ℝolliℵg M∀th Thr∑a∂

Makes me wish they just taught me Analysis in 9th and 10th grade, it would have all seemed less directionless.

I have a question re: the hairy ball theorem.

We know that combing the hairs on one ball flat will leave at least one tuft of hair sticking out unable to be flattened. But what about two hairy balls that are touching? Can we perform a smooth combing over them?

i hate podcasts but this is good imo

http://www.csbookclub.com/

crüt: depends what you mean by "touching" and how you define a vector field on the resulting space.

If by touching you mean, cut a hole in each ball and glue them together along the boundary, then the resulting surface is still a sphere so the hairy ball theorem still applies

If by touching you mean identified at a single point, then it gets a bit tricky.

You can comb each ball in the following way, such that they have only one "pole" or cowlick:

https://upload.wikimedia.org/wikipedia/commons/6/6e/Hairy_ball_one_pole_animated.gif

then you can identify the pole on each ball. The resulting space is no longer a manifold, so the question then becomes, how do you define a tangent vector at that point (the wedge)?

One possibility is to just ignore the "bad point", the complement of which *is* a manifold. If you mean that, then by construction (taking a vector field on each sphere whose only zero is at the bad point), you do indeed get a non-vanishing vector field.

On the other hand: if the two spheres are touching tangentially, then the tangent planes to the two spheres at the bad point line up, so we can still talk about a tangent vector at that point, and so it still makes sense to talk about a vector field on the whole thing.

OK i have a math question. something I should know how to do and surely learned at some pt at school but I forgot and I don't know what topic to look up

Let's say you have a function F from integers between 1 and 100 to R. generally a well-behaved function but locally can get spiky

I want to approximate it by a function G that is

1) as similar to F as possible
2) as smooth as possible
3) integrates to the same value as F: G(1) + G(2) + ... + G(100) = F(1) + F(2) + ... + F(100)

I know we did this stuff in Numerical Analysis and had to write Matlab scripts that did this sort of thing all the time. Condition (3) is straightforward, but (1) and (2) seem to be in tension; maybe I have to minimize some loss function of MSE and some measure of spikiness?

This seems as good a place as any to mention that Don Knuth has announced his experimental work for pipe organ.

http://www-cs-faculty.stanford.edu/~uno/fant.html

flopson: maximum entropy seems like one tack to take? http://www.lacan.upc.edu/arroyo/Site1/Research/Entries/2012/9/12_Maximum_entropy_approximation.html

let me restate to be sure i understand tho -- "i have an assignment of the integers from 1 to 100 to 100 respective values in R. i would like to make a new assignment of the same form N_[1-100] -> R, but with the condition that the sum of values in the codomain agree with the prior one, and with some sort of smoothing applied."

If that's correct, yeah, you need to decide the tradeoff yr willing to make, towards what end, between similarity and smoothness. i mean a lazy and decent thing to do is just to take some sort of quadratic or cubic interpolation on the points, then "bump" it into shape with a second pass that makes the sum tie out while minimizing MSE as you describe.

that's correct. looking for something simple and quick

friend linked me to: https://en.wikipedia.org/wiki/Mollifier

neat!

was led by this tweet

https://twitter.com/sigfpe/status/811704836622159872

to this book:

https://arxiv.org/abs/1612.06373

wiki:

In mathematics, singularity theory studies spaces that are almost manifolds, but not quite. A string can serve as an example of a one-dimensional manifold, if one neglects its thickness. A singularity can be made by balling it up, dropping it on the floor, and flattening it.

lol, awesome

You mean like orbifolds?

<3 sigfpe

https://en.wikipedia.org/wiki/Cox%E2%80%93Zucker_machine

Man if I had endless time I would read that Ghys book linked above

otm

why need endless time? do you read novels in your spare time? think of it as one of those!

maybe you can read math books at the same speed as you do novels, not me

why does it have to be at the same speed?

at one point in my life I had the habit of reading a page of Hilbert's Geometry and the Imagination a day. I never finished it but it doesn't matter; it was a blast

:)

https://sites.tufts.edu/gerrymandr/

"A 5-day summer school will be offered at Tufts University from August 7-11, 2017, with the principal purpose of training mathematicians to be expert witnesses for court cases on redistricting and gerrymandering."

that's hot. I wish I could audit that

Is there in iPhone app that I can use to solve or graph or factor algebraic equations? Basically do what my old TI-89 could do back in my high school days, (except in color obv.)?

love to read stats bantz http://davegiles.blogspot.com/2011/09/micronumerosity.html

asically do what my old TI-89 could do back in my high school days, (except in color obv.)?

this is a betrayal

Mr. Snurb, Wolfram Alpha is useful. It even has an iPhone app. In addition to factoring, it can be used to integrate or even solve mildly complicated PDE/sODEs. The syntax, IMO, is much nicer than a calculator you can use natural language or TeX-like formatting.

https://www.youtube.com/watch?v=AmgkSdhK4K8

Very sad: http://ifpnews.com/exclusive/iran-math-genius-die-cancer/

:(

voevodsky dead at 51 https://www.ias.edu/news/2017/vladimir-voevodsky

fields metalist dropping like flies

a terrible tragedy for math. he supervised to a close friend of mine, who told me they had been making progress on some problems as recently as this summer. needless to say friend is devastated

RIP

Must admit I was thinking this revive would be about Monty Hall problem.

extremely worthwhile Voevodsky content:

https://www.youtube.com/watch?v=O45LaFsaqMA

that's what you get for inventing calculus bitch lmao https://t.co/qwE6dL6fNH

— daddy. (@TayWest) December 12, 2017

https://en.wikipedia.org/wiki/Spherical_coordinate_system#Integration_and_differentiation_in_spherical_coordinates

does anybody know the line elements in 4D hyperspherical coordinates or how to go about deriving them? i need to know for a school project and didn't realize this information would be so hard to find lol

ok i guess this is the way to do it:

https://math.stackexchange.com/questions/74442/line-element-dl-in-spherical-coordinates-derivation-diagram

wow that was easy

Really?

Seems like you “just” stick another sin and cos on the end and then “just” do your calculations

So about those octonions

they squirt ink and make you cry

Lol

https://upload.wikimedia.org/wikipedia/commons/thumb/2/2d/FanoPlane.svg/250px-FanoPlane.svg.png

PY DERE! Have a Jupyter notebook question that I may hit you with in a little bit

Okay, been trying to make slides using nbconvert. It ends up creating an html version but no slides.

Oh, looks like a copy of reveal.js is needed.

https://www.youtube.com/watch?v=hslrD0Y837c

going to be learning random graph theory this spring, maybe some random matrix theory