Meetup summary

2024-11-29 - Generatingfunctionologypracticum - Classical Edition

(Upcoming)

Recommended reading:

  • Wikipedia’s entry on formal power series, particularly the section on operations on formal power series.
  • Don Knuth’s treatment of power series in TAOCP (Volume 2, section 4.7). I believe this is only available starting in the third edition.
  • My supplemental post on formal power series composition. This is a surprisingly difficult operation, so I recommend prereading and thinking about it.
  • Knots!

Agenda:

The last few sessions have run over and not allowed nearly enough time to get through the original plan. We’ll try paring it down to a narrow subject this time.

  • Knot show and tell. This is not your mother’s mathematical embedding of the S1S^1 into 33-space, but rather something you can tie with a physical rope segment. “Bring” your favorite knot and be prepared to show us how to tie it and expound on its virtues and vices. I use the word “knot” generously. Feel free to choose a knot proper, a loop knot, a hitch, a bend, or any other knot-like structure that interests you. Google for something interesting if you don’t already have one. Optionally, bring some rope or cord for demonstration (though I’ll have some available for this purpose as well).
  • Discuss how to numerically pick out GF coefficients with simple programming techniques. Sadly, it turns out that working directly with GFs on real, complex problems quickly becomes infeasible on paper (modulo asymptotics), so we need a way to actually compute answers to the problems we care about. My hope is to get through the following:
    • Multiplication (of a series by a constant or by a series)
    • Division (by a series)
    • Series composition
  • Work through a simple Python program that I’ve written to extract power series coefficients using “classical” (imperative generator) techniques. Verify that some canonical examples work (finite series, geometric series, well-known Taylor expansions, basic compositions of these).
  • If time allows: Go through some basic generating function problems with simple constructions.

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