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Mathematics |

- Mathematics possesses not only truth, but supreme beauty--
a beauty cold and austere, like that of sculpture, without appeal to any part of our
weaker nature… sublimely pure, and capable of a stern perfection such as only
the greatest art can show.
*Bertand Russell* - The Solution to the Generalized Helmholtz Equation in Matrix Form
- A Transformation of a Generalized Helmholtz Equation
- The Functional Derivative: Its Natue and Use
- Cayley's Theorem and its Proof
- The Existence of Creator and Annihilator Operators in the Canonical Quantization of a Field
- The Spatial Average of the Probability Density Function Derived from the Solution to a Generalized Helmholtz Equation of One Dimension
- Lagrangian Field Theory
- The Solution to a Generalized Helmholtz Equation of One Dimension
- Decimal Numbers of Concatenated Blocks Which are Fractional Numbers
- Decimal Numbers of Concatenated Blocks Which are Reciprocals of Integers
- The Binomial Theorem: A Proof in One Step
- Finding the Unit Digits of Powers of Integers
- Benjamin's Theorem and the Power of Algebra
- The d'Alembert Solution to the Wave Equation with Initial Conditions
- Determinants and Area and Volumes of Parallelotopes (Parallelograms and Parallelopipeds)
- The Change of Variables Theorem for Integrals
- The Definition and Properties of a Mathematical Field
- A Lemma on the Associativity of the Composition of Functions
- The Helmholtz Equation and Its Solution
- The Similarity of a Square Matrix to a Jordan Canonical Form
- The Matrix Representation of Sets of Linear Combinations of Vectors
- The Generalized Eigenvectors of a Matrix and their Linear Indepence
- The Similarity of Square Matrices
- Decomposition of the Vector Space Associated with a Matrix in Terms of the Null Spaces Associated with the Eigenvalues of that Matrix.
- The Linear Independence of the Eigenspaces of a Matrix
- The Cardinality of the Algebraic Numbers
- The Irrationality of the Square Root of 2
- The Theorem of George Green and its Proof
- The Components of Mandelbrot Sets Associated with Limit 2-Cycles
- The Powers of a Jordan Canonical Form Matrix
- The Square of a Jordan Canonical Form Matrix
- The Theorem on the Canonical Form of Matrices by Camille Jordan
- The Theorem that the Sum of the Eigenvalues of a Matrix is Equal to its Trace
- The Symmetries of the General Structures of Mandelbrot Sets
- A Beautiful Theorem Concerning Determinants
- The Relationship Between Group Representations and Their Generators
- The General Structures of the Fully General Mandelbrot Sets
- The Barest Introduction to Category Theory
- The Amazing, Incredible Life of the Mathematician, Alexander Grothendieck
- SU(2), the Special Unitary Group of Order 2
- The Representation of Rational Numbers as Decimal Numbers
- The Sets of Repeating Decimals
- The Properties of Integers and Their Extension to Integral Domains
- Some Applications of the Notion of an Infinitesimal in Geometry
- Infinitesimals and the Extention of the Real Number System
- The Uncertainty Principle for Energy and Time
- The Derivation of the Uncertainty Principle
- Quantum Theory and the Interactions of Electrons in Multi-Electron Atoms
- The Discrete Approximations of the 2nd Derivatives of a Function of Two Variables
- The Digit Sum Arithmetic of Exponentiation
- The Digit Sum of a Number to Base 10 is Equivalent to Its Remainder Upon Division by 9
- The Real Numbers Are Not Enough: The Need for More Numbers (the Hyperreals) to Label Ordered Sets
- The DigitSum of a Number and Its Remainder for Division by Nine
- Remainders and Representations: Simple Applications of Number Theory
- Fermat's Little Theorem
- The Interesting Properties of 1001, Scheherezade's Number
- The Cayley-Hamilton Theorem: Its Nature and Its Proof
- An Euler-type Theorem for Exponentially Weighted Homogeneous Functions
- Polyhedra with Three Types of Faces and One Type of Vertex
- The Regularities of the Platonic and Archimedean Polyhedra
- The Algebraic Determination of the Structures of Polyhedra
- On the Algebraic Determination of the Structure of Polyhedra
- The Analogues of the Dual Theorem of Euler's Twelve Pentagon Theorem
- Euler-type Theorems Concerning the Number of Polygons of Polyhedra
- The Euler-Poincare Characteristic of Conjoined Polyhedra
- Proof That There Are Only Five Platonic Polyhedra
- The Theorem Dual to Euler's Twelve Pentagon Theorem
- Euler's Theorem Concerning Polyhedra Composed of Pentagons and Hexagons: There Must Be Exactly 12 Pentagons
- The Virial Theorem and the Matter of the Origin of the Position Vectors of the Particles
- The Relative Variance of a Binary Function of Random Variables Compared to that of the Corresponding Unary Function
- Élie Cartan's Generalization of the Frenet-Serret Formulas
- The Virial Theorem: Its Explanation, Proof, Application and Limitations
- Real Numbers as Dedekind Cuts
- The Area of an (n-1)-Dimensional Subsphere in Terms of the Arc Distance in the n-Dimensional Sphere
- Real Numbers as Partitions of the Ordered Field of Rational Numbers (Dedekind Cuts)
- The Characteristic Function of Algebraic Combinations of Random Variables and cumulative Sums of Random Disturbances
- Simple, Yet Amazing Theorems Concerning the Cumulative Curvature of a Curve Between Two Points
- The Estimation of the Parameters of the Characteristic Function of a Levy-Stable Probability Distribution
- The Euler-Poincare Characteristic of Strip and Straps: Open, Closed and Mobius
- The Evaluation of the
integral ∫
_{r}^{∞}(exp(-z)/z²)dz - Noether's Theorem and its Proof
- Self-Similar Mathematical Structures
- Srinivasa Ramanujan: A Mathematician Brilliant Beyond Comparison
- The Euler Poincaré Characteristic for Different Dimensions of Geometric Figures
- The Euler Poincaré Characteristic of Polyhedral Solids
- The Euler Characteristic of Polyhedra
- An Introduction to Tensor Analysis
- The Spectra of Various Transformations of White Noise
- The Effect of Averaging Variables Which are the Cumulative Sum of Random Disturbances
- Digit Sum
Arithmetic

- The Real Numbers Are Not Enough: The Need for More Numbers (the Hyperreals) to Label Ordered Sets
- Remainders and Representations: Simple Applications of Number Theory
- Fermat's Little Theorem
- The Cayley-Hamilton Theorem: Its Nature and Its Proof
- The Interesting Properties of 1001, Scheherezade's Number
- Theorems Concerning the Existence of Solutions to Systems of Ordinary Differential Equations
- Polyhedra with Three Types of Faces and One Type of Vertex
- The Polyhedral Structure of Nuclear Shells
- The Regularities of the Platonic and Archimedean Polyhedra
- The Algebraic Determination of the Structures of Polyhedra
- On the Algebraic Determination of the Structure of Polyhedra
- The Analogues of the Dual Theorem of Euler's Twelve Pentagon Theorem
- Euler-type Theorems Concerning the Number of Polygons of Polyhedra
- The Euler-Poincare Characteristic of Conjoined Polyhedra
- Proof That There Are Only Five Platonic Polyhedra
- The Theorem Dual to Euler's Twelve Pentagon Theorem
- Euler's Theorem Concerning Polyhedra Composed of Pentagons and Hexagons: There Must Be Exactly 12 Pentagons
- Élie Cartan's Generalization of the Frenet-Serret Formulas
- Real Numbers as Partitions of the Ordered Field of Rational Numbers (Dedekind Cuts)
- An Introduction to Tensor Analysis
- Spherical Geometry and Trigonometry
- Bezout's Theorem
- The Envelope Theorem
- The Central Limit Theorem
- Area-Volume Formulas
- Pontryagin's Maximum Principle
- Second Order Conditions for Extremes
- Mandelbrot's
*Variation of Certain Speculative Prices* - Foundations of Algebraic Topology
- Georg Cantor and Cantor's Theorem
- Characteristic Functions
- Inversion of a Characteristic Function
- Complex Numbers
- Convex Sets
- Differential Forms
- Differential Geometry from

a Differential Forms Point of View - Principal Component Analysis
- Quadratic Equation Solver
- The Nature of the Real Numbers
- Stable Distributions
- Stochastic Processes
- The Generalized Stokes Theorem
- Perturbation Theory
- Infinitely Divisible Random Variables
- Infinitesimal Random Variables
- Fuzzy Sets
- Georg Cantor and Cantor's Theorem
- The Arithmetic of Cardinal Numbers
- The Arithmetic of Ordinal Numbers
- Fixed Point Theorems
- The Lambda Calculus
- Conditional Expectations and Random Walks of Security Prices
- Foundations of Point Set Topology
- Kuratowski's Closure Operation in Topology
- An Introduction to Cohomology in Topology
- The Hausdorff Separation Axiom and Other Separation Axioms in Topology
- The Euler-Poincaré Formula in Topology
- The Factor Group of a Group With Respect to a Normal Subgroup
- Axial Vectors in Rotating Coordinate Systems
- Euler's Theorem and Its Generalization
- Basic Theory of Groups
- The Converse of Euler's Theorem
- Fundamentals of Lie Algebras
- The Structure of the Set of Lie Algebras
- The Gauss-Bonnet Theorem and Its Generalization
- The Gauss-Bonnet Formula for Curves
- Mikusinki's Operational Calculus
- Spherical Geometry and Trigonometry
- The Nature of Spinors
- Levi-Civita's Concept of the Parallel
Transport

of Vectors on a Surface - The Implicit Function Theorem
- The Second Order Conditions for Extremes:

Constrained and Unconstrained - The Gamma Function: Euler's Generalization

of the Factorial Function - Extensions of the Central Limit Theorem
- Filters and Ultrafilters
- The Calculus of Infinitesimals
- An Introduction to Elementary Algebra
- Traveling Wave Solutions to the Regularized Long Wave Equation (RLWE)
- Traveling Wave Solutions to the Korteweg de Vries Equation (KdV)
- The Hyperbolic Secant Solution a·sech²(bz).
- Secondary Solitary Waves in the Collision of Solutions to the Regularized Long Wave Equation
- The Summation of Series Using the Anti-Differencing Operation
- The Nature of Differential Forms
- Kakutani's Fixed Point Theorem
- Real Numbers as Equivalence Classes of Cauchy Sequences of Rational Numbers
- The Structures of the Mandelbrot Sets
- The Elastic Collision of Spheres
- Digit Sum Arithmetic
- The Lagrange Theorem for Groups

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