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Mathematics all-in-one cheat-sheet (2013) [pdf]

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Re: Mathematics all-in-one cheat-sheet (2013) [pdf]

#91

Post by mkl »

xwin wrote: post #79
andrepd wrote: post #72

I really didn't want to be that guy, and props for the author for such comprehensive reference work, but god, what an appalling typography. Bad tables with too many lines bitmap formulas with misadjusted size, poor spacing, ugly font... Almost everything that could be wrong is wrong.

I came here to say the same thing. Looks like the author did not use a good authoring tool to produce good math output. I generally take notes like this with LaTeX and compile them to PDF but PDFs can be large and if committing to a Git repo can lead to large .git directories. The BasicTeX package for Mac (brew install basictex) has worked well for me: http://www.tug.org/mactex/morepackages.html These days I also tak…

Why commit the output files? I generally put "*.pdf" in .gitignore.

Re: Mathematics all-in-one cheat-sheet (2013) [pdf]

#92

Post by crdrost »

AstralStorm wrote: post #24
_2d30 wrote: post #18

Earlier quoted context omitted.

Aren't most proofs of the CLT pretty extensive?

There are a few short ones going from area integral comparisons.

The easiest is to look at characteristic functions and cumulants; for a random variable T with PDF p(t) we say T ~ p and define

φ_T[f] = ∫ dt p(t) exp(-2πift) = ⟨ exp(-2πifT) ⟩

If two variables X ~ r and Y ~ s are independent then you can prove [from ⟨f(X) g(Y)⟩ = ⟨f(X)⟩ ⟨g(Y)⟩ or X+Y ~ q where q(z) = ∫ dx r(x) s(z — x)] that their sum has a characteristic function

φ_{X+Y} = φ_X + φ_Y

And therefore the “sample mean” M of n IID variables is itself a random variable with characteristic function

φ_M[f] = ( φ[f/n] )^n.

So we find that

log φ_M[f] = n log φ[f/n] ≈ 0 + i a f – b f²/n + O(f³/n²).

These terms [a, b] from expanding the log of the characteristic function constitute the cumulant expansion and for large n the other terms shrink to zero, so that the characteristic function is to first order in 1/n a Gaussian.

The characteristic function was a Fourier transform of a PDF, so an inverse Fourier transform gets it back:

p(t) = ∫ df φ_T[f] exp(2πift)

But the Fourier transform of a Gaussian is just a Gaussian.

Re: Mathematics all-in-one cheat-sheet (2013) [pdf]

#95

Post by lotaezenwa »

supernova87a wrote: post #75

Wow, I have to say I feel somewhat bad for the author who spent so much time (months?) compiling this semi-comprehensive reference work. There's so much information, but at the same time, so little useful information to any particular reader. It's so broad as to be a hindrance to using it in any sort of daily reference. Who would use this? Wouldn't you probably resort to a reference more specific to your field? Is th…

This comment reminds me of the archetypal high-school student who asks "when are we ever going to use this stuff?"

If you have to ask...

Re: Mathematics all-in-one cheat-sheet (2013) [pdf]

#96

Post by digitalsushi »

killjoywashere wrote: post #48

Ah, 10% of the first section of the CRC Handbook of Chemistry and Physics, with amateur typography. Sort of like Greenspun's tenth rule, but for math.

I've heard of mathematicians handwriting proofs in the margins of books. Like an MVC, I can appreciate the data without appreciating the presentation.

Re: Mathematics all-in-one cheat-sheet (2013) [pdf]

#97

Post by snazz »

mkl wrote: post #91
xwin wrote: post #79

Earlier quoted context omitted.

I came here to say the same thing. Looks like the author did not use a good authoring tool to produce good math output. I generally take notes like this with LaTeX and compile them to PDF but PDFs can be large and if committing to a Git repo can lead to large .git directories. The BasicTeX package for Mac (brew install basictex) has worked well for me: http://www.tug.org/mactex/morepackages.html These days I also tak…

Why commit the output files? I generally put "*.pdf" in .gitignore.

Probably so that someone can download it without compiling it? GitHub Releases or an equivalent would probably be better, though.

Re: Mathematics all-in-one cheat-sheet (2013) [pdf]

#99

Post by [deleted] »

coldtea wrote: post #86
andrepd wrote: post #72

I really didn't want to be that guy, and props for the author for such comprehensive reference work, but god, what an appalling typography. Bad tables with too many lines bitmap formulas with misadjusted size, poor spacing, ugly font... Almost everything that could be wrong is wrong.

If I had only read this comment and didn't look into the PDF, I'd have thought it indeed have some appalling typography -- as opposed to some minor nitpicks I wouldn't even care about...

[deleted]

Re: Mathematics all-in-one cheat-sheet (2013) [pdf]

#100

Post by dual_basis »

rando171717 wrote: post #98

Slightly off-topic: from a security perspective, how safe/not safe is it to be clicking open random pdf's? Wouldn't this be a possible avenue for malware?

Sure, but so is clicking open websites unless you have Javascript disabled, in which case half the web is broken for you.
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