MathPages Combined Contents List

 



©1994-2008 Kevin Brown



Sequences Formed By Rounding
The Twelve-Step Cycle of 1/(sin(x)cos(x))
Symmetric Pseudoprimes
Digit Reversal Sums Leading to Palindromes
Min-Energy Configurations of Electrons On A Sphere
The Sum of the Prime Factors of N
Smith Numbers and Rhonda Numbers
Sequence Partitionable Into Powers of 2 or 3
Fermat's Last Theorem for Cubes
Tangents, Exponentials, and PI
Marginalia
On Eulerian Numbers
Can Schrodinger's Cat Factor Numbers?
Why z Is Not a Prime Power in z^p = x^p + y^p
Zeisel Numbers
Identities for Linear Recurring Sequences
Summations and Recurrences
Optimizing Your Wife
Thinking Outside the Triangle
Sum-of-Digits Iterations 
Higher Order Dynamics
Miscellaneous Diophantine Equations
Random High-Low Searching
What Happened to Dingle?
The Dartboard Sequence
Poisson Processes and Queues
Finite Subgroups of the Mobius Group
The Dullness of 1729
Highly Wilsonian Primes
Dedekind's Problem
Permutation Loops
Legendre's Prime Number Conjecture
Probabilities With Renaming
Discrete Circular Distributions
Multiple Linear Regression
At Least One Girl
Parabola Through Four Points
Archimedes and the Square Root of 3
Polyomino Enumerations
Divisors of an n-term Geometric Series
Nicolas Fatio and the Cause of Gravity
Ratio Populations
Piano Keys
No Four Squares In Arithmetic Progression
Integrating the Bell Curve
Differences Between Normal Samples
The Collector's Problem
Sum of Divisors Equals a Power of 2
Bertelsen's Number
Sandford's Elevator
Tables of 1 to N with Equal Columns
The Coin in Three Fountains
Euclid's Plan and Proposition 6
Geophysical Altitudes
Generalized Mediant
The Fundamental Theorem of Algebra
Generalized Birthday Problem (N Items in M Bins)
Numeri Idonei
Sequences With No Arithmetic Progressions
Cutting Self-Similar Pentagons
A Primality Criterion
Meeting Combinations and Interval Graphs
The Limit Paradox
Can n Divide !n ?
Asymptotic Approach to 2D Arrays
Prisca Sapientia
Iterated Logarithmic Functions
A Mass and Spring System in Motion
Pythagoras On Dot and Cross Products
Diophantine Walk-a-thon
Sums of Three Cubes
Why Calculus?
Congruences Involving the Totient Function
The Mystery of the Grazing Goat
The Jewel of Arithmetic: Quadratic Reciprocity
Isospectral Point Sets in Higher Dimensions
Expansion as Gravity and Many Worlds
Periods of Fibonacci Sequences mod m
Curvature of Linear Interpolation
Mayan Numeration
Cyclic Divisibility
Independence and Negation
Infinitely Many Rhondas
The Gambler's Ruin
Geometric Dot Products and Digit Reversals
Root-Matched Recurrences For DiffEQs
A Procrustean Protocol
Infinite Products and a Tangent Fan
Boolean Expansion as Linear Interpolation
Sighting Gravity
Additive and Multiplicative Partitions
Powers of Primes Dividing Factorials
Dice Rolling A Given Sum
Generating the Monotone Boolean Functions
Coloring The Edges of an Icosahedron
Platonic Solids and Plato's Theory of Everything
Sums of Powers in Terms of Symmetric Functions
Cumulative Permutation Sequences
On Ptolmey's Theorem
Binary Games
Evidence for Goldbach
Berkeley and the Infidel
Maxwell's Displacement and Einstein's Trace
Powers Differing By 1 (Catalan)
Barlow's Observation
The Quantity of Motion
Partitions into Consecutive Integers
Fibonacci, 1/89, And All That
The Speed of Sound
Perpendicular Regression Of A Line
Polynomials For Sums of Square Roots
Hypocycloidal Engine
Apollonius' Tangency Problem
Gravitational Slingshot
Minimum Difference Function
Intrinsic Wobble of Spinning Objects
Inverse-Square Forces and Orthogonal Polynomials
Gauss' Lemma Without Divisibility Arguments
Dual Failures with General Densities
Sliding Ladders in the Abstract
Newton's Birth Date and the Anni Mirabiles
Weibull Analysis
Fuzzy Logic
Meeting Probabilities
A Better Lottery?
Product Divisible By Sum of Squares
Lucas Pseudoprimes
Eigenvalue Problems and Matrix Invariants
Center of Gravity With Integer Coordinates
Iterated Means
Lesage's Shadows
Evaluating Probabilities of Boolean Events
Something Like Quantum Mechanics
On Relativistic Precession
Continuous Turing Machines
The Guest Star
Recurrences For Harmonic Sums
Saturnday, Sunday, Moonday
Newton's Proposition LXXI
Fractal Logic
On Olber's Paradox
What Mirrors Do
Bayesian Balls
Infinite Descent versus Induction
Ptolemy's Orbit
Square Roots by Pencil and Paper
Sums of Consecutive Nth Powers Equal to an Nth Power
Primitive Roots and Exponential Iterations
Quasi-Groups
Reverse Greed for Unit Fractions
Galileo's Anagrams and the Moons of Mars
Aliasing and Uncertainty
The Golden Pentagon
A Mersenne Coincidence
Franklin's Magic Squares
Did Archimedes Know Gauss-Bonnet?
A Removable Singularity in Lead-Lag Coefficients
Zeno and Uncertainty
Square Triangular Numbers
Did Einstein Misunderstand Aberration?
Planetary Alignment
A Quasi-Periodic Sequence
Volume of n-Spheres and the Gamma Function
The 450 Pound Problem (x^3 + y^3 = 6z^3)
The Factorial Number System
String Algebra
Is This a Prime?
The Half-Totient Tree
Sagnac and Fizeau
The Lemniscate and Spacetime
The Five Squarable Lunes
Interleaving Ad Infinitum
Loci of Equi-angular Points
Entangled Choices
Iterative Solutions of Homogeneous Linear Systems
The Statistics of Counterintelligence
High Order Integration Schemes
The No-Curvature Interpretation of GR
Frequency Response of First-Order Lag
Remembering Socrates
Fatio, Lesage, and the Camisards
The Bulging Earth
Pseudoprimes For x^2 - 4x - 9
Energy, Entropy, Enthalpy
In Defense of Base-Related Problems
Radical Expression For cos(2pi/7)
Interference or Refraction?
Generalized Lorentz Transformations and Relative Rotation
The Prismoidal Formula
Ancient Square Roots
On f(x^2+y^2) = f(x)^2 + f(y)^2 
Harmonia Mensurarum
Limit Cycles of xy (mod x+y)
Coprime Partitions
Fermat's Fallibility
Heron's Formula and Brahmagupta's Generalization
A Rotating Disk in Translation
Lead-Lag Algorithms
Balls In Bins
The Y10K Problem, Pascal's Wager, and Petersberg
Net Area and Green's Theorem
Sublime Numbers
Roemer's Hypothesis
More Results on the Form xy (mod x+y)
The Shortest Path To Trigonometric Identities
Do We Really Need Eigen Values?
Cramer's Paradox
Proving Algebraic Inequalities
Historical Assessments of the Fatio-Lesage Theory
Phase, Group, and Signal Velocity
Potential Flow and d'Alembert's Paradox
A Knot of Congruences
On  x^2 + y^3 = z^6 
The Laplace Equation and Harmonic Functions
Series Solution of Non-Linear Equation
Probabilities and Velocities
Dialogues
The Orbit of Triangles
Translating Aristotle
Integrating Factors
The Filter Of Observation
A Method Of Factoring Based On 1/N
The Distribution of Perfection
The Metamorphosis of Judas Iscariot
Choice Without Context?
Heron's Formula for Tetrahedrons
Rolling Spheres and Cones
Precession in a Circle
Round Trips and One-Way Speeds
On x^3 - x + y^3 - y = z^3 - z
Factoring Convex Figures
Restless Mass
Fermat's Last Theorem for Quadratic Integers
Generalizing Pythagoras and Carnot
Two Geophysical Coincidences
Quadratic Congruences
The 3m Halo
Quantum Interactions On Null Surfaces
Accidental Melodies
Magnetism and Earnshaw's Theorem
Why Was Michelson Surprised?
Huygens' Principle
Recurrences and Pell Equations
Whole Permutation Fractions
Compressor Stalls and Mobius Transformations
Photons, Neutrinos, And Their Anti-Particles
Series Solutions of the Wave Equation
Leibniz's Rule
Lead-Lag Frequency Response
Twin Times
The Wave Equation and Permutation of Rays
Benevolent Design and the Anthropic Principle
Leaning Ladders
Old Notes on Curvature
Differently Perfect
Iterative Isoscelizing
Reflecting on the Geometric Mean
Analytic Functions, The Magnus Effect, and Wings
Tilting Pencils
A Barrel of Colors and Null Outcomes
No Equilateral Triangles on a Chess Board
Reconstructing Brouwer
Simple Complex Quadratic Fields
Exponential Partitions
Detecting Squares
The Four Color Theorem
Combining Probabilities
Convoluting Arrays
Casting Out Nines
Napoleon's Theorem
Regular Polygonal Arrangements
A Special Property of 151
Retrograde Moon?
Mercurius in Sole Visus
Numbers Expressible As (a^2 - 1)(b^2 - 1)
The Shape of Coincidence
Angular Angst
Newtonian Gravity In Closed or Curved Space
Sums of Powers
Orbits and Perturbations
Hero and Fermat on Receding Mirrors
True, Equivalent, and Calibrated Airspeeds
Mean Distance from Vertex to Interior of Plane Figures
Bernoulli Numbers and Harmonic Series
Interleaving Fibonacci Numbers
Concordant Forms
The e in Petersburg
Fermat's Infinite Descent
If ab+1, ac+1, bc+1 are squares,... 
Galois Groups
Squares in Arithmetic Progression (mod p)
Hail Encounters
Inverse Square Weighted Interpolation 
The Fundamental Theorem For Palindromic Polynomials
Solving Magic Squares
Reducing Quartics to Cubics 
Annuities With Inflation 
Negative Numbers 
Random Triangles 
Two In the Shop
Relative Rain
Benford's Law
On 13 Bridges
Polynomials From Pascal's Triangle
Poincare Contemplates Copernicus
Hermitian Matrices
A Primer On Special Relativity
A Budget of Barn Poles
The Complex Domain of Probability
The Determinants of 4x4 Magic Squares
Acute Problem
Self-Similar Reverse-Sum Sequences
Proof That PI is Irrational
Intersections of Polynomials
Unit Fractions and Fibonacci
Waring's Problem
Herbert Dingle and the Twins
Convergence of Series (How NOT to Prove PI Irrational)
The Two Ohm Problem
Feynman's Ants
Importance of Being Urnest (B Balls, K Colors, N Urns)
Spiritualism and Electromagnetism
Cross-Linear Interpolation
Distances In Bounded Regions
Timely Confessions
Infinite Parallel Redundancy
N Points On Sphere all in One Hemisphere
Main Daigonals and Euler's Totient
Pent Up Ratios
Differential Operators and the Divergence Theorem
Two Properties of Markov Models
Unit Fraction Partitions
Laminar Transformations
N = (x^2 + y^2)/(1+xy) is a Square
Leibniz On Computers
Markov Models of Dual-Redundant Systems
Balls In Bins With Limited Capacity
Evaluate the Infinite Sum of n^2/(1+n^3)
Black Holes, Event Horizons, and the Universe
Egyptian Unit Fractions
Bernoulli Trials
Embedding Non-Euclidean Spaces in Euclidean Spaces
Archimedes on Spheres and Cylinders
Two-Dimensional Runs in Bernoulli Trials
Perrin's Sequence
Proof of Generalized Little Theorem of Fermat
Is Arithmetic Consistent?
Minimizing the Denominators of Unit Fraction Expansions
Sphere Packing in Curved 3D Space
Open-Loop and Closed-Loop Markov Models
Equation of a Triangle
414298141056 Quarto Draws Suffice!
Four Cubed (Complete Projection Cubes)
A Mirror to Physics
Bit-String Orbits Under Rotate-XOR
Partition Transform Cycles
Reliability with Mixed Periodic Repairs
Pythagorean Graphs
On General Palindromic Numbers
Sequences of Characters in Random Strings
Recurrence Relations for Ordinary Differential Equations
Reflective and Cyclic Sets of Primes
Fermat and Publication
Representing Sets of Pure Order
Interfering With PI
The Longitude of Lewis and Clark
On Case 1 of Fermat's Last Theorem
Uniform Squares
A Random Walk Through PI
Botches, Failures, and Successes
Cantor's Diagonal Proof
Certainty In Mathematics
Machin's Merit
Is Mathematical Knowledge Cumulative?
Weighing the Evidence
Morley's Trisection Theorem
Dialogue On Mathematical and Physical Knowledge
Permutation Puzzles
Quintisection of an Angle
Quadrilateral In a Circle
Integer Sequences Related To PI
The Ordering of Interactions
Computing the Partitions of n
Explanation of the Euclidean Algorithm
Series Within Parallel Resistance Networks
What is an Inertial Coordinate System?
Two Proofs for a Five Point Star
Golf Tournaments, Polyhedra, and Latin Squares
The Pigeon Hole Principle
Generating Functions for Point Set Distances
Harmonic Sums of Integers With k Binary 1's
Are All Triangles Isosceles?
Volume Under a Triangle
Clouds, Shy Squares, and Diophantus
Infinite Distance In Finite Time
White Ravens and Black Swans
Hipparchus on Coumpound Statements
Invariance, Contravariance, Covariance
Smallest Quad With Integer Sides, Perp Diags
Proof that e is Irrational
Constructible Points and Coverable Points
Gravity of a Torus
Anti-Carmichael Pairs
Mock-Rational Numbers
The Moon Always Falls Toward the Sun
Irreducibility Criteria
Newtonian and Relativistic Conservation Laws
Napoleonic Vectors
Accelerating In Place
Is the World Provably Indeterminate?
On the Density of Some Exceptional Primes
Path Lengths and Coordinates
Preserving the Wave
The Fundamental Anagram of Calculus
Higher-Order Wave Equations and Matter Waves
Cellular Automata
Magic Square of Squares
A Quarky Formula for Volume of Parallelpiped
On the Cumulative Results of Quantum Measurements
Bell Tests and Biased Samples
Impossible Strings for Cellular Automata
Born Rigidity, Acceleration, and Inertia
Round-Robin Ties
Piero della Francesca's Tetrahedron Formula
From Euclid to Gregory
Omni-Directional Flux
Orthomagic Square of Squares
On Random Chords
Automedian Triangles, Eigen Vectors, and Magic Squares
Derangements and Generalizations
Gamaliel's Principle
Trees On A Complex Plain
Equal Bisectors and Isosceles Triangles
Continued Fractions and Characteristic Recurrences
Hamilton Cycles on McCauley Graphs
The Expectation of Posterior Beliefs
Collecting k Complete Sets of Coupons
Double Equations from Triangles in Squares
Global Reversibility of Cellular Automata
Complete Solutions of Linear Systems
Reflections on The Mirror Question
Menelaus and Ceva
More on Exhibited Conservation in QM
On the Solution of the Cubic
One In The Chamber
The Projection Postulate In Quantum Mechanics
Kummer's Objection
Omar Khayyam On Cubics
Then There Were None
Christmas Ornaments
Measuring the Speed of Gravity
Generating Random Numbers In Order
Markov Models with Variable Transition Rates
The Greed Algorithm for Unit Fractions
Irrationality of Quadratic Sums
The Phase Effect of Acceleration
Tangent To PI
Cyclic Redundancy Checks
Average of Sigma(n)/n
Rational Sines of Rational Multiples of PI
The Super-Symmetric Mean
The Ten Means of Ancient Greece
Catch of the Day (153 Fishes)
Linear Fractional Transformations
Loxodromic Aliasing
Carnot, Organizer of Transversals
An Optical Illusion
Frequency and Energy
Weighing the Moon
An Algebraic Duality
Newton And The Sun
This Business of Interpretation
Lucas's Primality Test With Factored N-1
Highly Heronian Ellipses
Galois's Analysis of Analysis
Reactionless Propulsion (Not)
How Leibniz Might Have Anticipated Euler
Odd Greedy Unit Fraction Expansions
Factoring Zeta
Some Properties of the Lucas Sequence
Four Squares From Three Numbers
Ringing of the Changes (The Shape of 4!)
Meandering Convergence of a Dirichlet Series
Geodesic Diophantine Boxes
Determining the Galois Group of a Polynomial
Random Tiling of a Sphere
Constructing the Heptadecagon
Fourier Transforms and Uncertainty
Least Significant Non-Zero Digit of n!
Function 34776
Adam's Gold
Zeno's Mice and the Logarithmic Spiral
Unreasonable Effectiveness
Relativistic Speed Composition Formulas
Rigidly Re-Orienting An Extended Body
Redundant Systems With A Common Threat
Biased and Anti-Biased Variance Estimates
Failure Rates, MTBFs, and All That
Mean Partial Sums of Non-Convergent Series
Formal-Numeric Series
The Cube Unfolded
Constant Headings and Rhumb Lines
Markov's Marbles
Enveloping Circular Arcs
Normals From A Point To An Ellipse
Discrete Fourier Transforms
f(f(x))=exp(x) Generates the Multinomials
Laplace Transforms
Expulsion Sets
Normal Shock Waves
Discordance Impedes Square Magic
No Progression of Four Rectangles On A Conic?
Bi-Rational Substitutions Giving Squares
N-Fold Polygonal Spirals
Progress on Dedekind's Problem
Expected Area of Random Polygon in a Circle
The Tetratorus and Other Multi-Layered Polyhedra
The Stars In The Sky
Is e Normal?
The Curvatures of Hypersurfaces
Quantum Entanglement and Bell's Theorem
The Thought of a Thought - Edgar Allan Poe
Lagrangian and Hamiltonian Mechanics
Time for a Rocking Chair
Arranging the Solutions of f(x+y+z)=xyz
Continuity and the Void
Gerber's Gravity
Does a Uniformly Accelerating Charge Radiate?
From the Geometric Series to Stirling Numbers
An Infinite Wall
Platonions
Conditions for Relativity
Reasoning with Reversibility
Potential Energy, Inertia, and Quantum Coherence
Attractive Forces From Quantum Exchanges
Coherent Arrays of Squares
Zulu Hitler
The Hydrogen Atom
Non-Periodic Tilings With N-fold Symmetry
Aperiodic Tilings
Tilings Based on Perrin's Sequence
Pappus' Theorem
Pascal's Mystic Hexagram
The Resultant and Bezout's Theorem
A Unique Intersection Pattern for Plane Curves
Revisiting the Four-Point Parabola
The Crystallographic Restriction
Wonder En Is Gheen Wonder
Phased Summations
Harmonia Praestabilita
Is Entropy Quantized?
Reversibility and Entropy
Guessing Faster Than Light
World's Oldest Man Dies
Spacetime Coordinates and Frames
Partitions into Distinct Parts
An Anomaly In Translation
Hold Your Horses
Asymptotic Rate of an Open Loop Markov Model
Reliability of Series-Parallel Systems
Stokes' Mistake
Aberration of Forces and Waves
Electrodynamics - The Curl of the Curl
Noether's Theorem
Maxwell and Second-Degree Commutation
Central Forces from Retarded Action
Retarded and Advanced Potentials
Galileo's Law of Planetary Motion
Propagation of Pressure and Waves
Mnemonic Coins and Pattern Avoidance
Whittaker and the Aether
The Virial Theorem
Gravity, Entropy, and Coherence
Electoral Draws
The Electric Potential of a Moving Charge
The Electric and Magnetic Fields of a Moving Charge
Undetected Failures of Threat Protection
Pitch and Color Recognition
Color Space, Physical Space, and Fourier Transforms
Probability of Intersecting Intervals
Efficient and Final Causes
Gravitational and Inertial Mass in Newton's Principia
Series Solution of Relativistic Orbits
Continuous From Discrete Transfer Functions
If the Sun Were Suddenly to Explode
Geodesics by Differentiation
The Doppler Effect
General Relativity and the Principle of Inertia
Timed Markov Models
Periodic and Continuous Repair Models
Hawking Radiation as Delayed Choice
Hawking Radiation and Super-Acoustic Diffusion
Rotation Matrices
Hierarchial Repair and Quantum State Reduction
The Portraits of Emilie
Spacetime Inversion
Complete Markov Models and Reliability
Conquering the Perihelion
Markov Models with Boolean Transition Logic
Einstein on the Inertia of Energy
Another Derivation of Mass-Energy Equivalence
Binary Reverse-Sum Automata
Resonance and Motion
Null Cones in Cylindrical Dimensions
The Effect of Variations in Speed
The Ergodic Hypothesis and Equipartition of Energy
The Necessity of the Quantum
Zur Farbenlehre
Maxwell's Paradox of Attraction
Kinetic Pressure and Tetrode's Star
The Twelfth Rainbow
The Spacetime Hole and the EPR Arguments
Why Maxwell Couldn't Explain Gravity
Spherical Waves in Higher Dimensions
Causality and the Wave Equation
Fractional Calculus
Waves in Ascending and Descending Dimensions
Euclid's Proposition III,20 and its Converse
Dirac's Belt
Exponential Spiral Tilings
On de Bruijn Grids and Tilings
Controversies Over the Equivalence Principle
On Euler's Pentagonal Theorem
Stochastic Rhombic Tiling
Inverse Functions
Markov Models with Aging Components
Guillaume, Guillaume, and Einstein
Age Distributions in Continuous Markov Models
On The Many Principles of Equivalence
The Moszkowski Affair
On the Ellipse
Dialogue on the Foundations of String Theory
Centripetal and Centrifugal Forces
The Amanda Arrangement
Does Relativistic Mass Imply Special Relativity?
On Diffraction
On the Origins of Quantum Mechanics
Bras, Kets, and Matrices


Reflections on Relativity
Preface
Experience and Spacetime
Systems of Reference
Inertia and Relativity
The Relativity of Light
Corresponding States
A More Practical Arrangement
Staircase Wit
Another Symmetry
Null Coordinates
The Spacetime Interval
Force Laws and Maxwell's Equations
The Inertia of Energy
Doppler Shift for Sound and Light
Stellar Aberration
Mobius Transformations of the Night Sky
The Sagnac Effect
Refraction Between Moving Media
Accelerated Travels
The Starry Messenger
Thomas Precession
Postulates and Principles
Natural and Violent Motions
De Mora Luminis
Stationary Paths
A Quintessence of So Subtle a Nature
The End of My Latin
Zeno and the Paradox of Motion
A Very Beautiful Day
Constructing the Principles
Immovable Spacetime
Inertial and Gravitational Separations
Free-Fall Equations
Force, Curvature, and Uncertainty
Conventional Wisdom
The Field of All Fields
The Inertia of Twins
The Breakdown of Simultaneity
Absorbing Acceleration
Tensors, Contravariant and Covariant
Curvature, Intrinsic and Extrinsic
Relatively Straight
Schwarzschild Metric from Kepler's 3rd Law
The Equivalence Principle
Riemannian Geometry
The Field Equations
An Exact Solution
Anomalous Precession
Bending Light
Radial Paths in Spherically Symmetrical Fields
Proper Time in Circular Orbits
Flying Clocks
Acceleration in Schwarzschild Coordinates
Sources in Motion
Is the Universe Closed?
The Formation and Growth of Black Holes
Falling Into and Hovering Near A Black Hole
Curled-Up Dimensions
Packing Universes In Spacetime
Multiple Hubble Constants?
Boundaries and Symmetries
Global Interpretations of Local Experience
Kepler, Napier, and the Third Law
Newton's Cosmological Queries
The Helen of Geometers
Refractions On Relativity
Scholium
On Gauss' Mountains
Strange Meeting
Who Invented Relativity?
Paths Not Taken
In The Neighborhood
Up To Diffeomorphism
Higher-Order Metrics
Spin and Polarization
Entangled Events
Von Neumann's Fifth Postulate
Bell's Freedom
The Gestalt of Determinism
Imaginary Explanations
Locality and Temporal Asymmetry
Spacetime Mediation of Quantum Interactions
Conclusion
Mathematical Miscellany

Washed Away
Recovery
Practice Rooms
Steps
Hymn J
Temporary
Rain
Hymn C
Of Strangers
Student
Mercy
Hymn Y
Missing
1918
Prospect 6
Ideas
The Light Cubed
13 Aprils
Losing Hartford
Carol
Descent
Over There
Eta Carinae
Macbeth