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There were a lot of folks chiming in with alternatives due to the sad demise of Fry's electronics.
So what are we all doing with all those fun components?
What's your project, Soylent?
Doesn't need to be related to electronics, if you're painting a portrait or drafting a treatise on the heliocentric nature of the Solar System let's hear about it!
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What's Your Project, Soylentils?
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(Score: 3, Interesting) by khallow on Saturday February 27 2021, @05:07AM (6 children)
So anyway, I'm looking at a weird mathematical structure (the regularization which I think is a rudimentary example of a regularity structure [soylentnews.org] (or perhaps a "renormalization" transform over a regularity structure) and hence, part of the reason I'm using the term regularization) that I haven't seen before and trying to figure out its significance. For example, it appears that some regularizations can be extended from maps of polynomials to polynomials, to maps of continuous functions to continuous functions, but most can't.
The idea is that if I have an unknown set of orthogonal polynomials, is there a regularization to a known classical set, like the Legendre polynomials, so that the coefficients of the unknown polynomial set be readily calculated from the regularization and our knowledge of the classical set that we regularize to?
Now back to your regularized channel.
(Score: 1, Informative) by Anonymous Coward on Saturday February 27 2021, @03:31PM (2 children)
You should definitely look into pissing contests. Far more useful.
(Score: 1) by khallow on Saturday February 27 2021, @11:21PM (1 child)
(Score: 0) by Anonymous Coward on Sunday February 28 2021, @09:08AM
And no one if forcing us not to care, besides khallow him self, who seems to be doing all he can to force us, against all our well-brought up manners, to not care about his constant arguments in bad faith. Stop, khallow, please stop, lest we have to not pay attention to you, anymore.
(Score: 2) by hubie on Sunday February 28 2021, @05:30PM (2 children)
What do you mean by orthogonal polynomials with arbitrary inner product? Isn't the definition of an orthogonal polynomial one where its inner product is well defined?
I'm not quite sure I understand what you mean by regularization. Do you mean that if you have one set of polynomials, like Hermite (the favorite of physics students studying QM!), that you want to transform them into another kind?
It sort of sounds to me in your last paragraph that you are wanting to implement the Gram-Schmidt Method [hmc.edu].
(Score: 1) by khallow on Monday March 01 2021, @04:01AM (1 child)
The problem is that there's a lot of well-defined inner products possible. A typical one has the form of the product of two functions (for which this calculation will be the inner product), times a positive weight function (which can be scaled so that the inner product of the constant 1 function with itself is 1 and which, if necessary tapers off so any polynomial inner product yields a non-infinite value), integrated over an interval (which can include +/- infinity as limits of integration).
Gram-Schmidt is the default way of finding these polynomials. From above scaling of the weight function, you'll always have 1 as the first. Then for a given degree n, if you've calculated the n-1 orthonormal basis vectors, you can calculate the next by taking x^n, subtracting off projections of the previous n-1 polynomials (the usual Gram-Schmidt process), and scaling appropriately.
What's interesting is that I think I might have a mapping of these arbitrary systems to one of a small set of classical systems. The classical systems have a bunch of weird relationships that allow for faster computation of the polynomials than one would get through Gram-Schmidt (using the Legendre polynomials as an example): via a generating function [wikipedia.org] (and a recursive formula), differential equation [wikipedia.org], and via Rodrigues' formula [wikipedia.org]. It looks to me like the regularization may allow for these formulas to be pulled over into weird operators on the original space.
But there's a few big obstacles at present. First, I think it's important that continuous functions are preserved under the regularization (since for bounded intervals of integration, I can calculate inner products of arbitrary continuous functions under an arbitrary continuous weight function so any regularization shouldn't be breaking that). I'm still figuring that out. Second, it may end up being more computational difficult to perform the regularization computations than to do Gram-Schmidt. Again, an issue that can be answered once I figure out what the regularization is actually doing.
(Score: 2) by hubie on Tuesday March 02 2021, @09:30PM
Thank you for your explanation. If you make progress I would be very much interested in what you find out. You've tickled a few of my neurons that have been idle for a few decades. Coincidentally, I was watching a blackpenredpen YouTube video [youtube.com] last night having to do with the generating function for Chebyshev polynomials.