Mathematics

Notes and Papers by Philip J. Erdelsky

Please e-mail comments, corrections and additions to the webmaster at pje@efgh.com.

Approximate Matrix Inverses and the Condition Number: A note on approximate matrix inverses and the condition number, which I published without proof circa 1969. Proof included.
The Birthday Paradox: If there are at least 23 people in a randomly-selected group, the probability is greater than 50 percent that at least two were born on the same month and day (but not necessarily in the same year). A little C/C++ programming quickly demonstrates this well-known and counterintuitive result.
A Piece of Pi: Mathematics contains many interesting transcendental constants, but only one of them was known to the ancients. It is pi, the ratio of the circumference to the diameter of a circle. We use Archimedes' method and a modern computer to get a very accurate value for pi.
A Conjecture on Collections of Subsets: A little combinatorial problem that bugged me for years. It was finally resolved by searching the Internet.
The Cayley-Hamilton Theorem: My favorite proof of the beautiful Cayley-Hamilton Theorem of matrix algebra.
The Bridges of Königsberg: A classic problem solved by the Swiss mathematician Leonhard Euler in the eighteenth century. It is easy to state and fairly easy to solve.
The Fibonacci Sequence: The Fibonacci sequence {0, 1, 1, 2, 3, 5, 8 ...}, in which each term is the sum of the two preceding terms, has a complicated closed form that is easy to derive.
Things Computers Can Never Do: First published in Dr. Dobb's Journal May 1987. A simplified explanation of uncomputable and undecidable problems.
Modern Algebra: A treatise on elementary set theory, groups, rings, integral domains and fields. A work in progress, which is being augmented and improved from time to time.
Philip J. Erdelsky Ph.D. Thesis: The thesis I wrote in 1969 for my Ph.D. at the California Institute of Technology.