Readings in Mathematics

San José State University

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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 Asymptotic Limit of the Smallest Magnitude Root of a Polynomial Equation as the Constant Term Goes to Zero
The Asymptotic Limits of the Roots of a Quadratic Equation as the Constant Term Goes to Zero
The Relativistic Hamiltonian Function for a Particle in a Poteneial Field
Quotients in Digit Sum Arithmetic for Any Base and Any Divisor
Relationships Involving the "Volume" of a Region, the "Area" of its Boundary Surface, and the Average Distance from an Interior Point to that Boundary
Quotients in Digit Sum Arithmetic for Any Base
Remainders and the Weighted Digit Sums of Numbers
The Weighted Digit Sums of Multiples of Digits
Simple, Elegant yet Neglected Theorems Concerning Areas and Volumes That Includes Circles and Spheres as Limits
A Function Relevant for Quantum Theoretic Analysis
The Expected Value of Weighted Averages is the Expected Value of the Original Variable
The Asymptotic Sine Integral of Functions
Means and Variances of Probability Distributions, Quantum and Classical
Fourier Transforms of the Time Independent Schroedinger Equation
The Discovery by the Indian Mathematician Brahmagupta
that the Product Two Negative Numbers is a Positive Number
Some Simple Theorems Concerning Monoids
The Associativity of Function Composition
The Transformation of the Time Dependent Schroedinger Equation into Two Linked Equations Involving Only Real Dependent Variables
The Computation of Functions of Square Matrices
The Baker-Campbell-Hausdorff Formula for Exponential Matrices
An Alternate Solution to the Generalized Helmholtz Equation of One Dimension
A Solution to the Generalized Helmholtz Equation of One Dimension
Noether's Theorem for Fields
Noether's Theorem and Quantized Parameters
Noether's Theorem: Its Explanation and Proof
Digit Sum Arithmetic for Numbers with Nonintegral Bases
Illustration of the Quotient Theorem of Digital Arithmetic
Studies Concerning the Solution of the Generalized Helmholtz Equation
A property of the Solution of a Generalized Helmholtz Equation of Any Dimension
The Solution of a Generalized Helmholtz Equation of Two Dimensions
The Nature and History of the Bayes Rule for Computing Inverse Probabilities
Some Remarkable Properties of Polynomials
An Exponential of a Purely Imaginary Matrix and the Trigonometric Identities
The relative extrema of the solution to a generalized Helmholtz equation
The Yoneda lemma in Category Theory
The Concept of Functor in Category Theory
Studies Concerning the Weighted Sums of the Coefficients of Polynomials
Some Properties of the Sum of Digits Function
The Nature and History of the Bayes Rule for Computing Inverse Probabilities
Divisibility, Quotients and Remainders for Polynomials over a Commutative Ring
Divisibility, Quotients and Remainders for Polynomials over a Field
Quotients and Remainders in Digit Sum Arithmetic
Weighted Summations of Polynomial Coefficients and Remainders
The Scheme of Perturbation Analysis
Divisibility and Remainders for Polynomials over a Field
Six Lemmas on the Remainders for the Division of Polynomials
The Net Sum of a Decimal Representation of a Number Resulting From an Alternation of Addition and Subtraction
Coefficient Sums for Infinite Polynomial Series
Two General Theorems Concerning the Coefficient Sums of Polynomials over a Field
Some general properties of polynomials over a field
Some Interesting General Properties of Polynomials
The Wong-Zakai Adjustment to a Stochastic Differential Equation
The Introduction to Matrix Calculus: The Extention of Calculus Operations to Matrices
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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