Mathematics - 2020-01-15 (page 2 of 2)

luchonacho

19:01

Thanks. Will ask in site, to see if I get something else.

Semiclassical

part of what's confusing here is that you're not really composing $g$ with $f$. the composition is really with $g\times \text{id}_B$

typo in what I said above: should hvae been $f(x,h(x,y))=y$ at the end

luchonacho

0

Q: Is there a standard notation to express the "partial inverse" of a multivariate function?

Consider the function $$ y = f(x,w) $$ I am looking for "proper" notation for the "partial inverse" of $f(\cdot)$ with respect to each variable. Something like: $$ x = f^{-1}(y,w) $$ $$ w = f^{-1}(x,y) $$ But this does not differentiate them. Perhaps $$ x = f_x^{-1}(y,w) $$ $$ w = f_w^{-1...

thanks all

Semiclassical

maybe the better view is that $g(y,w)=(y/w,w)$ so that $f(g(y,w))=f(y/w,w)=y$

But that seems sorta arbitrary

Semiclassical

19:31

A question, based on one I just saw on main: Suppose $f(x)/x\to 1$ and $g(x)/x\to 1$ as $x\to 0$. Then it is certainly true that $f(x)/g(x)\to 1$ as $x\to 0$. Must $f^{n}(x)/g^{n}(x)\to 1$ as $x\to 0$?

Thorgott

If it exists, yes. Not sure if it necessarily does.

Semiclassical

for comparison, it's this question/answer: math.stackexchange.com/questions/3510410/…

Thorgott

It feels as if there should be a counterexample at the very least for n>1

Semiclassical

the notation $\displaystyle \lim_{f(x)\to 0} \frac{\sin f(x)}{f(x)}=1$ seems suspicious

I suspect it's actually fine, on the grounds that $x\mapsto \sin(x)$ is everywhere continuous and thus any iteration of it is as well

But the level of detail seems poor.

Thorgott

This should be a counter-example to your question.

Semiclassical

19:43

oof

though, wouldn't $f(x)/x\to 1$ as $x \to 0$ necessitate both $f(0)=0$ and $f'(0)=1$?

If $f(0)\neq 0$, then the limit wouldn't exist. And if the limit exists, then it's the definition of the derivative.

Thorgott

Oh wait, the example I linked is for $x\rightarrow\infty$, but it should work with some modification.

And yes, you do get $f^{\prime}(0)=1$, so any counterexample would need some discontinuity in the derivative (in particular, if you're assuming $C^1$, then the statement should ohld).

Semiclassical

yeah

So what ensures that $\sin x$ is a good example is presumably not just that it's everywhere continuous but that it's analytic

Thorgott

So, by L'Hospital, we get the statement is true for higher $n$ if $f$ is sufficiently differentiable and its higher derivatives zeros don't accumulate in $0$, though the second condition seems rather awkward.

Semiclassical

yeah, it's a bit gross

I don't know how to engineer a reasonably-simple counterexample tho

Thorgott

I was trying to throw something together with x^2sin(1/x), but I didn't get it to work. Now I'm trying to think of other examples of functions with discontinuous derivatives, but I actually don't know many.

It suffices to find a differentiable function $f$ with $f(x)/x\rightarrow1$ as $x\rightarrow0$ and $f^{\prime}$ discontinuous at $0$. Then we can take $g(x)=x$ and it will be a counterexample.

Semiclassical

20:00

hmm

I agree that that would suffice, but can that possibly occur?

MyWrathAcademia

@Thorgott I now understand exactly that a root of a polynomial is simply the zero of the corresponding polynomial function. Apologies for my previous questions, they were coming from a lack of understanding of polynomials and roots of polynomials. Your example of plugging 5 into that polynomial and the Wikipedia article got me to perfectly understand what a root of a polynomial is. Thanks a lot for your clear and concise explanations.

Thorgott

no problem, glad you understand it now

Wait, we can just take $f(x)=x+x^2\sin\frac{1}{x}$ (extended)

$\frac{f(x)}{x}=1+x\sin\frac{1}{x}\rightarrow1$ as $x\rightarrow0$, it's differentiable as sum of differentiable functions and its derivative is discontinuous at $0$, because the derivative of $x^2\sin\frac{1}{x}$ at $0$ is discontinuous, but the one of $x$ is not.

Semiclassical

nice

except that Mathematica claims that $f(f(x))/x\to 1$ as $x\to 0$

Thorgott

Why are you looking at $f(f(x))/x$? The claim is that $f^{\prime}(x)/g^{\prime}(x)\not\rightarrow1$

Semiclassical

um

oh

36 mins ago, by Semiclassical

A question, based on one I just saw on main: Suppose $f(x)/x\to 1$ and $g(x)/x\to 1$ as $x\to 0$. Then it is certainly true that $f(x)/g(x)\to 1$ as $x\to 0$. Must $f^{n}(x)/g^{n}(x)\to 1$ as $x\to 0$?

$f^n$ is meant as composition, not differentiation

Thorgott

20:08

Oh.........

Now that's unfortunate

Semiclassical

yeah.

I write nth derivatives as $f^{(n)}$

but $f^{\circ n}$ would probably have been better

Thorgott

Hmm, so $f(x)$ and $g(x)$ both tend to $0$ as $x\rightarrow0$, so for $f(f(x))/g(g(x))$ to not tend to $1$, $f(x)$ and $g(x)$ would need to tend to $0$ at different rates, but this may be ruled out because $f(x)/x$ and $g(x)/x$ tend to $1$, so they should tend to rate both roughly at the rate of $x$ itself.

Let's see if that can be turned into epsilons and deltas

Semiclassical

mostly I'm paranoid whether the answers given here can actually be generalized

for instance, the Taylor series approach falls apart if $f(x)$ is not analytic near the origin

and the other one is suspicious to me because the notation $\lim_{f(x)\to 0}\frac{\sin f(x)}{f(x)}=0$ seems highly dubious

Thorgott

If we can guarantee $f(f(x))/x\rightarrow1$, we're good to go, but I'm not sure if that's true

Semiclassical

I'm putting together a question for this

Maybe I should just stick with the simplest version, where $g(x)=x$

Given $f:\mathbb{R}\to \mathbb{R}$ such that $x^{-1}f(x)\to 1$ as $x\to 0$, does it follow that $$x^{-1}f^{\circ n}(x)=x^{-1}(f\circ f\circ\cdots \circ f)(x)\to 1$$ as $x\to 0$?

Thorgott

20:28

This implies the general version, because $f^{\circ n}(x)/g^{\circ n}(x)=(f^{\circ n}(x)/x)/(g^{\circ n}(x)/x)$

Semiclassical

Oh, nice

then I'll just stick with that

Thorgott

Doing a similar expansion iteratively, I think it should suffice that $f^{\circ n}/f^{\circ n-1}$ tends to $1$

Lukas Heger

@TedShifrin are you around?

Thorgott

It feels correct in my heuristic as above, but I can't seem to prove it and analysis counterexamples can be scary

Alessandro Codenotti

Hi @Lukas

Lukas Heger

20:32

Ciao @Alessandro

Alessandro Codenotti

Come va?

Lukas Heger

Va bene, grazie. Sono preso dal concorso per ALGANT.

E tu? Come stai? Come va la tesi di specializzazione (come si dice "master thesis" in italiano?)?

Alessandro Codenotti

Capito, entro quando devi fare la domanda per il concorso?

@LukasHeger Tesi magistrale

Ora sto studiando per gli esami più che altro, riprenderò a lavorare seriamente alla tesi a Febbraio

Lukas Heger

@AlessandroCodenotti prima di 1. Febbraio

Alessandro Codenotti

dell'1 Febbraio *

Lukas Heger

20:38

grazie

Semiclassical

@Thorgott Another way to pose this: If $f(x)\sim x$ as $x\to 0$, must one have $f^{\circ n}(x)\sim x$ for all $n$?

Lukas Heger

Devo fare troppe cose per il concorso: curriculum, una lettera di motivazione, due lettere di raccomandazione, un test di inglese (IELTS) ecc.

Thorgott

Yeah; that phrasing makes me feel it should be true at least for $f$ Lipschitz

Alessandro Codenotti

@LukasHeger Posso immaginare, io sto iniziando a guardarmi intorno per il dottorato ed è deprimente...

jjepsuomi

Hi, does anyone happen to have an idea what this notation means: NI(0,sigma^2). I'm reading about sampling statistics by Fuller and I don

dont recall seeing this notation before. NI(0, sigma^2)

x ~ NI(0, \sigma^2)

Thorgott

20:44

Looks like a normal distribution except for the I

jjepsuomi

Exactly, was thinking the same, the 'I' was where I got confused...

I thought it was a typo at first, but then he used it many times over, I'll see if I can upload a screenshot

Here you can see an example

Alessandro Codenotti

Maybe it is an I for Independent since you're actually considering more normal distributions?

jjepsuomi

Perhaps, thank you, I will add another screenshot

Thorgott

Sounds reasonable

It says bivariate normal in the text, so it must be something along those lines

Lukas Heger

@Alessandro domani farò una presentazione della geometria rigida e dopodomani sostenerò gli esami di IELTS (comprensione orale, comprensione scritta, composizione scritta ed esame orale), è un po' stressante

jjepsuomi

20:50

Alessandro Codenotti

@LukasHeger Oh, in bocca al lupo per entrambi!

Semiclassical

Fuller seems to introduce the NI notation in Theorem 1.2.4, judging from Google Books

Lukas Heger

@AlessandroCodenotti grazie!

jjepsuomi

Here's another example. Thank you @AlessandroCodenotti, @Thorgott and @Semiclassical

Thorgott

This is certainly consistent with Alessandros interpretation

jjepsuomi

20:51

Thank you :) I will take a look

Semiclassical

where $NI(\mu,\sigma^2)$ is a set of $n$ normal independent random variables, each with mean $\mu$ and variance $\sigma^2$

Alessandro Codenotti

I just realized that "in bocca al lupo" sounds very weird when translated into English @Lukas

Lukas Heger

it sounds very weird in German as well

but I learned it in one of my courses

jjepsuomi

excellent, thank you all for your help!

Semiclassical

@Thorgott what I've got so far as a draft:

"Suppose $f:\mathbb{R}\to\mathbb{R}$ and let $f^{n}$ be its $n$-fold iterate. If
$f(x)\sim x$ as $x\to 0$, is $f^{n}(x)\sim f^{n-1}(x)$? If not, what is a counterexample and what conditions on $f$ are sufficient?

This problem is motivated by the following recent question https://math.stackexchange.com/q/3510410/137524, which reduced to arguing that
$\sin(f(x))\sim f(x)$ where $f(x)$ is some iterate of $\sin x$ (and similarly for $\tan x$.)"

Thorgott

20:54

Sounds good

Semiclassical

0

Q: Asymptotic behavior of $n$-fold iterate near the origin

Suppose $f:\mathbb{R}\to\mathbb{R}$ and let $f^{n}$ be its $n$-fold iterate. If $f(x)\sim x$ as $x\to 0$, is $f^{n}(x)\sim f^{n-1}(x)$? If not, what is a counterexample and what conditions on $f$ are sufficient? This problem is motivated by the recent question $\lim_{x\to 0}\frac{\sin(\sin(\sin...

real-analysis calculus limits asymptotics examples-counterexamples

Thorgott

I got a partial affirmative: If $f$ is $L$-Lipschitz, then $|\frac{f^n(x)}{f^{n-1}(x)}-1|=\frac{|f^n(x)-f^{n-1}(x)|}{|f^{n-1}(x)|}\le L^{n-1}\frac{|f(x)-x|}{|f(x)|}\rightarrow0$ as $x\rightarrow0$.

Semiclassical

Nice.

Thorgott

I think we should be able to do better though

Semiclassical

I'm just going to leave that question up and let it simmer for a bit.

Thorgott

21:03

Also I think we need to assume that $f$ does not vanish at any point near zero (except zero itself of course) for the question to be well-posed

topologicalmagician

Given topological space $X_i$, what topology do I need on $X_i^*$ $=$ $\{$ $(x_i): x\in X_i$ $\}$ to make $X_i$ and $X_i^*$ homeomorphic?

Semiclassical

so no sequence of zeros accumulating at the origin

Alessandro Codenotti

@topologicalmagician I don't quite understand your definition of $X_i^\ast$

Thorgott

Yeah. Otherwise the limit won't exist.

Semiclassical

I feel like that's implied by $f(x)\sim x$ in the first place, though

Thorgott

21:05

Oh, you're right

topologicalmagician

@AlessandroCodenotti its the image of the canonical injection from $X_i$ to the disjoint union of $X_i's$

Semiclassical

The assumption $f(x)\sim x$ is pretty strong, is what this seems to amount to

topologicalmagician

ooooh wait

I meanat

Alessandro Codenotti

It is always a homeomorphism then if you give it the subspace topology inherited from the disjoint union

topologicalmagician

meant*

Thorgott

21:07

Yeah, it implies differentiability at $0$ if you have continuity at $0$, which seems strong

topologicalmagician

Given topological space $X_i$, what topology do I need on $X_i^*$ $=$ $\{$ $(x,i): x\in X_i$ $\}$ to make $X_i$ and $X_i^*$ homeomorphic

I'm supposed to construct a topology independent of the disjoint union

I meant $(x,i)$ not $(x_i)$

@AlessandroCodenotti

Alessandro Codenotti

The $i$ is irrelevant, if $U$ is open in $X_i$ then U^\ast=\{(u,i)\mid u\in U\}$ should be open in $X_i^\ast$

Lukas Heger

hi @ShineOnYouCrazyDiamond

Shine On You Crazy Diamond

Hello, is that Paul Hans Fischer?

Lukas Heger

no, it's MatheinBoulomenos

the author of without loss of universality :)

Shine On You Crazy Diamond

21:14

Oh, hi

:D

I'm waiting for someone to ping me here, someone that want's to work on BananaCats with me

Thorgott

I do know a Paul Fischer, but not a Paul Hans Fischer

Lukas Heger

@ShineOnYouCrazyDiamond I recently published a second part on adjunctions, though I'll need a third part on adjunctions as well

Shine On You Crazy Diamond

Nice! link?

Lukas Heger

wlou.blog/2020/01/11/…

Shine On You Crazy Diamond

I see you've removed the grey backing of LaTeX. Looks nicer

Lukas Heger

21:17

yeah

LaTeX didn't work nicely with the theme I had

Thorgott

(A Hölder condition obviously works with the same argument)

@Semiclassical Oh wow, the question has been answered

Semiclassical

yeah, was about to say

and remarkably simply

i'm not going to accept immediately, but if no one offers anything else I will

Thorgott

Indeed, I expected it to be uglier

Semiclassical

second answer as well

Ultradark

@ShineOnYouCrazyDiamond hi

Shine On You Crazy Diamond

21:25

hey

Ultradark

the more I learn, the less I know. I feel like I'm constantly losing ground lol

Semiclassical

Get used to it, I'm afraid.

Shine On You Crazy Diamond

That's true.

Ultradark

I'm used to it

Semiclassical

The more you know, the more aware you are of how much there is to know (and therefore how ignorant you are)

Ultradark

21:27

well said

Shine On You Crazy Diamond

The grass is always greener on the other side of the fence.

The bigger they are the harder they fall.

Ultradark

yeah

Define the diffeomorphism, $f:M\to N,$ with $(x,y)\in M$ and $(e^x,e^y)\in N$ s.t. $(x,y)$ is a point in quadrant I or III

I have the metric for $M$ but not for $N.$ Is this enough information to deduce the metric on $N?$

Alessandro Codenotti

You can always pull a metric through an homeomorphism of two spaces. Just measure the distance between the (pre)images of two points

Ultradark

oh okay

Thorgott

21:47

This does in no way determine a metric on $N$, your setup doesn't even rely on the notion of a metric being present.

user193319

1

Q: Minkowski Functional Evaluated at a Point Outside the Convex Set

Let $K$ be a convex set in a vector space such that $0$ is an internal point of $K$, let $p_K$ be the associated Minkowski functional, and let $z \notin K$. Why must $p_K(z) \ge 1$ hold? If it were true that $z \in p_K(z) \cdot K$, then if $p_K(z) < 1$ were the case, I could write $z = p_K(z) \cd...

linear-algebra functional-analysis vector-spaces convex-analysis

Ultradark

@Thorgott oh, so is alessandro wrong?

Ultradark

22:04

I said the manifolds are homeomorphic, and the pre-image already has a metric. That seems as if it would be enough to pull the metric through by measuring the distance between the pre-images of two points as alessandro said

Thorgott

I was assuming $M\subseteq\mathbb{R}$ (what sense does $e^x,e^y$ make otherwise?). In either case, Alessandros construction certainly gives a way to define a metric on $N$, but it's in general not unique (and "deduce the metric" sounds like you want it to be uniquely determined).

Ultradark

oh I see

yeah I actually meant, that I just wanted to find a metric, doesn't have to be uniquely determined, and then try to prove that it's equivalent to the metric on $M$

Thorgott

What does it mean for metrics on different spaces to be equivalent?

Ultradark

but I was thinking $M:=\Bbb R^{1,1}$ equipped with the corresponding poincare group

well a priori, I don't know if they are different

I'm trying to show that in fact they are the same

by comparing the pseudo-euclidean metrics and checking if they are the same

Thorgott

What do you mean "the same"? They're not literally the same.

Ultradark

22:18

isomorphic I guess

They can't be isometric I suppose

Thorgott

Forget all your manifold structure, this is topology: If $X,Y$ are topological spaces and $f\colon X\rightarrow Y$ is a homeomorphism and $d$ is a metric on $X$, then you can transport the metric via $f$ to get a metric $d^{\prime}$ on $Y$; this metric will induce the topology on $Y$ and $f$ will then be a homeomorphism (in fact, an isometry) by construction.

topologicalmagician

are adjunction spaces usually taught in an undergrad curriculum?

Ultradark

@Thorgott okay that was pretty helpful

Alessandro Codenotti

@topologicalmagician Glueing along a map is both common and useful

psa

How would you parametrize a curve like |x| (or y = -x if $x \leq 0$ and y = x if $x \geq 0$) AND force the parametrization to be differentiable everywhere?

Ultradark

22:31

I feel like that argument can be directly applied to the geometry (when $X$ $Y$ are pseudo-euclidean manifolds with the lorentz metric) @Thorgott

but that cleared a lot of stuff up :)

Thorgott

that's nice

@psa you can't

psa

I'm supposed to give an example of one, but that's what I thought too...

Thorgott

@psa sorry, I'm mistaken

it is possible

I was thinking about a regular parametrization (and that indeed would not be possible)

Think about tangents at (0,0) and what they necessarily have to look like for differentiability

Lucas Henrique

22:54

How crazy is this

arxiv.org/pdf/1904.00215.pdf

Exactly two rational points

Is it possible to have a curve in $\mathbb R^2$ with no rational points?

Akiva Weinberger

https://ai.facebook.com/blog/using-neural-networks-to-solve-advanced-mathematics-equations/

Look at thisss

Facebook has an AI division apparently

They managed to get a neural network, of the type originally designed for machine translation, to solve math equations

like differential equations

They generated problems by choosing a solution and working backwards

This was used as the training set and the testing/validation set

Shine On You Crazy Diamond

I don't know why no one's made an integer identity search algorithm. That would be neat. @AkivaWeinberger

Or identities with other spaces.

Akiva Weinberger

> Our model took the equations on the left as input — equations that both Mathematica and Matlab were unable to solve — and was able to find correct solutions (shown on the right) in less than one second.

Shine On You Crazy Diamond

I guess they get another billion dollars for that. While we get $0 for farting around

psa

@Thorgott well they'd have to be identical on either side of the origin. I can't imagine visually how that'd be possible.

Akiva Weinberger

23:00

I suppose the way the problems generated affected the priors - the network got to assume it was a trapdoor problem, while CASs like Mathematica were working under the assumption that it was the sort of thing that came up naturally

Not that that diminishes anything

Thorgott

@Lucas (x^2+1)(y-sqrt(2))=0

Akiva Weinberger

(And I'm not even sure if that changes anything)

psa

@Thorgott is there a way to parametrize it without using a piecewise function?

Lucas Henrique

@Thorgott idk if this should be called a curve

Thorgott

@psa well, identical, but flipped along the y-axis. if you want your parametrization to be differentiable, those two tangents should agree however. this can only be the case if?

Akiva Weinberger

23:02

Machine Learning been called "fancy statistics", but with results like this, that seems harder to defend

Or at least it significantly broadens what statistics can do

Statistical metamathematics

Thorgott

@Lucas it's an algebraic curve

Lucas Henrique

Oh, it’s just the line $y = \sqrt2$

Thorgott

yup

a rather trivial example, i admit

psa

the direction vector of the curve is identically $\mathbf{0}$ in a neighbourhood of (0,0)?

Lucas Henrique

What about a curve that has no points $(a,b):\ a\in\mathbb Q \lor b\in \mathbb Q$

$(a,b) \in \mathbb R^2$

Thorgott

23:05

not a neighborhood of (0,0), but definitely at (0,0)

Shine On You Crazy Diamond

@AkivaWeinberger can they use this to find "closed forms" of integer sequences?

psa

so just define the function piecewise to be 0 at the origin?

Thorgott

that won't guarantee you differentiability

@Lucas shouldn't work with any sensible definition of curve because of the IVT

Akiva Weinberger

@ShineOnYouCrazyDiamond Maybe. The same methodology would work, I imagine - you can generate the training and testing data the same way

Make "trapdoor problems" by starting with the solution (the formula) and computing the problem (the sequence)

Shine On You Crazy Diamond

@AkivaWeinberger interesting!

Akiva Weinberger

23:08

You should read the link, it's fascinating

Shine On You Crazy Diamond

They should get it to find something new (and interesting to us)

There are probably an infinity of equational identities out there, just waiting for Ramanujan types or an AI to discover them

How I would do this is first have a collection of known identities, of which there are many on wikipedia.

It shouldn't have to prove those, or alternatively it could prove those and find the shortest most elegant proof

Akiva Weinberger

I don't think neural networks can generate proofs

In the Facebook paper, it only generated solutions

Lucas Henrique

@Thorgott by curve I mean a continuous $\gamma: [a,b] \to B$

Adam

I just saw the word heuristic proof someone please explain what this means if no my current definition "half assing a proof drunk based on inductive reasoning of a collection of lemmas"

Akiva Weinberger

The "proofs" would consist of computing the derivative and manipulating the algebra, which I imagine would be best left to a traditional computer algebra system

Shine On You Crazy Diamond

23:13

Lol the AI sounds like a Ramanujan then. Didn't like creating the formal proofs

Akiva Weinberger

Haha wow

Shine On You Crazy Diamond

(according to the movie)

Adam

Yeah are you guys talking about eureaka? yeah I tried it when it became available it is pretty impressive

Shine On You Crazy Diamond

@Adam what's that

Akiva Weinberger

If I remember right, Ramanujan learned math from a big book of identities and Theorems without proof (basically a reference text)

so Ramanujan would read it and try to see why the identities were true and where they came from

Thorgott

23:15

@Lucas then the only such will be the constant curve

Shine On You Crazy Diamond

Why were people so smart back in the day. I think it's just the overwhelming complexity of all the available knowledge we have today

Akiva Weinberger

I guess he probably proved what he could, but learned intuition from all of them

Adam

its a program that takes empirical data and finds algebraic expressions (essentially curve fitting like all the CAS have been doing since the 90s) and then attempts to make logically consistent relations

Thorgott

it's fairly straightforward

Shine On You Crazy Diamond

@Adam link?

Lucas Henrique

23:16

@ShineOnYouCrazyDiamond I don’t think that’s true

Shine On You Crazy Diamond

@Lucas why then, was there an Euler, Gauss, Hilbert, Erdos, etc? It's because they took all the low hanging fruit from the tree of knowledge so to speak. And we're left with stuff that's nearly impossible (even for them) to discover.

Adam

far out that was link 2015 2014 ish and yeah something not cool happened to the laptop so yeah im not going to use this as a personal tragedy blog thingy but ill take a look, but ur search is as good as mine atm

I think you mean you're left with this is my excuse for half assing the rest of my life but ok

everything would be so boring if something wasn't impossible according to our present world view

any hoo

Akiva Weinberger

@ShineOnYouCrazyDiamond How many lived at the same time

Shine On You Crazy Diamond

Euler, Gauss, Hilber = 1800 - 1900's I think. Then Erdos = 1940 +

I half assed that estimate

Akiva Weinberger

Euler died in 1783 according to Google

Shine On You Crazy Diamond

23:21

It is true, however, that not using a computer to solve these complex math problems nowadays would be a bad idea

Akiva Weinberger

Gauss was born in 1777

They overlapped for six years

Shine On You Crazy Diamond

It makes me wonder what Euler would have done if he had a PC

or Emily Noether for that matter

Akiva Weinberger

Gauss died in 1855

Hilbert was born in 1862

Shine On You Crazy Diamond

Would they be better or worse than how they performed without a computer

@Akiva +/- 100 years

Lukas Heger

Hey @Ted

Ted Shifrin

23:23

Greetings.

Akiva Weinberger

Hilbert died when Erdős was 30 so they overlapped quite a bit I guess

But Euler, Gauss, and Hilbert didn't overlap

Lukas Heger

@TedS I sent you a mail with my motivation letter

psa

@Thorgott if they're flipped along the y-axis (so say one looks like $\mathbf{e_1} + \mathbf{e_2}$ and the other looks like $\mathbf{e_1} - \mathbf{e_2}$), the only way the tangents could be equal is if $\mathbf{e_2} = -\mathbf{e_2}$, which obviously can't be true... I guess I'm missing something obvious here though.

Ted Shifrin

My main comment is that it confuses me because you mention two places. The essay should normally be individualized for each school, if you can.

Lukas Heger

no, the ALGANT programme as a whole works so that if you get accepted, you go to two schools

Shine On You Crazy Diamond

23:25

@AkivaWeinberger are you a coder?

Lukas Heger

I don't apply to individual schools

Akiva Weinberger

No

I learned some Python and Java

but never built anything with them

Lukas Heger

one year at the first school (in my case Regensburg) and one year at the second school (in my case Padua)

Ted Shifrin

Oh, I see.

Cool.

Lukas Heger

and other than that?

Shine On You Crazy Diamond

23:26

@AkivaWeinberger Python is cool for GUI stuff. But C++ is still the king of speed

Python will have you finishing your app 10x faster, but in C++ your critical code will run 100x faster

You can use both at the same time (linking with a DLL from python)

I don't know why I'm telling you all this :D

Ted Shifrin

Probably should put teaching stuff at the end, @Lukas.

Lukas Heger

at the end of the first paragraph or at the end of the whole letter?

Ted Shifrin

Talk less about marks and more about some specifics from the seminar talks.

Lukas Heger

okay, maybe I'll mention the talk where I generalized everything from the source material which was on algebraic groups to group schemes

Ted Shifrin

Toward the end of the whole thing. I don't know if you even have to teach in this Masters program. But you can use it to say how much belief the faculty there had in you.

Shine On You Crazy Diamond

23:29

@AkivaWeinberger what about this idea. Do you know how some apps use a node / connection approach to doing stuff. Such as Orange (Data analysis) or NodeBox (graph-driven art). Why not do the same where each node is a theorem say. A theorem would take as inputs objects that meet the required specification. Output would be a conclusion

Akiva Weinberger

I feel like it would blow up

Shine On You Crazy Diamond

And then you do combinatorial math with it (throw a bunch of nodes into a box and randomely try to connect them) similarly to combinatorial chemistry

Akiva Weinberger

You'd basically be searching all possible proofs until you find one that either proves or disproves what you want

That's way to many for a computer

Shine On You Crazy Diamond

Not quite. The above, since each input only takes what makes sense to the theorem, this is actually way more efficient than the Facebook NN solution

Supposing that their domain of application overlaps...

Lukas Heger

@TedShifrin I don't know, it messes with the logical structure if I move it around too far towards the end

Ted Shifrin

23:32

OK, no big deal.

Thorgott

@psa It can. The derivative at (0,0) will (necessarily) be $0$. Instead of doing the obvious parametrization (t,|t|) pick a function f such that (f(t),|f(t)|) will do the job

Shine On You Crazy Diamond

If you wanted to apply a NN to it. You could make the NN model what a user does with those blocks and arrows

Akiva Weinberger

I'm not sure I understand

Shine On You Crazy Diamond

I don't either lol :)

psa

ahh okay

Akiva Weinberger

23:33

psa

that sounds like arc length parametrization, no?

Akiva Weinberger

Say I want to solve the differential equations on the left

How would I do them @ShineOnYouCrazyDiamond

Shine On You Crazy Diamond

You would have a bunch of nodes at least with the algebraic axioms, for example

Then some nodes would be identities known to be true

Think of a data flow network but with math objects as the data

Akiva Weinberger

So your program will essentially do a tree search through all algebraic manipulations that it thinks will simplify the equation?

Honestly that's probably what Mathematica and Matlab do

and the problems in the image stumped them

Shine On You Crazy Diamond

If Mathematica did this, then we would be presented with a user interface that shows this, but we're not

Akiva Weinberger

23:36

In any case this probably blows up way more than board games do

(I.e. chess algorithms essentially use tree search)

(and no Go algorithm uses tree search because it's too big)

(Well - AlphaGo uses a neural network together with Monte Carlo tree search, but the point is it's not tree search on its own)

This is a "solitaire" game rather than a two-player game but still

But, try it

Or even try it by hand for a bit

By the way

I notice that the solutions in the image aren't simplified

For example, $\exp(\sinh^{-1}(x))=\sqrt{x^2+1}+x$

Perhaps $\exp\circ\sinh^{-1}$ was easier to guess

It's certainly more likely to appear as the solution to a "trapdoor problem" than the other one

Ultradark

I asked a dumb question on mathoverflow like six months ago and it got 3 upvotes!

Akiva Weinberger

Recently got 3 upvotes?

Ultradark

I guess nobody could tell that it was not a good question

no not all at once

gradually

Akiva Weinberger

I still occasionally get upvotes for this quickie

45

A: "sup" in an equation

Sup ("supremum") means, basically, the largest. So this: $$\sup_{k\ge0}T^{(k)}(N)$$ refers to the largest value $T^{(k)}(N)$ could get to as $k$ varies. It's technically a bit different than the maximum—it's the smallest number that is greater-than-or-equal to every number in the set. So, for e...

I guess people are searching "What does sup mean" and finding it

Ultradark

yeah haha

Lucas Henrique

23:52

@ShineOnYouCrazyDiamond oh, I didn’t read the last part well. I mean, I think that we don’t have the same rate of discovery because of the complexity.

Ted Shifrin

Howdy @Lucas

Lukas Heger

@TedShifrin I sent you a revised version

I tried to go into some more specifics and added some stuff on my undergrad thesis and the research assistant position I will have over the summer

Ted Shifrin

Yeah, it looks good. Just make paragraphs into paragraphs.

Indentation, some vertical space between paragraphs. :)

Alessandro Codenotti

Hi @Ted @Lukas

Ted Shifrin

Hi, demonic @Alessandro

Alessandro Codenotti

23:55

fixing your CV? @Mathei

Lukas Heger

Hi @Alessandro

oh yeah, I shouldn't use \\ so much

Ted Shifrin

Also skip the heading before the date, @Lukas. This is included with the application, right?

Alessandro Codenotti

@LukasHeger This is one of the capital sins of latex

Ted Shifrin

You can reset parindent and parskip :P

Lukas Heger

@TedShifrin right

Thorgott

23:59

I sometimes do line breaks with negative height, I think I'll go to hell

Akiva Weinberger

The brachistochrone problem was solved in 1697. The Basel problem was solved in 1734.

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$Mathematics$ Mathematics