This website is doubtlessly one of the best introductions to higher math available online. I've gained a lot by both finding and reading the website posts. A weekly email with some sort of order would be great and I would sign up immediately. Keep it up!!!!
Math ∩ Programming
31–40 of 40 posts
Re: Math ∩ Programming
#32Earlier quoted context omitted.
Huh. I always thought n! grew faster than c^n which would be even worse than exponential. Maybe enough cancels out to make it simpler than it appears. edit Actually, for even dimensions it's pretty clear. n = dimension/2 pi^n / n! factorial wins. The problem is worse than exponentiation.
At some point, I wondered if n! is proportional to n^n. Turns out, it is: n! ~= (2 * pi * n)^1/2 * (n/e)^n ( https://en.wikipedia.org/wiki/Stirling%27s_approximation )
Not exactly "proportional", because you have that pesky e in the denominator. But that's the right idea. Basically, to get n! you're multiplying together n things that are sort of n-ish, so you'd expect n! ~ n^n. (When the numbers get really big, like in statistical mechanics, I've seen the approximation log n! ~ n log n.) The next step is to figure that you're multiplying together n things that are on average n/2, so n! ~ (n/2)^n. But then it really turns out that you should have been using a geometric average (since you're multiplying), not an arithmetic one, so n! must be smaller yet. (I don't know a way to get (n/e)^n without doing an integral, though.)
Re: Math ∩ Programming
#33This is really, really awesome. I have a long train ride coming up, so I just put together a quick wget script to get all the pages for when I have zero internet. :)
Re: Math ∩ Programming
#34@j2kun: Just curious, any chance you'd do an explainer on the Curry-Howard equivalence[1] sometime? Forgive me if you already have, but a search didn't turn anything up. Seems pretty relevant to math ∩ programming and also of interest to a lot of generalists.
[1] https://en.wikipedia.org/wiki/Curry%E2%80%93Howard_correspon...
Re: Math ∩ Programming
#35Earlier quoted context omitted.
At some point, I wondered if n! is proportional to n^n. Turns out, it is: n! ~= (2 * pi * n)^1/2 * (n/e)^n ( https://en.wikipedia.org/wiki/Stirling%27s_approximation )
Not exactly "proportional", because you have that pesky e in the denominator. But that's the right idea. Basically, to get n! you're multiplying together n things that are sort of n-ish, so you'd expect n! ~ n^n. (When the numbers get really big, like in statistical mechanics, I've seen the approximation log n! ~ n log n.) The next step is to figure that you're multiplying together n things that are on average n/2, s…
Right, proportional in the sense that for both n! and n^n, the fastest-growing component is ~n^n. (I was curious in the context of which grew faster for larger values of n in the big O notation).
Re: Math ∩ Programming
#36QUOTE: > Like in a non-dimensional number?. For example the Reynolds number in fluid flow.
No, the meaning dimension there is different. In the Reynolds number it's a ratio of measures where the units cancel. Here we're talking about mapping data into a high dimensional space, then trying to project that down to the minimum dimension space that can preserve the relevant information. In the example we don't have any reason to even think all our dimensions are orthogonal, we're assigning a new dimension for…
> Here we're talking about mapping data into a high dimensional space, then trying to project that down to the minimum dimension space that can preserve the relevant information.
Isn't this what M/PCA is all about? https://en.wikipedia.org/wiki/Principal_component_analysis and https://en.wikipedia.org/wiki/Multilinear_principal_componen...
Re: Math ∩ Programming
#37Math ∩ Programming = Math. Just sayin'. :P
That would imply Math is a subset of Programming, when in fact it's pretty clearly a strict superset of Programming.
Re: Math ∩ Programming
#38Re: Math ∩ Programming
#39@j2kun: Just curious, any chance you'd do an explainer on the Curry-Howard equivalence[1] sometime? Forgive me if you already have, but a search didn't turn anything up. Seems pretty relevant to math ∩ programming and also of interest to a lot of generalists. [1] https://en.wikipedia.org/wiki/Curry%E2%80%93Howard_correspon...
I think I have a mildly different opinion about the correspondence than most other people on the internet. That's mostly why I have decided to write about other things. But then again maybe that's a good reason to write about it.
Re: Math ∩ Programming
#40I'd like to find a way to reorganize Math ∩ Programming eventually. As alfonsodev points out, it's a bit hard to find stuff for newer and less mathy readers, though there is a handful of articles aimed at them. [1,2,3,4] As other users point out, they don't like the layout. Suggestions? [1]: http://jeremykun.com/2011/06/26/teaching-mathematics-graph-t... [2]: http://jeremykun.com/2014/05/26/learning-to-love-complex-n…
Off topic - thank you for producing such an awesome blog!