Mathematics - 2017-06-23 (page 1 of 5)

Daminark

00:06

Yo @Avantgarde and @PVAL!

Avantgarde

@Daminark Yo Denmark

Daminark

:P

Avantgarde

:D

Yesterday I saw a guy wearing a swastika tshirt

I'm not sure if he knew what it's about

Daminark

Hopefully, and hopefully he finds out soon

Maybe he's rolling with its older religious connotations?

Avantgarde

I don't think so. Because it was tilted at 45 deg kinda like the nazi one

There's no nazism here, so I doubt he even knew what it was about

anyway, that's the only thing other than the usual that happened in the past couple of weeks

Daminark

00:12

That would make sense. I know in Morocco it wouldn't surprise me if some people didn't know what I meant

Avantgarde

yeah many people do not know. Particularly those who were not in WWII, I'd think

Daminark

00:27

shrug

user400188

01:14

does anyone have a symbolic definition for sentential interperatation? I can't seem to find one.

Typhon

@Avantgarde uuhh... 65-95% of the population were not in World War II.

user400188

@Typhon TheGreatDuck: did you just change your name or am I mistaking you for the Great Duck?

Typhon

i changed my name a long time ago

user400188

ah I see

Avantgarde

@Typhon I know. My country wasn't. Though the British dragged us into a war we had nothing to do with

@Typhon I'm not sure if your % figure is accurate though.

ALannister

02:29

Is there anything that can be said about the composition of an injective function and a surjective one?

Akiva Weinberger

Don't think so

ALannister

I'm trying to prove that the composition of the inclusion homomorphism and the quotient homomorphism is surjective.

Typhon

@Avantgarde I was referring to the literal number of human beings who were alive during World War II and also served in the war and are also still alive today.

holy shit

@Dodsy the collatz conjecture isn't special math

you just have to write it as a series

a sum

well... once you start peeling away the recursion

i have to think of a way to represent a formula to count the number of a certain factor

but I think that is just a trivial logarithm

user78103

02:53

f(x+x0)≃f(x)+x0∂xf(x0)
does this mean that x is a variable while x0 is a value?

Typhon

use mathjax, not special symbols. All I see are a bunch of squares.

user78103

f(x+x_0) \simeq f(x) + x_0 \partial_x f(x_0)

Typhon

@user78103 also... context

without any context I see no reason to say that x and x0 are not variables

Akiva Weinberger

@user78103 Yes, x usually is a variable and x_0 a constant

Typhon

but then I cannot see how I am justified in saying they are variables.

user78103

02:55

I'm trying to understand a part of a functional derivative, where the single variable function equals what I posted

Typhon

@AkivaWeinberger but without context we cannot make that conclusion.

Akiva Weinberger

@Typhon I know the context.

It's a well-known approximation.

Typhon

@AkivaWeinberger tangent line approximation?

@user78103 well... if it is a tangent line approximation then yes. However, I am not justified in trying to assume what the author had in mind when writing whatever surrounded that.

0

Q: Is there a multivariate integer function f(x,y) that returns the number of factors of y in x with a closed form?

I'm looking for a simple function in terms of the elementary and exponential functions that when passed an integer x and integer y, it returns the number of times that y can be divided from x such that the result is an integer. I believe that it is something along the lines of $\lfloor\log_y(x)\r...

user78103

I was looking at the mathematics of physics question on math stack exchange

Akiva Weinberger

Given $f(x)$ and $f'(x)$, approximate $f(x+x_0)$

user78103

02:57

1

Q: The basic of functional derivative

I've just started to learn mathematical physics, and I read Stone and Goldbart's Mathematics for Physics. But right at the beginning when they introduce the functional derivative, I couldn't understand their explanation: 1.2.1. The functional derivative: We restrict ourselves to expressi...

derivatives

Typhon

@Dodsy @Semiclassical will find my question intriguing. It actually provides a means of expressing the collatz conjecture via one case as well as the "Dodsy Conjecture"

Akiva Weinberger

(Incidentally, you can turn on LaTeX through the link in the room description, on the top-right ^^>>)

user78103

what I'm having a hard time understanding is that its possible to take the partial derivative instead of a normal derivative

Semiclassical

My own guess is that there's no such function. @Typhon

Akiva Weinberger

The way you've written it, $f$ is a single-variable function

Typhon

02:59

umm

@AkivaWeinberger um what? my question is a function of x y

Akiva Weinberger

Oh, wait

Semiclassical

Which question were you responding to, @akiva?

Akiva Weinberger

Never mind, sorry. $x$ and $x_0$ could be vectors

Typhon

@AkivaWeinberger Yeah... f(x+x_0) \approx f(x) + x_0*f'(x)

Akiva Weinberger

@Semiclassical, @user78103's

Typhon

03:00

my point was that the author might not have intended x_0 to be a parameter

it could be a variable without changing the truthfulness of the approximation

Akiva Weinberger

Oh, I take back my retraction actually

Typhon

?

user78103

didn't even think about seeing them as vectors, Akiva

Akiva Weinberger

They have to be scalars, or else $x_0\partial_x f(x_0)$ makes no sense

Typhon

@Semiclassical Well no matter what one does. I just defined it. It's a question of whether it has a closed form.

user78103

03:04

so x_0 is just the change of the variable x?

Akiva Weinberger

I think that's usually written $\nu_y(x)$ or something @Typhon

@user78103 Yeah

Especially when $y$ is prime @Typhon

user78103

Thanks Akiva, Typhon, for the answers to my poorly constructed question

Typhon

03:19

@AkivaWeinberger interesting. Does it have a closed form?

because I think the collataz conjecture function can be written as one single application of a function repeatedly

at which point.... it's just like factorial

sort of

Semiclassical

Yeah, it's one of the functions here: en.wikipedia.org/wiki/…

Typhon

@Semiclassical but does it have a closed form?

Semiclassical

I doubt it.

Typhon

hmm

Just win baby

\o @Daminark

Daminark

03:25

Oh @Akiva I saw your stuff, and at least in my tired state I think it checks out alright

Hey @Justwinbaby!

And everyone!

Semiclassical

Terry Tao has some results on asymptotics/bounds for the number-of-divisors function here: terrytao.wordpress.com/2008/09/23/the-divisor-bound

2

(it's not what you're looking for, but it gives a sense of what people have actually been able to prove)

Typhon

@Semiclassical except my question is about a formula, not the properties of the function.

damnit. sniped.

Just win baby

Getting "sniped" by Tao is something to be proud of pal :-)

Typhon

no

i meant that semi's comment about "(it's not what you're looking for, but it gives a sense of what people have actually been able to prove)" sneaked past my reply.

Just win baby

I see.

Typhon

03:35

@Justwinbaby unless you mean to claim that semi is Terry Tao?

Just win baby

He could be.

or Tim Gowers

Typhon

unlikely

and I can show why it is unlikely

@Justwinbaby no because Terry Tao didn't write this paper and Semi did. arxiv.org/pdf/1303.6386.pdf

Semiclassical

I was one of the authors, at any rate.

Daminark

Nah pretty sure you're actually all the authors simultaneously

Typhon

@Semiclassical while you are add it, add "semiclassical" to the list of authors.

Semiclassical

03:39

Yes, because my internet handle on this site is how I want to be known professionally.

@Daminark I wish, then I'd actually understand the part that Peter wrote at the end about Seiberg-Witten.

and noooope

Just win baby

Are you the youngest?

Semiclassical

I think so, but only by a few months.

(It was my advisor + another grad student about my age + an older one)

Just win baby

cool

Daminark

Nice

Typhon

@Semiclassical well you have four identities according to Demonark and it cannot lie to me because I am its master. So... at this point your "professional identity" isn't my concern and clearly isn't your concern.

XD

Semiclassical

03:42

riiiight

Daminark

Looking at the title it definitely seems like your type of stuff, given how you seem to be much more on the mathematical side of physics

Semiclassical

yeah.

not on the formal side of that, though.

Typhon

@Daminark Bow before your master. Prove Semiclassical wrong.

Daminark

Um, ah, erm, does something in an attempt to prove Semi wrong but forgets to bow

fails at proving Semi wrong anyway

Semiclassical

There are still some nice results in that paper, though they're hardly very accessible

Typhon

03:45

bah!

Semiclassical

We did come across some generalized hypergeometric function identities, so that was neat.

Typhon

my minion is too incompetent to follow orders

XD

Daminark

Lol, I mean, I know nearly nothing about Riemann surfaces and even less about physics so I probably couldn't follow well

fails to realize that I was not supposed to assassinate my boss

becomes ruler ironically

Semiclassical

(By which I mean: We found a differential equation, plugged it into Mathematica subject to certain boundary conditions, and found that those identities came out as a consistency condition.)

Typhon

contemplates how exactly Demonark tried to assasinate boss

Semiclassical

03:47

(I still have no idea if those identities were already known or how to prove them by other means.)

Typhon

anyways

@Semiclassical was thinking about those spherical matrices and I think that they will be related to my concept of continuous matrices in some way. I was trying to think of infinitely dense matrices. Essentially becoming continuous grids rather than discrete grids.

Semiclassical

The funny bit of that paper is that one of the main results and impetus for the paper was inspired by me doing a certain calculation and not realizing how I was "supposed" to do it.

Which turns out to work better than the usual way, lol

Typhon

I guess that would become "linear calculus"

XD

Semiclassical

@Typhon I have a sneaking suspicion that that's essentially what an integral transform corresponds to. Not sure I'd be able to justify that though.

Typhon

hmm

determinants is what intrigue me though

since the matrix is still integer labeled in terms of size

Semiclassical

03:50

heh, then you'll dig this:

Functional determinant

In functional analysis, a branch of mathematics, it is sometimes possible to generalize the notion of the determinant of a square matrix of finite order (representing a linear transformation from a finite-dimensional vector space to itself) to the infinite-dimensional case of a linear operator S mapping a function space V to itself. The corresponding quantity det(S) is called the functional determinant of S. There are several formulas for the functional determinant. They are all based on the fact that the determinant of a diagonalizable finite-dimensional matrix is equal to the product of t...

Typhon

but isn't a step function if we start thinking of functions as matrices

THANK YOU

Semiclassical

That thing is actually a pretty big deal in QFT.

Typhon

I might hold onto that for now though as i do wish to make one solid attempt to come up with something myself before reading up on it. It's fun to try and conceive (even old) ideas.

@Semiclassical heh. I was just thinking "what is matrices were functions". Though I suppose the actual math would be tricky.

Semiclassical

@Typhon It very much is.

Typhon

@Semiclassical matrix multiplication wouldn't be though if one can set up a manner of composing them. That's just an integral.

or series of integrals

Semiclassical

03:53

Well, the analogy I had in mind was this.

The way to write matrix multiplication (specifically a matrix acting on a column vector) is as $\sum_k A_{jk}v_k = u_j$

Typhon

@Semiclassical regardless the spherical surface matrix would be a continuous matrix made from the multiplication of a ring continuous vector and a ring continuous vector I think. Depends on whether it is inner or outer product.

I was thinking carefully and i believe that the rows of a matrix are essentially the parallel geodesics within some surface assigned points like a function

so it might be that only euclidean or surfaces with parallel lines can have matrices shaped like them.

Semiclassical

So quite schematically the integral analogue would be $\int f(x')K(x,x')\,dx'$

Typhon

I figured something like that.

Semiclassical

Yeah. Which is exactly the form of an integral transform.

Typhon

O.O

head explodes

Semiclassical

03:56

Integral transform

In mathematics, an integral transform is any transform T of the following form: ( T f ) ( u ) = ∫ t 1 t 2 K ( t , u ) f ( t ) d t {\displaystyle (Tf...

Typhon

like the differential equations kind?

Semiclassical

Right.

Typhon

holy ****

Semiclassical

This be functional analysis territory.

Typhon

so by my reasoning... continuous matrices are what we use to solve differential equations...

I guess I was write about "linear calculus", lol.

Semiclassical

03:57

And, I'll be quite honest, that's not a field I know well enough to speak rigorously.

Typhon

fair point

but thanks for the info

Semiclassical

np

Typhon

mostly I just figured it would be a silly game.

like simple art's big numbers

Semiclassical

QM uses functional analysis a lot, though not always with infinite-dimensional stuff.

But we tend to be quite informal about it.

Typhon

i mean... I had to find something new now that I represented that one operator I was seeking out as a closed form. Kind of lost its appeal at that point. XD

@Semiclassical btw, does the term integral equation in reference to a differential equation mean something like...

Semiclassical

03:59

integral equations are a branch of math about which I know less than I should.

Typhon

$y' = f(x)$ and therefore the associated integral equation is $y = \int_c^x f(t) dt$

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