Some Open Questions about Random Walks, Involutions, Limiting Distributions, and Generating Functions
✍ Scribed by Lou Shapiro
- Publisher
- Elsevier Science
- Year
- 2001
- Tongue
- English
- Weight
- 116 KB
- Volume
- 27
- Category
- Article
- ISSN
- 0196-8858
No coin nor oath required. For personal study only.
✦ Synopsis
I want to present a few questions that I think are of general interest and which I hope to see solved in the next five years. These problems were first presented at the Foatafest in the fall of 2000. Since then I have had the benefit of many people's interest and there has been progress on several of the problems as well as suggestions for some other excellent problems. The solutions are due to Frank Schmidt, a referee, and Naiomi Cameron and Lynnell Matthews, graduate students at Howard Unviersity. The problems appear in three categories, limiting distributions, involutions, and oddball. Let's start with an oddball problem, the missing resistor problem.
These circuits, where all resistors are one ohm, have resistances 2/1 5/3 13/8 34/21 F 2n /F 2n-1 where F n n≥0 = 1 1 2 3 5 8 13 21 , the Fibonacci numbers. This leads to our first open question. Q1. What simple family of circuits will have resistances C 2n /C 2n-1 (or something similar) where C m = 1 m+1 2m m is the mth Catalan number?
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