# Mathematics

#### Notes and Papers by Philip J. Erdelsky

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.
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.