Mathematics - 2024-09-03

« first day (5144 days earlier) ← previous day next day → last day (172 days later) »

copper.hat

02:31

i had a really short proof of the twin crimes conjecture but it was stolen.

Jakobian

02:56

@copper.hat by Mochizuki?

2 hours later…

Joako

04:56

Hi everyone.... I noticed today that some tag filters in questions have stopped working.. Does anybody else having issues with them?... as example the following selection show no questions and last month it due show questions:

math.stackexchange.com/questions/tagged/…

leslie townes

is it really all that common for a single post to be tagged with all four of those?

i would take 'fourier transform' out of that if you want 'pure math' results, and maybe keep 'fourier transform' and remove some of the others if you want more 'applied math' oriented results

if you search by 'or'ing those together instead of anding them (concatenation means and on MSE), tons of results. math.stackexchange.com/questions/tagged/…

Soumik Mukherjee

Using the other three tags except the Fourier transform tag gives only 6 results. I have no idea but probably those 4 tags don't get along that much.

leslie townes

yeah, the example that came to my mind was, i don't know why someone would use 'fourier transform' and 'fourier analysis' at the same time.

sometimes a tag will have people dedicated to removing (perceived) improper instances of that tag, and it wouldn't surprise me if e.g. someone were constantly removing "fourier analysis" from posts that are more about a calculation

copper.hat

05:45

@Jakobian bless you

tokyo revengers?

5 hours later…

nickbros123

11:11

Have a newfound interest in numerical analysis

Seems like a good use of analysis

1 hour later…

Soumik Mukherjee

12:34

@nickbros123 get well soon :"(

Jk

nickbros123

12:48

@SoumikMukherjee lol

the method for finding roots using fixed point iteration method happens to be the exact proof for banach fixed point theorem lol. the same is the case for the existence of roots theorem. The algorithm proves the theorem, which is cool

2 hours later…

Soumik Mukherjee

14:37

What would be an example that satisfies all the conditions of being a monoidal category except the pentagram axiom?

Ben Steffan

...why do you need something like that

feels like something that might be annoying to come by

Soumik Mukherjee

@BenSteffan I am just a beginner at category theory, so asking out of curiosity :)

Ben Steffan

I see :)

Sine of the Time

15:02

@peek-a-boo sorry for pinging you, I'm working with pullback of distribution (in particular with the delta), but I'm struggling to find material regarding this topic. Can you suggest some books, if you know any? I took a look at Hormander and Shilov-Gelfand

Jakobian

5

Q: Examples of "Monoidal Categories" without Pentagon Axiom

One of the main goals of asserting the pentagon axiom for monoidal categories is to get the coherence theorem. I wondered how a typical example would look like that does not satisfy the theorem. More concretely, I'm looking for a nice enough example of a "monoidal category" that has all the data ...

category-theory monoidal-categories

Thorgott

you can take $B(\mathbb{Z}/2\mathbb{Z})$ with the tensor product doing the only possible thing on object and addition on morphisms

if you take the identity as associator, this is a monoidal category, but if you take $-1$ as associator, the pentagon axiom fails

looks like the answer Jakobian linked got the same idea

Soumik Mukherjee

15:23

Thank you @Jakobian @Thorgott

Joako

16:18

@leslietownes thnks... looks like before they were considered as "or" conjuctions and now as "and" conjuctions by default... its good to know from the shared link it has an easy fix.

Xander Henderson

16:30

I am teaching a class this semester in which I have to teach the order of operations. Most books give students some kind of mnemonic, like BODMAS or PEMDAS. I introduced them to GEMA (grouping, exponents, multiplication, addition), and worked very hard to emphasize that subtraction is "just" addition of the opposite, and division is "just" multiplication of the inverse. I've had four students (out of a class of around 50) email to me to express how much clearer this is.

It feels like a minor victory. Yay!

Soumik Mukherjee

nice!

Sine of the Time

@XanderHenderson next quest: new term for LIATE

Pizza

16:53

Hi 👋

Sine of the Time

sup

Jakobian

17:12

@XanderHenderson whats grouping

monoidaltransform

In a metric space say locally compact, if $K$ is compact and $U$ is open and $K\cap U= \varnothing$, then can we separate the two sets by open sets?

Jakobian

@monoidaltransform no

monoidaltransform

what about separating them by closed sets?

Jakobian

also no

monoidaltransform

hmmmm, okay suppose $U$ is an open set in $X$, and $K$ is some compact open set in $X$. Suppose $X$ is locally compact and $U\cap K = \varnothing$. Then is the set $\{f \in C(U,X): f(U)\cap K = \varnothing, f is open map \}$ open in the compact open topology, where $C(U,X)$ is continuous functions from $U$ to $X$

nickbros123

17:25

@SineoftheTime ILATE is subjective though cuz it depends on a "feeling" of which is easy to differentiate, which is easy to integrate

for wolfram alpha, everything is easy to integrate

Pizza

17:44

@SineoftheTime $\int \frac{1}{\log(x)} dx = \text{li}(x)$?

Jakobian

@monoidaltransform Let $U = \mathbb{R}$ and $X = \mathbb{R}$ and $K = \emptyset$. There exists non-open surjections $f_n:\mathbb{R}\to\mathbb{R}$ such that $f_n$ converges uniformly to $\text{Id}_\mathbb{R}$. So that set isn't open

monoidaltransform

@Jakobian what if $X$ is compact?

Jakobian

the same thing can be done for $[0, 1]$

so in this setting both $U, K$ are compact open and $X$ is compact

Sine of the Time

Logarithmic integral function

In mathematics, the logarithmic integral function or integral logarithm li(x) is a special function. It is relevant in problems of physics and has number theoretic significance. In particular, according to the prime number theorem, it is a very good approximation to the prime-counting function, which is defined as the number of prime numbers less than or equal to a given value x {\displaystyle x} . == Integral representation == The logarithmic integral has an integral representation defined for all positive real numbers x ≠ 1 by the definite integral...

@nickbros123 I've never liked these "tricks", just do a lot of exercises and build intuition

Xander Henderson

@SineoftheTime Meh. I've never taught integration by parts that way.

I prefer that students use trial and error, and eventually come up with their own heuristics.

@Jakobian Parentheses, braces, $\lfloor\cdot \rfloor$, things above and below the vinculum in a fraction, etc. Symbols used to group expressions into a single expression.

monoidaltransform

17:53

thank you @Jakobian.

Joe

I am reading a proof of the following statement: Let $A$ be a commutative ring, $M$ a free module, and $m_1,\dots,m_p$ a linearly independent family in $M$. Then, for every nonzero element $a$ of $A$, there is a $p$-linear alternating form $f$ on $M$ such that $af$ does not vanish at $(m_1,\dots,m_p)$. The proof is by induction on $p$.

user image

Xander Henderson

Ugh... it is 11am. I've been at my desk since 7am. I have just finished responding to all of the email that came in over the three day weekend. I have grading that I should do, and prep work for tomorrow's classes, but... I DON'T WANNA!

Email is dumb.

:(

It is so enervating.

Joe

The proof then claims that $f'$ is a $p$-linear alternating form. As far as I can tell, this is not true.

Ben Steffan

@Joe why would it not be?

It looks like a $p$-linear alternating form to me :)

Joe

Maybe I made a mistake when trying to think of some examples for small $p$. I will check.

Ben Steffan

18:06

I think the proof is rather straightforward

if you scale one of the $x_i$ by some $a$, then you can pull the $a$ out of the sum: for $i \neq j$ you use multilinearity of $f$, and for $i = j$ the summand is already of the right form

additivity in each variable is similar, so it's bilinear

$p$-linear, rather

Joe

Hmmm I think I did make a mistake in my calculation. Thanks for your comments. I am reading through them now...

Joe

18:35

Maybe I am just tired, but I don't even know what $f(x_1,\dots,\hat{x_j},\dots,x_p)x_j$ is supposed to mean.... Isn't $x_j$ a member of $M$, and $f(x_1,\dots,\hat{x_j},\dots,x_p)$ also a member of $M$?

Ben Steffan

$f$ takes values in $A$

so it's just scalar times element of $M$

Joe

18:50

Oh I see.

peek-a-boo

@SineoftheTime if pulling back by diffeomorphisms, then it is obvious how to define it using the change of variables theorem (see Folland). Otherwise Hormander is where I learnt from.

Sine of the Time

@peek-a-boo I'm working in particular on the representation of manifolds using delta distribution (expecially the formula $\int \delta(f(x))\varphi(x)dx=\int \varphi / \sqrt{\det (Df \cdot (Df)^t)} d\sigma$)

I'm trying to see if we can simplify some computation of surface integrals using distribution so I wa searching for some other properties

leslie townes

19:15

surface integrals can get really involved, i am not sure i would naturally look to distributions as a tool for simplifying them. if there is a self contained example that captures some of the relevant difficulty that you are trying to simplify, maybe worth an MSE post

Sine of the Time

I'm working with my advisor and we are trying to find if this way of doing surface integral can be useful (i.e. if we can simplify surface integrals without using parametrizations). So far we have establish some general rules, like what happens with product and composition and I know Gauss-Green formula can be proven using the pullback of the delta

the problem is that there's not much material

as far as I know, this is useful sometimes when dealing with PDEs

peek-a-boo

Dieudonne’s chapter 17 is about currents (and hence distributions) on manifolds, maybe you find that helpful. It has lots of nice problems :)

Sine of the Time

ok thanks

peek-a-boo

You always have the “surface measure” which is well, a measure, and hence can be made a distribution in a natural wa, but I don’t know enough abt your situation to see how this can be helpful.

Sine of the Time

I'm afraid I don't know enough about measure theory

peek-a-boo

19:31

Btw it’s section 17.5 problem 9 which deals with pullback of currents under subermsions… but yea it’s an exercise so idk how useful it is for you

Sine of the Time

I'll take a look and let you know. thanks

Silly Goose

Is a stochastic process always equivalent to a family of probability distributions?

I am reading about an example of a stochastic differential equation, the Langevin equation. It is then stated that the Langevin equation is equivalent to a normal partial differential equation, a Fokker-Planck equation, whose solution is a probability density function in terms of the random variable.

so I am wondering how can these two equations be equivalent if one gives stochastic processes as solution and the other gives a PDF as solution

1 hour later…

ILikeMathematics

20:59

Btw @BenSteffan are the lectures for Bonn undergrad at the Mathematik Zentrum in Endenicher Allee 60?

Ben Steffan

no

the MZ doesn't have lecture halls

ILikeMathematics

Then where are they?

Ben Steffan

the lectures are in Wegelerstr. 10

just a short walk away

it's an older building, most of which belongs to the physicists

ah, there's also the institute for discrete mathematics

ILikeMathematics

What is the Mathematik Zentrum for then?

Ben Steffan

everything else

exercise groups, seminars, ...

it's where all the staff have their offices

ILikeMathematics

21:01

Ah

Ben Steffan

the library's there

and so on

...but yeah the institute for discrete mathematics exists; it's very fancy; it has its own lecture hall

but it's also in the city, pretty far away from the MZ

ILikeMathematics

@BenSteffan I'm looking for a Studierendenwohnheim (Appartment) so that I have to walk as little as possible, do you know which ones that would be?

There is one at Endenicher Allee 21

Ben Steffan

there's one right opposite the MZ

doesn't get any closer

ILikeMathematics

I think that one isn't an Appartment

Just WG

Ben Steffan

hum, really

that's news to my ears, but I don't know

maps.app.goo.gl/ok7LUYm8J5Y7vRDE8

this one's pretty close, and huge

ILikeMathematics

21:06

Oh, you're right, sry

Ben Steffan

these are pretty much the only ones close to the MZ that I know off

ILikeMathematics

user image

Ben Steffan

?

ILikeMathematics

@BenSteffan Oh, you didn't mean that this one is opposite to the MZ

Ben Steffan

yeah

that's the next closest I know

ILikeMathematics

21:09

Alright, thanks a lot!

Ben Steffan

you're welcome :)

ILikeMathematics

21:24

@BenSteffan I will be done with my Abitur in july next year. Does this look right then?

user image

It's the furthest it goes - can't go past 01.09.2025

Maybe I have to wait some more

Ben Steffan

you're very early

but 1.9. would probably be alright

the term starts 1.10.

ILikeMathematics

Alright, thanks a lot

user image

I'm not sure if I should say '3', I will be done with analysis 1, 2 and linear algebra 1, 2 but not algorithmische Mathematik 1, 2

@BenSteffan Would you input 3 here?

Ben Steffan

I would put 1

even with the Fruehstudium

ILikeMathematics

Alright, thanks

@BenSteffan Sorry, last question and then I'll stop annoying you:

user image

If I know I want to stay there for MSc, should I add that into it?

Ben Steffan

21:39

is there a limit on how long you can stay there?

usually there's some limit

if you don't plan on moving I would just put the maximum

ILikeMathematics

<= 36

Ben Steffan

ah, so they may kick you out after 3 years

ILikeMathematics

Alright, then I'll just put 36, thanks!

Ben Steffan

you're welcome, as always

if you have more questions just ask!

ILikeMathematics

Thanks!

psie

22:25

Studying the Schröder-Bernstein theorem. We have two injections $f:X\to Y$ and $g:Y\to X$. We divide $X$ into three disjoint subsets $X_X, X_Y, X_{\infty}$ with different mapping properties as follows.

First, let $x_1 \in X$. If $x_1$ is not in the range of $g$, then we say $x_1 \in X_X$. Otherwise there exists $y_1 \in Y$ such that $g(y_1)=x_1$. If $y_1$ is not in the range of $f$, then we say $x_1 \in X_Y$. Otherwise there exists a $x_2 \in X$ such that $f(x_2)=y_1$. Continuing in this way, we generate a sequence of points $$x_1, y_1, x_2, y_2, \ldots, x_n, y_n, x_{n+1}, \ldots.$$

I wonder, each $x\in X$ generates a unique sequence by injectivity, right? Hence the reason why $X_X, X_Y, X_{\infty}$ are disjoint.

Ben Steffan

since you seem to be putting $x$ as $x_1$ the answer is rather trivially yes :^)

but yes, even if you drop the $x_1$ the generated sequences is unique

psie

ok, thanks for clearing up that doubt :)

Ben Steffan

it's not even injectivity either: if $y_1$ is such that $g(y_1) = x_1$, it cannot also satisfy $g(y_1) = x'_1$ for some other $x'_1$

you do need injectivity to say that there is a unique choice of $y_1$ in the sequence, however

psie

@BenSteffan and why does that matter?

Ben Steffan

I don't know because I'm not familiar with the proof you're reading

maybe it doesn't matter at all

maybe it's completely essential

you tell me :)

Derso

22:35

Hi, guys! I have a "basic" ode theory's question

Posted a few minutes ago

If you have a vector field $X:I\times \mathbb{R}\to \mathbb{R}^2$ of the form $X(s,t)=(1,f(s,t))$

Can it have finite-time explosion integral curves?

Ben Steffan

what's finite-time explosion mean?

Derso

Well, it means basically that the solution "goes to infinity", even though for a finite "time" (finite parameter of the solution).

Say $\alpha(s)$ is an integral curve, i.e. $\alpha'(s) = X(\alpha(s)) = (1, f(\alpha(s)))$. Then we must have basically $\alpha(s) = (s,y(s))$, for some $y$ with $y'(s) = f(s,y(s))$. Then finite explosion means $\lim_{s\to s_0} y(s)=\pm \infty$ for some finite $s_0$, you see?

Here's the full question, btw math.stackexchange.com/questions/4966781/…

Sine of the Time

$y(s)$ has a vertical asymptote

Derso

Yes! That's what my intuition says. But how's that possible?

$X$ is defined on that supposedly vertical line, but has horizontal component $1$

Sine of the Time

I was explaining the meaning of finite-time explosion

Derso

22:46

Oh, ok

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