s_euclidean_plane [axiom] Euclidean plane ℝ²
s_euclidean_3_space [axiom] Euclidean 3-space (sphere-packing context) | aliases: s_euclidean_solid_geometry
s_real_numbers [axiom] Real numbers ℝ | aliases: 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 | aliases: 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) | aliases: 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 | aliases: totally additive function
s_generalized_divisor_function_tau_k [axiom] Generalized divisor function τ_k(n) | aliases: Piltz divisor function, d_k(n), τ_k
t_selberg_delange_method [technique] Selberg–Delange method | aliases: Selberg-Delange method
s_smoothed_perron_formula [theorem] Smoothed Perron formula | aliases: smooth Perron formula
t_dyadic_decomposition [technique] Dyadic decomposition | aliases: dyadic subdivision
t_smooth_partition_of_unity [technique] Smooth partition of unity | aliases: smooth dyadic partition
t_heath_brown_identity [technique] Heath-Brown's identity | aliases: Heath-Brown decomposition
t_linnik_identity [technique] Linnik's identity | aliases: Linnik decomposition
s_type_i_sum [axiom] Type I sum | aliases: Type I estimate, linear sum
s_type_ii_sum [axiom] Type II sum | aliases: Type II estimate, bilinear sum
s_wirsing_theorem [theorem] Wirsing's theorem | aliases: Wirsing mean value theorem
s_shiu_bound [theorem] Shiu's bound | aliases: Shiu's theorem
s_general_l_function_ik [axiom] General L-function (Iwaniec–Kowalski framework) | aliases: standard L-function, IK axiomatic L-function
s_completed_l_function [axiom] Completed L-function | aliases: Lambda function, Λ-function
s_root_number [axiom] Root number | aliases: epsilon factor, sign of functional equation
s_gamma_factors_functional_equation [axiom] Gamma factors in functional equations | aliases: archimedean gamma factor
s_conductor_l_function [axiom] Conductor of an L-function | aliases: arithmetic conductor, q(π)
s_analytic_conductor [axiom] Analytic conductor | aliases: C(π), effective conductor
s_degree_l_function [axiom] Degree of an L-function | aliases: d(π)
s_convexity_bound_l_functions [theorem] Convexity bound for L-functions | aliases: convexity estimate
s_subconvexity_bound [theorem] Subconvexity bound | aliases: subconvex estimate, breaking convexity
s_grand_lindelof_hypothesis [axiom] Grand Lindelöf hypothesis | aliases: GLH, generalized Lindelöf hypothesis
s_approximate_functional_equation [theorem] Approximate functional equation | aliases: AFE
s_mean_value_theorem_dirichlet_polynomials [theorem] Mean value theorem for Dirichlet polynomials | aliases: Montgomery-Vaughan mean value theorem
s_montgomery_large_values [theorem] Montgomery's mean value theorem for large values | aliases: large values estimate
s_huxley_jutila_large_values [theorem] Huxley–Jutila large values estimate | aliases: Jutila large values
s_fourth_moment_zeta [theorem] Fourth moment of the Riemann zeta function | aliases: Ingham fourth moment
s_second_moment_l_functions_characters [theorem] Second moment of L-functions over characters | aliases: mean square of L(1/2,χ)
s_moments_l_functions_conjectural [axiom] Moments of L-functions (conjectural asymptotics) | aliases: CFKRS moment conjecture, Keating-Snaith conjecture
s_hecke_l_function_grossencharakter [axiom] Hecke L-function | aliases: Hecke Grössencharakter L-function
s_rankin_selberg_l_function [axiom] Rankin–Selberg L-function | aliases: Rankin-Selberg convolution
s_kronecker_limit_formula [theorem] Kronecker's limit formula | aliases: first Kronecker limit formula
s_sifting_function [axiom] Sifting function S(A, P, z) | aliases: sieve sum, S(A,z)
s_sieve_dimension [axiom] Sieve dimension | aliases: sifting density
s_buchstab_identity [theorem] Buchstab's identity | aliases: Buchstab recursion
s_linear_sieve [theorem] Linear sieve (Iwaniec's formulation) | aliases: Iwaniec linear sieve
s_sieve_limit_functions [axiom] Sieve upper/lower bound functions F_κ(s), f_κ(s) | aliases: sieve adjoint functions
t_rosser_iwaniec_sieve [technique] Rosser–Iwaniec sieve (β-sieve) | aliases: beta-sieve, β-sieve
s_parity_problem_sieve [axiom] Parity problem in sieve theory | aliases: parity barrier, Selberg parity phenomenon
t_semilinear_sieve [technique] Semilinear sieve (κ = 1/2) | aliases: half-dimensional sieve
t_bombieri_asymptotic_sieve [technique] Bombieri's asymptotic sieve | aliases: Bombieri sieve
t_asymptotic_sieve_for_primes [technique] Asymptotic sieve for primes (Bombieri–Friedlander–Iwaniec) | aliases: BFI sieve
s_level_of_distribution [axiom] Level of distribution | aliases: exponent of distribution
s_well_factorable_function [axiom] Well-factorable function | aliases: well-factorable weights
t_switching_principle_sieve [technique] Switching principle in sieve theory | aliases: Buchstab switching
t_van_der_corput_a_process [technique] Van der Corput's A-process | aliases: van der Corput second derivative test, A-process
t_van_der_corput_b_process [technique] Van der Corput's B-process | aliases: B-process, Poisson transform for exponential sums
t_method_of_exponent_pairs [technique] Method of exponent pairs | aliases: exponent pair method
s_exponent_pair [axiom] Exponent pair (k, ℓ) | aliases: van der Corput exponent pair
t_vinogradov_method_exponential_sums [technique] Vinogradov's method of exponential sums | aliases: Vinogradov exponential sum method
s_salie_sum [axiom] Salié sum | aliases: Salié-Kloosterman sum
s_hyper_kloosterman_sum [axiom] Hyper-Kloosterman sum | aliases: generalized Kloosterman sum
t_stepanov_method [technique] Stepanov's method | aliases: Stepanov auxiliary polynomial method
s_bombieri_proof_rh_curves [theorem] Bombieri's proof of RH for curves | aliases: Bombieri-Stepanov proof
t_square_sieve [technique] Square sieve (Heath-Brown) | aliases: Heath-Brown square sieve
t_bombieri_iwaniec_method [technique] Bombieri–Iwaniec method | aliases: Bombieri-Iwaniec first spacing method
s_duality_principle_bilinear [theorem] Duality principle for bilinear forms | aliases: bilinear duality, norm duality
s_dual_large_sieve_inequality [theorem] Dual large sieve inequality | aliases: dual form of the large sieve
s_large_sieve_characters [theorem] Large sieve inequality for Dirichlet characters | aliases: multiplicative large sieve
t_deshouillers_iwaniec_method [technique] Deshouillers–Iwaniec method | aliases: Deshouillers-Iwaniec spectral method
t_linnik_dispersion_method [technique] Linnik's dispersion method | aliases: dispersion method
s_halasz_montgomery_inequality [theorem] Halász–Montgomery inequality | aliases: Montgomery-Halász inequality
s_motohashi_formula [theorem] Motohashi's formula | aliases: Motohashi spectral formula
s_gallagher_larger_sieve [theorem] Gallagher's larger sieve | aliases: Gallagher sieve, the larger sieve
s_barban_davenport_halberstam_theorem [theorem] Barban–Davenport–Halberstam theorem | aliases: BDH theorem
s_elliott_halberstam_conjecture [axiom] Elliott–Halberstam conjecture | aliases: EH conjecture
s_modular_group_sl2z [axiom] Modular group SL₂(ℤ) | aliases: SL(2,Z), full modular group
s_congruence_subgroup_gamma0 [axiom] Congruence subgroup Γ₀(N) | aliases: Gamma_0(N), Hecke congruence subgroup
s_congruence_subgroup_gamma1 [axiom] Congruence subgroup Γ₁(N) | aliases: Gamma_1(N)
s_principal_congruence_subgroup [axiom] Principal congruence subgroup Γ(N) | aliases: Gamma(N)
s_nebentypus_character [axiom] Nebentypus character | aliases: Nebentypus
s_fundamental_domain_sl2z [axiom] Fundamental domain for SL₂(ℤ) | aliases: standard fundamental domain
s_cusps_of_congruence_subgroup [axiom] Cusps of a congruence subgroup | aliases: cusps, parabolic vertices
s_hyperbolic_laplacian [axiom] Hyperbolic Laplacian on H | aliases: Laplace-Beltrami operator on H
s_cusp_form [axiom] Cusp form | aliases: cuspidal modular form, Spitzenform
s_slash_operator [axiom] Slash operator |_k | aliases: slash operator, weight-k action
s_poincare_series_modular [axiom] Poincaré series P_m(z, k, Γ) | aliases: Poincaré series for modular forms
s_petersson_inner_product [axiom] Petersson inner product | aliases: Petersson scalar product
s_fourier_expansion_at_cusps [axiom] Fourier expansion at cusps | aliases: q-expansion
s_atkin_lehner_theory [theorem] Atkin-Lehner theory | aliases: newform-oldform decomposition
s_atkin_lehner_involution [axiom] Atkin-Lehner involution W_Q | aliases: Atkin-Lehner operator
s_fricke_involution [axiom] Fricke involution W_N | aliases: Fricke operator
s_oldform [axiom] Oldform | aliases: old form
s_conductor_of_modular_form [axiom] Conductor of a modular form | aliases: level of a newform
s_dimension_formula_mk_sl2z [theorem] Dimension formula for M_k(SL₂(ℤ)) | aliases: dimension of modular forms for SL₂(ℤ)
s_dimension_formula_sk_gamma0 [theorem] Dimension formula for S_k(Γ₀(N)) | aliases: Riemann-Roch dimension formula for cusp forms
s_theta_function_of_quadratic_form [axiom] Theta function of a quadratic form | aliases: theta series of a lattice
s_maass_form [axiom] Maass form | aliases: Maass wave form
s_maass_cusp_form [axiom] Maass cusp form | aliases: Maass cuspidal form
s_spectral_parameter_maass [axiom] Spectral parameter r | aliases: Maass spectral parameter
s_ramanujan_petersson_conjecture [theorem] Ramanujan-Petersson conjecture | aliases: Ramanujan conjecture for modular forms
s_real_analytic_eisenstein_series [axiom] Real-analytic Eisenstein series E(z,s) | aliases: non-holomorphic Eisenstein series
s_scattering_matrix_automorphic [axiom] Scattering matrix for Γ\H | aliases: scattering matrix, phi function
s_spectral_decomposition_l2_gamma_h [theorem] Spectral decomposition of L²(Γ\H) | aliases: Roelcke-Selberg spectral decomposition
s_discrete_spectrum_gamma_h [axiom] Discrete spectrum of Γ\H | aliases: cuspidal plus residual spectrum
s_continuous_spectrum_gamma_h [axiom] Continuous spectrum of Γ\H | aliases: Eisenstein contribution
s_pre_trace_formula [theorem] Pre-trace formula | aliases: automorphic kernel identity
s_selberg_zeta_function [axiom] Selberg zeta function Z_Γ(s) | aliases: Selberg zeta function
s_maass_selberg_relations [theorem] Maass-Selberg relations | aliases: Maass-Selberg formula
s_petersson_trace_formula [theorem] Petersson trace formula | aliases: Petersson-Kuznetsov formula
s_eichler_selberg_trace_formula [theorem] Eichler-Selberg trace formula | aliases: Selberg trace formula for Hecke operators
s_kuznetsov_trace_formula [theorem] Kuznetsov trace formula | aliases: Kuznetsov sum formula, Bruggeman-Kuznetsov formula
s_voronoi_summation_formula [theorem] Voronoi summation formula (GL(2)) | aliases: Voronoi formula
s_l_function_of_modular_form [axiom] L-function of a modular form | aliases: standard L-function of f
s_euler_product_hecke_eigenform [theorem] Euler product for Hecke eigenform L-function | aliases: Hecke Euler product
t_rankin_selberg_method [technique] Rankin-Selberg method | aliases: Rankin-Selberg unfolding
s_symmetric_square_l_function [axiom] Symmetric square L-function L(sym²f, s) | aliases: symmetric square lift
s_jacquet_langlands_correspondence [theorem] Jacquet-Langlands correspondence | aliases: Jacquet-Langlands transfer
s_gelbart_jacquet_lift [theorem] Gelbart-Jacquet lift | aliases: symmetric square lift to GL(3)
s_gl_n_automorphic_form [axiom] GL(n) automorphic form | aliases: automorphic form on GL(n)
s_kirillov_model [axiom] Kirillov model | aliases: Kirillov space
s_strong_multiplicity_one_gl2 [theorem] Strong multiplicity one for GL(2) | aliases: strong multiplicity one theorem
s_twisted_l_function [axiom] Twisted L-function L(f⊗χ, s) | aliases: character twist of L-function
s_satake_parameters [axiom] Satake parameters | aliases: Langlands parameters at unramified primes
s_hecke_multiplicativity [theorem] Hecke multiplicativity relations | aliases: Hecke recursion
t_amplification_method [technique] Amplification method | aliases: amplification technique
s_spectral_large_sieve [theorem] Spectral large sieve inequality | aliases: large sieve for automorphic forms
s_subconvexity_gl2 [theorem] Subconvexity for GL(2) L-functions | aliases: GL(2) subconvexity
t_spectral_mean_value_estimates [technique] Mean value estimates via spectral methods | aliases: spectral moment method
s_spectral_reciprocity [theorem] Spectral reciprocity formula | aliases: reciprocity formula
s_nonvanishing_l_functions_at_center [theorem] Nonvanishing results for central L-values | aliases: nonvanishing at the central point
s_singular_integral_circle_method [axiom] Singular integral (circle method) | aliases: singular integral in Waring's problem
s_hua_inequality [theorem] Hua's inequality | aliases: Hua's lemma
s_hardy_littlewood_asymptotic_waring [theorem] Hardy-Littlewood asymptotic for Waring's problem | aliases: Waring asymptotic formula
t_farey_dissection [technique] Farey dissection | aliases: Farey arc decomposition
t_kloosterman_refinement [technique] Kloosterman refinement (HLK method) | aliases: HLK method
t_delta_method [technique] Delta method (Duke-Friedlander-Iwaniec) | aliases: DFI delta method
s_huxley_zero_density_estimate [theorem] Huxley's zero density estimate | aliases: Huxley density estimate
s_jutila_zero_density_estimate [theorem] Jutila's zero density estimate | aliases: Jutila density estimate
s_log_free_zero_density_estimate [theorem] Log-free zero density estimate | aliases: log-free density estimate
t_zero_detecting_polynomial [technique] Zero-detecting polynomial | aliases: zero detector
t_turan_power_sum_method [technique] Turán's power sum method | aliases: power sum method
t_halasz_montgomery_method [technique] Halász-Montgomery method | aliases: Halász-Montgomery zero density method
s_gue_hypothesis [axiom] GUE hypothesis for ζ(s) zeros | aliases: random matrix theory connection, Katz-Sarnak philosophy
t_mollifier_method [technique] Mollifier method | aliases: mollification technique
t_levinson_method [technique] Levinson's method | aliases: Levinson's theorem on zeros
s_deuring_heilbronn_repulsion [theorem] Deuring-Heilbronn repulsion phenomenon | aliases: Deuring-Heilbronn phenomenon, zero repulsion
s_linnik_constant [axiom] Linnik's constant | aliases: Linnik constant L
s_fouvry_theorem [theorem] Fouvry's theorem | aliases: Fouvry-Iwaniec level of distribution
s_duke_equidistribution [theorem] Duke's theorem (equidistribution of CM points) | aliases: Duke's equidistribution theorem
s_subconvexity_implies_equidistribution [theorem] Subconvexity implies equidistribution | aliases: subconvexity-equidistribution paradigm
s_erdos_turan_inequality [theorem] Erdős-Turán inequality | aliases: Erdős-Turán discrepancy inequality
s_discrepancy_of_sequence [axiom] Discrepancy of a sequence | aliases: discrepancy, star discrepancy
s_friedlander_iwaniec_theorem [theorem] Friedlander-Iwaniec theorem | aliases: primes of the form a²+b⁴
t_half_dimensional_sieve [technique] Iwaniec's half-dimensional sieve | aliases: half-dimensional sieve
s_shifted_convolution_sums [axiom] Shifted convolution sums | aliases: shifted convolution problem
s_additive_divisor_problem [axiom] Additive divisor problem | aliases: binary additive divisor problem
s_hecke_converse_theorem [theorem] Hecke's converse theorem | aliases: Hecke's inverse Mellin theorem
s_weil_converse_theorem [theorem] Weil's converse theorem | aliases: Weil's converse theorem for GL(2)
s_holomorphic_modular_form_formal [axiom] Holomorphic modular form (weight k, level N) | aliases: classical modular form
s_eisenstein_series_nebentypus [axiom] Eisenstein series with Nebentypus | aliases: Eisenstein series with character
s_ramanujan_conjecture_gl2 [axiom] Ramanujan conjecture for GL(2) | aliases: 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 | aliases: 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 | aliases: pro-p-group, p-profinite group
s_cm_field [axiom] Complex multiplication (CM) field | aliases: CM field
s_linear_algebraic_group [axiom] Linear algebraic group | aliases: algebraic group
s_semi_simple_algebraic_group [axiom] Semi-simple algebraic group | aliases: semisimple algebraic group
s_simply_connected_algebraic_group [axiom] Simply connected algebraic group | aliases: simply connected semisimple group
s_restricted_topological_product [axiom] Restricted topological product (restricted direct product) | aliases: restricted direct product, restricted product
s_group_homology [state] Group homology H_n(G, M) | aliases: homology of groups
s_absolute_galois_group [state] Absolute Galois group Gal(K^sep/K) | aliases: absolute Galois group
s_supernatural_number [state] Supernatural number | aliases: Steinitz number
s_free_profinite_group [state] Free profinite group
s_continuous_cohomology_profinite [state] Continuous cohomology of profinite groups | aliases: profinite group cohomology, Galois cohomology
s_cm_type [state] CM type
s_abelian_variety_with_cm [state] Abelian variety with complex multiplication | aliases: CM abelian variety
s_reflex_field [state] Reflex field (reflex CM type) | aliases: reflex CM type, dual CM type
s_algebraic_hecke_character [state] Algebraic Hecke character (Grössencharakter of type A₀) | aliases: algebraic Grössencharakter, type A₀ Hecke character
s_l_extension [state] ℓ-extension | aliases: l-extension, p-extension
s_z_l_extension [state] ℤ_ℓ-extension | aliases: Z_l-extension, Z_p-extension
s_global_zeta_integral_tate [state] Global zeta integral (Tate) | aliases: Tate's global zeta function
s_quasi_character [state] Quasi-character of K_v* | aliases: quasi-character
s_self_dual_measure [state] Self-dual Haar measure | aliases: self-dual measure
s_kroneckers_jugendtraum [state] Kronecker's Jugendtraum | aliases: Kronecker's youth dream, Hilbert's 12th problem (partial)
s_kneser_tits_conjecture [state] Kneser–Tits conjecture | aliases: Kneser–Tits problem
s_golod_shafarevich_theorem [theorem] Golod–Shafarevich theorem | aliases: 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) | aliases: 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 | aliases: 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 | aliases: 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 | aliases: Euler product from local factors
s_global_local_compatibility_reciprocity [theorem] Global–local compatibility of reciprocity maps | aliases: compatibility of local and global Artin maps
s_hilbert_different_formula [theorem] Hilbert's different formula | aliases: Hilbert's formula for the different
s_first_inequality_cft [theorem] First inequality (global class field theory) | aliases: first inequality
s_second_inequality_cft [theorem] Second inequality (global class field theory) | aliases: second inequality
s_krull_theorem_infinite_galois [theorem] Krull's theorem (infinite Galois theory) | aliases: fundamental theorem of infinite Galois theory
s_herbrand_theorem_upper_numbering [theorem] Herbrand's theorem (upper numbering of ramification groups) | aliases: 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 | aliases: Farey series
s_mediant [axiom] Mediant of two fractions | aliases: 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 | aliases: bounded partial quotients
s_durfee_square [axiom] Durfee square
s_stern_brocot_tree [axiom] Stern–Brocot tree | aliases: Stern-Brocot tree, Calkin-Wilf tree
s_visible_lattice_point_density [state] Density of visible lattice points | aliases: 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 | aliases: Borel's theorem on normal numbers
s_kempner_series_convergence [theorem] Kempner series convergence | aliases: Kempner's series
s_lagrange_periodic_cf_theorem [theorem] Lagrange's periodic continued fraction theorem | aliases: 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 | aliases: Kronecker's theorem
s_kronecker_simultaneous_approximation [theorem] Kronecker's simultaneous approximation theorem | aliases: Kronecker's theorem in k dimensions
s_rogers_ramanujan_continued_fraction [theorem] Rogers–Ramanujan continued fraction | aliases: 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 | aliases: 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 | aliases: Estermann's proof
t_elliptic_curve_chord_tangent_addition [technique] Chord-and-tangent addition on elliptic curves | aliases: 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 | aliases: 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 | aliases: 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) | aliases: 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) | aliases: equational class, variety of algebras
s_birkhoff_hsp_theorem [theorem] Birkhoff's HSP theorem | aliases: 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 | aliases: semisimple module
s_composition_length [state] Composition length
s_krull_dimension_of_ring [state] Krull dimension | aliases: Krull dimension of a ring
s_prufer_domain [axiom] Prüfer domain | aliases: 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 | aliases: smooth algebra, formally smooth morphism
s_formally_unramified_algebra [axiom] Formally unramified algebra | aliases: unramified algebra
s_formally_etale_algebra [axiom] Formally étale algebra | aliases: étale algebra, formally étale morphism
s_g_ring [axiom] G-ring (Grothendieck ring) | aliases: Grothendieck ring, G-ring
s_quasi_excellent_ring [axiom] Quasi-excellent ring | aliases: quasi-excellent
s_n_1_ring [axiom] N-1 ring | aliases: N-1 domain
s_buchsbaum_ring [axiom] Buchsbaum ring | aliases: Buchsbaum local ring
s_geometrically_regular_formal_fibres [axiom] Geometrically regular formal fibres | aliases: regular formal fibres
s_attached_primes [axiom] Attached primes | aliases: attached prime ideals, Att(M)
s_dualizing_complex [axiom] Dualizing complex | aliases: residual complex
s_singular_locus [state] Singular locus | aliases: non-regular locus, Sing(R)
s_formal_fibre [state] Formal fibre | aliases: formal fiber
s_henselization [state] Henselization | aliases: Henselization of a local ring, R^h
s_strict_henselization [state] Strict Henselization | aliases: strict Henselization, R^{sh}
s_local_criterion_for_flatness [theorem] Local criterion for flatness | aliases: local flatness criterion
s_equational_criterion_for_flatness [theorem] Equational criterion for flatness | aliases: equational flatness criterion, Lazard criterion
s_generic_flatness_theorem [theorem] Generic flatness theorem | aliases: generic freeness, generic flatness lemma
s_grothendieck_non_vanishing_theorem [theorem] Grothendieck's non-vanishing theorem | aliases: non-vanishing of local cohomology at dimension
s_hochster_big_cm_modules [theorem] Hochster's theorem on big Cohen-Macaulay modules | aliases: existence of big CM modules, Hochster's big CM theorem
s_openness_of_regular_locus [theorem] Openness of the regular locus | aliases: regular locus is open
s_fibre_dimension_formula_flat [theorem] Fibre dimension formula for flat local homomorphisms | aliases: dimension formula for flat extensions
s_finitistic_dimension_theorem [theorem] Finitistic dimension theorem | aliases: finitistic projective dimension equals depth
s_cohen_structure_theorem_mixed_char [theorem] Cohen structure theorem (mixed characteristic) | aliases: 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 | aliases: theorem on formal functions
t_cohen_macaulay_approximation [technique] Cohen-Macaulay approximation | aliases: Auslander-Buchweitz approximation
t_secondary_representation [technique] Secondary representation | aliases: secondary decomposition
s_quasi_isomorphism [axiom] Quasi-isomorphism | aliases: quasi-iso, quasi isomorphism, homology isomorphism
s_mapping_cylinder_chain [state] Mapping cylinder (chain complexes) | aliases: mapping cylinder of a chain map
s_split_exact_sequence [axiom] Split exact sequence | aliases: split short exact sequence, splitting
s_homotopy_equivalence_complexes [state] Homotopy equivalence of chain complexes | aliases: chain homotopy equivalence
s_erasable_functor [axiom] Erasable (effaceable/coeffaceable) functor | aliases: erasable functor, effaceable functor definition
s_flat_resolution [state] Flat resolution | aliases: flat resolution of a module
s_weak_global_dimension [axiom] Weak (global) dimension | aliases: weak dimension, Tor-dimension, w.gl.dim
s_yoneda_ext [state] Yoneda Ext | aliases: Yoneda extension group, Yoneda product
s_baer_sum [state] Baer sum of extensions | aliases: Baer addition
s_lim1_derived_inverse_limit [state] lim¹ (first derived functor of inverse limit) | aliases: lim^1, R^1 lim, derived inverse limit
s_auslander_buchsbaum_serre_theorem [theorem] Auslander–Buchsbaum–Serre theorem | aliases: regularity and global dimension theorem
s_grade_of_module [axiom] Grade of a module | aliases: j-number, grade
s_derived_couple [state] Derived couple | aliases: derived exact couple
s_transgression_spectral_sequence [state] Transgression | aliases: transgression map, transgressive element
s_cartan_eilenberg_resolution [state] Cartan–Eilenberg resolution | aliases: CE resolution
s_hyperhomology [state] Hyperhomology | aliases: hyper-homology, left hyper-derived functor, hypertor
s_hypercohomology [state] Hypercohomology | aliases: hyper-cohomology, right hyper-derived functor, hyperext
s_acyclic_assembly_lemma [theorem] Acyclic assembly lemma
s_group_invariants [axiom] G-invariants M^G | aliases: invariants, fixed points of G-module, M^G
s_coinvariants [axiom] G-coinvariants M_G | aliases: coinvariants, module of coinvariants, M_G
s_normalized_bar_resolution [state] Normalized bar resolution | aliases: normalized standard resolution, reduced bar resolution
s_five_term_exact_sequence_group [theorem] Five-term exact sequence (group cohomology) | aliases: inflation-restriction exact sequence, five-term exact sequence
s_principal_crossed_homomorphism [state] Principal crossed homomorphism | aliases: principal derivation, inner derivation (group cohomology)
s_factor_set [axiom] Factor set (group extension) | aliases: 2-cocycle for group extension, factor system
s_tate_cohomology [axiom] Tate cohomology | aliases: Tate cohomology groups, Ĥ*(G,M), complete cohomology
s_complete_resolution [state] Complete (Tate) resolution | aliases: Tate resolution, complete projective resolution
s_norm_map_group [state] Norm map (group cohomology) | aliases: norm element, norm homomorphism
s_cup_product_group_cohomology [state] Cup product in group cohomology | aliases: cup product on H*(G,M)
s_cohomological_dimension_group [axiom] Cohomological dimension of a group | aliases: cd(G), cohomological dimension
s_group_type_fp [axiom] Group of type FP | aliases: type FL, type FP_n, finiteness property of groups
s_lie_algebra_homology [axiom] Lie algebra homology H_n(𝔤, M) | aliases: Lie algebra homology, H_n(g,M)
s_lie_algebra_cohomology [axiom] Lie algebra cohomology H^n(𝔤, M) | aliases: Lie algebra cohomology, H^n(g,M)
s_chevalley_eilenberg_complex [state] Chevalley–Eilenberg complex | aliases: CE complex, Koszul–Chevalley–Eilenberg complex
s_casimir_element [state] Casimir element | aliases: Casimir operator, Casimir invariant
s_relative_lie_algebra_cohomology [state] Relative Lie algebra cohomology H^n(𝔤, 𝔨; M) | aliases: relative cohomology of Lie pair
s_simplicial_object [axiom] Simplicial object | aliases: simplicial object in a category
s_cosimplicial_object [axiom] Cosimplicial object | aliases: cosimplicial object in a category
s_simplicial_set [axiom] Simplicial set | aliases: simplicial set, semi-simplicial set
s_face_map_simplicial [axiom] Face map (simplicial) | aliases: face operator, face morphism
s_degeneracy_map_simplicial [axiom] Degeneracy map (simplicial) | aliases: degeneracy operator
s_moore_complex [state] Moore complex | aliases: Moore complex of a simplicial group
s_normalized_chain_complex_simplicial [state] Normalized chain complex (simplicial) | aliases: normalized Moore complex, N(A)
s_dold_kan_correspondence [theorem] Dold–Kan correspondence | aliases: Dold–Kan theorem, Dold–Kan equivalence, Dold–Puppe theorem
s_alexander_whitney_map [state] Alexander–Whitney map | aliases: AW map, Alexander–Whitney diagonal
s_shuffle_map [state] Shuffle map (Eilenberg–Zilber) | aliases: Eilenberg–Zilber map, shuffle product, ∇
s_eilenberg_maclane_space [axiom] Eilenberg–MacLane space K(π, n) | aliases: K(π,n), Eilenberg–MacLane space
s_comonad [axiom] Comonad (cotriple) | aliases: cotriple, comonad
s_cotriple_resolution [state] Cotriple resolution | aliases: comonadic resolution, comonad resolution
s_andre_quillen_homology [state] André–Quillen homology | aliases: AQ homology, cotangent homology
s_cotangent_complex [state] Cotangent complex | aliases: L_{A/k}, Illusie cotangent complex
s_bar_construction [state] Bar construction | aliases: bar construction B(T,A), simplicial bar construction
s_kan_complex [axiom] Kan complex | aliases: Kan fibrant simplicial set, fibrant simplicial set
s_hochschild_homology [axiom] Hochschild homology HH_n(A, M) | aliases: Hochschild homology, HH_*(A,M)
s_gerstenhaber_bracket [state] Gerstenhaber bracket | aliases: Gerstenhaber algebra structure, Gerstenhaber Lie bracket
s_formal_deformation_algebra [axiom] Formal deformation of an algebra | aliases: infinitesimal deformation, deformation quantization
s_cyclic_homology [axiom] Cyclic homology HC_n(A) | aliases: cyclic homology, HC_*(A)
s_mixed_complex [axiom] Mixed complex | aliases: mixed complex (b, B)
s_connes_b_operator [state] Connes' B operator | aliases: Connes operator, Connes boundary B
s_sbi_exact_sequence [theorem] SBI exact sequence (Connes) | aliases: Connes' exact sequence, Connes' SBI sequence, periodicity exact sequence
s_cyclic_bicomplex [state] Cyclic bicomplex | aliases: Connes' bicomplex, CC(A)
s_periodic_cyclic_homology [state] Periodic cyclic homology HP_n(A) | aliases: periodic cyclic homology, HP_*(A)
s_negative_cyclic_homology [state] Negative cyclic homology HN_n(A) | aliases: negative cyclic homology, HN_*(A)
s_hkr_theorem [theorem] Hochschild–Kostant–Rosenberg theorem | aliases: HKR theorem, HKR isomorphism
s_loday_quillen_tsygan_theorem [theorem] Loday–Quillen–Tsygan theorem | aliases: LQT theorem
s_dennis_trace_map [state] Dennis trace map | aliases: Dennis trace
s_morita_invariance_hochschild [theorem] Morita invariance of Hochschild and cyclic homology | aliases: Morita invariance of HH and HC
s_homotopy_category_chain [axiom] Homotopy category K(A) | aliases: homotopy category of chain complexes, K(A)
s_distinguished_triangle [axiom] Distinguished triangle | aliases: exact triangle, distinguished triangle in triangulated category
s_octahedral_axiom [axiom] Octahedral axiom (TR4) | aliases: TR4, octahedral axiom
s_verdier_quotient [state] Verdier quotient | aliases: Verdier localization, localization of triangulated category
s_ore_conditions [axiom] Ore conditions (calculus of fractions) | aliases: calculus of fractions, left Ore condition, right Ore condition
s_derived_category [axiom] Derived category D(A) | aliases: derived category, D(A), D^b(A), bounded derived category
s_bounded_derived_category [axiom] Bounded derived category D^b(A) | aliases: D^b, D^+(A), D^-(A), bounded above/below derived category
s_total_left_derived_functor [state] Total left derived functor LF | aliases: LF, left hyper-derived functor, 𝕃F
s_total_right_derived_functor [state] Total right derived functor RF | aliases: RF, right hyper-derived functor, ℝF
s_rhom_complex [state] RHom complex | aliases: RHom, derived Hom, ℝHom
s_derived_tensor_product [state] Derived tensor product ⊗^L | aliases: ⊗^L, derived tensor, left derived tensor product
s_way_out_functor [axiom] Way-out functor | aliases: way-out right, way-out left
s_way_out_lemma [theorem] Way-out lemma (Hartshorne) | aliases: Hartshorne's way-out lemma
s_t_structure [axiom] t-structure on a triangulated category | aliases: t-structure, truncation structure
s_heart_of_t_structure [state] Heart of a t-structure | aliases: core of a t-structure
s_truncation_functors [state] Truncation functors τ≤n and τ≥n | aliases: truncation, τ≤n, τ≥n, canonical truncation
s_grothendieck_ab_axioms [axiom] Grothendieck's AB axioms | aliases: 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 | aliases: boundary map, boundary operator
s_cycles_of_chain_complex [state] Cycles | aliases: n-cycles, cycle group
s_boundaries_of_chain_complex [state] Boundaries | aliases: n-boundaries, boundary group
s_cocycles_of_cochain_complex [state] Cocycles | aliases: n-cocycles
s_coboundaries_of_cochain_complex [state] Coboundaries | aliases: n-coboundaries
s_bounded_chain_complex [axiom] Bounded chain complex | aliases: bounded complex, bounded below complex, bounded above complex
s_chain_homotopy [axiom] Chain homotopy | aliases: 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 | aliases: 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 | aliases: Ch(A)
s_homotopy_category_K_A [state] Homotopy category K(A) | aliases: K(A)
s_homology_is_functor [theorem] Homology is a functor
s_hom_complex [state] Hom complex | aliases: 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 | aliases: Baer's criterion
t_balancing_via_double_complexes [technique] Balancing via double complexes
s_flat_dimension [axiom] Flat dimension | aliases: 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 | aliases: 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 | aliases: hyper-Tor
s_hyper_derived_functors_right [axiom] Right hyper-derived functors | aliases: 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 | aliases: comparison theorem for spectral sequences
s_degeneration_at_E2 [state] Degeneration at E_2 | aliases: 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 | aliases: augmentation homomorphism
s_coinvariants_M_G [axiom] Coinvariants M_G | aliases: module of coinvariants
s_invariants_M_G [axiom] Invariants M^G | aliases: 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 | aliases: Hopf's formula
s_inhomogeneous_bar_resolution [axiom] Inhomogeneous bar resolution | aliases: unnormalized bar resolution
s_homogeneous_bar_resolution [axiom] Homogeneous bar resolution
s_coinduced_module [axiom] Coinduced module Coind_H^G N | aliases: 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 | aliases: LHS spectral sequence
s_five_term_exact_sequence_group_cohomology [theorem] Five-term exact sequence in group cohomology | aliases: 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 | aliases: Stallings-Swan theorem
s_complete_tate_resolution [axiom] Complete (Tate) resolution | aliases: 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 | aliases: 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 | aliases: 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 | aliases: representable simplicial set
s_singular_simplicial_set [state] Singular simplicial set | aliases: singular complex
s_geometric_realization [axiom] Geometric realization
s_simplicial_map [axiom] Simplicial map | aliases: 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 | aliases: AW map
t_shuffle_map [technique] Shuffle map (Eilenberg-MacLane map) | aliases: EML map, Eilenberg-MacLane map
s_shuffle [axiom] Shuffle | aliases: (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) | aliases: BG
s_cotriple_comonad [axiom] Cotriple (comonad) | aliases: comonad
s_cotriple_homology [state] Cotriple homology
s_andre_quillen_cohomology [state] Andre-Quillen cohomology | aliases: AQ cohomology
s_bisimplicial_object [axiom] Bisimplicial object | aliases: 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) | aliases: 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) | aliases: extra degeneracy argument
s_hochschild_complex [axiom] Hochschild complex (bar complex for algebras) | aliases: 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 | aliases: 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 | aliases: 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) | aliases: TR1-TR4
s_RHom_complex [axiom] RHom complex | aliases: 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) | aliases: Grothendieck's axioms
s_grothendieck_category [axiom] Grothendieck category | aliases: AB5 category with generator
s_inverse_limit_functor_lim [axiom] Inverse limit functor lim | aliases: 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 | aliases: abstract category
s_small_category [axiom] Small category
s_locally_small_category [axiom] Locally small category
s_functor [axiom] Functor | aliases: 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 | aliases: monic, mono
s_epimorphism [axiom] Epimorphism | aliases: 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) | aliases: 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 | aliases: dual category, C^op
t_duality_principle [technique] Duality principle (categorical)
s_product_category [axiom] Product category
s_comma_category [axiom] Comma category | aliases: comma category (F ↓ G)
s_slice_category [axiom] Slice category | aliases: over category
s_coslice_category [axiom] Coslice category | aliases: under category
s_arrow_category [axiom] Arrow category | aliases: 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) | aliases: presheaf, set-valued presheaf
s_limit_of_functor [axiom] Limit of a functor | aliases: projective limit, inverse limit
s_colimit_of_functor [axiom] Colimit of a functor | aliases: 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 | aliases: adjoint pair, adjoint functors
s_special_adjoint_functor_theorem [theorem] Special adjoint functor theorem (SAFT) | aliases: 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 | aliases: 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) | aliases: separator
s_cogenerator [axiom] Cogenerator (coseparator) | aliases: coseparator
s_well_powered_category [axiom] Well-powered category
s_ends [axiom] End (category theory) | aliases: end, wedge
s_coends [axiom] Coend (category theory) | aliases: 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) | aliases: density formula, Yoneda density
s_category_of_elements [axiom] Category of elements | aliases: 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) | aliases: T-algebra, Eilenberg-Moore algebra
s_eilenberg_moore_category [axiom] Eilenberg-Moore category | aliases: 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) | aliases: Beck's precise monadicity theorem, precise tripleability theorem
s_beck_monadicity_crude [theorem] Beck's monadicity theorem (crude form) | aliases: 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 | aliases: 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) | aliases: 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 | aliases: CCC
s_internal_hom [axiom] Internal hom (exponential object) | aliases: 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 | aliases: 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 | aliases: 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) | aliases: 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 | aliases: 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 | aliases: SMC
s_braided_monoidal_category [axiom] Braided monoidal category | aliases: 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 | aliases: V-category, V-enriched category
s_2_category [axiom] 2-category | aliases: strict 2-category
s_elementary_topos [axiom] Elementary topos | aliases: topos, elementary topos
s_subobject_classifier [axiom] Subobject classifier
s_grothendieck_topos [axiom] Grothendieck topos | aliases: 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 | aliases: contragredient representation, dual representation
s_hom_representation [state] Hom representation | aliases: Hom(V,W) representation
s_space_of_invariants [state] Space of invariants V^G | aliases: G-invariants, fixed subspace
s_second_orthogonality_relation [theorem] Second orthogonality relation for characters | aliases: column orthogonality
s_character_determines_representation [theorem] Character determines representation
s_projection_formula_representation [state] Projection formula for representations | aliases: 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 | aliases: standard representation of symmetric group
s_sign_representation [state] Sign (alternating) representation | aliases: alternating representation, signature representation, sgn representation
s_character_of_induced_representation [theorem] Character of induced representation | aliases: induced character formula
s_conjugate_partition [state] Conjugate (transpose) partition | aliases: transpose partition, dual partition
s_row_group_column_group [state] Row group and column group of a Young tableau | aliases: row stabilizer, column stabilizer
s_young_symmetrizer [state] Young symmetrizer | aliases: Young idempotent
s_specht_module [state] Specht module V_lambda | aliases: Specht representation
s_branching_rule_S_d [theorem] Branching rule for S_d | aliases: 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 | aliases: Murnaghan-Nakayama formula
s_youngs_rule [theorem] Young's rule
s_semistandard_young_tableau [axiom] Semistandard Young tableau | aliases: SSYT, column-strict tableau
s_young_subgroup [state] Young subgroup | aliases: 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 | aliases: class sums
s_idempotents_in_group_algebra [state] Primitive idempotents in the group algebra | aliases: primitive central idempotents
s_schur_weyl_duality [theorem] Schur--Weyl duality | aliases: Schur-Weyl duality
s_plethysm [state] Plethysm
s_matrix_lie_group [axiom] Matrix Lie group | aliases: 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 | aliases: Lie group representation
s_lie_bracket_of_matrices [state] Lie bracket of matrices | aliases: 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 | aliases: Lie's third theorem for representations
s_sl2_C [state] The Lie algebra sl_2(C) | aliases: sl(2,C), sl_2
s_standard_basis_sl2 [state] Standard basis {H,E,F} of sl_2
s_heisenberg_algebra [state] Heisenberg algebra | aliases: Heisenberg Lie algebra
s_irreducible_representations_of_sl2 [theorem] Classification of irreducible representations of sl_2 | aliases: irreps of sl(2)
s_highest_weight_vector [axiom] Highest weight vector | aliases: maximal vector, primitive vector
s_sl3_C [state] The Lie algebra sl_3(C) | aliases: sl(3,C), sl_3
s_root_and_root_space [axiom] Root and root space | aliases: root of a Lie algebra
s_weight_diagram [state] Weight diagram | aliases: weight lattice diagram
s_partial_order_on_weights [state] Partial order on weights | aliases: dominance order on weights
s_weight_lattice [state] Weight lattice | aliases: integral weight lattice
s_weyl_group_action_on_weights [theorem] Weyl group action on weights
s_weyl_vector_rho [state] Weyl vector rho | aliases: half-sum of positive roots
s_rank_of_lie_algebra [state] Rank of a Lie algebra | aliases: 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 | aliases: accidental isomorphisms
s_spin_representation_odd [state] Spin representation of so_{2m+1} | aliases: spinor representation of type B
s_half_spin_representations [state] Half-spin representations of so_{2m} | aliases: chiral spinor representations, Weyl spinors
s_triality_so_8 [theorem] Triality for so_8 | aliases: D_4 triality
s_weyl_denominator_formula [theorem] Weyl denominator formula | aliases: Weyl denominator identity
s_steinbergs_formula [theorem] Steinberg's formula for tensor product decomposition | aliases: Steinberg tensor product multiplicity formula
s_klimyks_formula [theorem] Klimyk's formula | aliases: 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 | aliases: real forms of semisimple Lie algebras
s_split_real_form [state] Split real form | aliases: maximally split real form
s_satake_diagram [axiom] Satake diagram | aliases: Tits-Satake diagram
s_longest_element_w0 [state] Longest element w_0 of the Weyl group | aliases: 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 | aliases: PRV theorem, PRV conjecture
s_exceptional_lie_algebra_G2 [state] Exceptional Lie algebra G_2 | aliases: G2
s_exceptional_lie_algebra_F4 [state] Exceptional Lie algebra F_4 | aliases: F4
s_exceptional_lie_algebra_E6 [state] Exceptional Lie algebra E_6 | aliases: E6
s_exceptional_lie_algebra_E7 [state] Exceptional Lie algebra E_7 | aliases: E7
s_exceptional_lie_algebra_E8 [state] Exceptional Lie algebra E_8 | aliases: E8
t_decomposition_via_inner_products [technique] Decomposition via character inner products
t_weight_string_analysis [technique] Weight string analysis | aliases: 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) | aliases: sl_n, type A_n Lie algebra
s_classical_lie_algebras_bcd [axiom] Classical Lie algebras (types B, C, D) | aliases: 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 | aliases: simple reflection, Weyl reflection
s_root_string_through_beta [state] Root string | aliases: α-string through β
s_height_of_root [state] Height of a root
s_fundamental_weyl_chamber [state] Fundamental Weyl chamber | aliases: 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 | aliases: 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) | aliases: irreducible highest weight module
s_dot_action_shifted_weyl [state] Dot action (shifted Weyl group action) | aliases: 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 | aliases: Kostant integral form
s_root_subgroup [state] Root subgroup of a Chevalley group
s_chevalley_group_steinberg_relations [state] Steinberg relations in Chevalley groups | aliases: Chevalley commutator formula
s_dual_coxeter_number [state] Dual Coxeter number
s_highest_root_maximal [state] Highest root | aliases: 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 | aliases: 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 | aliases: probability triple
s_probability_measure [axiom] Probability measure
s_random_variable [axiom] Random variable | aliases: measurable function to ℝ
s_law_of_random_variable [axiom] Distribution (law) of a random variable | aliases: law of X, push-forward measure
s_independence_of_events [axiom] Independence of events | aliases: mutual independence of events
s_independence_of_random_variables [axiom] Independence of random variables
s_almost_sure_convergence [axiom] Almost sure convergence | aliases: convergence with probability one, a.s. convergence
s_convergence_in_probability [axiom] Convergence in probability
s_convergence_in_distribution [axiom] Convergence in distribution | aliases: weak convergence of random variables, convergence in law
s_characteristic_function [axiom] Characteristic function | aliases: Fourier transform of a distribution
s_stable_distribution [axiom] Stable distribution | aliases: α-stable law, Lévy stable distribution
s_domain_of_attraction [axiom] Domain of attraction | aliases: domain of normal attraction
s_tightness_of_measures [axiom] Tightness of a family of probability measures | aliases: uniform tightness
s_poisson_limit_theorem [theorem] Poisson limit theorem | aliases: law of rare events, Poisson approximation theorem
s_local_central_limit_theorem [theorem] Local central limit theorem | aliases: local CLT, local limit theorem
s_strong_markov_property_rw [theorem] Strong Markov property for random walks | aliases: strong Markov property (discrete)
s_arc_sine_law [theorem] Arc sine law | aliases: arc sine law for random walks, Lévy's arc sine law
s_gambler_ruin [theorem] Gambler's ruin | aliases: gambler's ruin problem
s_ladder_variables [axiom] Ladder variables | aliases: 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 | aliases: law of total expectation, iterated expectations
s_ui_and_l1_convergence [theorem] Uniform integrability and L¹ convergence | aliases: UI characterization of L¹ convergence
s_doob_decomposition [theorem] Doob's decomposition theorem (discrete) | aliases: Doob decomposition
s_backward_martingale_convergence [theorem] Backward martingale convergence theorem | aliases: reversed martingale convergence
s_levy_zero_one_law [theorem] Lévy's 0-1 law
s_branching_process [axiom] Branching process (Galton-Watson) | aliases: Galton-Watson process
s_branching_process_extinction [theorem] Extinction criterion for branching processes | aliases: Galton-Watson extinction theorem
s_martingale_clt [theorem] Martingale central limit theorem | aliases: CLT for martingale differences
s_exponential_martingale [state] Exponential martingale | aliases: Wald's exponential martingale
s_markov_chain [axiom] Markov chain | aliases: discrete-time Markov chain, DTMC
s_transition_matrix [axiom] Transition matrix | aliases: stochastic matrix, transition probability matrix
s_chapman_kolmogorov_equations [theorem] Chapman-Kolmogorov equations
s_markov_property [theorem] Markov property | aliases: 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 | aliases: irreducible chain
s_recurrence_mc [axiom] Recurrence for Markov chains | aliases: recurrent state
s_transience_mc [axiom] Transience for Markov chains | aliases: transient state
s_recurrence_is_class_property [theorem] Recurrence is a class property
s_positive_recurrence [axiom] Positive recurrence | aliases: positive recurrent state, ergodic state
s_null_recurrence [axiom] Null recurrence | aliases: null recurrent state
s_period_of_state [axiom] Period of a state | aliases: 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 | aliases: basic limit theorem for Markov chains
s_detailed_balance [axiom] Detailed balance | aliases: reversibility condition, time-reversibility
s_ratio_limit_theorem [theorem] Ratio limit theorem
s_mean_return_time [state] Mean return time | aliases: expected return time
s_absorbing_state [axiom] Absorbing state | aliases: absorbing barrier
s_ergodicity_definition [axiom] Ergodicity | aliases: metric transitivity, ergodic transformation
s_maximal_ergodic_lemma [theorem] Maximal ergodic lemma | aliases: Hopf maximal ergodic lemma
s_strong_mixing [axiom] Strong mixing | aliases: mixing
s_stationary_process [axiom] Stationary process | aliases: strictly stationary process
s_shannon_mcmillan_breiman [theorem] Shannon-McMillan-Breiman theorem | aliases: asymptotic equipartition property, AEP
s_kac_lemma [theorem] Kac's lemma | aliases: Kac's recurrence theorem
s_kolmogorov_centsov_continuity [theorem] Kolmogorov-Čentsov continuity theorem | aliases: 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 | aliases: Lévy's arc sine laws
s_nowhere_differentiability_bm [theorem] Nowhere differentiability of Brownian motion | aliases: Paley-Wiener-Zygmund theorem
s_quadratic_variation_bm [theorem] Quadratic variation of Brownian motion
s_bm_harmonic_functions [theorem] Brownian motion and harmonic functions | aliases: 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 | aliases: BM inversion
s_bm_scaling_property [theorem] Scaling property of Brownian motion | aliases: 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 | aliases: Lévy-Ciesielski construction, Wiener's construction
s_hitting_times_bm [state] Hitting times of Brownian motion | aliases: 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 | aliases: Chebyshev-Lobatto points, Chebyshev extreme points, Chebyshev grid
s_chebyshev_three_term_recurrence [theorem] Chebyshev three-term recurrence | aliases: Recurrence for Chebyshev polynomials
s_chebyshev_orthogonality [theorem] Chebyshev orthogonality | aliases: Orthogonality of Chebyshev polynomials
s_chebyshev_series [state] Chebyshev series | aliases: Chebyshev expansion (series object)
s_equivalence_chebyshev_fourier_cosine [theorem] Equivalence of Chebyshev and Fourier cosine series | aliases: Chebyshev-Fourier correspondence, cos-substitution equivalence
s_chebyshev_projection [state] Chebyshev projection (truncated Chebyshev series) | aliases: Truncated Chebyshev series, Chebyshev L2 projection
s_chebyshev_aliasing [theorem] Aliasing of Chebyshev coefficients | aliases: Chebyshev aliasing, Aliasing relation interpolant vs projection, Trigonometric aliasing on Chebyshev grid
t_barycentric_interpolation_formula [technique] Barycentric interpolation formula | aliases: Second barycentric formula, Barycentric Lagrange interpolation
s_chebyshev_barycentric_weights [state] Barycentric weights for Chebyshev points | aliases: Chebyshev barycentric weights
s_numerical_stability_barycentric [theorem] Numerical stability of the barycentric formula | aliases: Backward stability of barycentric interpolation
s_bernstein_ellipse [axiom] Bernstein ellipse | aliases: Bernstein regularity ellipse, ρ-ellipse
t_joukowski_map [technique] Joukowski map | aliases: Joukowsky map, Zhukovsky transform
s_chebyshev_coeff_decay_bv [theorem] Chebyshev coefficient decay for bounded-variation functions | aliases: BV Chebyshev coefficient bound
s_chebyshev_coeff_decay_differentiable [theorem] Chebyshev coefficient decay for differentiable functions | aliases: Algebraic Chebyshev coefficient decay
s_chebyshev_coeff_decay_analytic [theorem] Chebyshev coefficient decay for analytic functions | aliases: Geometric Chebyshev coefficient decay
s_geometric_convergence_analytic_chebyshev [theorem] Geometric convergence of Chebyshev approximation for analytic functions | aliases: Spectral/geometric convergence of Chebyshev interpolation
s_convergence_rate_differentiable_chebyshev [theorem] Algebraic convergence of Chebyshev approximation for differentiable functions | aliases: Finite-smoothness Chebyshev convergence rate
s_optimal_chebyshev_rate_rho [theorem] Estimate of optimal Chebyshev convergence rate ρ | aliases: Root test for Chebyshev convergence rate
s_lebesgue_constant_interpolation [state] Lebesgue constant (interpolation operator norm) | aliases: Lebesgue constant for interpolation
s_lebesgue_constant_chebyshev_points [theorem] Lebesgue constant for Chebyshev points | aliases: Logarithmic growth of Chebyshev Lebesgue constant
s_lebesgue_constant_equispaced_points [theorem] Lebesgue constant for equispaced points | aliases: Exponential growth of equispaced Lebesgue constant
s_lebesgue_constant_bound [theorem] Lebesgue constant interpolation error bound | aliases: Near-best interpolation bound, Operator-norm inequality for interpolation
s_conditioning_polynomial_interpolation [theorem] Conditioning of polynomial interpolation | aliases: Stability of polynomial interpolation
s_existence_uniqueness_best_approximation [theorem] Existence and uniqueness of best polynomial approximation | aliases: Uniqueness of minimax polynomial
s_de_la_vallee_poussin_alternation_lemma [theorem] de la Vallée Poussin lower bound (alternation lemma) | aliases: Vallée-Poussin alternation lemma, de la Vallée Poussin lower bound
s_haar_condition [axiom] Haar condition (Haar/Chebyshev system) | aliases: Chebyshev system, Haar system
s_linear_approximation_scheme [axiom] Linear approximation scheme | aliases: Linear approximation
s_nonlinear_approximation [axiom] Nonlinear approximation | aliases: Nonlinear approximation theory
s_node_polynomial [state] Node polynomial | aliases: Nodal polynomial
s_hermite_integral_formula [theorem] Hermite integral formula (Cauchy interpolation error formula) | aliases: Cauchy interpolation formula, Hermite-Cauchy error formula
t_potential_theory_of_approximation [technique] Potential theory of polynomial approximation | aliases: Potential-theoretic convergence analysis
s_green_function_complement_log_singularity [state] Green's function of the complement with logarithmic pole | aliases: Green's function with logarithmic singularity, complement Green's function
s_convergence_rate_from_green_function [theorem] Convergence rate from Green's function | aliases: Geometric convergence region (potential theory)
s_equispaced_equilibrium_potential [state] Equispaced equilibrium potential | aliases: Uniform-density logarithmic potential
s_impossibility_stable_equispaced_approximation [theorem] Impossibility of stable high-order equispaced approximation | aliases: No stable geometric equispaced reconstruction, Platte-Trefethen-Kuijlaars theorem
s_interlacing_orthogonal_polynomial_roots [theorem] Interlacing of orthogonal polynomial roots | aliases: Separation/interlacing of orthogonal polynomial zeros
s_colleague_matrix [state] Colleague matrix | aliases: Chebyshev companion matrix
t_rootfinding_via_eigenvalues [technique] Rootfinding via eigenvalues (colleague/companion method) | aliases: Eigenvalue rootfinding, companion matrix rootfinding
s_max_degree_exactness_gauss_optimality [theorem] Maximal algebraic degree of exactness (Gauss optimality) | aliases: Gauss quadrature optimality
t_golub_welsch_algorithm [technique] Golub-Welsch algorithm | aliases: Golub-Welsch, Jacobi matrix quadrature
s_comparable_accuracy_cc_gauss [theorem] Comparable accuracy of Clenshaw-Curtis and Gauss quadrature | aliases: Trefethen's observation on Clenshaw-Curtis vs Gauss
t_caratheodory_fejer_approximation [technique] Carathéodory-Fejér (CF) approximation | aliases: CF approximation, Caratheodory-Fejer method
s_cf_hankel_singular_value_identity [theorem] CF singular value / Hankel eigenvalue identity | aliases: Hankel singular value = best approximation error
s_aak_theory [theorem] AAK (Adamyan-Arov-Krein) theory | aliases: Adamyan-Arov-Krein theorem, AAK theorem
s_rational_best_approximation_type_mn [state] Rational best approximation of type (m,n) | aliases: Best rational minimax approximant, R_{mn} best approximation
s_rational_equioscillation_characterization [theorem] Rational equioscillation characterization theorem | aliases: Rational Chebyshev characterization
s_defect_of_rational_approximant [axiom] Defect of a rational approximant | aliases: Rational approximation defect
s_existence_uniqueness_rational_best_approximation [theorem] Existence and uniqueness of rational best approximation | aliases: Rational best approximation existence/uniqueness
s_root_exponential_convergence_rational [theorem] Root-exponential convergence of rational approximation | aliases: exp(−C√n) rational convergence
t_pole_clustering [technique] Pole clustering near singularities | aliases: Exponential pole clustering, lightning/AAA pole placement
s_newman_theorem_abs_x [theorem] Newman's theorem (rational approximation of |x|) | aliases: Newman rational approximation of |x|
s_newman_construction_points [state] Newman's construction (Newman points) | aliases: Newman points, Newman rational
s_stahl_theorem_sharp_rates [theorem] Stahl's theorem (sharp |x| and exp rational rates) | aliases: Stahl asymptotics, Halphen constant rate
s_zolotarev_first_problem [theorem] Zolotarev's first problem (best rational approximation of sign) | aliases: Zolotarev sign approximation
s_zolotarev_problems [theorem] Zolotarev problems (elliptic extremal rational functions) | aliases: Four Zolotarev problems
s_chebyshev_pade_approximation [state] Chebyshev-Padé (Clenshaw-Lord) approximation | aliases: Clenshaw-Lord approximation, Chebyshev-Pade
s_convergence_pade_pole_detection [theorem] Convergence of Padé approximants and pole detection | aliases: Montessus de Ballore / Stahl convergence, Padé pole detection
t_analytic_continuation_via_rational_approximation [technique] Analytic continuation via rational approximation | aliases: Rational analytic continuation, Padé continuation
s_froissart_doublets [axiom] Froissart doublets (spurious poles) | aliases: Spurious poles, Froissart doublets
s_quasimatrix [axiom] Quasimatrix | aliases: Continuous-analogue matrix
s_chebfun [state] Chebfun (numerical function representation) | aliases: Chebfun representation
t_chebfun_adaptive_chopping [technique] Chebfun adaptive degree selection (chopping) | aliases: Chopping, standardChop, resolution test
s_spectral_differentiation_matrix [state] Spectral differentiation matrix | aliases: Chebyshev differentiation matrix, spectral derivative matrix
s_chebyshev_differentiation_matrix_entries [theorem] Chebyshev differentiation matrix entries formula | aliases: Negative sum trick, explicit Chebyshev D matrix
s_spectral_accuracy [theorem] Spectral accuracy (spectral convergence) | aliases: Spectral convergence, exponential accuracy of spectral methods
t_spectral_collocation_method [technique] Spectral collocation method for differential equations | aliases: 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) | aliases: TC(x)
s_burali_forti_paradox [theorem] Burali-Forti paradox | aliases: Burali-Forti
s_ordinal_arithmetic [state] Ordinal arithmetic | aliases: ordinal addition, ordinal multiplication
s_cantor_normal_form [theorem] Cantor normal form | aliases: CNF
s_epsilon_numbers [state] Epsilon numbers (ε-numbers) | aliases: ε-numbers, ε₀
s_order_type [state] Order type | aliases: otp
s_rank_function [state] Rank function | aliases: set-theoretic rank
t_epsilon_induction [technique] ∈-induction (Foundation induction)
s_well_founded_relation [axiom] Well-founded relation | aliases: well-founded
t_well_founded_recursion [technique] Well-founded recursion
s_hereditarily_small_sets_H_kappa [state] Hereditarily small sets H(κ) | aliases: H(κ), H_κ
s_aleph_function [state] Aleph function ℵ_α | aliases: ℵ_α
s_initial_ordinal [state] Initial ordinal
s_cardinal_arithmetic [state] Cardinal arithmetic | aliases: κ+λ, κ·λ, κ^λ
s_hessenberg_kappa_squared [theorem] Hessenberg's theorem (κ²=κ) | aliases: κ²=κ
s_godel_pairing_function [state] Gödel pairing function | aliases: canonical well-ordering of pairs
s_cofinality [state] Cofinality cf(α) | aliases: cf(α)
s_regular_cardinal [axiom] Regular cardinal | aliases: regular
s_singular_cardinal [axiom] Singular cardinal | aliases: singular
s_successor_cardinals_regular [theorem] Successor cardinals are regular
s_continuum_function [state] Continuum function | aliases: 2^κ
s_hausdorff_formula [theorem] Hausdorff formula
s_gimel_function [state] Gimel function ℷ(κ) | aliases: ℷ, gimel
s_beth_function [state] Beth function ℶ_α | aliases: ℶ_α, beth numbers
s_generalized_continuum_hypothesis [state] Generalized Continuum Hypothesis (GCH) | aliases: GCH
s_tukey_lemma [theorem] Tukey's lemma
s_trichotomy_of_cardinals [theorem] Trichotomy of cardinals | aliases: cardinal comparability
s_ideal_set_theory [axiom] Ideal (set-theoretic) | aliases: dual ideal
s_kappa_complete_filter [axiom] κ-complete filter | aliases: κ-complete
s_principal_filter [state] Principal vs non-principal ultrafilter | aliases: principal ultrafilter, free ultrafilter
s_boolean_prime_ideal_theorem [theorem] Boolean Prime Ideal Theorem | aliases: BPIT, ultrafilter theorem
s_complete_boolean_algebra [axiom] Complete Boolean algebra | aliases: cBa
s_regular_open_algebra [state] Regular open algebra RO(P) | aliases: RO(P)
s_club_set [state] Closed unbounded (club) set | aliases: club, closed unbounded set
s_club_filter [state] Club filter
s_normal_filter [state] Normal filter | aliases: normal
s_club_intersection_theorem [theorem] Intersection of clubs is club
s_tree_set_theoretic [axiom] Tree (set-theoretic) | aliases: set-theoretic tree
s_tree_branch_level_height [state] Branch, level, and height of a tree | aliases: branch, level, height
s_aronszajn_tree [state] Aronszajn tree | aliases: 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) | aliases: 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 | aliases: Kurepa's hypothesis, KH
s_special_aronszajn_tree [state] Special Aronszajn tree
s_partition_calculus [state] Partition calculus (arrow notation) | aliases: arrow notation, κ→(λ)^n_m
s_square_principle [axiom] Square principle □_κ (Jensen) | aliases: □_κ, Jensen square
s_club_principle [state] Club principle ♣ | aliases: ♣, clubsuit
s_diamond_implies_ch [theorem] ◊ implies CH
s_singular_cardinal_hypothesis [state] Singular Cardinal Hypothesis (SCH) | aliases: SCH
s_levy_hierarchy [state] Lévy hierarchy (Σ_n/Π_n) | aliases: 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 | aliases: <_L
s_ac_in_L [theorem] AC holds in L
s_relative_constructibility_L_A [state] Relative constructibility L[A] | aliases: L[A]
s_L_of_A [state] Constructible closure L(A) | aliases: 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 | aliases: J_α, J-hierarchy
s_sigma_n_projectum [state] Σ_n-projectum ρ_n | aliases: 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 | aliases: forcing poset, forcing notion
s_dense_predense [state] Dense and predense sets | aliases: dense set, predense
s_compatible_conditions [state] Compatible and incompatible conditions | aliases: 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 | aliases: 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) | aliases: 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 | aliases: 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 | aliases: 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 | aliases: 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 | aliases: MA_κ
s_martins_axiom [axiom] Martin's Axiom (MA) | aliases: 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 | aliases: Mahlo
s_weakly_compact_cardinal [state] Weakly compact cardinal | aliases: weakly compact
s_weakly_compact_tree_property [theorem] Tree property characterization of weak compactness
s_normal_measure [state] Normal measure | aliases: normal ultrafilter
t_ultrapower_Ult_V_U [technique] Ultrapower Ult(V,U)
s_measurable_embedding_characterization [theorem] Measurable cardinals and elementary embeddings | aliases: critical point characterization
t_iterated_ultrapowers [technique] Iterated ultrapowers
s_ramsey_cardinal [state] Ramsey cardinal | aliases: Ramsey
s_erdos_cardinal [state] Erdős cardinal κ→(α)^{<ω} | aliases: 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 | aliases: strongly compact
s_huge_cardinal [state] Huge cardinal | aliases: huge
t_extenders [technique] Extenders
s_vopenka_principle [axiom] Vopěnka's principle | aliases: Vopěnka
s_proper_forcing [state] Proper forcing | aliases: proper
s_cantor_space [state] Cantor space 2^ω | aliases: 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] | aliases: [T], tree on ω
s_well_founded_tree_rank [state] Well-founded tree and its rank | aliases: tree rank
s_borel_hierarchy [state] Borel hierarchy (Σ⁰_α,Π⁰_α,Δ⁰_α) | aliases: Σ⁰_α, Π⁰_α
t_borel_codes [technique] Borel codes
s_property_of_baire [state] Property of Baire | aliases: Baire property, BP
t_suslin_operation [technique] Suslin operation 𝒜
s_coanalytic_set [state] Coanalytic set Π¹₁ | aliases: Π¹₁
s_cantor_bendixson_rank [technique] Cantor–Bendixson derivative and rank | aliases: CB derivative, CB rank
s_perfect_set_property [state] Perfect set property (PSP) | aliases: 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 | aliases: Σ¹₁-bounding
s_projective_hierarchy [state] Projective hierarchy (Σ¹_n,Π¹_n,Δ¹_n) | aliases: 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 | aliases: PWO
s_scale_property [state] Scale property | aliases: scale
s_periodicity_theorems [theorem] Periodicity theorems (Moschovakis) | aliases: 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 | aliases: ℝ∩L
s_infinite_game [state] Infinite game G(A) and strategy | aliases: G(A), winning strategy
s_determinacy_of_a_set [state] Determinacy of a set Det(A) | aliases: Det(A)
s_gale_stewart_theorem [theorem] Gale–Stewart theorem | aliases: 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 | aliases: Σ¹₁-determinacy
s_sharps_analytic_determinacy_equivalence [state] Sharps and analytic determinacy equivalence | aliases: x#
t_homogeneous_tree [technique] Homogeneously Suslin set / homogeneous tree
s_martin_steel_pd [theorem] Martin–Steel theorem (projective determinacy) | aliases: 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 | aliases: ≤_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 | aliases: transitive set
s_delta_0_formula [state] Δ₀ (bounded) formula | aliases: Σ₀ formula, Π₀ formula, bounded formula, Delta-0 formula
t_relativization [technique] Relativization φᴹ
s_diamond_plus [axiom] ◊⁺ (diamond-plus) | aliases: diamond-plus, ◊+
s_diamond_plus_kurepa [theorem] ◊⁺ implies a Kurepa tree (Jensen) | aliases: 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 | aliases: language, vocabulary
s_domain_universe [axiom] Domain (universe) | aliases: 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) | aliases: 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) | aliases: literal diagram
s_canonical_term_structure [state] Canonical (term) structure | aliases: 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 | aliases: closed formula
s_parametrically_definable_set [state] Parametrically definable set | aliases: 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) | aliases: potential isomorphism
s_cantor_dlo_theorem [theorem] Cantor's theorem on DLO
s_ehrenfeucht_fraisse_game [state] Ehrenfeucht–Fraïssé game | aliases: EF game
s_ef_quantifier_rank_theorem [theorem] Ehrenfeucht–Fraïssé theorem | aliases: Fraïssé's theorem on games
s_quantifier_rank [state] Quantifier rank | aliases: quantifier depth
s_hintikka_formulas [state] Hintikka formulas | aliases: Hintikka normal form
s_age_of_structure [state] Age of a structure
s_hereditary_property [axiom] Hereditary property (HP) | aliases: HP
s_joint_embedding_property [axiom] Joint embedding property (JEP) | aliases: JEP
s_amalgamation_property [axiom] Amalgamation property (AP) | aliases: AP
s_fraisse_limit [state] Fraïssé limit
s_ultrahomogeneous_structure [axiom] Ultrahomogeneous structure | aliases: homogeneous structure
s_fraisse_theorem [theorem] Fraïssé's theorem
s_weak_homogeneity [axiom] Weak homogeneity
s_random_graph [state] The random graph (Rado graph) | aliases: Rado graph
s_extension_property_graphs [axiom] Extension property (graphs) | aliases: 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 | aliases: topology of pointwise convergence
s_closed_subgroup_aut_theorem [theorem] Closed-subgroup characterization of automorphism groups
s_k_transitivity [state] k-homogeneity / k-transitivity | aliases: k-homogeneity
s_imaginary_elements [state] Imaginary elements | aliases: 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 | aliases: Meq
s_elimination_of_imaginaries [axiom] Elimination of imaginaries
s_pseudo_elementary_class [axiom] Pseudo-elementary class (PC) | aliases: PC class
t_word_construction_interpretation [technique] Word construction interpreting groups/fields
s_completeness_theorem [theorem] Gödel completeness theorem | aliases: Gödel's completeness theorem
s_direct_reduced_product [state] Direct and reduced product | aliases: 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 | aliases: 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 | aliases: ℵ₀-saturated model
s_aleph0_categorical_theory [axiom] ℵ₀-categorical theory | aliases: omega-categorical theory
s_oligomorphic_permutation_group [axiom] Oligomorphic permutation group
s_existentially_closed_structure [axiom] Existentially closed structure | aliases: e.c. structure
s_inductive_theory [axiom] Inductive theory | aliases: inductive class
s_existence_ec_models [theorem] Existence of existentially closed models
s_model_companion [state] Model companion / model completion | aliases: model completion
s_existence_uniqueness_model_companion [theorem] Existence and uniqueness of the model companion
s_quantifier_elimination [axiom] Quantifier elimination | aliases: 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 | aliases: stable theory
s_stable_superstable_omega_stable [state] Stable / superstable / ω-stable hierarchy | aliases: superstable theory
s_order_property [axiom] Order property | aliases: no order property
s_indiscernible_sequence [axiom] Indiscernible sequence | aliases: order-indiscernibles
s_set_indiscernibles [axiom] Set indiscernibles
s_ramsey_erdos_rado [theorem] Ramsey / Erdős–Rado theorem | aliases: Ramsey's theorem
s_em_model_functor [state] EM model / functor | aliases: EM functor
t_em_construction [technique] EM construction technique
s_forking [state] Forking | aliases: 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 | aliases: partial recursive function, φ_e
s_total_computable_function [axiom] Total computable function | aliases: total recursive function, computable function
s_primitive_recursion_schema [axiom] Primitive recursion schema | aliases: primitive recursion
s_mu_operator_minimization [axiom] μ-operator (unbounded minimization) | aliases: minimization, unbounded search, μ
s_partial_recursive_functions [axiom] Partial recursive functions | aliases: μ-recursive functions
s_register_machine [axiom] Register machine | aliases: unlimited register machine, URM, counter machine
s_church_turing_thesis [axiom] Church–Turing Thesis | aliases: Church's thesis, Turing's thesis
s_computability_thesis_soare [axiom] Computability Thesis (Soare) | aliases: Soare's thesis
s_oracle_turing_machine [axiom] Oracle Turing machine | aliases: o-machine, relativized Turing machine
s_use_function [axiom] Use function | aliases: use, u(A;e,x)
s_godel_index [axiom] Gödel number / index | aliases: index, program number, e
s_turing_jump_A_prime [axiom] Turing jump A′ | aliases: jump, A'
s_arithmetical_hierarchy [axiom] Arithmetical hierarchy | aliases: Kleene arithmetical hierarchy, Σ0n/Π0n/Δ0n
s_many_one_reducibility [axiom] Many-one (m) reducibility | aliases: m-reducibility, ≤m
s_one_one_reducibility [axiom] One-one (1) reducibility | aliases: 1-reducibility, ≤1
s_truth_table_reducibility [axiom] Truth-table (tt) reducibility | aliases: tt-reducibility, ≤tt
s_weak_truth_table_reducibility [axiom] Weak truth-table (wtt) reducibility | aliases: 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 | aliases: K_0, general halting set
s_hypersimple_set [state] Hypersimple set
s_hyperimmune_set [state] Hyperimmune set | aliases: h-immune set
s_hyperhypersimple_set [state] Hyperhypersimple set | aliases: hhsimple set
s_maximal_set [state] Maximal set
s_stage_approximation_Phi [state] Stage approximation Φ_{e,s}^A(x) | aliases: finite-stage approximation, Φ_{e,s}
s_enumeration_theorem [theorem] Enumeration Theorem | aliases: Universal function theorem
s_kleene_normal_form_theorem [theorem] Kleene Normal Form Theorem | aliases: Normal form theorem, Kleene T-predicate
s_ce_equals_sigma01 [theorem] c.e. = Σ0_1 characterization | aliases: 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 | aliases: Complementation theorem, Post's theorem (computable iff c.e. and co-c.e.)
s_recursion_theorem_with_parameters [theorem] Recursion Theorem with parameters | aliases: uniform recursion theorem, parametrized fixed-point theorem
s_myhill_isomorphism_theorem [theorem] Myhill Isomorphism Theorem | aliases: Myhill's theorem
s_myhill_creative_characterization [theorem] Myhill characterization of creative sets | aliases: creative = m-complete
s_m_completeness_of_K [theorem] m-completeness of K | aliases: K is m-complete
s_jump_theorem [theorem] Jump Theorem | aliases: properties of the jump, jump monotonicity
s_posts_theorem [theorem] Post's Theorem | aliases: Post's theorem (hierarchy/jump)
s_arithmetical_hierarchy_theorem [theorem] Hierarchy Theorem (arithmetical hierarchy is proper) | aliases: arithmetical hierarchy proper, strictness of the arithmetical hierarchy
s_limit_lemma [theorem] Shoenfield Limit Lemma | aliases: Limit lemma, Shoenfield's lemma, Δ02 = computable in 0'
s_modulus_lemma [theorem] Modulus Lemma | aliases: modulus of convergence lemma
s_post_simple_set_existence [theorem] Existence of a simple set (Post) | aliases: Post's simple set
s_productive_not_ce [theorem] Productive sets are not c.e.
s_relativization_principle [theorem] Relativization principle | aliases: relativization
t_dovetailing [technique] Dovetailing | aliases: interleaving computations
t_pairing_function [technique] Pairing function / coding of tuples | aliases: Cantor pairing, sequence coding, tupling
t_construction_by_stages_markers [technique] Construction by stages with movable markers | aliases: movable markers, stage construction
t_permitting [technique] Permitting method | aliases: permitting argument, C-permitting
t_reduction_to_K [technique] Reduction to K (proving undecidability) | aliases: reduction from the halting problem
t_computable_approximation_delta02 [technique] Computable approximation (Δ0_2 method) | aliases: finite approximation method, stage approximation method
s_turing_equivalence [axiom] Turing equivalence (≡_T) | aliases: ≡T
s_turing_degree [axiom] Turing degree | aliases: degree of unsolvability, T-degree
s_degree_structure_D [axiom] Degree structure D | aliases: the degrees, D, global degree structure
s_join_of_degrees [state] Join of degrees (a∨b) | aliases: least upper bound of degrees, a∨b, A⊕B
s_degree_0 [state] Degree 0 | aliases: zero degree, the computable degree
s_degree_0_prime [state] Degree 0′ (zero jump) | aliases: 0', the halting degree, zero jump
s_jump_hierarchy [state] Jump hierarchy (0,0′,0″,…) | aliases: iterated jumps, 0,0',0''
s_ce_degrees_R [state] c.e. degrees R | aliases: R, the c.e. degrees, r.e. degrees
s_low_set [state] Low set / degree | aliases: low degree, low
s_lown_degree [state] Low_n degree | aliases: low_n, Low_n
s_high_set [state] High set / degree | aliases: high degree, high
s_highn_degree [state] High_n degree | aliases: high_n, High_n
s_high_low_hierarchy [state] High/Low hierarchy | aliases: high/low hierarchy
s_superlow_degree [state] Superlow degree | aliases: superlow
s_hyperimmune_free_degree [state] Hyperimmune-free degree | aliases: computably dominated degree, hyperimmune-free
s_hyperimmune_degree [state] Hyperimmune degree | aliases: hyperimmune degree
s_GL1_degree [state] GL1 degree (generalized low 1) | aliases: generalized low, GL_1, GL1
s_GL2_degree [state] GL2 degree (generalized low 2) | aliases: generalized low 2, GL_2, GL2
s_minimal_degree [state] Minimal degree | aliases: degree minimal over 0
s_1_generic_set [state] 1-generic set / degree | aliases: 1-generic, weakly 1-generic
s_n_generic_degree [state] n-generic set / degree | aliases: n-generic
s_ershov_hierarchy [state] Ershov hierarchy | aliases: difference hierarchy, Boolean hierarchy over c.e. sets
s_n_ce_set [state] n-c.e. set | aliases: n-r.e. set, n-c.e.
s_dce_set [state] d.c.e. set / degree | aliases: difference c.e., 2-c.e. set, d-c.e.
s_e_state [state] e-state | aliases: e-state of an element
s_posts_problem [state] Post's Problem | aliases: Post problem
s_kleene_post_theorem [theorem] Kleene–Post Theorem | aliases: Kleene–Post incomparable degrees below 0'
s_incomparable_ce_degrees [theorem] Existence of incomparable c.e. degrees | aliases: incomparable c.e. degrees
s_low_basis_theorem [theorem] Low Basis Theorem (Jockusch–Soare) | aliases: Jockusch–Soare low basis theorem
s_hyperimmune_free_basis_theorem [theorem] Hyperimmune-free Basis Theorem | aliases: Jockusch–Soare hyperimmune-free basis theorem
s_minimal_pair_theorem [theorem] Minimal Pair Theorem (Lachlan–Yates) | aliases: Lachlan–Yates minimal pair
s_spector_minimal_degree [theorem] Existence of a minimal degree (Spector) | aliases: Spector minimal degree theorem
s_exact_pair_theorem [theorem] Exact Pair Theorem | aliases: exact pair (Kleene–Post / Spector)
s_lattice_embedding_into_ce_degrees [theorem] Embedding lattices into the c.e. degrees | aliases: distributive lattice embedding theorem
s_nondistributivity_of_ce_degrees [theorem] Nondistributivity of the c.e. degrees (Lachlan) | aliases: Lachlan nondistributivity
s_nondiamond_theorem [theorem] Nondiamond Theorem (Lachlan) | aliases: Lachlan nondiamond theorem
s_robinson_low_splitting [theorem] Robinson Low Splitting / interpolation | aliases: Robinson splitting theorem, Robinson interpolation
s_friedberg_jump_inversion [theorem] Friedberg Jump Inversion Theorem | aliases: Friedberg completeness criterion, jump inversion (Friedberg)
t_finite_injury_priority_method [technique] Finite-injury priority method | aliases: finite injury method, Friedberg–Muchnik method
t_priority_requirements_and_injury [technique] Priority requirements and injury | aliases: injury, priority ordering
t_kleene_post_finite_extension [technique] Kleene–Post finite extension method | aliases: finite extension method, forcing the jump
t_length_of_agreement_function [technique] Length-of-agreement function | aliases: agreement length, ℓ(e,s)
t_restraint_function [technique] Restraint function | aliases: restraint, r(e,s)
t_hat_trick [technique] Hat trick | aliases: hatting, Â_s
t_infinite_injury_priority_method [technique] Infinite-injury priority method | aliases: infinite injury method, 0''-priority argument
t_tree_of_strategies [technique] Tree of strategies | aliases: strategy tree, tree method
s_true_path [state] True path | aliases: true path of the construction
t_thickness_lemma [technique] Thickness Lemma | aliases: thickness requirement
t_pinball_machine [technique] Pinball machine | aliases: pinball construction
t_0_triple_prime_priority_argument [technique] 0‴-priority argument | aliases: 0''' argument, triple-jump priority method
t_minimal_pair_construction [technique] Minimal pair construction | aliases: minimal pair method, gap-cogap argument
t_spector_tree_forcing [technique] Spector's construction (tree forcing) | aliases: perfect tree forcing, Spector forcing
t_cohen_forcing_in_arithmetic [technique] Cohen forcing in arithmetic | aliases: 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] (noname)
t_gauge_holonomy_transport [technique] (noname)
t_simplicial_euler_computation [technique] (noname)
t_homotopy_classification_order_parameter [technique] (noname)
t_virasoro_central_charge_computation [technique] (noname)
s_reduced_root_system [axiom] Reduced root system | aliases: reduced root system
s_existence_of_cartan_subalgebras [theorem] Existence of Cartan subalgebras
s_compact_lie_algebra [axiom] Compact Lie algebra | aliases: 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 | aliases: 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 | aliases: 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 | aliases: 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) | aliases: 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 | aliases: 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 | aliases: 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 | aliases: Borel-de Siebenthal procedure
s_maximal_abelian_subspace_of_p [state] Maximal abelian subspace a of p | aliases: maximal split abelian subalgebra
s_real_rank [state] Real rank of G | aliases: split rank, R-rank
s_restricted_roots [state] Restricted roots | aliases: relative roots, a-roots
s_restricted_root_space_decomposition [theorem] Restricted root space decomposition | aliases: a-root space decomposition
s_restricted_root_multiplicity [state] Restricted root multiplicity
s_restricted_root_system [theorem] Restricted root system Sigma | aliases: 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) | aliases: 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 | aliases: Iwasawa nilpotent subalgebra
s_iwasawa_decomposition_algebra [theorem] Iwasawa decomposition of the Lie algebra | aliases: g = k + a + n
s_iwasawa_decomposition_group_kan [theorem] Iwasawa decomposition G = KAN | aliases: 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) | aliases: 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 | aliases: 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 | aliases: minimal parabolic P = MAN
s_standard_parabolic_subgroups_real [theorem] Standard parabolic subgroups from subsets of simple restricted roots | aliases: standard parabolic subgroups
s_langlands_decomposition [theorem] Langlands decomposition of a parabolic | aliases: Levi decomposition of a real parabolic
s_iwasawa_projection_H [state] Iwasawa projection H(g) | aliases: Iwasawa a-projection, H(g) function
s_half_sum_of_positive_restricted_roots [state] Half-sum of positive restricted roots rho | aliases: restricted rho
s_integral_formula_kan [theorem] Integral formula for the KAN decomposition | aliases: Iwasawa integration formula
s_integral_formula_kak [theorem] Integral formula for the KAK decomposition | aliases: 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 | aliases: modular function of AN
s_unimodularity_of_reductive_groups [theorem] Unimodularity of semisimple and reductive groups | aliases: semisimple groups are unimodular
s_hermitian_symmetric_space [axiom] Hermitian symmetric space | aliases: 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 | aliases: Cartan domain, classical domain
s_harish_chandra_decomposition [theorem] Harish-Chandra decomposition g^C = p- + k^C + p+ | aliases: g^C = p- + k^C + p+
s_harish_chandra_embedding [theorem] Harish-Chandra embedding | aliases: Harish-Chandra realization
s_strongly_orthogonal_roots [state] Strongly orthogonal roots | aliases: strongly orthogonal noncompact roots
s_kostant_linear_convexity_theorem [theorem] Kostant's linear convexity theorem | aliases: Kostant convexity theorem, linear convexity theorem
s_opposite_nilpotent_subalgebra [state] Opposite nilpotent subalgebra n-bar = theta(n) | aliases: opposite nilradical, n-bar
s_open_bruhat_cell [theorem] Open Bruhat cell (big cell) | aliases: 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] | aliases: symmetric space curvature formula
s_siegel_domain_realization [state] Siegel domain (tube domain) realization | aliases: tube domain, Siegel domain of type I/II
t_cayley_transform_lie [technique] Cayley transform (Lie-theoretic) | aliases: 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