Mathematics - 2024-11-25

Ben Steffan

00:00

yeah, an metrizable space is an AN iff it is an ANR and contractible

I don't know anything about the theory for normal spaces, sorry

we care more about ANRs to begin with, and even that is not really my neck of the woods

Jakobian

@Thorgott maybe you know? If I recall correctly, you studied AR's and ANR's a lot

copper.hat

00:17

kick in the ars

Ben Steffan

haha, very funny

how does the quote go

"Only Thorgott deals in absolutes" -- Star Wars

Xander Henderson

@BenSteffan But... that's an absolute!

Jakobian

@BenSteffan on the same topic, is there some elementary argument that the circle is an ANR?

Thorgott

@Frusciante factors through in what sense?

@XanderHenderson that's a fantastic story lol

@Jakobian only really the metrizable case, which was all I needed for applications, but I can point you to the appendix of Lundell-Weingram's "The Topology of CW complexes", where a few things about the general theory are set up, I believe

Jakobian

00:33

that's a long book

Thorgott

@Jakobian elementary, yes, but annoying

it's the union of two open subsets which are ANRs and that makes it an ANR

@BenSteffan :P

Jakobian

sorry, not that long, only 225 pages, earlier springer was showing me its 500 pages for some reason, or maybe I just misread

@Thorgott it looks like for neighbourhood retracts at least, there is a characterization of what they call ANRN's as ANR's which are completely metrizable

(here ANR refers to for separable metrizable spaces, which is the same as ANR for metrizable spaces; in particular those are separable metrizable spaces)

so $S^1$ being a complete metric space and an ANR, it is clearly an ANRN

@Thorgott I want to prove its an ANRN, but I think this will lead me to a good direction, thanks

Frieren

01:04

My book solves the following problem using the Cauchy-Schwarz inequality: "Let $a_n \ge 0$ and let the series $\sum a_n$ be convergent. Prove that the series $\sum \sqrt{a_n}/n^\alpha$ converges." I tried another approach:
we have $0 \le (\sqrt{a_n}-1/n^\alpha}^2 = a_n-2\sqrt{a_n}+1/n^{2\alpha}$, so $\sqrt{a_n}/n^\alpha \le (a_n+1/n^{2\alpha})/2$. Since $\sum(a_n+1/n^{2\alpha})$ converges because of the hypotheses on $\sum a_n$ and $\alpha>1/2$, by the comparison theorem (being $a_n \ge 0$) we have proved the result. Is this correct?

ModularMindset

Let $X=[0,1]^n$. For all $n>2$ does $X$ admit a unique (up to permutation) codimension one surface of revolution, $L$, with a complete metric (away from the cone points) and an embedding $e :L\hookrightarrow X$, which maximizes volume while retaining constant positive sectional curvature?

Assume the cone points $p,q$ satisfy $\mathrm{sup~dist}_n(p,q)=\sqrt{n}$ where $p,q \in L$.

I understand the case $n=3$, where metrics of constant curvature correspond to quadratic polynomials

$n=3$ gives a unique $L$

Intuitively this seems like it should work in all dimensions

ModularMindset

01:33

This is just sphere's in R^n at the end of the day

copper.hat

01:53

that's a lot of balls

pie

I recently came across this book, and it looks intriguing. Has anyone read it? Would you recommend it? A 1000-page book on analysis seems quite extensive—it reminds me of those bulky calculus textbooks.

ModularMindset

Tackleball - sign the petition to change the name for Football.

change.org/p/…

Xander Henderson

@ModularMindset I prefer to call it "Concussion Ball".

@pie I do not know the book.

ModularMindset

chat.stackexchange.com/transcript/message/66680831#66680831 There is a major need to prove uniqueness in my case. In the sphere in R^n case volume blows up

therefore these are different problems

Xander Henderson

02:08

Looking over the table of contents, I do think that I like the approach. I like the ordering of sequences and series before differentiation and integration (if I had the time, I would probably write a calculus text which does the theory of polynomials (differentiation and integration of polynomials), and then does stuff with series, connecting the two things with Taylor's Theorem; the idea of starting with sequences and series is not quite the same, but sympathetic).

Jakobian

@pie a 1000 page book sounds like something you don't want to read in its entirety

Xander Henderson

That being said, it doesn't look like it is mean to be an introductory text. It seems to assume that students have already taken calculus, and are now looking to take analysis (or that they have a very skilled guide to help them through the book).

And I wonder at how much time one is expected to take in order to get through the text. It looks like at least three semesters worth of material.

@Jakobian Given the table of contents, and how many pages are spent on each topic, I get the impression that there is a lot of exposition. Fundamentally, I don't think that the text covers all that much more material than Baby Rudin. It is just a lot more verbose (for better or worse).

Jakobian

> I must say that I've gained insights from this book that I did not get from any other text. Prof. Johar manages to deliver the perfect balance between rigor and eloquence, producing a text that is both enjoyable and rigorous, inspiring and exhaustive. The subjects are well concatenated, and I never felt lost due to the structure of the book (only due to the fact that it is a hard subject!).

> With this book, I felt that you have all-in-one; you will not need to refer to another book for a long time. It really goes from numbers to measures. This one takes the crown in my analysis book collection

from a review

author of the review also calls Rudin the "best terse Real Analysis textbook"

they have read a lot of books on the topic (for some reason), and decided this one is the best, apparently

I suppose, real analysis is a topic where you learn a lot of fundamental stuff, so 1000 pages for an exposition to a subject maybe isn't entirely surprising

especially if its exhaustive

the description on springer also mentions that:

> the book is fully self-contained

pie said before that they have no teachers or anything of the sort, so this might be good for them

Jakobian

03:11

@Thorgott I've looked up the proof of this by the way, and it seems to me that the statement holds a) for normal spaces and finite unions, b) for metric spaces and countable unions, c) for separable metric spaces and arbitrary unions

The proof of this also seems to go from a to b to c

The proof of a) seems to also be the most convoluted, but I'll read it 😅

copper.hat

04:06

Rudin PMA is good, R&C awful, FA a mess.

Thomas Finley

07:21

While studying about PDEs I found that we can calculate the general solution of a PDE from it's complete solution.

The method goes like this:

If say, a PDE $F(x,y,z,p,q)=0$ where $p=z_x,q=z_y$ is given, and $f(x,y,z,a,b)=0$ is a complete solution of $F,$ then assume $b$ to be an unknown function of $a$ say, $b=\phi(a).$

So, the complete solution $f$ becomes, $f(x,y,z,a,\phi(a))=0\tag 1$

Now, differentiating the above equation wrt $a$ we get, $f_a(x,y,z,a,\phi(a))+f_b(x,y,z,a,\phi(a))\phi'(a)=0\tag 2$

PM 2Ring

08:57

@pie You might like The Princeton Companion to Mathematics

The Princeton Companion to Mathematics

The Princeton Companion to Mathematics is a book providing an extensive overview of mathematics that was published in 2008 by Princeton University Press. Edited by Timothy Gowers with associate editors June Barrow-Green and Imre Leader, it has been noted for the high caliber of its contributors. The book was the 2011 winner of the Euler Book Prize of the Mathematical Association of America, given annually to "an outstanding book about mathematics". == Topics and organization == The book concentrates primarily on modern pure mathematics rather than applied mathematics, although it does also cover...

psie

09:09

Let $\{E_r\}_{r>0}$ be a family of Borel subsets of $\mathbb R^n$. They are said to "shrink nicely" to $x\in\mathbb R^n$ if

1) $E_r\subset B(r,x)$ for each $r$,

2) there is a constant $\alpha>0$, independent of $r$, such that $m(E_r)>\alpha m(B(r,x))$.

Let $U$ be a Borel subset of $B(1,0)$ such that $m(U)>0$. Then $E_r=\{x+ry:y\in U\}$ shrinks nicely to $x$. How? My attempt: $E_r$ is the translation by $x$ of the set $rU$, so $$m(E_r)=m(rU)=r^nm(U),$$but I don't know how to continue to either show 1) or 2).

psie

09:28

Well, 1) is pretty straightforward, but 2) I struggle with.

Frusciante

10:06

@Thorgott factors in the sense that if $f \circ \alpha: p \Rightarrow \text{const}_c'$ is a cone under $p$, then $(f \circ \alpha) = (\bar{p}: p \Rightarrow \text{const}_c \Rightarrow \text{const}_c')$.

Basically as in the definition of coCartesian lift

Sha Vuklia

12:13

@Astyx Nice thanks! This morning when I woke up I thought I should probably look at the minimal polynomial, and I regret feeling hesitant to try that :')

Sine of the Time

@Soumik Guk panicking :|

Xander Henderson

13:38

@PM2Ring That's really more a reference than a book to learn out of...

PM 2Ring

14:03

@XanderHenderson Sure, but from math.stackexchange.com/q/4994330/207316 I think pie would be interested in an "extensive overview of mathematics".

PM 2Ring

14:14

A pretty self-organizing system by Steven Dollins from mathstodon.xyz/@[email protected]/113490752614970456

Jakobian

after looking more concretely, the proof I am reading doesn't seem to be able to be applied to normal spaces

I am looking at Some theorems on absolute neighborhood retracts by Hanner, and they extend a function relative to $U_0\cup Y_1$ which is not necessarily a normal space

however, there is a theorem of Borsuk that if $X = A\cup B$ where $A, B$ and $A\cap B$ are ANR's then $X$ is an ANR, so maybe this translates to normal spaces

however, the article is in German

actually it seems that Borsuk assumes $X$ is compact, and I forgot to say but $A, B$ are closed

it looks like there is a more complex version of the proof in Theory of retracts by Hu

its proposition 8.1 there

the book deals with ANR's and AR's with respect to a lot of different categories of topological spaces, and I read its a good reference

proposition 8.1 is for ANE's, but its proven that ANE's and ANR's agree for normal spaces

Soumik Mukherjee

14:58

@SineoftheTime yeah

Pizza

15:18

Hi

@SineoftheTime What do you think of the book: Mathematical Methods for Physics and Engineering - 3rd Edition

I don't know if I saw correctly but in the PDF you recommended to me , Is the Fourier transform missing?

I mean shaum's book

Sine of the Time

15:37

@Pizza author of the book?

@Pizza no, you are correct. That's a book about complex analysis

For Fourier and Laplace transform, there's also a Schaum, but I don't know if it's adapt for you, I need more details on your course

Pizza

16:04

@SineoftheTime Riley, Hobson, Bence

@SineoftheTime Ok I'm sending now

Complex numbers. Algebraic, trigonometric, exponential form. Properties of the modulus and argument. De Moivre and n-th root formulas.

Elementary functions in the field of complex numbers: exponential, sine and cosine, hyperbolic sine and cosine, logarithm, power. Sequences and series in the field of complex numbers.

Power series: radius of convergence and properties, term-by-term derivation.

Analytical functions. Holomorphy and Cauchy-Riemann conditions. Line integrals of functions of complex variables. Cauchy's theorem and formula. Taylor series development. Laurent series development. Zeroes of analytic functions and identity principles. Classification of isolated singularities. Liouville theorem.

Integration. Notes on measurement and the Lebesgue integral. Addable functions. Theorems of passage to the limit under the integral sign. Integrals in the sense of the principal value according to Cauchy. Spaces of summable functions.

Residues. Residue theorem. Calculation of residues at the poles. Calculation of integrals with the residual method. Jordan lemmas. Breakdown into simple fractions. Difference equations. Z-transform: definition and properties. Z-antitransform. Successions defined by recurrence.

Laplace transformation. Signals. General information on signals. Periodic signals. Convolution. Definition and domain of the bilateral Laplace transform. Infinite analyticity and behavior. Notable examples of Laplace transform. Formal properties of the Laplace transform. Unilateral Laplace transform and properties. Initial and final value theorems. Antitransformed (s.d.). Use of the Laplace transform in linear differential models.

Fourier series. Notes on Banach and Hilbert spaces. Energy of a periodic signal. Trigonometric polynomials. Exponential and trigonometric Fourier series. Convergence in the punctual sense and in the energy sense. Fourier transform. Definition of Fourier transform. Formal properties of the Fourier transform. Antitransformed. The Fourier transform and the heat equation.

Sine of the Time

that's a lot of things :D

Pizza

Distributions. Linear functionals. Limits in the sense of distributions. Derivative in the sense of distributions. Derivation rules. Notable examples: Dirac's δ, v.p. 1/t. Convolution of distributions. Space of rapidly decreasing functions and related topology. Temperate distributions and slow-growing functions. Fourier transform of temperate distributions. Laplace transform of distributions. Fourier transform of the Dirac δ, of the pulse train. Fourier transform of periodic signals.

Boundary problems Self-adjoint equations. Green's function, the alternative theorem. The Sturm-Liouville problem, orthogonality eigenfunctions. Partial differential equations General information. Laplace and Poisson equations, harmonic functions, Dirichlet and Neumann problems. Solving the Dirichlet problem for the Laplace equation in a circle. Heat equation, Cauchy problem in the half-plane. Wave equation, Cauchy problem in the half-plane, mixed problem in the half-strip.

Thats all the programm :(

Sine of the Time

I don't know the book you tolde me about, in this situation is best to follow the one suggested by your professor

Pizza

Sorry for the spam

Sine of the Time

The problem with these course is that they're meant to give you mathematical tools besides linear algebra and analysis 1/2

So it's hard to find a book, because the programs are always different. The professor usually choses the topics

For exercises on the complex analysis part, I'd use Schaum

Pizza

16:16

Is there anything about distributions and the Laplace transform?

These are the topics that come up the most

Sine of the Time

for Laplace transform try to see: Schaum Differential equations, chapters 21-25

for the exercises I mean

For distribution theory, I don't know

Pizza

Is there also the Fourier transform in that book?

Sine of the Time

@Pizza no

Pizza

Oh no ok, so there is another book aside

Sine of the Time

Yes, Fourier analysis (Schaum)

I never used these books, I've only took a look at complex variables

So I can't assure you'll find what you're searching

Pizza

16:25

@SineoftheTime The one with the pink cover?

Sine of the Time

yes

Pizza

@SineoftheTime I think it should be fine

Thanks !

Sine of the Time

np

psie

16:38

A Borel measure $\nu$ on $\mathbb R^n$ will be called regular if it is a positive measure that is finite on compact sets $K$, i.e. $\nu(K)<\infty$. Silly questions maybe, but

1) does it imply that it is finite on bounded Borel sets?

2) is it $\sigma$-finite?

Jakobian

@psie If $B$ is a bounded Borel set, then what is $\overline{B}$

psie

@Jakobian it is the closure of $B$, which is closed and bounded, so compact, right?

Jakobian

@psie can you write $\mathbb{R}^n$ as a countable union of compact sets?

@psie yes

psie

@Jakobian ok 👍 yes, hmm, so the closed balls with rational center?

Jakobian

Note that this is not the usual definition of a regular measure, but its equivalent to the standard definition for space like $\mathbb{R}^n$

@psie you're overthinking it

psie

16:43

alright, let me think some less

Xander Henderson

@psie I actually kind of like this characterization, as it generalizes to any metric space with a countable dense subset.

psie

I don't know what else to think of :)

Xander Henderson

Though your description is a little bit of a problem, as there are uncountably many balls with any given center. Perhaps "closed balls of radius 1 with rational centers" could be better.

psie

ah yes, that's much better

Jakobian

@XanderHenderson does it? What do you mean

@psie $X = \bigcup_{n\in\mathbb{N}}B(0, n)$

Xander Henderson

16:52

@Jakobian I guess you have to assume the Heine-Borel condition, as well.

@Jakobian Ah, that's what you had in mind. Yeah, that works, too.

Jakobian

@XanderHenderson I think this is equivalent to being locally compact and separable

I remember there was some kind of characterization like that of metrizable spaces admitting a metric which satisfies Heine-Borel

psie

@Jakobian ok, nice, so the family of sets would be the closure of $B(0,n)$. Great, thanks.

Frieren

tMy book solves the following problem using the Cauchy-Schwarz inequality: "Let $a_n \ge 0$ and let the series $\sum a_n$ be convergent. Prove that the series $\sum \sqrt{a_n}/n^\alpha$ converges." I tried another approach:
we have $0 \le (\sqrt{a_n}-1/n^\alpha)^2 = a_n-2\sqrt{a_n}/n^\alpha+1/n^{2\alpha}$, so $\sqrt{a_n}/n^\alpha \le (a_n+1/n^{2\alpha})/2$. Since $\sum(a_n+1/n^{2\alpha})$ converges because of the hypotheses on $\sum a_n$ and $\alpha>1/2$, by the comparison theorem (being $a_n \ge 0$) we have proved the result. Is this correct?

Jakobian

yeah, a locally compact Lindelof space is exhaustible by compact sets, so for any locally compact separable metric space, if a measure on it is finite on compact sets, then that measure is $\sigma$-finite

Sine of the Time

@Frieren Which step are you not sure about?

Jakobian

16:59

you just need to cover your space by open sets with compact closures and take countable subcover

psie

interesting, there's a chapter on Radon measures later on, I guess that's where "sh*t's going down"

Jakobian

and from what I remember, if you have an exhaustion by compact sets of your metric space then you can obtain an equivalent metric with Heine-Borel property from that

been a while since I've researched the topic, but I think this is how it goes

@XanderHenderson web.math.utk.edu/~freire/teaching/m561f20/…

might interest you since you deal with metrics in your fractal considerations

Xander Henderson

@Jakobian Yes, that seems right.

Man, I wish I still got to do actual math in my work. I forgotten more than half the people in this room even know. :(

Jakobian

17:18

I wonder, normal and Tychonoff spaces are spaces with enough real-valued continuous functions

I wonder what would be a "space with enough Borel measures"

Jakobian

17:33

Maybe lets say that a measure $\mu$ separates $A, B$ if there exist open sets $U, V$ containing $A, B$ with $\nu(U\cap V) = 0$

ZaWarudo

@XanderHenderson How much do you believe that capitalism has to do with this condition? Genuine question, I always wondered how much being in this economical system takes away from what we really want to do, but maybe I am just wrong and there are other reasons why.

Jakobian

And we can test with $A, B$ being points and $\nu$ going over, say, all Borel measures on $X$

Xander Henderson

@ZaWarudo Honestly? Very little.

As a mathematician, you kind of have to choose between research and teaching (if you want to stay in academia, anyway) and, for a number of reasons, I chose teaching. I enjoy teaching, but (as with any job) there are things that I like and things that I don't like.

ZaWarudo

@XanderHenderson As far as I know, correcting written exams is one of the tasks no one wants to do :D

Xander Henderson

@ZaWarudo Yeah, it's a pain.

But, like, someone has to do it.

Ben Steffan

17:47

do you not recruit tutors and TAs to do it?

Xander Henderson

Unless one wants to argue that, absent capitalism, there would be no need for teachers to provide feedback to students.

@BenSteffan I am at a small community college. No.

We have a total student population of around 2,000. There are a total of five full-time mathematics faculty. Five.

Ben Steffan

ok, but then you probably also don't have to correct that many exams to begin with (?)

ZaWarudo

About this, I was reading about Lean software for proof assistance. In my opinion, it would be awesome for students as well because a lot of times I find myself looking for a check of proof and come here asking for a feedback. I understand that one should learn to verify his own proofs, but especially in the beginning it would be cool to have a reliable (and that's the hard feature) software that can avoid the boring part of reading a proof written by a student just to give a feedback.

Xander Henderson

@BenSteffan Yes and no. In any one class, I typically don't have more than 30 students. But I often have multiple preps, which means writing multiple exams, writing multiple exam keys, and then grading those exams.

Ben Steffan

at the small small cost of having everyone learn how to formalize proofs in lean

@XanderHenderson oof

Xander Henderson

17:51

This semester, I only have three preps! Happy days!

Ben Steffan

also I somehow doubt that lean feedback is very useful for a beginning student

Xander Henderson

But one of those was only given to me three days before classes started, and I'd never taught the class before, so I have been struggling since day 1 to keep up. In other news, that class starts in 10, so I'd better go deal with it.

Ben Steffan

have... fun?

ZaWarudo

@BenSteffan Yeah, that was an example, maybe a simpler version :D

Xander Henderson

Just sent the following to my dean:

> I am capable of being a real adult and doing real adult things like setting up calendar reminders for myself (even if I don't want to admit that I am capable of "adulting"). :P

It has been a long semester, and I am feeling punchy...

And now I'm really off.

leslie townes

17:55

@BenSteffan this is often a pain point in US universities, as there is a lot of emphasis on assignments and written evaluation and yet usually little/no room in a budget specifically for that. even at otherwise solvent and well funded schools individual instructors are often just expected to do a lot of written evaluation on top of everything else.

which surprisingly enough can mean that it is done somewhat poorly

which is weird, comparatively speaking. it seems really common in the US for there to be way more points of evaluation of student work than in other places, and yet what's the point if it's just someone spending 5 seconds on it

Jakobian

actually never mind my question about a space having enough measures. That sounds like a whole ant nest that I don't want to touch

Ben Steffan

@leslietownes I'm a little surprised because, well, this is just part of my job. I teach an exercise class during the term and when finals are here all of us + TA + Prof. sit down and spend an afternoon grading and that's that

I don't get paid extra for it or anything

leslie townes

there's probably something going on with terminology here. what i'm saying is that at a lot of places in the US, what you describe as "all of us + TA + prof" would be a smaller number of people, not infrequently just a prof.

at least, i think that's what i'm saying. i'm not sure what you're saying :)

without the paperwork necessarily scaling down to match. instructors are often a position of "i would like to hire someone to do things like grade homework and exams, but there is no money to do that, and for whatever departmental reason i can't just not give homework"

Ben Steffan

do your courses not have exercise classes? are they also taught by the prof/ta?

leslie townes

what's an "exercise class"? there is no fixed system in the US, every department would do it differently. large universities will often have sections that parallel a lecture (not separate classes, but a required part of the lecture) taught by grad students or TAs. this would be most common with larger classes, like, "calculus" and "linear algebra" but maybe not anything that only a math major would take.

so yes in that type of situation a TA would probably handle a lot of that grading. smaller schools might not have an abundant resource of cheap grad student labor though, although the scale of their calculus offerings might roughly the same as it would be at a school that did.

ModularMindset

18:10

I think exercise class might be called recitation class in the US?

leslie townes

there's no fixed term for it! "recitation" sounds like something some schools would use. "discussion section" is another one.

Ben Steffan

I see

that sounds rather terrible

leslie townes

so like when i was at my undergrad (which had a huge math department) there was that kind of infrastructure around a limited number of large course offerings and usually nothing for more advanced classes (which admittedly were smaller, on the scale of xander's 30 students, but also had homework that was more difficult to grade). it would change based on the budget and the year

they were always trying to change this (it was, i think, perceived of as a problem that many instructors felt like they were in the position of assigning more homework than they could reasonably grade) and never quite doing it

for whatever reason the obvious solution of just not assigning homework, or assigning it but making it optional and letting students decide what they want evaluated and what they don't, is regarded as more untenable than it actually is

ModularMindset

arxiv.org/pdf/math/0411106

I was looking at similar works to my problem and found this

but I don't think I should spend time on this one

Ben Steffan

why anybody felt the need to upload that to arxiv is a mystery

ModularMindset

18:25

-2

Q: "A Symplectic Look At Surfaces of Revolution," by Dr. Hwang.

I am looking at the document "A Symplectic Look..." by Dr. Hwang. I have a more general, but related question about this document: Let $X=[0,1]^n$. For all $n>2$ does $X$ admit a unique (up to permutation) codimension one surface of revolution, $L$, with a complete metric (away from the cone poi...

this website is a joke

copper.hat

you get what you pay for

i'm still waiting for my orange mse jumpsuit

Ryder Rude

hi

ModularMindset

hi (let's keep it going)

copper.hat

lo

think_meaning_buildß

medium

Ben Steffan

18:30

and beehive

think_meaning_buildß

🐝🐝🐝

Sine of the Time

@copper.hat is still there a reward for hitting 100k rep?

copper.hat

@SineoftheTime i don't know. i got an mse t shirt, a stack exchange mug and some other bits & pieces.

Sine of the Time

nice, but 100k is a lot :\

copper.hat

i don't have many hi rep answers, mostly grind

ModularMindset

18:38

me too it's the grindset

I get 2 downvotes - get ready for 2^3 questions as revenge

copper.hat

my two highest are an elucidation of algebraic/geometric multiplicity and the derivative of a quadratic form. i just know that Terrence is shaking in his boots.

ModularMindset

good questions that you can't in good conscience downvote

Sine of the Time

If I keep with this trend, it'll take me 24/25 years to get to 100k

copper.hat

i'm not sure that rep on mse counts for much :-)

Sine of the Time

pretty useless

copper.hat

18:41

but it keeps my brain active and the chat rm can be amusing :-)

Sine of the Time

I feel I'm too addicted, maybe I should take a break

3

copper.hat

i do like my mse t shirt, but only my kids know what it is

it serves an one of my primary sources of procrastination

ModularMindset

You can buy one of those

without having the rep

or does the shirt actually say the rep on it

psie

I'm havin' a basic doubt. In $\mathbb R^n$, can every unbounded set be written as a countable union of bounded sets?

Ben Steffan

yes

$X = \bigcup_{i = 1}^\infty X \cap B_i(0)$, say

psie

18:49

Ok 👍

think_meaning_buildß

@SineoftheTime admitting it could be a problem is the first step

Ryder Rude

The funhouse is a good horror movie

ModularMindset

Assume $X=(0,1)^3$ and $\partial X=[0,1]^3-(0,1)^3$ and our foliation $\mathcal F$ satisfies the condition:
$$
\lim_{p \to \partial X} L_\alpha(p) = \{v_i, v_j\},
$$
where $ L_\alpha $ are the leaves in $ \mathcal{F}$ and $v_i=(1,1,1)$ and $v_j=(0,0,0)$ where $v_i,v_j\subset \partial X.$ Then in some rigorous sense the $L_{\alpha}$ decay to zero rapidly, if we choose a "direction" for example a flow that maps points on the $L_{\alpha}$ from $v_i$ to $v_j$.

This is similar to what's called a Schwartz space, which is the space of rapidly decreasing functions. And rapidly decreasing here can …

Is there an adaptation of this to rapidly decreasing surfaces, specifically does the example I gave work?

The question really becomes - "how useful is my definition"

1

Q: Rapidly decreasing surface? Adaptation from Schwartz functions?

A foliation $\mathcal{F}$ on a smooth manifold $ X $ is a smooth decomposition of $ X $ into disjoint connected submanifolds $ \{L_\alpha\}_{\alpha \in A} $, called leaves, where each leaf $ L_\alpha $ is locally diffeomorphic to $ \mathbb{R}^k $ for some $ k \leq n $. The local trivialization of...

reference-request differential-topology surfaces schwartz-space foliations

Here's the full question if you want to take a look

Sine of the Time

19:37

@think_meaning_buildß If you have a problem, double it and give it to the next person

think_meaning_buildß

19:50

yes, that's one way of looking at it

Thorgott

20:21

@Jakobian there also is a theorem (due to Hanner) that says that being an ANE is a local property for paracompact spaces

@Jakobian this is true (indeed, assuming $A,B$ closed) even if $X$ is not compact as long as you replace ANR with ANE

both of these facts are in Sakai

though Sakai only sketches the proof IIRC and I recall struggling a bit to work out the details

Jakobian

@Thorgott the issue is that Sakai does it in the category of metrizable spaces, where I am asked to do it in category of $T_4$ spaces

there might be slight differences (like with Hanner where the proof seemed to translate well, but it didn't)

either way the proof I am reading now, from Hu, works for $T_4$ spaces

Thorgott

yeah I don't think everything translates to T4 spaces

Jakobian

the theorem about finite unions of open ANR's, and the Borsuk theorem translate to $T_4$ spaces it seems

I read some kind of history on this subject, and they recommended this book for general treatment of AR's and ANR's

maths.ed.ac.uk/~v1ranick/papers/mardesic.pdf

leslie townes

20:43

the biographical notes in there are pretty cool

> S. Lefschetz was born in Moscow and educated in Paris. After working for some years in industry, he turned to mathematics (following an industrial accident in which he lost both hands)

3

today i learned

think_meaning_buildß

:O

Thorgott

I did not know this either!

copper.hat

wow.

think_meaning_buildß

> R.H. Bing (1914-1986) was a student of the legendary topology teacher Robert Lee Moore (1882-1974) at the University of Texas in Austin. Bing obtained his Ph.D. in Austin in 1945. He did pioneering work concerning decomposition spaces and homeomorphisms in 3-dimensional manifolds [22]. The first systematic study of homology manifolds is due to another student of Moore, Raymond Louis Wilder (1896-1982) [247].

copper.hat

just gets wilder and wilder

Ben Steffan

20:55

Is a sheaf of rings the same thing as a commutative algebra object in the category of sheaves of abelian groups? If this is true, is a sheaf of modules over a sheaf of rings the same thing as a module over that algebra?

Thorgott

@Frusciante hmm, I'm not fully sure right now, but I think the difference becomes apparent when considering $K=\emptyset$, an $f$-initial object $x$ has a map $x\rightarrow y$ determined up to equivalence by its image $fx\rightarrow fy$, but the mere existence of a map $fx\rightarrow fy$ (a factorization over the empty diagram) does not imply anything in and of itself

Frusciante

21:26

@Thorgott right

I don't know, if thought about it quite a bit and I'm still convinced that by unraveling the equivalence in the definition, and by your answer, that this should be the right intuition

But in any case tomorrow I'll have to talk about such things in front of an audience, so I guess I'll find out lol

copper.hat

21:47

apparently stackexchange for "for non-technical questions and answers such as cooking, biology, and more". and all this time i thought i was being technical...

IThinkHighlyOfEiligh

How did they write with no hands?

leslie townes

copper: not with all those answers to PSQs about how to compute in polar coordinates, or whatever

Xander Henderson

22:02

@IThinkHighlyOfEiligh Certainly with some form of prosthesis. Not having hands does not prevent one from manipulating chalk or pen. It just makes it a bit harder.

copper.hat

22:23

@leslietownes i have finally reached my computational level. to wit, see my comment on math.stackexchange.com/questions/5003427/…

not suggesting an equivalent here, Pontryagin was blind at 14. Pontryagin's maximum principle is no walk with full eyesight, and that's just trying to follow the proof.

leslie townes

first he should pull UN demographic data on the average height and weight of the people in the country. that will determine the zoom. not the area, your phone can only see so much of that at one time.

copper.hat

:-)

leslie townes

he should ask about the areas of certain politically disputed nations on one of the more active politics SEs.

:)

copper.hat

i am very pleased at how restrained i was. no ffs, etc, no suspension, just a slight polite dig.

leslie townes

guys i keep trying to find the area of [country]. wikipedia says [area] but that's wrong, i think they're including [region] which is not part of that country. yes before you ask i am cross posting from math.SE.

Ben Steffan

22:35

maybe we should invite Quin to the chat room :^)

so copper can show them some applied mathematics

leslie townes

there were a few 'guess the country using x facts about it' web games that amply demonstrate why this is not a well formed question. some such games absolutely do use very rough indicators of size, like area (or some other measure of extent, maybe not land area, think archipelagos) and sometimes they get it wildly visually off because they are hitting a corner case of what their algebra says ought to work and what looks right.

putting aside all the difficulty of dealing with what devices people are on. that might be everything but a math question

copper.hat

it might hurt Quin to know that many of the devices he/she-it uses on a daily basis were designed/verified using software that i (and others, of course) wrote.

Sine of the Time

Area of the country=$\frac 12\int_{\partial \text{country}}(x\,dy-y\,dx)$

copper.hat

indeed, and what if they were to zoom on Great Ireland?

leslie townes

from experience playing those games it feels visually wrong to think in terms of normalized area. like, the mind expects a map of china to look bigger than a map of ecuador. it just does. it won't look right to scale them the same

Ben Steffan

22:40

they should collect data and train a neural network that spits out the right zoom factor

isn't that the modern way to do it

copper.hat

and the left zoom factor too

leslie townes

let's ask chatGPT what it thinks.

copper.hat

'cause they are far apart

i am training my crapGPT model as we speak

make of that what you will

didn't quite come out the way i intended, and neither did this sentence

Derso

Hi, guys

copper.hat

nice zoom

Derso

22:44

The geodesics in $\widetilde{SL(2,\mathbb{R})}$ I was talking about

Blue ones are the toocute geodesics @leslietownes

copper.hat

i thought it was a peacock playing baseball

ModularMindset

Why are there no arrow points on the back left and the back right and the bottom center?

The disk shouldn't be blue as well

psie

Let $\nu$ be a regular signed or complex measure (meaning that $|\nu|$ is regular, although I don't think this matters). Let $d\nu=d\lambda+f\,dm$ be its Lebesgue-Radon-Nikodym representation. It is then claimed that we have $d|\nu|=d|\lambda|+|f|\,dm$. Why is that true?

For signed measures I think I see why; if $\nu=\mu_1+\mu_2$, where $\mu_1,\mu_2$ are two signed measures, then $|\nu|=|\mu_1|+|\mu_2|$ by the Jordan decomposition. But what about complex measures? The definition there is a bit more...complex. If we can write $d\nu=f\,dm$, then $d|\nu|=|f|\,dm$. I don't know how to proceed.

Derso

@copper.hat Maybe it is, I think you can check from another angles geogebra.org/classic/cb5hbbzw

@ModularMindset Why shouldn't be blue?

copper.hat

@Derso pretty neat!

ModularMindset

22:50

@Derso On second thought I think it is okay

Derso

lol

I thought maybe Poincaré hated blue...

Or something

(because, that's his disk in the figure, ofc)

ModularMindset

this reminds me of something

copper.hat

@psie is that Folland?

psie

@copper.hat yes :)

First sentence (in the proof).

There are some details around this theorem that I think are irrelevant for what I'm asking.

Derso

@ModularMindset This really looks like $\widetilde{SL(2,\mathbb{R})}$! Where did you get that image from?

And looking from above...

psie

22:59

I simply wonder why $d\nu=d\lambda+f\,dm \implies d|\nu|=d|\lambda|+|f|\,dm$.

Derso

I haven't proved but it seems like blue geodesics go to two points in the boundary (one above and other below the Poincaré disk). The red one (lightlike separating geodesic), seems to be asympotic to the boundary

And green ones (timelike), their projections look like circles entirely contained in the disk

So maybe I could also call them hyperbolic, parabolic and elliptic geodesics @leslietownes

X4J

Is the notion of conjugation in group actions can be thought as ghg^{-1} describes h in g's language analogous to similar matrices? I mean it looks very much the same if we take G as a subgroup of Sn

copper.hat

@psie offhand i don't think that is true. i think it should be $\le$.

i take that back, $\lambda$ and $m$ are mutually singular

psie

23:19

@copper.hat $\lambda$ and $m$ are mutually singular. So any set can be partioned into subsets $E$ and $F$ such that $\lambda(E)=0$ and $m(F)=0$. On $E$, $d\nu=f\,dm$, so $d|\nu|=|f|\,dm$. On $F$, $d\nu=d\lambda$, but here I'm stuck. I don't think this implies $d|\nu|=d|\lambda|$, or does it?

well, maybe it does...

ModularMindset

@Derso I got that image from google images - keywords: AdS space, Poincaré patch, universal cover of special linear group

Derso

@ModularMindset Makes sense

ModularMindset

yeah so it's very similar to your diagram - perhaps the one I gave is more mathematical physics related

I have a question actually

copper.hat

23:37

@psie on $E$ we have $\nu = \mu$ (that is $\nu(A \cap E) = \mu A$. Now take $\sup$ over all disjoint measurable partitions.

note that the variation of $\nu$ can be split since they are singular into the the variation over $E$ and $F$.

in general with complex measures, the variation is a norm, so you just have the triangle inequality.

psie

ok, mutual singularity, makes life so much easier

ModularMindset

How does one construct a configuration of four 2-disks in $\Bbb R^3$ that yield coordinate projections, s.t. the boundaries of the disks map to 2 orthogonal quartic curves 🤔

four 2-disks centered at the origin that is.

assume the disks have unit radius.

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