## 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.