_maslov_asymptotics | technique | Short-wave (WKB) asymptotics with Maslov index
s_extended_phase_space | state | Extended phase space
s_torsion_coefficients_homology | state | Torsion coefficients of homology
s_euler_poincare_theorem | theorem | Euler–Poincaré theorem
s_maurer_cartan_form | state | Maurer–Cartan form
s_codifferential_operator | state | Codifferential (adjoint exterior derivative)
s_dolbeault_operators | state | Dolbeault operators ∂ and ∂̄
s_chern_connection | state | Chern connection
s_structure_group_reduction | state | Reduction of the structure group
s_universal_bundle | state | Universal bundle
s_fundamental_vector_field | state | Fundamental vector field
s_vertical_subspace_bundle | state | Vertical subspace of a principal bundle
s_ehresmann_connection | state | Ehresmann connection
s_local_connection_form_gauge_potential | state | Local connection form (gauge potential)
s_local_field_strength | state | Local field strength
s_topological_instanton_charge | state | Topological instanton charge
s_berry_phase | state | Berry phase
s_berry_connection_curvature | state | Berry connection and curvature
s_spin_structure | state | Spin structure
s_analytical_index | state | Analytical index
s_topological_index | state | Topological index
s_chiral_abelian_anomaly | state | Chiral (abelian) anomaly
s_non_abelian_gauge_anomaly | state | Non-abelian gauge anomaly
s_gravitational_anomaly | state | Gravitational anomaly
s_anomaly_polynomial | state | Anomaly polynomial
s_consistent_vs_covariant_anomaly | state | Consistent versus covariant anomaly
s_mckean_singer_formula | theorem | McKean–Singer formula
s_nambu_goto_action | state | Nambu–Goto action
s_polyakov_action | state | Polyakov action
s_worldsheet_diff_weyl_invariance | state | Worldsheet diffeomorphism and Weyl invariance
s_conformal_gauge | state | Conformal gauge
s_conformal_anomaly_central_charge | state | Conformal anomaly (central charge)
s_critical_dimension | state | Critical dimension
s_faddeev_popov_ghosts | state | Faddeev–Popov ghosts
s_string_mass_spectrum_tachyon | state | String mass spectrum and tachyon
s_moduli_space_riemann_surfaces | state | Moduli space of Riemann surfaces
s_topological_defect_classification | theorem | Homotopy classification of topological defects
t_fujikawa_method | technique | Fujikawa method
t_descent_equations_stora_zumino | technique | Stora–Zumino descent equations
t_susy_proof_index_theorem | technique | Supersymmetric proof of the index theorem
t_brst_quantization_string | technique | BRST quantization of the string
t_heat_kernel_supertrace | technique | 
t_gauge_holonomy_transport | technique | 
t_simplicial_euler_computation | technique | 
t_homotopy_classification_order_parameter | technique | 
t_virasoro_central_charge_computation | technique | 
s_reduced_root_system | axiom | Reduced root system ;; reduced root system
s_existence_of_cartan_subalgebras | theorem | Existence of Cartan subalgebras
s_compact_lie_algebra | axiom | Compact Lie algebra ;; Lie algebra of compact type
s_killing_form_negative_definite_compact | theorem | Negative-definiteness of the Killing form on a compact semisimple Lie algebra
s_weyl_theorem_finite_fundamental_group | theorem | Weyl's theorem on finiteness of the fundamental group ;; Weyl's finiteness theorem ;; compactness of the universal cover of a compact semisimple group
s_surjectivity_of_exp_compact_connected | theorem | Surjectivity of the exponential map for compact connected Lie groups
s_conjugation_relative_to_real_form | state | Conjugation relative to a real form ;; conjugation with respect to a real form
s_existence_of_compact_real_form | theorem | Existence of a compact real form
s_uniqueness_of_compact_real_form | theorem | Uniqueness of the compact real form up to conjugacy
s_complexification_of_compact_lie_group | state | Complexification of a compact Lie group ;; universal complexification of a compact group
s_existence_of_cartan_involution | theorem | Existence of a Cartan involution
s_conjugacy_of_cartan_involutions | theorem | Conjugacy of Cartan involutions
s_borel_weil_theorem | theorem | Borel-Weil theorem
s_reductive_lie_group_harish_chandra_class | axiom | Reductive Lie group (Knapp's class) ;; Harish-Chandra class ;; reductive group in the sense of Knapp
s_cartan_subalgebra_of_real_semisimple | axiom | Cartan subalgebra of a real semisimple Lie algebra
s_theta_stable_cartan_subalgebra | state | Theta-stable Cartan subalgebra ;; theta-stable CSA
s_existence_of_theta_stable_cartan_subalgebra | theorem | Existence of a theta-stable Cartan subalgebra
s_maximally_compact_noncompact_cartan_subalgebras | state | Maximally compact and maximally noncompact Cartan subalgebras ;; fundamental Cartan subalgebra ;; maximally split Cartan subalgebra
s_finiteness_of_conjugacy_classes_of_cartan_subalgebras | theorem | Finiteness of conjugacy classes of Cartan subalgebras
s_real_imaginary_complex_roots | state | Real, imaginary, and complex roots
s_compact_and_noncompact_imaginary_roots | state | Compact and noncompact imaginary roots
s_vogan_diagram | state | Vogan diagram
s_vogan_diagram_theorem | theorem | Vogan diagram theorem
s_borel_de_siebenthal_theorem | theorem | Borel-de Siebenthal theorem ;; Borel-de Siebenthal procedure
s_maximal_abelian_subspace_of_p | state | Maximal abelian subspace a of p ;; maximal split abelian subalgebra
s_real_rank | state | Real rank of G ;; split rank ;; R-rank
s_restricted_roots | state | Restricted roots ;; relative roots ;; a-roots
s_restricted_root_space_decomposition | theorem | Restricted root space decomposition ;; a-root space decomposition
s_restricted_root_multiplicity | state | Restricted root multiplicity
s_restricted_root_system | theorem | Restricted root system Sigma ;; relative root system
s_centralizer_m_of_a_algebra | state | Centralizer m = Z_k(a)
s_centralizer_M_of_a_group | state | M = Z_K(a) ;; M = centralizer of A in K
t_ordering_of_restricted_roots | technique | Positivity and ordering of restricted roots
s_nilpotent_subalgebra_n | state | Nilpotent subalgebra n ;; Iwasawa nilpotent subalgebra
s_iwasawa_decomposition_algebra | theorem | Iwasawa decomposition of the Lie algebra ;; g = k + a + n
s_iwasawa_decomposition_group_kan | theorem | Iwasawa decomposition G = KAN ;; KAN decomposition
s_uniqueness_in_iwasawa_decomposition | theorem | Uniqueness in the Iwasawa decomposition
s_vector_group_A | state | A = exp(a) vector group
s_nilpotent_group_N | state | N = exp(n) nilpotent group
s_weyl_group_of_restricted_roots | state | Weyl group of the restricted root system W(Sigma) ;; relative Weyl group ;; small Weyl group
s_weyl_group_as_NK_over_ZK | theorem | W(Sigma) = N_K(a)/Z_K(a)
s_kak_decomposition | theorem | KAK (Cartan/polar) decomposition ;; Cartan decomposition of the group ;; polar decomposition G = KAK
s_closed_positive_weyl_chamber_a_plus | state | Closed positive Weyl chamber a+ and A+
s_minimal_parabolic_subgroup | state | Minimal parabolic subgroup P = MAN ;; minimal parabolic P = MAN
s_standard_parabolic_subgroups_real | theorem | Standard parabolic subgroups from subsets of simple restricted roots ;; standard parabolic subgroups
s_langlands_decomposition | theorem | Langlands decomposition of a parabolic ;; Levi decomposition of a real parabolic
s_iwasawa_projection_H | state | Iwasawa projection H(g) ;; Iwasawa a-projection ;; H(g) function
s_half_sum_of_positive_restricted_roots | state | Half-sum of positive restricted roots rho ;; restricted rho
s_integral_formula_kan | theorem | Integral formula for the KAN decomposition ;; Iwasawa integration formula
s_integral_formula_kak | theorem | Integral formula for the KAK decomposition ;; sinh Jacobian formula ;; Cartan integration formula
s_integral_formula_over_G_mod_K | theorem | Integral formula over the symmetric space G/K
s_non_unimodularity_of_AN | theorem | Non-unimodularity of AN ;; modular function of AN
s_unimodularity_of_reductive_groups | theorem | Unimodularity of semisimple and reductive groups ;; semisimple groups are unimodular
s_hermitian_symmetric_space | axiom | Hermitian symmetric space ;; Hermitian symmetric space of noncompact/compact type
s_criterion_for_hermitian_type | theorem | Criterion for Hermitian type
s_bounded_symmetric_domain | state | Bounded symmetric domain ;; Cartan domain ;; classical domain
s_harish_chandra_decomposition | theorem | Harish-Chandra decomposition g^C = p- + k^C + p+ ;; g^C = p- + k^C + p+
s_harish_chandra_embedding | theorem | Harish-Chandra embedding ;; Harish-Chandra realization
s_strongly_orthogonal_roots | state | Strongly orthogonal roots ;; strongly orthogonal noncompact roots
s_kostant_linear_convexity_theorem | theorem | Kostant's linear convexity theorem ;; Kostant convexity theorem ;; linear convexity theorem
s_opposite_nilpotent_subalgebra | state | Opposite nilpotent subalgebra n-bar = theta(n) ;; opposite nilradical ;; n-bar
s_open_bruhat_cell | theorem | Open Bruhat cell (big cell) ;; big cell ;; open dense Bruhat cell
s_invariant_metric_on_G_mod_K_from_B_theta | state | G-invariant metric on G/K from B_theta
s_curvature_of_symmetric_space_bracket_formula | theorem | Curvature formula R(X,Y)Z = -[[X,Y],Z] ;; symmetric space curvature formula
s_siegel_domain_realization | state | Siegel domain (tube domain) realization ;; tube domain ;; Siegel domain of type I/II
t_cayley_transform_lie | technique | Cayley transform (Lie-theoretic) ;; Cayley transform of a Cartan subalgebra
t_harish_chandra_projection_rho_shift | technique | Harish-Chandra projection (rho-shift)
s_analytic_manifold | axiom | Analytic manifold
s_lie_third_theorem_existence | theorem | Lie's third theorem (existence)
s_ad_exp_naturality | theorem | Naturality of exp: Ad(exp X) = exp(ad X)
s_canonical_coordinates | state | Canonical coordinates of the first and second kind
t_taylor_expansion_product_map | technique | Taylor expansion of the product map
s_symmetrization_map_pbw | state | Symmetrization map S(g) -> U(g)
s_infinitesimal_character | state | Infinitesimal (central) character
t_bost_slope_formalism | technique | Bost slope formalism for Euclidean lattices
s_reverse_minkowski_transference | theorem | Reverse-Minkowski / transference inequalities for lattices
s_fekete_points_equidistribution | theorem | Equidistribution of Fekete points via pluripotential theory
s_coulomb_gas_pluripotential | state | Coulomb gas / determinantal pluripotential point process
s_eigenvector_normality_random_regular | theorem | Asymptotic normality of eigenvectors of random regular graphs
t_factor_of_iid_regular_tree | technique | Typical processes on the d-regular tree (factor-of-i.i.d. method)
s_coarse_embeddability_nonexact_groups | theorem | Coarse embeddability of non-exact groups into Hilbert space
s_property_a_coarse_amenability | axiom | Property A (coarse amenability)
s_crystalline_companions_theorem | theorem | Crystalline companions theorem
s_abe_padic_langlands_function_fields | theorem | Abe's p-adic Langlands correspondence for function fields
s_furstenberg_boundary_cstar_simplicity | theorem | Furstenberg-boundary criterion for C*-simplicity
s_cstar_simplicity | axiom | C*-simplicity
s_bab_boundedness_fano | theorem | Boundedness of singular Fano varieties (BAB conjecture)
t_theory_of_complements | technique | Theory of complements (boundedness of complements)
s_ksba_moduli_stable_varieties | theorem | Moduli of stable (KSBA) varieties of general type
s_higher_teichmuller_theory | state | Higher Teichmüller theory
s_anosov_representation | axiom | Anosov representation
s_homfly_from_hilbert_schemes | theorem | HOMFLY polynomial from Hilbert schemes of a plane curve singularity
s_oblomkov_shende_conjecture | state | Oblomkov–Shende / Maulik conjecture
s_max_zeta_short_intervals | theorem | Maximum of the Riemann zeta function in short intervals
t_log_correlated_field_zeta | technique | Log-correlated field / branching random walk heuristic for zeta
s_sharpness_phase_transition | theorem | Sharpness of the phase transition (Duminil-Copin–Raoufi–Tassion)
t_osss_sharp_thresholds | technique | OSSS inequality method for sharp thresholds
s_disproof_triangulation_conjecture | theorem | Disproof of the Triangulation Conjecture
t_pin2_sw_floer_homology | technique | Pin(2)-equivariant Seiberg–Witten Floer homology (Manolescu invariants)
s_homological_stability_hurwitz | theorem | Homological stability for Hurwitz spaces (Cohen–Lenstra over function fields)
t_homological_stability_scanning | technique | Homological stability method (scanning / group completion)
s_infinitely_many_minimal_hypersurfaces | theorem | Infinitely many minimal hypersurfaces (Yau's conjecture, dim 3–7)
t_almgren_pitts_min_max | technique | Almgren–Pitts min-max theory / volume spectrum Weyl law
s_herman_positive_metric_entropy | state | Herman's positive metric entropy conjecture
s_berger_turaev_theorem | theorem | Berger–Turaev theorem (positive metric entropy)
s_stability_lamb_oseen_vortex | theorem | Asymptotic stability of the Lamb–Oseen vortex
t_pseudospectral_enhanced_dissipation | technique | Pseudospectral / enhanced-dissipation estimates
s_c_infinity_closing_lemma | theorem | C^∞ closing lemma (Asaoka–Irie)
t_ech_spectral_invariants_weyl | technique | Embedded contact homology spectral invariants (Weyl law)
s_resolvent_estimates_non_self_adjoint | theorem | Resolvent estimates for non-self-adjoint semiclassical operators
s_definability_of_period_maps | theorem | Definability of period maps
t_definable_chow_ominimal_gaga | technique | Definable Chow / o-minimal GAGA (Peterzil–Starchenko)
s_algebraicity_of_hodge_loci | theorem | Algebraicity of Hodge loci (via o-minimality)
s_maximal_isotopy_transverse_foliation | state | Maximal isotopy & transverse singular foliation
t_le_calvez_tal_forcing | technique | Le Calvez–Tal forcing theory
s_nodal_set_volume_estimates | theorem | Nodal-set volume estimates (Yau's nodal conjecture)
t_doubling_index_frequency_function | technique | Doubling index / frequency-function method
s_ratner_infinite_volume | theorem | Ratner-type rigidity in infinite-volume hyperbolic manifolds
s_quantitative_inverse_gowers | theorem | Quantitative inverse theorem for the Gowers norms
t_higher_order_fourier_analysis | technique | Higher-order Fourier analysis (inverse-theorem method)
s_reconstruction_from_zariski_topology | theorem | Reconstruction of a variety from its Zariski topology
s_zagier_polylogarithm_conjecture | state | Zagier's polylogarithm conjecture
s_goncharov_rudenko_zeta_f4 | theorem | Goncharov–Rudenko theorem on ζ_F(4)
s_goncharov_motivic_polylog_complexes | state | Goncharov motivic polylogarithm complexes
t_motivic_correlators_cluster_polylog | technique | Motivic correlators / cluster polylogarithm method
s_fractal_uncertainty_principle | state | Fractal uncertainty principle
s_essential_spectral_gap_fractal | theorem | Essential spectral gap via fractal uncertainty
s_finiteness_totally_geodesic_subvarieties | theorem | Finiteness of totally geodesic subvarieties in moduli of curves
s_asymptotic_counting_minimal_surfaces | theorem | Asymptotic counting of minimal surfaces in hyperbolic 3-manifolds
t_equidistribution_minimal_surfaces_frame_flow | technique | Equidistribution of minimal surfaces (frame-flow mixing)
s_roelcke_precompact_polish_group | axiom | Roelcke precompact Polish group
s_property_t_roelcke_precompact | theorem | Property (T) for Roelcke-precompact Polish groups
s_classification_nip_fields_johnson | theorem | Classification of NIP fields (Johnson's theorem)
s_nip_dependent_theory | axiom | NIP (dependent theory)
t_dp_rank_valued_fields | technique | dp-rank / dp-finiteness analysis of valued fields
s_high_dimensional_expanders | state | High-dimensional expanders
t_local_spectral_coboundary_expansion | technique | Local spectral expansion / coboundary expansion
s_metric_john_theorem | theorem | Metric John theorem (Naor)
s_nonlinear_spectral_gap | state | Nonlinear spectral gap
s_local_marked_length_spectrum_rigidity | theorem | Local marked length spectrum rigidity
t_microlocal_geodesic_xray_transform | technique | Microlocal geodesic X-ray transform method
s_finite_time_implosion | theorem | Finite-time implosion for compressible fluids and NLS
t_self_similar_imploding_profile | technique | Self-similar imploding profile construction
s_kls_conjecture | state | Kannan–Lovász–Simonovits (KLS) conjecture
t_stochastic_localization_eldan | technique | Stochastic localization (Eldan)
s_binary_additive_problems_function_fields | theorem | Binary additive problems for polynomials over finite fields
t_sheaf_theoretic_counting_function_fields | technique | Sheaf-theoretic counting over function fields
s_crystalline_measure | state | Crystalline measure
s_kpi1_affine_artin_groups | theorem | K(π,1) conjecture for affine Artin groups
t_dual_braid_garside_method | technique | Dual braid monoid / Garside structure method
s_improved_bounds_three_term_aps | theorem | Improved bounds for sets without three-term arithmetic progressions
t_slice_rank_polynomial_method | technique | Slice rank / polynomial method (cap sets)
s_unbounded_denominators_conjecture | theorem | Unbounded denominators conjecture
t_arithmetic_holonomicity_bound | technique | Arithmetic holonomicity bound
s_fluctuation_theory_boltzmann_grad | theorem | Fluctuation theory for the Boltzmann–Grad limit
t_bbgky_cumulant_expansion | technique | BBGKY hierarchy / cumulant expansion for fluctuations
s_pointwise_ergodic_polynomial_sequences | theorem | Pointwise ergodic theorems along polynomial sequences
s_direct_summand_conjecture | theorem | Direct summand conjecture
t_perfectoid_almost_mathematics | technique | Perfectoid / almost-mathematics method
s_strong_convergence_random_permutation_matrices | theorem | Strong convergence of spectra of random permutation matrices
t_strong_freeness_non_backtracking | technique | Strong asymptotic freeness / non-backtracking operator method
s_mip_star_equals_re | theorem | MIP* = RE
s_refutation_connes_embedding | state | Refutation of the Connes embedding conjecture
s_non_simplicity_area_preserving_homeo_s2 | theorem | Non-simplicity of the group of area-preserving homeomorphisms of S²
s_periodic_floer_homology_spectral_invariants | state | Periodic Floer homology spectral invariants
s_pointwise_convergence_schrodinger_carleson | theorem | Pointwise convergence for the Schrödinger equation (Carleson's problem)
t_polynomial_partitioning_schrodinger_maximal | technique | Polynomial partitioning / decoupling for the Schrödinger maximal operator
s_well_ordering_growth_rates_hyperbolic | theorem | Well-ordering of exponential growth rates of hyperbolic groups
s_aspero_schindler_mm_implies_star | theorem | Asperó–Schindler theorem (MM⁺⁺ implies (∗))
t_pmax_stationary_tower_forcing | technique | Pmax / stationary-tower forcing
s_classification_symmetric_tensor_categories_char_p | theorem | Classification of symmetric tensor categories in positive characteristic
s_verlinde_category_ver_p | state | Verlinde category Ver_p
s_non_uniqueness_leray_hopf_forced_ns | theorem | Non-uniqueness of Leray–Hopf solutions to forced Navier–Stokes
t_unstable_self_similar_profile_method | technique | Unstable self-similar profile / spectral instability method
s_twisted_lang_weil_estimate | theorem | Twisted Lang–Weil estimate
s_rotational_invariance_critical_percolation | theorem | Rotational invariance of critical planar percolation
t_rsw_star_triangle_method | technique | RSW / star-triangle (Yang–Baxter) transformation method
t_quadratic_chabauty | technique | Quadratic Chabauty
t_kim_nonabelian_chabauty | technique | Kim's Nonabelian Chabauty Method
s_coleman_effective_chabauty_bound | theorem | Coleman's Effective Chabauty Bound
t_lawrence_venkatesh_method | technique | Lawrence-Venkatesh Method
t_skolem_p_adic_method | technique | Skolem's Method (p-adic, integral points)
s_croot_lev_pach_lemma | theorem | Croot-Lev-Pach lemma
s_ellenberg_gijswijt_cap_set | theorem | Ellenberg-Gijswijt cap set theorem
s_slice_rank_of_a_tensor | axiom | Slice rank of a tensor
s_sunflower_free_tricolored_slice_rank | theorem | Sunflower-free / tri-colored sum-free bounds via slice rank
s_analytic_l_function_axiomatic | axiom | Analytic L-function (axiomatic)
s_stably_rational_variety | axiom | Stably Rational Variety
s_luroth_problem | state | Lüroth Problem
t_decomposition_of_the_diagonal | technique | Decomposition of the Diagonal Method
s_non_stable_rationality_hypersurfaces | theorem | Non-Stable-Rationality of Very General Hypersurfaces
s_big_polynomial_ring_uniformity | state | Big Polynomial Ring / Uniformity (Erman-Sam-Snowden)
s_stillmans_conjecture | theorem | Stillman's Conjecture (Theorem)
s_strength_of_a_polynomial | axiom | Strength of a Polynomial
s_tangent_developable_surface | axiom | Tangent Developable Surface
s_greens_conjecture | theorem | Green's Conjecture
s_koszul_cohomology_property_np | axiom | Koszul Cohomology / Property N_p
t_degeneration_to_tangent_developable | technique | Degeneration to Tangent Developable Surfaces
s_compactified_jacobian_universal_picard | state | Compactified Jacobian / Universal Picard Variety
t_tropical_dual_graph_boundary | technique | Tropical / Dual-Graph Combinatorics of the Boundary
s_space_of_diagonal_harmonics | axiom | Space of Diagonal Harmonics
s_nabla_operator | axiom | Nabla Operator
s_parking_function | axiom | Parking Function (combinatorics)
s_shuffle_conjecture | theorem | Shuffle Conjecture (Theorem)
s_compositional_shuffle_conjecture | theorem | Compositional Shuffle Conjecture (Theorem)
s_overconvergent_modular_form | axiom | Overconvergent Modular Form
s_up_operator_slope_decomposition | axiom | U_p Operator and Slope Decomposition
s_coleman_classicality_theorem | theorem | Coleman Classicality Theorem
s_rigid_meromorphic_cocycles | state | Rigid Meromorphic Cocycles / p-adic Singular Moduli
s_classical_siegel_weil_formula | theorem | Classical Siegel-Weil Formula
s_special_cycles_shimura | axiom | Special Cycles on Shimura Varieties
s_kudla_modularity_conjecture | theorem | Kudla's Modularity Conjecture (Theorem)
s_geometric_siegel_weil_formula | theorem | Geometric Siegel-Weil Formula
s_arithmetic_siegel_weil_formula | theorem | Arithmetic Siegel-Weil Formula
s_kudla_rapoport_conjecture | theorem | Kudla-Rapoport Conjecture (Theorem)
s_rapoport_zink_space | axiom | Rapoport-Zink Space
s_arithmetic_inner_product_formula | theorem | Arithmetic Inner Product Formula
s_beilinson_bloch_conjecture | state | Beilinson-Bloch Conjecture
t_macaulay_inverse_system | technique | Macaulay Inverse System
s_embedding_q_finitely_presented | theorem | Embedding Q into a Finitely Presented Group (Theorem)
s_twisted_brin_thompson_group | axiom | Twisted Brin-Thompson Group
s_space_time_white_noise | axiom | Space-time white noise
s_dynamical_phi4_equation | axiom | Dynamical Phi^4 equation
s_parabolic_anderson_model | axiom | Parabolic Anderson Model
t_renormalization_counterterm_spde | technique | Renormalization (counter-term subtraction) for singular SPDEs
t_da_prato_debussche_trick | technique | Da Prato-Debussche trick
s_primary_uncertainty_principle | theorem | Primary uncertainty principle
s_k_hadamard_matrix | axiom | k-Hadamard matrix
s_donoho_stark_uncertainty | theorem | Donoho-Stark support-size uncertainty principle
s_meshulam_nonabelian_uncertainty | theorem | Meshulam's non-abelian uncertainty principle
s_hardy_uncertainty_principle | theorem | Hardy's uncertainty principle
s_dynamical_hardy_reformulation | state | Dynamical (Schrödinger) reformulation of Hardy's uncertainty principle
t_liouville_proof_hardy_uncertainty | technique | Liouville-theorem proof of Hardy's uncertainty principle
s_sine_kernel_fredholm_determinant | axiom | Sine-kernel Fredholm determinant
s_widom_large_gap_asymptotics | theorem | Widom's large-gap asymptotics for the sine-kernel determinant
s_tracy_widom_painleve_representation | theorem | Tracy-Widom Painlevé representation
s_integrable_fredholm_operator | axiom | Integrable Fredholm operator
t_nonlinear_steepest_descent_rh | technique | Riemann-Hilbert / nonlinear steepest descent method
s_arnold_geodesic_hydrodynamics | axiom | Arnold's geodesic formulation of ideal hydrodynamics
s_newton_equations_diffeo_density | state | Newton's equations on diffeomorphism groups and density spaces
s_fisher_rao_metric_infinite | axiom | Fisher-Rao information metric (infinite-dimensional)
s_madelung_transform_symplectomorphism | theorem | Madelung transform as a symplectomorphism
t_symplectic_poisson_reduction_hydro | technique | Symplectic and Poisson reduction for hydrodynamics
t_frequency_envelopes | technique | Frequency envelopes
t_rough_solutions_as_limits | technique | Rough solutions as limits of smooth solutions
t_quasilinear_energy_estimates | technique | Quasilinear energy estimates
t_convex_integration_hydrodynamics | technique | Convex integration in hydrodynamics
s_onsagers_conjecture | state | Onsager's conjecture
s_flexible_side_onsager | theorem | Resolution of the flexible side of Onsager's conjecture
s_nonuniqueness_weak_navier_stokes | theorem | Nonuniqueness of weak solutions of Navier-Stokes
s_intermittency_convex_integration | state | Intermittency in convex integration
s_polya_fourier_transform_gaussian | axiom | Polya-Fourier transform with Gaussian factor
s_de_bruijn_newman_constant | state | de Bruijn-Newman constant
s_polya_debruijn_newman_bounds | theorem | Polya monotonicity / De Bruijn upper bound / Newman lower bound
s_rodgers_tao_theorem | theorem | Rodgers-Tao theorem (Λ_DN ≥ 0)
s_newman_wu_weak_convergence | theorem | Newman-Wu weak-convergence theorem for real-zero distributions
s_euler_alignment_equations | axiom | Euler alignment (hydrodynamic alignment) equations
s_p_alignment_hydrodynamics | axiom | p-alignment hydrodynamics
s_entropic_pressure_tensor | state | Entropic pressure tensor
s_flocking_p_alignment_entropic | theorem | Flocking of p-alignment with entropic pressure
t_energy_fluctuation_decay_flocking | technique | Energy-fluctuation decay method for flocking
s_non_euclidean_elasticity | axiom | Non-Euclidean (incompatible) elasticity
s_metric_frustration_biological_shape | state | Metric frustration as the origin of biological shape
s_gamma_convergence_plate_shell_hierarchy | theorem | Gamma-convergence hierarchy of thin elastic plate/shell theories
t_convex_integration_prestrained_immersions | technique | Convex-integration construction of low-regularity isometric immersions of prestrained films
s_monge_ampere_constrained_energy | state | Monge-Ampère constrained energy for weak prestrains
s_boolean_gaussian_noise_stability | axiom | Boolean / Gaussian noise stability
s_majority_is_stablest | theorem | Majority is Stablest theorem
s_gaussian_quantitative_arrow | theorem | Quantitative Arrow theorem
s_quantitative_gibbard_satterthwaite | theorem | Quantitative Gibbard-Satterthwaite theorem
t_invariance_principle_low_influence | technique | Invariance principle for low-influence functions
s_boolean_social_choice_function | axiom | Boolean social choice function
s_dolds_theorem_equivariant | theorem | Dold's theorem (equivariant map nonexistence)
t_test_map_configuration_space | technique | Test map / configuration space scheme
s_mass_partition_problem | axiom | Mass partition problem
s_general_position_point_set | axiom | General position (point set)
s_necklace_splitting_theorem | theorem | Necklace splitting theorem (Hobby-Rice / Alon)
t_moment_curve_lifting | technique | Moment curve lifting (necklace splitting proof)
s_grunbaum_hadwiger_ramos_problem | state | Grünbaum-Hadwiger-Ramos problem
s_nandakumar_ramana_rao_problem | state | Nandakumar-Ramana-Rao problem
s_convex_equipartition_theorem | theorem | Convex equipartition theorem (spicy chicken theorem)
s_holmsen_kyncl_valculescu_conjecture | state | Holmsen-Kynčl-Valculescu conjecture
s_center_transversal_theorem | theorem | Center Transversal theorem
s_yao_yao_partition_theorem | theorem | Yao-Yao partition theorem
s_same_type_lemma | theorem | Same-type lemma
s_ppa_completeness_ham_sandwich | theorem | PPA-completeness of discrete Ham Sandwich
t_categorification | technique | Categorification
s_reduced_khovanov_homology | state | Reduced Khovanov homology
s_functoriality_khovanov_cobordism | theorem | Functoriality of Khovanov homology under link cobordisms
s_lee_spectral_sequence_homology | theorem | Lee spectral sequence and Lee homology
s_rasmussen_s_invariant | state | Rasmussen s-invariant
s_rasmussen_slice_genus_bound | theorem | Rasmussen's slice-genus bound
s_combinatorial_milnor_conjecture | theorem | Combinatorial proof of the Milnor conjecture (torus knots)
s_khovanov_stable_homotopy_type | state | Khovanov stable homotopy type (Lipshitz-Sarkar spectrum)
s_odd_khovanov_homology | state | Odd Khovanov homology
s_kronheimer_mrowka_unknot_detection | theorem | Kronheimer-Mrowka unknot detection
s_symplectic_khovanov_homology | state | Symplectic Khovanov homology
t_spectral_sequences_to_khovanov | technique | Spectral sequences from gauge/Floer theories to Khovanov homology
s_teichmuller_curve | axiom | Teichmüller curve
s_veech_group_lattice_surface | axiom | Veech group / lattice surface SL(X, ω)
s_primitive_teichmuller_curve | axiom | Primitive Teichmüller curve
s_optimal_dynamics_lattice_billiards | theorem | Optimal dynamics of billiards in lattice polygons
s_regular_polygons_primitive_teichmuller | theorem | Regular polygons generate primitive Teichmüller curves
s_masurs_criterion | theorem | Masur's criterion
t_cylinder_decomposition_parabolic | technique | Cylinder decomposition and parabolic affine automorphisms
s_weierstrass_curve_wd | state | Weierstrass curve W_D (genus 2 classification)
s_classification_primitive_teichmuller_genus2 | theorem | Classification of primitive Teichmüller curves in genus 2
s_finiteness_primitive_teichmuller_genus3 | theorem | Finiteness of primitive Teichmüller curves in genus 3 outside W_D
s_infinitude_primitive_teichmuller_open | state | Infinitude of primitive Teichmüller curves in higher genus (open problem)
s_slice_knot | axiom | Slice knot
s_slice_genus | axiom | Slice genus
s_knot_mutation | axiom | Knot mutation
s_knot_trace | axiom | Knot trace X(K)
s_conway_knot_not_slice | theorem | The Conway knot is not slice
s_trace_embedding_lemma | theorem | Trace embedding lemma
t_handle_rearrangement_trace_equivalent | technique | Handle-rearrangement construction of trace-equivalent knots
s_trace_invariance_nu | theorem | Trace invariance of nu (knot Floer slice obstruction)
s_vanishing_classical_slice_obstructions_conway | state | Vanishing of classical slice obstructions for the Conway knot
s_topological_helly_theorem | theorem | Topological Helly theorem
s_colorful_caratheodory_theorem | theorem | Colorful Caratheodory theorem
s_upper_bound_theorem_convex_sets | theorem | Upper bound theorem for convex sets
s_topological_tverberg_conjecture | state | Topological Tverberg conjecture
s_sierksma_conjecture | state | Sierksma's conjecture
s_cascade_conjecture | state | Cascade conjecture
t_nerve_complex_method | technique | Nerve complex method
s_impagliazzo_wigderson_theorem | theorem | Impagliazzo-Wigderson theorem (hardness vs randomness)
t_hardness_amplification | technique | Hardness amplification (Yao XOR lemma / direct product)
s_zigzag_product_of_graphs | state | Zig-zag product of graphs
s_sl_equals_l_theorem | theorem | SL = L theorem (Reingold)
t_operator_scaling | technique | Operator scaling / noncommutative optimization
s_efficiency_gap | state | Efficiency gap
s_partisan_symmetry | axiom | Partisan symmetry
t_ensemble_outlier_analysis_districting | technique | Ensemble (outlier) analysis of districting plans
s_recombination_recom_markov_chain | state | Recombination (ReCom) Markov chain
s_chikina_frieze_pegden_local_outlier_test | theorem | Chikina-Frieze-Pegden local outlier test
s_nphardness_balanced_connected_partition | theorem | NP-hardness of balanced connected graph partition
s_sunflower_conjecture_open | state | Sunflower conjecture
s_spread_robust_sunflower_family | axiom | Spread (robust sunflower) family
s_kahn_kalai_expectation_threshold | theorem | Kahn-Kalai expectation-threshold conjecture (Park-Pham theorem)
t_random_restriction_with_spreadness | technique | Random restriction with spreadness
s_chow_ring_of_a_matroid | axiom | Chow ring of a matroid
s_kahler_package_for_matroids | theorem | Kahler package for matroids
s_heron_rota_welsh_conjecture | theorem | Heron-Rota-Welsh conjecture (theorem)
s_dowling_wilson_top_heavy | theorem | Dowling-Wilson top-heavy theorem
s_mason_log_concavity_conjecture | theorem | Mason's log-concavity conjecture (theorem)
t_hodge_riemann_relations_combinatorial | technique | Hodge-Riemann relations (combinatorial use)
s_classical_verification_quantum_computation | axiom | Classical verification of quantum computation
s_trapdoor_claw_free_function_family | axiom | Trapdoor claw-free function family
s_mahadev_classical_verification_theorem | theorem | Mahadev's classical verification theorem
t_cryptographic_measurement_protocol | technique | Cryptographic measurement protocol (qubit commitment)
s_quintic_calabi_yau_threefold | axiom | Quintic Calabi-Yau threefold ;; quintic threefold
s_virtual_fundamental_class | axiom | Virtual fundamental class ;; virtual class
s_moduli_space_stable_maps | axiom | Moduli space of stable maps ;; Kontsevich moduli space ;; moduli of stable maps
s_bcov_holomorphic_anomaly_equation | axiom | BCOV holomorphic anomaly equation ;; holomorphic anomaly equation ;; BCOV equation
s_yamaguchi_yau_finiteness_conjecture | state | Yamaguchi-Yau finiteness conjecture ;; Yamaguchi-Yau polynomial structure conjecture
s_structure_theorem_higher_genus_gw_quintic | theorem | Structure theorem for higher-genus GW invariants of the quintic
t_mixed_spin_p_field | technique | Mixed-Spin-P (MSP) field ;; MSP field
t_virtual_localization | technique | Virtual (Graber-Pandharipande) localization ;; Graber-Pandharipande localization ;; virtual localization formula
s_yau_conjecture_infinitely_many_minimal_hypersurfaces | state | Yau conjecture on infinitely many minimal hypersurfaces
s_marques_neves_infinitely_many_minimal_hypersurfaces | theorem | Marques-Neves theorem (infinitely many minimal hypersurfaces)
s_weyl_law_volume_spectrum | theorem | Weyl law for the volume spectrum (Liokumovich-Marques-Neves) ;; Liokumovich-Marques-Neves Weyl law
s_volume_spectrum_p_widths | state | Volume spectrum (p-widths) ;; p-widths ;; min-max widths
s_equidistribution_minimal_hypersurfaces_mns | theorem | Equidistribution of minimal hypersurfaces (Marques-Neves-Song)
s_generic_density_minimal_hypersurfaces_imn | theorem | Generic density of minimal hypersurfaces (Irie-Marques-Neves)
s_probabilistically_checkable_proof | axiom | Probabilistically checkable proof (PCP) ;; PCP
s_unique_games_conjecture | state | Unique Games Conjecture ;; UGC
s_two_to_two_games_theorem | theorem | 2-to-2 Games Theorem ;; 2-to-2 games conjecture (theorem) ;; two-to-two games theorem
s_grassmann_graph | axiom | Grassmann graph
s_grassmann_graph_expansion_theorem | theorem | Grassmann graph expansion theorem (characterization of non-expanding sets)
t_dictatorship_test | technique | Dictatorship test
s_arithmetic_spectral_transition_amo | theorem | Arithmetic spectral (localization) transition for the almost Mathieu operator ;; arithmetic phase transition ;; Jitomirskaya arithmetic localization transition
s_universality_critical_almost_mathieu | theorem | Universality of the critical almost Mathieu operator ;; critical almost Mathieu universality ;; hidden singularity
s_parity_sheaf | axiom | Parity sheaf
s_soergel_bimodule | axiom | Soergel bimodule
s_lusztig_modular_representation_conjecture | state | Lusztig modular representation conjecture ;; Lusztig character formula conjecture
s_williamson_counterexamples_modular_bounds | theorem | Williamson counterexamples to expected bounds in modular representation theory ;; Williamson torsion examples
s_p_canonical_basis | state | p-canonical basis
s_symmetry_protected_topological_phase | axiom | Symmetry protected topological (SPT) phase ;; SPT phase
s_gapped_ground_state_phase_equivalence | axiom | Gapped ground state phase equivalence
s_matrix_product_state | axiom | Matrix product state ;; MPS
s_group_cohomology_invariant_spt | state | Group cohomology invariant of SPT phases
s_ogata_classification_spt_phases | theorem | Ogata classification of SPT phases by H^2(G,U(1))
s_split_property_spin_chains | state | Split property of quantum spin chains
s_black_hole_stability_problem | state | Black hole stability problem ;; Kerr stability conjecture
s_kerr_de_sitter_spacetime | axiom | Kerr (Kerr-de Sitter) spacetime ;; Kerr-de Sitter black hole ;; Kerr metric
s_hintz_vasy_kerr_de_sitter_stability | theorem | Hintz-Vasy global stability of Kerr-de Sitter
t_microlocal_stationary_wave_operators | technique | Microlocal analysis of stationary wave operators (b-calculus / scattering calculus)
s_quasinormal_modes_black_holes | state | Quasinormal modes / resonances of black holes ;; ringdown frequencies ;; black hole resonances
t_redshift_effect_dafermos_rodnianski | technique | Red-shift effect (Dafermos-Rodnianski)
s_painleve_conjecture_noncollision_singularities | state | Painleve conjecture (noncollision singularities) ;; noncollision singularity conjecture
s_von_zeipel_theorem_singularities | theorem | Von Zeipel theorem on singularities
s_xia_noncollision_singularities_5body | theorem | Xia's example of noncollision singularities (5-body)
s_xue_painleve_4body | theorem | Xue's solution of the Painleve conjecture for the 4-body problem
t_near_collision_hyperbolic_dynamics | technique | Near-collision (regularization) analysis via hyperbolic dynamics
s_phi4_4_euclidean_field_theory | axiom | Phi^4_4 Euclidean field theory ;; phi^4 in four dimensions ;; Phi^4_4 model
t_random_current_representation | technique | Random current representation
s_triviality_scaling_limit_d_ge_4 | theorem | Triviality of the scaling limit in dimension d >= 4
s_aizenman_duminil_copin_marginal_triviality_4d | theorem | Aizenman-Duminil-Copin marginal triviality theorem in 4D
s_aizenman_frohlich_tree_graph_inequality | theorem | Aizenman-Frohlich (tree-graph) inequality ;; tree-graph inequality
s_reductive_p_adic_group | axiom | Reductive p-adic group ;; p-adic reductive group
s_smooth_representation_p_adic_group | axiom | Smooth representation of a p-adic group
s_admissible_representation_p_adic | axiom | Admissible representation (p-adic group)
s_supercuspidal_representation | axiom | Supercuspidal representation ;; supercuspidal ;; cuspidal representation of a p-adic group
t_parabolic_induction_p_adic | technique | Parabolic induction (p-adic group)
s_parahoric_subgroup_bruhat_tits_building | axiom | Parahoric subgroup and Bruhat-Tits building ;; Bruhat-Tits building ;; parahoric subgroup ;; Iwahori subgroup
s_moy_prasad_filtration | axiom | Moy-Prasad filtration
s_depth_of_a_representation | state | Depth of a representation
s_type_bushnell_kutzko | axiom | Type (Bushnell-Kutzko theory) ;; Bushnell-Kutzko type
s_tame_elliptic_regular_pair | axiom | Tame elliptic regular pair / generic cuspidal datum ;; generic cuspidal datum ;; Yu datum
s_twisted_levi_sequence | axiom | Twisted Levi sequence
t_yu_construction_supercuspidals | technique | Yu's construction of supercuspidal representations
s_supercuspidality_yu_representations_fintzen | theorem | Supercuspidality of Yu-constructed representations (Fintzen)
s_fintzen_exhaustion_tame_supercuspidals | theorem | Exhaustion theorem for tame supercuspidals (Fintzen) ;; Fintzen exhaustion theorem
s_fintzen_kaletha_spice_quadratic_twist | state | Fintzen-Kaletha-Spice quadratic twist
s_adler_spice_character_formula | theorem | Adler-Spice / DeBacker-Spice character formula
s_depth_zero_supercuspidal_representation | state | Depth-zero supercuspidal representation
s_local_langlands_p_adic_groups | state | Local Langlands correspondence for p-adic groups ;; local Langlands conjecture for general reductive p-adic groups
s_mixed_quantum_state_density_matrix | axiom | Mixed quantum state (density matrix) ;; density matrix ;; density operator ;; mixed state
s_quantum_state_tomography_problem | state | Quantum state tomography problem ;; quantum state tomography
s_quantum_spectrum_estimation_problem | state | Quantum spectrum estimation / testing problem
t_weak_schur_sampling | technique | Weak Schur sampling ;; Keyl-Werner spectrum estimation ;; Schur sampling
s_schur_weyl_distribution | state | Schur-Weyl distribution
s_rsk_realization_schur_weyl_distribution | theorem | RSK realization of the Schur-Weyl distribution ;; RSK-Schur-Weyl connection
t_empirical_young_diagram_algorithm | technique | Empirical Young Diagram (EYD) algorithm ;; EYD algorithm
s_optimal_sample_complexity_state_tomography | theorem | Optimal sample complexity of state tomography ;; O'Donnell-Wright / Haah et al. tomography bound
s_pure_state_tomography_sample_complexity | theorem | Pure-state tomography sample complexity
s_max_schur_weyl_row_top_eigenvalue | theorem | Convergence of the maximal Schur-Weyl row to the top eigenvalue
s_quantum_identity_testing_bounds | theorem | Quantum identity / maximal-mixedness testing bounds ;; quantum state certification
s_p_adic_galois_representation | axiom | p-adic Galois representation
s_de_rham_potentially_semistable_representation | axiom | de Rham / potentially semistable representation ;; de Rham Galois representation ;; potentially semistable representation ;; Hodge-Tate representation
s_algebraic_galois_representation | axiom | Algebraic Galois representation
s_fontaine_mazur_conjecture | state | Fontaine-Mazur conjecture
s_regular_algebraic_cuspidal_automorphic_representation | axiom | Regular algebraic cuspidal automorphic representation ;; RACAR ;; regular algebraic cuspidal form
s_symmetric_power_lifting_sym_m | state | Symmetric power lifting Sym^m ;; symmetric power functoriality lift ;; Sym^m lift
s_thorne_reciprocity_conjecture | state | Reciprocity conjecture (Langlands global reciprocity, Thorne) ;; global Langlands reciprocity conjecture
s_sato_tate_conjecture | theorem | Sato-Tate conjecture ;; Sato-Tate equidistribution
s_newton_thorne_symmetric_power_functoriality | theorem | Newton-Thorne symmetric power functoriality theorem ;; symmetric power functoriality for holomorphic newforms
s_automorphy_lifting_theorem | theorem | Automorphy lifting theorem ;; modularity lifting theorem
t_taylor_wiles_kisin_patching | technique | Taylor-Wiles-Kisin patching ;; Taylor-Wiles method ;; Taylor-Wiles-Kisin method
t_brauer_induction_l_functions | technique | Brauer induction reduction of L-functions
s_coleman_mazur_eigencurve | state | Coleman-Mazur eigencurve ;; eigencurve ;; eigenvariety
s_overconvergent_finite_slope_eigenform | axiom | Overconvergent finite-slope Hecke eigenform
t_analytic_continuation_functoriality_eigencurve | technique | Analytic continuation of functoriality along the eigencurve
s_pseudo_representation_pseudocharacter | axiom | Pseudo-representation (pseudocharacter) ;; pseudocharacter ;; determinant law
s_future_null_infinity | axiom | Future null infinity (scri-plus) ;; scri-plus ;; null infinity ;; I-plus
s_bondi_metzner_sachs_group | axiom | Bondi-Metzner-Sachs (BMS) group ;; BMS group
s_supertranslation | axiom | Supertranslation
s_bondi_sachs_energy_momentum | state | Bondi-Sachs energy-momentum ;; Bondi energy-momentum ;; Bondi four-momentum
s_news_tensor | state | News tensor
s_bondi_mass_loss_formula | theorem | Bondi mass loss formula
s_supertranslation_ambiguity_angular_momentum | state | Supertranslation ambiguity of angular momentum
s_wang_yau_quasilocal_mass | state | Wang-Yau quasi-local mass ;; Wang-Yau quasilocal mass ;; Wang-Yau quasi-local energy
s_optimal_isometric_embedding | state | Optimal isometric embedding
s_chen_wang_yau_angular_momentum | state | Chen-Wang-Yau angular momentum ;; Chen-Wang-Yau quasilocal angular momentum
s_angular_momentum_at_null_infinity | state | Angular momentum at null infinity
s_supertranslation_invariance_cwy_angular_momentum | theorem | Supertranslation invariance of Chen-Wang-Yau angular momentum
s_supertranslation_invariant_center_of_mass | state | Supertranslation-invariant center of mass at null infinity
s_basis_generating_polynomial_matroid | state | Basis generating polynomial of a matroid ;; bases generating polynomial
s_completely_log_concave_polynomial | axiom | Completely log-concave polynomial ;; completely log-concave ;; strongly log-concave polynomial
s_lorentzian_polynomial | axiom | Lorentzian polynomial ;; Lorentzian polynomials (Branden-Huh)
s_anari_liu_oveisgharan_vinzant_log_concavity | theorem | Anari-Liu-Oveis Gharan-Vinzant log-concavity theorem
s_matroid_expansion_conjecture_mihail_sudan | theorem | Mihail-Sudan / matroid expansion theorem ;; Mihail-Vazirani matroid expansion conjecture
s_local_spectral_expander | axiom | Local spectral expander ;; high-dimensional expander ;; local-spectral expander
t_spectral_independence | technique | Spectral independence
s_mihail_vazirani_matroid_sampling | theorem | Mihail-Vazirani matroid sampling consequence
s_random_lozenge_tiling | axiom | Random lozenge tiling ;; random rhombus tiling
s_dimer_model | axiom | Dimer model ;; perfect matching model
s_limit_shape_variational_principle_tilings | theorem | Limit shape (variational principle for tilings) ;; Cohn-Kenyon-Propp variational principle
s_arctic_circle_frozen_boundary | state | Arctic circle / frozen boundary ;; arctic curve ;; frozen boundary
s_gaussian_free_field_fluctuations_tilings | theorem | Gaussian free field fluctuations of tilings
s_airy_line_ensemble_edge_universality | theorem | Airy line ensemble edge universality ;; Airy process edge universality
s_sine_kernel_bulk_universality_tilings | theorem | Sine kernel bulk universality for tilings
s_prismatic_cohomology | axiom | Prismatic cohomology ;; prismatic site cohomology
s_prism | axiom | Prism ;; bounded prism
s_delta_ring | axiom | Delta-ring ;; delta-ring ;; lambda-ring of p-typical type
s_prismatic_f_crystal | axiom | Prismatic F-crystal
s_bhatt_scholze_crystals_equals_lattices | theorem | Bhatt-Scholze crystals equals lattices theorem
s_crystalline_cohomology | axiom | Crystalline cohomology
s_p_adic_hodge_comparison | theorem | p-adic Hodge theory comparison ;; p-adic comparison isomorphisms
s_p_adic_motive | state | p-adic motive
s_circular_unitary_ensemble_characteristic_polynomial | axiom | Circular Unitary Ensemble / CUE characteristic polynomial ;; CUE ;; CUE characteristic polynomial
s_log_correlated_gaussian_field | axiom | Log-correlated Gaussian field ;; log-correlated field
s_branching_random_walk_maximum | theorem | Branching random walk maximum (BRW) ;; BRW maximum asymptotics
s_maximum_cue_characteristic_polynomial | theorem | Maximum of CUE characteristic polynomial
t_arithmetic_algebraization | technique | Arithmetic algebraization ;; arithmetic holonomy method
s_borel_dwork_rationality_criterion | theorem | Borel-Dwork rationality criterion ;; Borel-Dwork criterion
s_schur_siegel_smyth_trace_problem | theorem | Schur-Siegel-Smyth trace problem result ;; Schur-Siegel-Smyth trace problem
s_kronecker_coefficient | state | Kronecker coefficient
s_complexity_class_sharp_p | axiom | Complexity class #P ;; #P ;; sharp-P
s_combinatorial_interpretation_in_sharp_p | state | Combinatorial interpretation as membership in #P
t_geometric_complexity_theory | technique | Geometric Complexity Theory ;; GCT
s_plethysm_coefficient | state | Plethysm coefficient
s_gct_obstruction_vanishing_rectangular_kronecker | theorem | Vanishing of rectangular Kronecker coefficients / GCT obstruction
s_saturation_theorem_littlewood_richardson | theorem | Saturation theorem for Littlewood-Richardson coefficients ;; Knutson-Tao saturation theorem
s_positivity_kronecker_coefficients_np_hard | theorem | Positivity of Kronecker coefficients is NP-hard
s_clique_width | axiom | Clique-width
s_hereditary_graph_class | axiom | Hereditary graph class
s_clique_width_h_free_dichotomy | theorem | Boundedness of clique-width for H-free graphs
t_cut_distance_convergence | technique | Cut distance and left convergence
t_flag_algebra_method | technique | Flag algebra method
s_independence_complex_connectedness | axiom | Topological connectedness of the independence complex
s_aharoni_haxell_topological_hall | theorem | Aharoni–Haxell topological Hall theorem
s_independent_transversal | state | Independent transversal
s_long_paths_cycles_in_expanders | theorem | Long paths and cycles in expanders
s_supersingular_isogeny_graph | axiom | Supersingular isogeny graph
s_isogeny_graph_hardness_sidh | state | Isogeny-graph hardness / SIDH
s_delta_matroid | axiom | Delta-matroid
s_ribbon_graph_delta_matroid | state | Ribbon graph / embedded graph correspondence
s_ordered_ramsey_turan | axiom | Ordered Ramsey/Turán theory
s_furedi_hajnal_interval_chromatic | theorem | Füredi–Hajnal theorem and interval chromatic number
s_spherical_building | axiom | Spherical building
t_combinatorial_construction_buildings | technique | Combinatorial construction of buildings
s_stanley_reisner_ring | axiom | Stanley–Reisner ring
s_cohen_macaulay_shellability | theorem | Shellable complexes are Cohen–Macaulay
s_planar_exceptional_polynomials | state | Planar and exceptional polynomial functions
s_graph_edge_decomposition | axiom | Graph edge decomposition
s_nash_williams_arboricity | theorem | Nash-Williams arboricity theorem
s_linear_star_arboricity_conjectures | state | Linear and star arboricity conjectures
t_switching_method | technique | Switching method
s_graph_isomorphism_problem | state | Graph isomorphism problem
s_babai_quasipolynomial_isomorphism | theorem | Babai's quasipolynomial graph isomorphism algorithm
t_weisfeiler_leman_algorithm | technique | Weisfeiler–Leman algorithm and dimension
s_hypergraph_decomposition_extremal | axiom | Graph and hypergraph decomposition (extremal)
s_keevash_existence_of_designs | theorem | Existence of designs (Keevash; Glock–Kühn–Lo–Osthus)
t_absorption_method | technique | Absorption method
t_iterative_absorption | technique | Iterative absorption
s_borel_combinatorics_of_graphs | axiom | Borel combinatorics of graphs
s_borel_measurable_chromatic_number | state | Borel/measurable chromatic number
s_johnson_graph | axiom | Johnson graph
s_codes_designs_johnson_graphs | theorem | Codes and designs in Johnson graphs
s_maximal_subgroups_finite_simple_groups | theorem | Maximal subgroups of finite simple groups
s_onan_scott_theorem | theorem | O'Nan–Scott theorem
s_fair_partition_nrr | state | Fair partition / Nandakumar–Ramana-Rao problem
t_equivariant_topology_test_map | technique | Equivariant topology test-map method
s_hypergraph_turan_problem | axiom | Hypergraph Turán problem
s_hypergraph_turan_l2_norm | theorem | Hypergraph Turán in the ℓ₂-norm
s_circuit_imbalance_measure | axiom | Circuit imbalance measure
s_circuit_imbalance_lp_complexity | theorem | Circuit imbalance and LP complexity
t_junta_method | technique | Junta method for intersection problems
t_finite_geometry_extremal_constructions | technique | Finite geometry constructions for extremal graphs
s_erdos_renyi_polarity_graph | state | Erdős–Rényi C₄-free polarity graph
s_rainbow_subgraph_transversal | axiom | Rainbow subgraph / transversal
s_aharoni_berger_rainbow_matching | theorem | Aharoni–Berger rainbow matching results
s_explicit_bounds_graph_minors | theorem | Explicit bounds for graph minors
s_tropical_geometry_combinatorial | axiom | Tropical geometry (combinatorial)
t_matroid_chow_intersection_theory | technique | Matroid Chow ring intersection theory
s_erdos_covering_system | axiom | Erdős covering system
s_minimum_modulus_hough | theorem | Minimum modulus problem (Hough's theorem)
t_cluster_expansion | technique | Cluster expansion
s_polymer_model | axiom | Polymer model
s_kotecky_preiss_criterion | theorem | Kotecký–Preiss criterion
s_sublinear_expander | axiom | Sublinear expander
s_komlos_szemeredi_sublinear_expander | theorem | Komlós–Szemerédi sublinear expander theorem
t_sublinear_expander_embedding | technique | Sublinear-expander embedding method
s_latin_square_transversal | axiom | Transversal in a Latin square
s_ryser_brualdi_stein_conjecture | state | Ryser–Brualdi–Stein conjecture
s_montgomery_transversal_theorem | theorem | Montgomery's transversal theorem
s_transshipment_over_time | axiom | Transshipment / flows over time
s_quickest_transshipment_hoppe_tardos | theorem | Quickest transshipment theorem (Hoppe–Tardos)
s_oriented_trees_paths_digraphs | axiom | Oriented trees and paths in digraphs
s_sumner_conjecture | state | Sumner's conjecture
s_gallai_roy_directed_path | theorem | Gallai–Roy and Burr–Erdős directed-path results
s_sidorenko_conjecture | state | Sidorenko's conjecture
t_gcd_graph_method | technique | GCD graph method
s_duffin_schaeffer_theorem | theorem | Duffin–Schaeffer theorem (Koukoulopoulos–Maynard) ;; Koukoulopoulos-Maynard theorem
t_o_minimal_point_counting | technique | O-minimal point counting (Pila–Wilkie)
s_andre_oort_conjecture | theorem | André–Oort conjecture (Pila–Shankar–Tsimerman)
t_betti_map_height_inequality | technique | Betti-map height inequality
s_uniform_mordell_lang | theorem | Uniform Mordell–Lang for subvarieties of abelian varieties
s_schinzel_zassenhaus_theorem | theorem | Schinzel–Zassenhaus theorem (Dimitrov)
s_picard_rank_jumps_k3 | theorem | Exceptional jumps of Picard ranks of K3 reductions
s_caraiani_scholze_torsion_vanishing | theorem | Caraiani–Scholze torsion vanishing
t_entropy_method_extremal | technique | Entropy (information-theoretic) method
s_ggmt_polynomial_freiman_ruzsa | theorem | Polynomial Freiman–Ruzsa theorem (Gowers–Green–Manners–Tao)
s_kelley_meka_3ap | theorem | Kelley–Meka bound for three-term progressions
s_sensitivity_conjecture | theorem | Sensitivity theorem (Huang)
t_signed_adjacency_interlacing | technique | Signed adjacency / eigenvalue interlacing
s_ringel_conjecture | theorem | Ringel's conjecture (Montgomery–Pokrovskiy–Sudakov)
s_multicolor_ramsey_lower_bound | theorem | Exponential improvement of multicolor Ramsey lower bounds (Conlon–Ferber)
s_erdos_faber_lovasz_theorem | theorem | Erdős–Faber–Lovász theorem (Kang–Kelly–Kühn–Methuku–Osthus)
s_random_symmetric_matrix_singularity | theorem | Singularity probability of a random symmetric matrix
s_high_girth_steiner_triple_systems | theorem | High-girth Steiner triple systems
s_union_closed_constant_bound | theorem | Union-closed sets constant bound (Gilmer)
s_diagonal_ramsey_exponential_improvement | theorem | Exponential improvement for diagonal Ramsey (Campos–Griffiths–Morris–Sahasrabudhe)
t_ramsey_book_algorithm | technique | Book algorithm for Ramsey embeddings
s_ramsey_r4t_asymptotics | theorem | Asymptotics of R(4,t) (Mattheus–Verstraete)
s_erdos_hajnal_conjecture | state | Erdős–Hajnal conjecture
s_erdos_hajnal_loglog_bound | theorem | First improvement to the Erdős–Hajnal bound (Bucić–Nguyen–Scott–Seymour)
s_talagrand_selector_process | theorem | Talagrand selector-process theorem (Park–Pham)
s_hadwiger_improved_bound | theorem | Improved bound for Hadwiger's conjecture (Delcourt–Postle / Norin–Postle–Song)
s_cheskidov_luo_sharp_nonuniqueness | theorem | Sharp non-uniqueness for Navier–Stokes (Cheskidov–Luo)
s_knotted_3_balls_s4 | theorem | Knotted 3-balls in S^4 (Budney–Gabai)
s_rectangular_peg_problem | theorem | Rectangular peg problem (Greene–Lobb)
s_arithmeticity_geodesic_submanifolds | theorem | Arithmeticity from totally geodesic submanifolds (Bader–Fisher–Miller–Stover)
s_generalized_smale_conjecture | theorem | Generalized Smale conjecture (Bamler–Kleiner)
t_pointed_nash_entropy | technique | Pointed Nash entropy method
s_noncollapsed_ricci_flow_structure | theorem | Structure of non-collapsed Ricci-flow limits (Bamler)
s_michael_simon_sobolev_brendle | theorem | Sharp Michael–Simon–Sobolev inequality (Brendle)
s_generic_mcf_singularities | theorem | Generic singularities of mean curvature flow (Chodosh–Choi–Mantoulidis–Schulze)
s_generic_regularity_minimizers | theorem | Generic regularity of minimizing hypersurfaces in dimensions 9 and 10
t_convex_hypersurface_theory | technique | Convex hypersurface theory (higher-dimensional)
t_floer_homotopy_theory | technique | Floer homotopy theory
s_arnold_conjecture_morava | theorem | Arnold conjecture over arbitrary fields (Abouzaid–Blumberg)
s_telescope_conjecture_disproof | theorem | Disproof of Ravenel's telescope conjecture (Burklund–Hahn–Levy–Schlank)
s_chromatic_redshift | theorem | Chromatic redshift for truncated Brown–Peterson spectra (Hahn–Wilson)
t_even_filtration | technique | Even filtration
s_chromatic_nullstellensatz | theorem | Chromatic Nullstellensatz (Burklund–Schlank–Yuan)
s_lqg_metric | theorem | Existence and uniqueness of the Liouville quantum gravity metric (Gwynne–Miller)
s_wave_kinetic_equation_derivation | theorem | Rigorous derivation of the wave kinetic equation (Deng–Hani)
s_directed_landscape | state | Directed landscape
s_lis_directed_landscape_convergence | theorem | Convergence of the longest increasing subsequence to the directed landscape (Dauvergne–Virág)
s_kerr_stability_small_a | theorem | Nonlinear stability of slowly rotating Kerr (Klainerman–Szeftel)
s_schwarzschild_nonlinear_stability | theorem | Nonlinear stability of the Schwarzschild family (Dafermos–Holzegel–Rodnianski–Taylor)
t_condensed_mathematics | technique | Condensed mathematics
t_absolute_prismatic_cohomology | technique | Absolute prismatic cohomology (Cartier–Witt stack)
s_fargues_scholze_geometrization | theorem | Geometrization of the local Langlands correspondence (Fargues–Scholze)
t_smith_treumann_theory | technique | Smith–Treumann theory
s_modular_tilting_character_formula | theorem | Character formula for tilting modules (Riche–Williamson)
t_fourier_interpolation | technique | Fourier interpolation (modular-form method)
s_universal_optimality_e8_leech | theorem | Universal optimality of E8 and Leech lattices
s_nuclear_dimension_one | theorem | Nuclear dimension one for classifiable C*-algebras (Castillejos–Evington–Tikuisis–White–Winter)
s_keisler_order_maximal_complexity | theorem | Maximal complexity of Keisler's order (Malliaris–Shelah)
s_mip_star_re_connes | theorem | MIP* = RE and refutation of the Connes embedding problem
