The shadowy world of umbral calculus [video] | Hacker News Reader

The shadowy world of umbral calculus [video]

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Excellent video.

Bill Heinemann, one of the three guys who created the original Oregon Trail game, taught me the basics of the umbral calculus when I was in middle school for the purpose of math competitions. Fitting polynomials to data via successive applications of the forward difference operator, etc. I remember being fascinated and applying difference operators to all kinds of number sequences to see what other rules I could figure out. Ended up majoring in math and now I'm doing my PhD. Strange to think how different my life would be if Bill had never introduced me to the good old forward difference...

Discrete Fractional Calculus changed my life too. It got me interested in mathematics in a way that the ordinary stuff never could.

https://books.google.com/books/about/Discrete_Fractional_Cal...

"As a matter of fact, the feeling of witchcraft that has hovered over umbral calculus is probably what kept it from dying altogether. ... At long last, it was realized that umbral calculus could be made entirely rigorous by using the language of Hopf algebras, and this was done in a lengthy treatment. However, although the notation of Hopf algebra satisfied the most ardent advocate of spic-and-span rigor, the translation of 'classical' umbral calculus into the newly found rigorous language made the method altogether unwieldy and unmanageable. Not only was the eerie feeling of witchcraft lost in the translation, but, after such a translation, the use of calculus to simplify computation and sharpen our intuition was lost by the wayside." (Gian-Carlo Rota, 1994)

Why does something have to be portrayed as spooky and magical. If it works it works, then it’s our job to explain clearly why.

That quote seems strange.

I have not seen the explanation in terms of Hopf algebras but if the translation makes it unwieldy that seems like it’s not a very good language to use to describe it, Most of these umbral calculations seem pretty clean.

It was spooky and magical because it worked so well and it was based on symbolic transformations -- not mathematical ones. It'd be liking having calculus' basic derivative rules without their proofs.

Without the rigorous backing, Umbral calculus was used for research and exploration but then additionally work had to be done to prove the results.

Now, with the rigorous backing, the results stand alone - as the umbral transformations have been proven to be rigorously valid.

I am not too knowledgeable on the proofs of validity of Umbral calculus but my guess would be that only part of the vast library of umbral tricks, hacks, and symbolic bashing have been formalized - so the rigorous subset of provably correct transformations is likely more limited and thus cumbersome.

Wow that was pretty mind-blowing.

His Phi operator almost felt like he was doing some sort of "diagonalization" step of continuous derivatives/integrals into discrete differences/sums in the same way you would diagonalize a matrix.

I wonder if there's something there, maybe phi is some sort of analog to eigenvectors of integrals/derivatives.

Yeah, the Fourier transform diagonalizes differentiation. Or in other words "differentiation is a Fourier multiplier". Integration is like a low-pass filter (you divide by the amplitude of a component by its frequency and so higher frequencies get quieter) and differentiation is "noisy" since high frequency components get louder.

d^2/dt^2(sin kt) = -k^2 sin(kt)

This idea is used in the study of partial differential equations, especially dispersive.

https://www.math.ucla.edu/~tao/preprints/chapter.pdf

This reminds of Doron Zeilberger's statement that "real" analysis is just a degenerate case of discrete analysis [0].

[0]: https://sites.math.rutgers.edu/~zeilberg/mamarim/mamarimPDF/...

I recently found Tom Copeland's blog 'Shadows of Simplicity', and liked the part I can discern. Here is the link to the Umbral Calculus tagged posts:

https://tcjpn.wordpress.com/tag/umbral-calculus/

Opinion: One of the best higher math videos I have seen in a long time!

Future to-do: Re-watch this video!

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