Mathematics

«Αριθμός, το Πάν»

Presentations^[1]

• Lambert's W Function, Infinite Exponentials, Powertowers, Iterated Exponentials and Tetration

  (Preliminary results, 10 years ago. For published versions of the majority of this section, consult Google scholar^[3]).
1. A Strange Encounter With Johann Heinrich Lambert.
2. Infinite Exponentials.
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 of the hyper4 Operator.
11. A Series Expansion of ^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 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.

• Group Theory

  1. The TetraMag Puzzle.

• Harmonic Analysis, Fourier Analysis, Spectral Theory

  1. Google Scholar and the Spectra of the Scientists.
  2. Web Site Performance.

• 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. A Program to (almost) Solve the 24, 3-color Tile Puzzle.
  3. Distinct Resistances Possible With At Most n Distinct Resistors.
  4. Match Puzzles.
  5. Find the Number!

Topology

1. The Author's View of the Universe.

• Philosophy of Mathematics

  1. Can Mathematics Explain What Is Real?
  2. Understanding Symbolic Levels.
  3. Is The Mind Infinite?
  4. The Importance of the Golden Ratio.

• Set Theory

  1. A Most Beautiful Construction In Mathematics.
  2. How Fast Can You Count?
  3. Is There a Veblen Hierarchy of Functions that Doesn't Cause an Untimely Collapse?

• 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. The Author's Webpage Statistics Projector.

• Complex Analysis, Dynamical Systems and Fractals

  1. Using Maple To Visualize the Complex w-Plane.
  2. Lichtenberg Figures.
  3. A Quick Comparison Between the Power Tower and the Mandelbrot Fractals.

• 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 Lossless Data Transmission.

• Numerical Analysis

  1. An Application of Numerical Analysis To Science Fiction.

• Miscellaneous

  1. From Mathematics To Color Perception.

Theses^[2]

1. "The Problem of the Crack in the 'Generally' Anisotropic Disk" (English Translation of the author's father's Ph.D. Thesis. Original Greek Title: "Το Πρόβλημα της Ρωγμής εις τον «Γενικώς» Ανισότροπον Δίσκον").
2. "The Problem of the Infinite Exponential", thesis submitted to the Mathematics Department of the Agricultural University of Athens. English, Greek.
3. Companion Maple code for some of the various exponential fractals of the thesis above, at: Maple Primes.

Notes

1. Function/data graphs done with Maple and THINK Pascal and geometrical schematics with EucliDraw.
2. Using LaTeX.

Citations

• Google Scholar.
• R^G.
• math.stackexchange for old answers. For recent answers, please consult: ^[26.8].
• Links
• Mathematics answers on Quora.