Tag: mathematics

Sun 30 August 2026 - A Tree Center Is One Vertex or One Edge
Mon 17 August 2026 - Euler's Pentagonal Theorem Expands the Partition Denominator
Fri 08 May 2026 - The Jacobian Criterion Detects Smooth Complete Intersections - updated: Mon 10 August 2026
Sat 08 August 2026 - Artin--Schreier Extensions Are Additive
Sun 25 June 2023 - Poisson Brackets Differentiate Observables along Hamiltonian Flow - updated: Wed 05 August 2026
Fri 31 July 2026 - Multiplicative Hilbert 90 Describes Norm-One Elements
Thu 07 May 2026 - Zero Divisors on a Finite Module Lie in Associated Primes - updated: Wed 22 July 2026
Thu 08 January 2026 - Convex Hull Trick Minimizes a Family of Affine Functions - updated: Sun 28 June 2026
Sun 24 May 2026 - Jacobi Fields Describe Infinitesimal Families of Geodesics - updated: Mon 22 June 2026
Wed 17 June 2026 - The Fixed Field of the Full Galois Group Is the Base
Tue 15 July 2025 - Zorn's Lemma Converts Chain Bounds into a Maximal Element - updated: Mon 15 June 2026
Thu 11 June 2026 - Gram--Schmidt Produces an Orthonormal Basis
Wed 11 September 2019 - Proj Glues Degree-Zero Localizations - updated: Wed 03 June 2026
Sat 28 December 2024 - Symplectic Euler Preserves Phase-Space Geometry Better than Energy - updated: Mon 18 May 2026
Sat 23 March 2024 - A Pullback Is a Fibre Product of Morphisms - updated: Sun 17 May 2026
Thu 04 February 2021 - A Modular Form Has a Weighted Transformation Law - updated: Sat 02 May 2026
Sat 04 April 2026 - The Matrix Exponential Solves a Constant Linear System
Tue 29 July 2025 - Euler's Formula Forces a Planar Edge Bound - updated: Fri 03 April 2026
Wed 30 April 2025 - The Law of Total Expectation Averages Conditional Means - updated: Thu 02 April 2026
Wed 01 April 2026 - Transcendence Degree Counts Algebraically Independent Parameters
Fri 14 February 2025 - Dihedral Groups Are Semidirect Products - updated: Wed 01 April 2026
Wed 25 March 2026 - Cholesky Factorization Characterizes Positive Definiteness
Tue 24 March 2026 - Universal Machines Encode Programs as Data
Wed 18 March 2026 - Seifert--van Kampen Builds a Fundamental Group from an Open Cover
Thu 10 February 2022 - The Torus Fundamental Group Has One Commutator Relation - updated: Tue 17 March 2026
Thu 21 November 2024 - Baker--Campbell--Hausdorff Corrects Noncommutative Addition - updated: Sun 15 March 2026
Thu 12 March 2026 - The sl2 Commutation Relations Fit in Three Generators
Mon 02 March 2026 - The Prime Number Theorem Is a Statement about Chebyshev's Function
Thu 19 February 2026 - Linearized Polynomials Encode F_q-Linear Maps
Mon 08 September 2025 - Bézout Counts Plane Intersections with Multiplicity - updated: Mon 16 February 2026
Sat 07 February 2026 - A Short Exact Sequence Is an Extension
Sat 30 August 2025 - Sobolev Norms Penalize High Frequencies - updated: Wed 21 January 2026
Mon 19 January 2026 - The Cycle Index Averages Monomials of Cycle Counts
Sat 17 January 2026 - Cantor's Theorem Makes Every Power Set Larger
Thu 29 December 2022 - Tilde Turns Modules into Quasi-Coherent Sheaves - updated: Sun 11 January 2026
Mon 05 January 2026 - Combinatorial Species Turn Constructions into Series
Tue 08 October 2024 - Odd p-Adic Squares Are Detected Modulo p - updated: Wed 17 December 2025
Mon 15 December 2025 - Faithfully Flat Base Change Detects Vanishing
Thu 20 November 2025 - Singular Value Decomposition Works for Every Matrix
Mon 27 October 2025 - Beta Is a Quotient of Three Gamma Values
Sun 19 October 2025 - Cauchy Products Encode Convolution
Thu 16 October 2025 - The Circle Has Fundamental Group Z by Winding Number
Wed 04 May 2022 - A Rational Function Has a Principal Divisor - updated: Wed 15 October 2025
Tue 30 April 2024 - The Argument Principle Counts Zeros Minus Poles - updated: Sat 04 October 2025
Sun 19 March 2023 - The Regular Representation Contains Every Irreducible - updated: Fri 03 October 2025
Wed 01 October 2025 - Factorisation Modulo p Reveals Cycle Types
Mon 01 September 2025 - Integrality Is Equivalent to a Finite Stable Module
Fri 08 August 2025 - Transfinite Induction Follows the Well-Ordering of Ordinals
Wed 19 July 2017 - Yoneda Identifies an Object through All Maps into It - updated: Thu 31 July 2025
Fri 23 June 2017 - q-Cyclotomic Cosets Factor Roots of Unity over Finite Fields - updated: Sun 27 July 2025
Sun 08 November 2020 - The Riemann Tensor Is a Commutator Corrected by the Lie Bracket - updated: Mon 14 July 2025
Mon 16 June 2025 - The Tutte Polynomial Is a Rank-Nullity Sum
Sun 26 July 2020 - A Splitting Field Is Generated by Every Root - updated: Sun 18 May 2025
Mon 21 March 2022 - The Riemann--Roch Space Bounds Allowed Poles - updated: Thu 08 May 2025
Mon 28 April 2025 - Binomial Inversion Undoes a Subset Sum
Tue 22 April 2025 - Morse Critical Points Have a Quadratic Normal Form
Mon 21 April 2025 - Cohen--Macaulay Rings Have No Depth Defect
Sat 26 November 2022 - The Cup Product Is Graded-Commutative - updated: Thu 17 April 2025
Thu 12 December 2024 - The Recursion Theorem Gives Programs Access to Their Own Description - updated: Mon 07 April 2025
Fri 23 February 2024 - The Derivative Decides Local Fixed-Point Stability - updated: Wed 12 March 2025
Tue 07 December 2021 - Cauchy's Integral Formula Reconstructs Interior Values - updated: Wed 26 February 2025
Tue 17 January 2023 - Kähler Differentials Represent Derivations - updated: Thu 20 February 2025
Wed 19 February 2025 - A Short Exact Sequence of Complexes Produces a Long Exact Sequence
Sat 19 November 2016 - Least Squares Projects b onto the Column Space - updated: Sun 02 February 2025
Sun 12 January 2025 - Normalization Repairs Missing Integral Functions
Sun 12 January 2025 - The Summatory Möbius Function Measures Cancellation
Sat 04 January 2025 - Kullback--Leibler Divergence Is Nonnegative
Sat 14 January 2023 - The Spectral Theorem Diagonalizes Self-Adjoint Operators - updated: Wed 04 December 2024
Sun 24 November 2024 - Galois Correspondence Reverses Inclusion
Fri 24 April 2020 - Bellman--Ford Detects Reachable Negative Cycles - updated: Wed 20 November 2024
Wed 20 November 2024 - Gauss's Lemma Counts Sign Crossings
Wed 13 November 2024 - Annihilators Reverse Inclusion
Sat 28 September 2024 - Turán's Theorem Maximizes Edges without a Clique
Tue 31 May 2016 - Integral Elements Form a Subring - updated: Mon 23 September 2024
Mon 16 September 2024 - The Dirac Delta Is a Distribution, Not an Ordinary Function
Sat 02 April 2016 - K5 and K3,3 Violate Planar Density in Different Ways - updated: Sun 15 September 2024
Tue 28 February 2023 - Planar Duality Exchanges Cycles and Cuts - updated: Wed 28 August 2024
Sat 23 December 2023 - Dirichlet L-Functions Twist the Zeta Function - updated: Fri 12 July 2024
Wed 10 July 2024 - Lagrange Inversion Extracts Coefficients from Recursive Series
Tue 30 August 2022 - The Legendre Family Displays a One-Parameter Moduli Coordinate - updated: Tue 11 June 2024
Wed 20 January 2021 - The Laplacian Quadratic Form Is an Edge-Energy Sum - updated: Tue 11 June 2024
Tue 04 June 2024 - The Euclidean Algorithm Writes a Rational as a Continued Fraction
Fri 05 January 2024 - Rooted-Tree Subtree Sizes Obey a Bottom-Up Recurrence - updated: Thu 23 May 2024
Mon 20 May 2024 - Jordan Form Separates Eigenvalues from Nilpotent Drift
Tue 23 August 2022 - Backward Euler Is A-Stable on the Left Half-Plane - updated: Tue 16 April 2024
Fri 29 March 2024 - The First-Moment Method Deletes All Bad Outcomes
Wed 28 February 2024 - An Adjunction Is a Natural Hom-Set Bijection
Tue 27 February 2024 - A Prime Is a Sum of Two Squares Exactly in the Split Case
Wed 31 January 2024 - Hoeffding Gives Subgaussian Tails for Bounded Sums
Mon 09 March 2015 - Multiplicative Order Divides Every Valid Exponent - updated: Mon 29 January 2024
Fri 26 January 2024 - A DFS Tree Edge Is a Bridge When No Back Edge Escapes
Tue 09 February 2021 - Linear-Programming Duality Prices Every Constraint - updated: Mon 15 January 2024
Fri 12 January 2024 - The Killing Form Takes a Trace of Two Adjoint Maps
Fri 12 May 2017 - Independence Factorizes Every Joint Event - updated: Wed 13 December 2023
Sat 18 May 2019 - The Picard Group Splits Divisors by Degree - updated: Tue 21 November 2023
Sat 18 June 2022 - Hilbert's Basis Theorem Adds One Polynomial Variable - updated: Mon 13 November 2023
Thu 02 November 2023 - Maschke Averaging Produces an Invariant Projection
Mon 23 October 2023 - Newton's Method Is Tangent-Line Root Finding
Thu 19 October 2023 - Optional Stopping Preserves Expectation under Safe Hypotheses
Thu 14 September 2023 - Cyclotomic Polynomials Have Abelian Galois Groups
Mon 26 June 2023 - The Wedge Product Is Alternating Multiplication of Forms - updated: Thu 14 September 2023
Mon 18 May 2015 - The Tower Formula Multiplies Extension Degrees - updated: Sun 10 September 2023
Sat 09 September 2023 - Naturality Is a Family of Commuting Squares
Sat 20 August 2022 - Direct Sums Are Unique Coordinate Decompositions - updated: Mon 28 August 2023
Mon 21 September 2015 - Hamilton's Equations Come from Contracting dH with omega - updated: Wed 23 August 2023
Mon 31 August 2015 - A Durfee Square Decomposes a Partition into Three Pieces - updated: Sun 13 August 2023
Mon 17 July 2023 - The Jacobi Identity Is a Cyclic Cancellation Law
Tue 27 June 2023 - Simple Random Walk Is Reversible with Degree Weights
Mon 01 November 2021 - Riesz Representation Identifies Hilbert-Space Functionals - updated: Sat 24 June 2023
Tue 20 June 2023 - Lower Ramification Groups Measure How Nearly Automorphisms Fix Integers
Tue 23 May 2023 - Every Nontrivial Tree Has at Least Two Leaves
Wed 18 May 2022 - A Noetherian Ring Need Not Be Artinian - updated: Wed 17 May 2023
Tue 31 May 2016 - Effective Resistance Is a Laplacian Pseudoinverse Distance - updated: Sun 07 May 2023
Fri 05 January 2018 - Möbius Inversion Extracts Primitive Necklaces - updated: Sun 30 April 2023
Mon 10 April 2023 - Character Inner Products Count Multiplicities
Tue 04 April 2023 - The Supplementary Laws Handle Minus One and Two
Sun 19 September 2021 - Order Ideals Form a Distributive Lattice - updated: Sun 02 April 2023
Sun 26 March 2023 - The Discriminant Changes by the Sign of a Root Permutation
Sun 19 March 2023 - Conjugacy Classes in S_n Are Indexed by Partitions
Tue 28 February 2023 - Algebraic Elements Form a Field
Wed 08 February 2023 - Five Characterizations of a Tree Collapse to One Object
Tue 29 October 2019 - Convergents Satisfy a Determinant Identity - updated: Sun 15 January 2023
Sun 23 October 2022 - Morse Inequalities Compare Critical Points with Homology - updated: Sun 04 December 2022
Mon 21 November 2022 - Connectedness Is the Absence of a Nontrivial Clopen Split
Mon 14 November 2022 - The Group Algebra Converts Representations into Modules
Fri 30 September 2022 - Sperner's Theorem Finds the Largest Boolean Antichain
Mon 26 November 2018 - Closed Immersions Are Surjections of Coordinate Rings - updated: Fri 23 September 2022
Tue 24 May 2022 - Orthogonal Projection Solves the Closest-Point Problem - updated: Tue 06 September 2022
Sat 24 August 2019 - Split Exact Sequences Produce Direct Sums - updated: Thu 18 August 2022
Tue 09 August 2022 - The Open Mapping Theorem Makes Surjective Operators Quantitatively Open
Tue 19 July 2022 - Ascending Chains and Finite Generation Are Equivalent
Sat 21 February 2015 - Formal Power Series Complete a Polynomial Ring at x - updated: Fri 08 July 2022
Sun 26 June 2022 - An Irreducible Cubic Has Galois Group A3 or S3
Thu 23 June 2022 - Cayley's Theorem Embeds Every Group into a Symmetric Group
Fri 26 April 2019 - Detailed Balance Makes a Chain Reversible - updated: Wed 22 June 2022
Fri 17 June 2022 - The Modular Discriminant Defines Ramanujan's Tau Function
Thu 17 February 2022 - The Hockey-Stick Identity Adds a Final Chosen Element - updated: Mon 06 June 2022
Fri 06 May 2022 - Euler Trails Are Controlled by Vertex Parity
Sat 19 June 2021 - A Discrete Valuation Ring Has One Uniformizer - updated: Sun 01 May 2022
Wed 27 April 2022 - Fatou's Lemma Gives the Safe Direction for liminf
Fri 17 May 2019 - S3 Is the Smallest Nonabelian Semidirect Product - updated: Wed 05 January 2022
Tue 13 September 2016 - Divisor-Count and Divisor-Sum Functions Factor Prime by Prime - updated: Mon 03 January 2022
Thu 11 November 2021 - Valuations Extend with Ramification Index - updated: Sun 19 December 2021
Thu 11 March 2021 - Dijkstra Finalizes the Smallest Tentative Distance - updated: Fri 10 December 2021
Thu 11 November 2021 - The Torsion Subgroup Is Functorial
Wed 27 October 2021 - The Adjoint Representation Is Bracketing by an Element
Sun 17 October 2021 - A Basis Generates Open Sets by Unions
Sat 20 March 2021 - Stokes' Theorem Contains Several Integral Theorems at Once - updated: Sun 19 September 2021
Mon 17 April 2017 - Compactness Converts Open Covers into Finite Data - updated: Mon 30 August 2021
Wed 19 October 2016 - Schur's Lemma Makes Irreducible Endomorphisms Scalar - updated: Thu 26 August 2021
Thu 21 December 2017 - The Trace Pairing over a Finite Field Is Nondegenerate - updated: Tue 10 August 2021
Sat 07 August 2021 - Full Digit DP Tracks Position, Tightness, and Automaton State
Wed 04 August 2021 - Gaussian Binomial Coefficients Count Finite Subspaces
Tue 03 August 2021 - Gaussian Curvature Is a Determinant of the Shape Operator
Mon 26 July 2021 - A Branching Process Goes Extinct at a Fixed Point
Tue 14 May 2013 - Artinian Rings Have Krull Dimension Zero - updated: Tue 13 July 2021
Thu 03 December 2015 - Commutators Build the Largest Abelian Quotient - updated: Tue 15 June 2021
Mon 31 May 2021 - Continued-Fraction Convergents Beat One over q Squared
Tue 15 November 2016 - Cellular Homology Uses One Generator per Cell - updated: Sat 01 May 2021
Tue 03 July 2012 - The Minimal Polynomial Has the Largest Jordan Exponents - updated: Sun 25 April 2021
Tue 16 March 2021 - Chebyshev Nodes Suppress Endpoint Oscillation
Fri 10 October 2014 - Dilworth Decomposes a Poset by Its Width - updated: Mon 01 March 2021
Tue 23 February 2021 - Associated Primes Are Annihilators of Elements
Tue 19 January 2021 - Classical RK4 Blends Four Slopes
Mon 04 January 2021 - Chernoff Bounds Optimize an Exponential Moment
Wed 30 December 2020 - Double Cosets Record Two-Sided Symmetry
Wed 09 December 2020 - Perron--Frobenius Gives a Positive Dominant Eigenvector
Sat 28 November 2020 - Rank--Nullity Splits the Dimension Budget
Tue 22 October 2019 - The Schur Complement Controls a Block Matrix - updated: Thu 26 November 2020
Tue 23 June 2020 - Nakayama's Lemma Removes Redundant Local Generators - updated: Fri 13 November 2020
Sun 04 October 2020 - The Affine Group of a Field Is a Semidirect Product
Mon 07 September 2020 - Brouwer's Fixed-Point Theorem Forbids a Retraction
Fri 21 August 2020 - A Linear Recurrence Produces a Rational Generating Function
Thu 20 August 2020 - The Laplace--Beltrami Operator Includes the Volume Density
Mon 03 August 2020 - Uniform Order Statistics Have Beta Densities
Mon 27 July 2020 - Geodesics Have Zero Covariant Acceleration
Wed 23 April 2014 - Poisson Summation Relates a Lattice Sum to Its Dual - updated: Sun 19 July 2020
Sat 04 July 2020 - The Tensor--Hom Adjunction Curries Bilinear Data
Thu 02 July 2020 - The Multiplication Law in a Semidirect Product
Fri 05 June 2020 - Uniform Convergence Permits Termwise Integration
Sat 30 May 2020 - The Krull Topology Uses Finite Fixed Fields as Neighbourhoods
Wed 20 May 2020 - Gödel Completeness Identifies Semantic and Syntactic Consequence
Fri 25 August 2017 - Forward Euler Uses One Tangent Step - updated: Fri 15 May 2020
Tue 14 April 2020 - Prüfer Codes Count Labeled Trees
Tue 07 April 2020 - Strong Components Form an Acyclic Condensation
Sun 29 March 2020 - Étale Morphisms Are Algebraic Local Isomorphisms
Mon 02 May 2016 - A Minimal Polynomial Is the Kernel of Evaluation - updated: Mon 02 March 2020
Sat 10 August 2013 - Uniform Boundedness Turns Pointwise Bounds into an Operator-Norm Bound - updated: Fri 28 February 2020
Wed 29 January 2020 - A p-Adic Integer Is an Infinite Base-p Expansion
Sun 11 November 2018 - LTE Computes Valuations of Exponential Differences - updated: Mon 16 December 2019
Sat 14 December 2019 - Kernels Are Normal Because Conjugation Cannot Leave Zero
Sat 29 September 2018 - The Hilbert Function Eventually Becomes a Polynomial - updated: Mon 09 December 2019
Thu 05 December 2019 - RSK Turns a Permutation into Two Tableaux of One Shape
Sun 17 November 2019 - Normal Extensions Contain Every Conjugate
Thu 25 May 2017 - Homogenization Adds the Hyperplane at Infinity - updated: Wed 06 November 2019
Thu 24 October 2019 - Additive Hilbert 90 Describes Trace-Zero Elements
Wed 03 October 2018 - Subset DP Makes the Chosen Set an Explicit Coordinate - updated: Fri 18 October 2019
Thu 17 October 2019 - Trace and Norm Are Additive and Multiplicative Shadows
Sun 29 September 2019 - The Gradient Is Metric-Dual to the Differential
Tue 24 September 2019 - The Möbius Function Is the Convolution Inverse of One
Tue 07 August 2018 - Conditional Expectation Is an L2 Projection - updated: Sat 27 July 2019
Sat 19 July 2014 - Discriminant and j-Invariant Separate Smoothness from Isomorphism - updated: Mon 15 July 2019
Tue 14 February 2012 - Cantor's Diagonal Argument Makes Binary Sequences Uncountable - updated: Fri 05 July 2019
Tue 25 June 2019 - The Weierstrass M-Test Controls a Function Series
Thu 06 June 2019 - Levi--Civita Is the Unique Metric Torsion-Free Connection
Thu 11 April 2019 - Riemann--Roch Balances a Divisor with Its Canonical Complement - updated: Fri 10 May 2019
Mon 29 April 2019 - A Generic Polynomial Can Have the Full Symmetric Group
Thu 09 January 2014 - Banach's Fixed-Point Theorem Gives a Unique Contractive Fixed Point - updated: Thu 25 April 2019
Wed 20 March 2019 - The Sign of a Permutation Is the Parity of Its Inversions
Sun 10 March 2019 - The Logistic Map Changes Stability at r Equals Three
Fri 22 February 2019 - Frobenius Is a Linear Operator over the Base Field
Thu 25 April 2013 - A DFS Low-Link Value Detects Back Edges - updated: Thu 07 February 2019
Tue 01 January 2019 - The Order of a Power in a Cyclic Group
Sat 01 December 2018 - Brownian Covariance Is the Minimum of Two Times
Fri 01 June 2018 - Infinite Galois Groups Are Inverse Limits - updated: Sun 18 November 2018
Sat 17 November 2018 - Exponential Generating Functions Remember Labels
Tue 13 October 2015 - A Normal Basis Is One Frobenius Orbit - updated: Wed 10 October 2018
Tue 04 September 2018 - Hahn--Banach Extends a Bounded Functional without Growing Its Norm
Tue 14 August 2018 - FFT Splits a Polynomial into Even and Odd Coefficients
Wed 08 August 2018 - The Canonical Divisor Has Degree Two g Minus Two
Mon 30 July 2018 - I-Adic Completion Is an Inverse Limit
Tue 24 July 2018 - The Universal Property of a Group Quotient
Tue 24 December 2013 - Cross Ratio Is the Projective Quantity Preserved by Möbius Maps - updated: Sat 21 July 2018
Sun 24 June 2018 - Quadratic Irrationals Have Periodic Continued Fractions
Sat 09 June 2018 - Articulation Points Separate DFS Child Subtrees
Sat 12 May 2018 - The Torus Cohomology Ring Is an Exterior Algebra
Sun 15 April 2018 - The Weak Law Follows from Variance Decay
Fri 18 September 2009 - Soundness Says Formal Proofs Preserve Truth - updated: Tue 20 March 2018
Tue 06 March 2018 - The Principle of Double Counting Proves an Average Identity
Tue 20 February 2018 - Löwenheim--Skolem Produces Smaller Elementary Models
Wed 24 January 2018 - The Poincaré Disk and Half-Plane Are Isometric
Wed 03 January 2018 - When a Direct Product of Cyclic Groups Is Cyclic
Wed 03 January 2018 - The First Isomorphism Theorem Turns Fibres into Cosets
Tue 12 December 2017 - Hall's Condition Is a Set-System Inequality
Wed 14 June 2017 - A Floor Sum Counts Divisors in a Hyperbola - updated: Sat 09 December 2017
Fri 17 November 2017 - Burnside Averaging Counts Orbits from Fixed Points
Tue 24 December 2013 - Stirling's Formula Is a Logarithmic Asymptotic Expansion - updated: Wed 08 November 2017
Tue 07 November 2017 - Characteristic Functions Turn Sums into Products
Fri 20 October 2017 - Parseval Turns L2 Energy into Fourier Coefficients
Thu 08 May 2014 - A Symplectic Form Is Closed and Nondegenerate - updated: Thu 12 October 2017
Tue 03 October 2017 - Laplacian Nullity Counts Connected Components
Wed 20 September 2017 - The Derivative Detects Repeated Irreducible Factors
Fri 24 August 2012 - Dirichlet Characters Have Finite Fourier Orthogonality - updated: Tue 15 August 2017
Thu 10 August 2017 - Azuma--Hoeffding Controls Martingales with Bounded Differences
Thu 03 August 2017 - Legendre's Formula Counts p in a Factorial
Mon 26 June 2017 - Sylvester's Law of Inertia Classifies Real Quadratic Forms
Tue 25 February 2014 - The Lovász Local Lemma Handles Sparse Dependence - updated: Wed 21 June 2017
Sun 28 May 2017 - Pullback Moves Covectors in the Opposite Direction
Sun 26 March 2017 - Kronecker Products Multiply Blockwise
Tue 21 March 2017 - Gauss--Bonnet Turns Total Curvature into Euler Characteristic
Sat 26 March 2011 - Exterior Powers Record All Minors at Once - updated: Fri 17 March 2017
Fri 17 March 2017 - A Stationary Distribution Is a Left Eigenvector
Sat 27 September 2014 - Menger Equates Disjoint Paths with Minimum Separators - updated: Sat 11 February 2017
Fri 10 February 2017 - The Zeta Euler Product Encodes Unique Factorization
Fri 18 November 2016 - Exponential Waiting Times Are Memoryless
Sat 05 November 2016 - A Lyapunov Exponent Measures Exponential Separation
Thu 07 January 2016 - Euler's Totient Counts the Units Modulo n - updated: Wed 24 August 2016
Wed 08 April 2015 - Bellman's Principle Replaces a Global Search by Optimal Suffixes - updated: Fri 19 August 2016
Sat 22 November 2014 - The Matrix--Tree Theorem Uses a Laplacian Cofactor - updated: Thu 28 July 2016
Tue 26 July 2016 - The Exponent of a Finite Abelian Group
Fri 06 March 2009 - Lying Over Makes an Integral Map of Spectra Surjective - updated: Sun 17 July 2016
Sun 19 June 2016 - A Random Coloring Gives an Exponential Ramsey Lower Bound
Fri 17 June 2016 - An Isogeny Is a Finite Group Morphism of Elliptic Curves
Wed 08 June 2016 - Gamma Extends the Factorial by an Integral
Mon 06 June 2016 - Chebyshev's Inequality Applies Markov to Squared Deviation
Sat 30 April 2016 - Complementary Slackness Identifies Which Constraints Bind
Fri 15 April 2016 - Pólya Substitution Counts Colorings up to Symmetry
Thu 07 April 2016 - A Knot Group Comes from the Complement
Wed 17 February 2016 - A Presentation Solves a Universal Mapping Problem
Sat 06 February 2016 - Doob's Maximal Inequality Controls a Submartingale Peak
Fri 29 January 2016 - The Associated Graded Ring Keeps Leading I-Adic Terms
Thu 17 September 2015 - Regular Local Rings Match Dimension with Tangent Dimension - updated: Fri 25 December 2015
Sun 25 October 2015 - Trace Is Cyclic but Not Fully Commutative
Wed 14 October 2015 - Poisson Increments Depend Only on Interval Length
Thu 17 September 2015 - Dominated Convergence Uses One Integrable Envelope
Sun 30 August 2015 - Unit and Counit Encode the Triangle Identities
Fri 21 August 2015 - The Halting Problem Defeats a Diagonal Decider
Sun 09 August 2015 - A Scheme Point Carries a Residue Field
Sun 14 November 2010 - Interval DP Chooses the Last Split inside a Segment - updated: Thu 30 July 2015
Mon 27 July 2015 - Many-One Reductions Transfer Undecidability
Thu 23 July 2015 - Exactness of Sheaves Can Be Checked on Stalks
Mon 22 June 2015 - NTT Performs the Fourier Transform inside a Finite Field
Sun 07 June 2015 - Kummer Extensions Come from Classes Modulo n-th Powers
Sat 01 September 2012 - Conjugates Are Images under Field Embeddings - updated: Fri 29 May 2015
Sun 17 May 2015 - Bayes' Formula Reverses a Conditional Direction
Thu 07 May 2015 - Rational Canonical Form Works without Splitting Polynomials
Fri 04 October 2013 - The Differential Pushes Tangent Vectors Forward - updated: Thu 16 April 2015
Sun 29 January 2012 - Primary Ideals Have One Radical Prime - updated: Mon 06 April 2015
Mon 04 March 2013 - Primitive Roots Exist for a Very Specific List of Moduli - updated: Tue 03 March 2015
Wed 10 March 2010 - Möbius Functions Invert Sums over Poset Intervals - updated: Wed 18 February 2015
Thu 12 February 2015 - Finitely Generated Modules over Noetherian Rings Are Noetherian
Sat 10 September 2011 - Lagrange Interpolation Uses Cardinal Basis Polynomials - updated: Thu 01 January 2015
Sun 25 July 2010 - A Centered Random Walk Is a Martingale - updated: Wed 19 November 2014
Sat 15 November 2014 - Compactness Reduces Infinite Satisfiability to Finite Pieces
Tue 14 October 2014 - The Multinomial Coefficient Counts Ordered Blocks
Wed 17 September 2014 - The Chinese Remainder Theorem Builds One Residue from Local Data
Sun 17 August 2014 - A Regular Sequence Uses Successive Non-Zero-Divisors
Sat 16 August 2014 - Uniform Limits of Continuous Functions Are Continuous
Mon 07 July 2014 - Eisenstein Series Turn Divisor Sums into Fourier Coefficients
Sun 09 October 2011 - Quotient Topology Makes a Surjection Final - updated: Fri 20 June 2014
Thu 29 May 2014 - Inclusion--Exclusion Counts Onto Functions
Thu 15 May 2014 - Arzelà--Ascoli Converts Equicontinuity into Compactness
Tue 15 April 2014 - Completion of Finite Modules Is Tensoring with the Completed Ring
Tue 11 February 2014 - Mayer--Vietoris Computes Homology from Two Pieces - updated: Sun 30 March 2014
Sun 28 July 2013 - The Degree of a Sphere Map Is Its Action on Top Homology - updated: Sat 01 March 2014
Wed 26 February 2014 - Frobenius Satisfies a Quadratic Equation on an Elliptic Curve
Thu 06 February 2014 - Quadratic Reciprocity Turns One Odd Prime around Another
Thu 09 January 2014 - Quasi-Coherent Sheaves on an Affine Scheme Recover Modules
Sat 04 January 2014 - The Functor of Points Tests a Scheme on Every Ring
Wed 23 October 2013 - Poincaré Duality Pairs Complementary Degrees
Sat 15 October 2011 - The Fundamental Matrix Sums Visits before Absorption - updated: Wed 28 August 2013
Mon 26 August 2013 - Intersection Multiplicity Is a Local Algebra Length
Fri 16 August 2013 - Partitions of an Exponent Classify Abelian p-Groups
Thu 08 August 2013 - The Exponential Map Sends Lie Algebra Directions to One-Parameter Subgroups
Thu 25 July 2013 - Triangle-Free Planar Graphs Have an Even Stronger Bound
Tue 23 July 2013 - Rook Polynomials Encode Nonattacking Placements
Fri 19 July 2013 - Dirichlet Convolution Multiplies over Divisor Splittings
Tue 02 July 2013 - Euler's Partition Product Is a q-Series Identity
Wed 08 May 2013 - Projective Modules Lift through Surjections
Thu 02 May 2013 - The Lifting Criterion Is a Subgroup Condition
Tue 16 April 2013 - Primary Decomposition Splits an Operator by Coprime Factors
Wed 13 March 2013 - Degree Is the Leading Coefficient of the Hilbert Polynomial
Sun 03 February 2013 - Line Bundles Form the Picard Group
Tue 02 October 2012 - Left Translation Identifies Every Tangent Space of a Lie Group - updated: Sun 03 February 2013
Mon 14 January 2013 - Tonelli Integrates Nonnegative Functions in Either Order
Tue 01 January 2013 - Alteration Repairs a Random Structure after Sampling
Mon 31 December 2012 - Möbius Inversion Recovers a Function from Its Divisor Sum
Sat 15 December 2012 - Fubini Requires Absolute Integrability
Thu 25 October 2012 - Convolution Becomes Multiplication after Fourier Transform
Sun 14 October 2012 - Choice, Well-Ordering, and Zorn Are Equivalent
Tue 17 April 2012 - Rouché's Theorem Preserves the Number of Zeros - updated: Mon 10 September 2012
Sun 05 August 2012 - Boundary of a Boundary Is Zero
Sun 29 July 2012 - Tensor Product Is Right Exact
Wed 13 June 2012 - A Primary Decomposition Separates Radical Components
Wed 13 June 2012 - The Weil Pairing Detects Two-Dimensional Torsion
Tue 29 May 2012 - Cauchy--Riemann Equations Encode Complex Differentiability
Sun 29 April 2012 - Tangent Vectors Are Derivations at a Point
Tue 10 April 2012 - Hom Is Left Exact
Sun 10 July 2011 - Krull's Intersection Theorem Separates I-Adic Orders - updated: Fri 06 April 2012
Wed 07 January 2009 - Ramanujan Congruences Hide in Partition Coefficients - updated: Tue 03 April 2012
Mon 02 January 2012 - The Tent Map Is Coded by Binary Itineraries - updated: Tue 03 April 2012
Tue 03 April 2012 - Fibre Products of Affine Schemes Are Tensor Products
Wed 22 February 2012 - The Correspondence Theorem Organises Subgroups above a Kernel
Sat 11 February 2012 - Cheeger's Inequality Links Expansion to the Spectral Gap
Wed 28 December 2011 - Carmichael's Function Gives the Exponent of the Unit Group
Tue 27 December 2011 - Normalization Separates the Two Branches of a Node
Sat 26 November 2011 - Slutsky's Theorem Combines Random and Deterministic Limits
Thu 17 November 2011 - The Dual Map Reverses a Linear Arrow
Mon 14 November 2011 - Minimal Generators Are a Basis Modulo the Maximal Ideal
Wed 05 October 2011 - The Zariski Tangent Space Is a Kernel of the Jacobian
Tue 20 September 2011 - The Chromatic Polynomial Satisfies Deletion--Contraction
Mon 29 August 2011 - Łoś's Theorem Evaluates Formulas Coordinatewise in an Ultraproduct
Thu 25 August 2011 - Kőnig's Theorem Equates Matching and Vertex Cover in Bipartite Graphs
Thu 23 September 2010 - Local Truncation Error Accumulates into Global Error - updated: Mon 22 August 2011
Tue 25 May 2010 - QR Iteration Reveals Eigenvalues without Forming a Polynomial - updated: Sun 14 August 2011
Tue 12 July 2011 - A Distribution Is the Pushforward of Probability
Sun 03 July 2011 - Coboundaries Reverse the Chain Direction
Sun 26 June 2011 - Sectional Curvature Assigns Curvature to a Two-Plane
Thu 23 June 2011 - Simple Roots Lift Uniquely by Hensel's Lemma
Tue 19 April 2011 - Pell Solutions Come from Convergents to a Square Root
Thu 14 April 2011 - The Five-Color Theorem Uses a Low-Degree Vertex
Sun 10 April 2011 - Laplace's Method Localizes an Integral near Its Maximum
Tue 22 February 2011 - Euler Characteristic Alternates Cell Counts
Sun 19 December 2010 - The Residue Theorem Sums Local Laurent Coefficients
Thu 16 December 2010 - A Quotient Space Removes a Subspace Direction
Sun 12 December 2010 - The Species of Sets of Nonempty Sets Gives Bell Numbers
Tue 23 November 2010 - The First Borel--Cantelli Lemma Needs No Independence
Thu 14 October 2010 - Bisection Trades Speed for a Guaranteed Bracket
Sun 17 January 2010 - Unramified Local Extensions Are Controlled by Residue Fields - updated: Mon 27 September 2010
Thu 23 September 2010 - The Real Projective Plane Attaches a Two-Cell by Degree Two
Mon 13 September 2010 - The Second Borel--Cantelli Lemma Uses Independence
Sun 22 August 2010 - Depth Is the Length of a Maximal Regular Sequence
Tue 18 May 2010 - The Hook-Length Formula Counts Standard Young Tableaux - updated: Tue 13 July 2010
Sat 26 June 2010 - Semirings Explain Why One Graph Recurrence Solves Many Problems
Thu 10 June 2010 - The Rees Algebra Remembers an Entire Filtration
Fri 23 April 2010 - Conditional Probability Renormalizes an Event
Fri 26 March 2010 - Max-Flow Equals Min-Cut
Thu 04 February 2010 - Support Is the Closed Set Defined by an Annihilator
Sat 23 January 2010 - Flat Modules Preserve Short Exact Sequences
Fri 22 January 2010 - A Transfer Matrix Counts Objects with Finite Memory
Fri 06 November 2009 - Euler's Criterion Computes the Legendre Symbol by Exponentiation
Sat 31 October 2009 - A Riemannian Metric Converts Velocity into Length
Wed 21 October 2009 - Monotone Convergence Interchanges an Increasing Limit and Integral
Tue 20 October 2009 - Reduced Homology Removes the Extra Connected Component
Thu 24 September 2009 - Gödel's First Incompleteness Theorem Produces an Undecidable Sentence
Mon 07 September 2009 - The Gaussian Is an Eigenfunction of the Fourier Transform
Sun 16 August 2009 - Affine Morphisms Reverse Ring Maps
Sun 02 August 2009 - Gaussian Quadrature Is Exact through Degree Two n Minus One
Mon 13 July 2009 - Uniform Derivative Convergence Recovers a Differentiable Limit
Thu 09 July 2009 - Going Up Lifts Chains of Prime Ideals
Thu 02 July 2009 - Markov's Inequality Uses Only Nonnegativity
Thu 21 May 2009 - Tree Metrics Satisfy the Four-Point Condition
Wed 29 April 2009 - Chapman--Kolmogorov Is Matrix Multiplication
Sun 05 April 2009 - Cayley--Hamilton Substitutes a Matrix into Its Characteristic Polynomial
Wed 25 March 2009 - Gambler's Ruin Solves a Discrete Harmonic Equation
Sun 01 March 2009 - Loop Concatenation Defines the Fundamental Group
Fri 20 February 2009 - Product Topology Is the Smallest Topology Making Projections Continuous
Wed 18 February 2009 - Absolute Stability Is Tested on y Prime Equals lambda y
Mon 26 January 2009 - Homology Quotients Cycles by Boundaries
Wed 21 January 2009 - The Upper Half-Plane Metric Is Möbius Invariant