San José State University
applet-magic.com Thayer Watkins
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 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
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
The Implicit Function Theorem The Second Order Conditions for Extremes:
The Gamma Function: Euler's Generalization
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
