Mathematics

Presentations

• Lambert's W Function, Infinite Exponentials, Powertowers, Iterated Exponentials and Tetration
  1. The Author's Strange Encounter With Johann Heinrich Lambert.
  2. Infinite Exponentials (often called infinitely iterated exponentials or exponential towers).
  3. A Slightly Deeper Analysis of Infinite Exponentials.
  4. A Maclaurin Expansion for Lambert's W function.
  5. Approximating the Principal Branch of the Complex Lambert's W Function.
  6. Approximating the Branches of the Complex Lambert's W Function Using Analytic Continuation.
  7. An Infinite Family of Hyper-Lambert Functions And Their Use In Solving Exponential Tower Equations.
  8. A Detailed Look at the Hyperroot Functions Using Lambert's W Function.
  9. A Continuous Extension of the hyper4 Operator.
  10. An Infinitely Differentiable Extension for the hyper4 Operator.
  11. A Series Expansion for ^m(e^x) Using Lambert's W Function.
  12. A More Consistent Definition of Tetration for Rational Exponents.
  13. The General Infinite Exponential and its Limit.
  14. The Analytic Continuation of the Infinite Hyperpower Function F(x) = ^+∞x.
  15. The Birth of the Infinite Tetration Fractal.
  16. The Exponential Fractal.
  17. A Strange Hyperexponential Series.
  18. Solving the First Auxiliary Real Exponential Equation Using Lambert's W Function.
  19. Solving the Second Auxiliary Real Exponential Equation Using Lambert's W Function.
  20. Solving the n-th Auxiliary Real Exponential Equation.
  21. A Family of Hyperpower/Hyperfactorial Series.
  22. File Signatures and Tetration of 2.
  23. A Collection of References for Infinite Exponentials and Tetration.
• Number Theory
  1. An Interesting Sequence of Cyclic Digits.
  2. The 3n+1 problem.
  3. Does Arc Light Flickering Interfere With Video?
  4. Trends in Euler's Totient φ Function.
  5. A Short Tutorial On Big Numbers.
• Analysis
  1. Suprema and Infima in Real Life.
  2. The Cantor Function.
  3. Temperatures on The Surface of the Earth.
  4. Is There a Limit on How Fast the Olympic 100 Meters Can Be Ran?
  5. Modelling Time Under Total Anesthesia.
  6. An Infinite Hierarchy of Hyperfactorial Functions.
  7. Overall Function Behavior.
  8. Modelling Relative Mind Power Using the ELO Rating System.
• Combinatorics
  1. A Program to Solve Loyd's Wooden Tile Puzzle.
  2. Distinct Resistances Possible With at Most n Distinct Resistors.
• Topology
  1. The Author's View of the Universe.
• Philosophy of Mathematics
  1. Can Mathematics Explain What Is Real?
  2. Understanding Symbolic Levels.
  3. Should New Mathematics Ideas Be Announced?.
  4. Is The Mind Infinite?
• Set Theory
  1. The Most Beautiful Construction in Mathematics.
  2. How Fast Can You Count?
• Analytic Geometry
  1. Estimating the Area and Perimeter of a Circle and π Using Archimedes' Method.
  2. The Golden Ratio and Other Metal Ratios.
  3. Mirrors!
• Financial Mathematics
  1. An Almost Impossible Puzzle Designed For The Author by His Father.
  2. The Author's Webpage Statistics Projector.
• Complex Analysis
  1. Using Maple to Visualize the Complex w-Plane.
• The Mathematics of Chess
  1. Every Chess Configuration.
  2. The Main Principles of a Chess Engine.
  3. The Attack-Force Metric in Chess.
• Computer Science and Cryptography
  1. How Does tinyurl Work?
  2. The Dynamics of Computer Programming.
  3. A Lossy Compression Scheme and its Fixed Points.
• Miscellaneous
  1. The Author's Puzzles.
  2. From Mathematics To Color Perception.
  3. Lichtenberg Figures.
  4. Numerology.

Papers

1. "On An Application Of Lambert's W Function to Infinite Exponentials", Complex Variables, 49(11) (Sep. 2004), 759-780.
2. "On Solving the p-th Complex Auxiliary Equation f^(p)(z)=z", Complex Variables, 50(13) (Oct. 2005), 977-997.
3. "Lambert's W Function and Convergence of Infinite Exponentials in the Space of Quaternions", Complex Variables, 51(12) (Dec. 2006), 1129-1152.
4. "Corrigendum for 3", Complex Variables, 52(4), (Apr. 2007), 351.
5. "Explicit Solution of the Kepler Equation" (with Alexandr Dubinov), Physics of Particles and Nuclei, Letters, 4(3) (May 2007), 213-216.
6. "On Some Applications of the Generalized Hyper-Lambert Functions", Complex Variables, 52(12) (Dec. 2007), 1101-1119.
7. "On the Signal, Spectrum and Some Fractal Properties of Video-Feedback" (to appear in Intl. J. of Signal Processing). (associated video 62 MB .avi).

1. "On The Convergence of Series With Infinite Exponentials and Iterated Factorials".
2. "On Extending hyper4 and Knuth's Up-arrow Notation to the Reals".
3. "On a Probabilistic Theorem for the Asymptotic Densities of φ, the Timing of the Euclidian GCD Algorithm and a Tetration Function".
4. "On Certain Design Aspects of the Implosion-Type Nuclear Weapon".

1. "The Problem of the Crack in the 'Generally' Anisotropic Disk" (the author's father's Ph.D. Thesis. Original Greek Title: "Το Πρόβλημα της Ρωγμής εις τον «Γενικώς» Ανισότροπον Δίσκον").

Function/data graphs were done with Maple and THINK Pascal and geometrical schematics with EucliDraw.

Citations

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