Mathematics - 2025-01-18

Thorgott

00:28

@BenSteffan wow, he's just like me

Ben Steffan

the joke goes that up until a few years ago he didn't know what the derivative of $\sin$ was

which I find very relatable

in fact from the top of my head I only know it up to a sign

Thorgott

00:57

ok I do think everybody should know that

Ben Steffan

If I imagine a circle I can figure it out

Thorgott

yeah, $e^{ix}=\cos(x)+i\sin(x)$ is how I think about all trigonometry

it's the only thing worth remembering, everything else can be rederived on the spot

Ben Steffan

fair

...and there's more power series for you

I really need to sit down with a proof of the algebraic quillen theorem

Ben Steffan

01:52

the proof is not very enlightening :(

Thorgott

02:07

which Quillen theorem is that

Ben Steffan

one of them

neither A nor B

:^)

it says that $\pi_* \mathrm{MU} \cong L$, where $L$ is Lazard's ring

equivalently that $\pi_* \mathrm{MU}$ classifies (1-dimensional, commutative) formal group laws

I would be very surprised if you knew about it

together with the topological Quillen theorem, which says that $\mathrm{MU}$ classifies complex orientations, it forms the foundation on which chromatic homotopy theory is built

the question is why/how the formal group laws get into the complex bordism, so to speak

Shaun

06:29

@OceansBleed Well, their internal logics are emergent phenomena, built out of classical logic. A truth value is a subobject of the terminal object, as it is for the category SET, which is the motivating example of a topos. See Goldblatt's "Topoi". They make me question the universality of the classical laws of logic.

I, too, am quite busy.

Colab Google

10:28

Ann and Ben are playing a game consisting of multiple rounds. The first person to reach $10$ points wins the game. Each round has only one winner. Assume the rounds are independent, and the probability of Ann winning each round is $0.6$.

a) Calculate the probability that Ann wins the game if the winner of each round earns $2$ points.

b) As in (a), but if the winner also won the previous round, they earn an additional $1$ point.

c) Calculate the expected number of rounds needed for Ann to win in scenarios (a) and (b).

Vladimir Lysikov

10:39

@ColabGoogle Do you know about Markov chains?

Colab Google

@VladimirLysikov yes, I had studied about it.

Vladimir Lysikov

You can model this as a markov chain
one round is a transition
the state contains current score and the winner of the last round
and two special states "Ann wins" and "Ben wins"

And then you can find the limiting distribution

Colab Google

@VladimirLysikov thanks for your idea of part (b), can you assist me to solve part (c) too?

Martin Brandenburg

13:43

Is the interval category 0 < 1 algebraic? I think so, empty signature, one axiom x = y. Can someone confirm?

Thorgott

@BenSteffan I've heard of this, but really only heard (almost attended a seminar on chromatic stuff last summer, but only almost)

@MartinBrandenburg which definition of algebraic are you using?

Martin Brandenburg

category of algebras for a finitary algebraic theory

I ask because this category doesn't feel algebraic at all to me - but there is a basic proof that it is :D

Ben Steffan

14:02

@Thorgott let it draw you in :)

Thorgott

I'm not familiar with the universal-algebra language. What should the category presenting the theory whose category of algebras is the interval be?

@BenSteffan all roads lead to Rome eventually

Martin Brandenburg

yes another def is to have a category with finite products P generated by a single object x. Then the associated algebraic category is the category of functors P ---> Set preserving finite products.

if A : P -> Set is such a functor, A(x) is the underlying set of the algebra, and morphisms x^n ---> x induce maps A(x)^n ---> A(x), the operations.

so in our case P will have objects 1,x,x^2,... but morphisms are a bit more boring ...

in particular, the two projections x^2 -> x will actually be equal

ah yes so P just has the objects 1,x, and the product of x with x is x. the projections x -> x should both be the identity.

then an algebra A : P ---> Set is a set U such that the two projections U^2 -> U are equal

(equivalently, the diagonal U ---> U^2 is an isomorphism)

so we just have the terminal and the initial set. giving the interval category 0 < 1.

right?

ok maybe a less "trivial" algebraic theory is the following one. one binary operation b. the axioms are: b(x,y) = x, b(x,y) = y.

Thorgott

14:19

ah, so your P is equivalent to the opposite of the simplex category

Martin Brandenburg

I think the simplex category has more morphisms

Thorgott

err, I meant interval

Martin Brandenburg

ah, yes

Thorgott

yeah ok, that sounds correct to me too

Martin Brandenburg

cool, thanks

the category of measurable spaces is probably not algebraic

also wondering how to prove that

Shaun

14:30

Hi :) There's a very real possibility I won't prove enough theorems in my PhD. I mean: I might leave with another MPhil. This is my second attempt at a PhD (mostly because of the pandemic). If needs be, I'll try a fourth or fifth time. It doesn't do my self-esteem any good though. I Love mathematics and I find research an extremely meaningful part of it. I have until December of supervision to get my act together. Then I'll have a write-up extension.

It's stressful as Hell.

Thorgott

@MartinBrandenburg do you know if it is exact?

Shaun

In all, my research in this project so far fits into 16 sides of A4.

If I explain the background material, that'll flesh things out a little more.

Xander Henderson

"Enough theorems" seems like a weird metric...

But 16 pages is pretty slim, even by the very terse standards of mathematics.

Though if you are using LaTeX defaults, it is single spaced, and most places in the US require double spacing, so maybe it is actually closer to 32 pages? Which, with another 10-15 pages of background, is getting into the typical length range...

Colab Google

15:38

Ann and Ben are playing a game consisting of multiple rounds. The first person to reach $10$ points wins the game. Each round has only one winner. Assume the rounds are independent, and the probability of Ann winning each round is $0.6$.

a) Calculate the probability that Ann wins the game if the winner of each round earns $2$ points.

b) As in (a), but if the winner also won the previous round, they earn an additional $1$ point.

c) Calculate the expected number of rounds needed for Ann to win in scenarios (a) and (b).

Xander Henderson

As for me, being overly verbose, my thesis clocked in at 122 pages (the .pdf is 136 pages, but I won't count the administrative frontmatter).

@ColabGoogle I feel like this was asked-and-answered on the main site yesterday...

Indeed: math.stackexchange.com/q/5024228 .

Is there something about the answer you find unsatisfactory? Can you narrow your question?

Sine of the Time

You can also ask for clarification in the comment section of the answer

Shaun

15:52

There's a lot I need to explain to put the theorems in context. For instance, I need to explain rational canonical form of matrices, which would take up a lot of space. So there's that.

I find myself reading a tonne of motivational material lately. I read two inspirational autobiographies over the winter break. I need them. The maths is often enough, but I'm only human.

I'll check the specifications for my institution regarding the spacing of text. That's s good idea. Thanks :)

I counted the front matter in those 16 pages.

I spent a lot of time coding for this project.

The data I gathered that way suggests a conjecture I've been working towards.

It's that a certain function takes on a particular value almost all the time, with the exceptions being finite (enough to count on one hand).

I have a lower bound two thirds of the time.

I have a technique that might work as an upper bound.

Xander Henderson

For a phd thesis, you are expected to do something novel. Beyond that, all the metric you point out seem to kind of miss the mark. Page counts, theorem counts, the amount of work you have done, etc---irrelevant.

Honestly, in a lot of ways, I think that my masters thesis is better than my PhD thesis. I think that the result is much more novel and interesting. Though another author posted a very similar result to the arXiv as I was scheduling my defense, so it was just time for the result, I suppose.

(The other author used different techniques. We later collaborated on a paper that ended up in Advances, so I'm happy.)

My PhD thesis, on the other hand, is pretty incremental. I think it is useful mostly for a new idea, i.e. that of "local complex dimensions". I suspect that there is a way of using local complex dimensions to to extract local spectral information which can then be stitched together into global spectral information. But I didn't get that far in the thesis, and have had all of five minutes to think about it in the last five years.

Shaun

If I do get another MPhil, then I'll definitely try again if I could. But who'd take me on board?

My contribution to MSE could be something I could leverage.

I have fears that the viva would end in major corrections. The corrections: "produce more research". A PhD is about a S.O.C.K.: a significant, original contribution to knowledge. It's the significance I need.

How would I fund a third attempt at a PhD? Would anyone take me seriously?

Xander Henderson

16:09

"Significance" is relative. You need to convince your advisor and committee that your result is "good enough". And keep in mind that a PhD is not an end---by the end of a PhD program, you should have demonstrated that you are capable of productively engaging in the research process as a semi-independent researcher.

By the end of a postdoc, you should be able to show that you really are an independent researcher, and that you can run your own research program.

@Shaun Depends on where you are. My recollection is that you are somewhere in the UK, and I know very little about the system there (though, fun fact, I was accepted into the PhD program at University of Warwick, but was not offered enough money to support myself and my family, so ended up declining).

Shaun

My end goal is: Professorship. That's by the UK standards.

@XanderHenderson Yeah, I'm in Scotland; Aberdeen. I had an offer to study at St Andrews back in 2015 but I had a breakdown and ended up spending 18 months in a mental institution.

Xander Henderson

@Shaun That is a pretty lofty goal. You either need a banger of a thesis, or one or two pretty good postdocs. The job market is brutal right now.

@Shaun Oh, my coauthor is at St Andrews! (well, one of them, anyway)

I really want to get out there at some point...

Shaun

Why aim low?

Xander Henderson

@Shaun Sure. Just make sure that you have a plan B. and C.

Ben Steffan

@Thorgott I've decided to basically give up on the torsion question, so I wrote up my progress and put a bounty on it instead math.stackexchange.com/questions/5022965/…

Shaun

16:17

Plan B is a lectureship. Plan C is, if I can't even get a postdoc, to work for a science based company doing mathematics, and studying for the odd degree part-time. If I'm not challenged intellectually, I don't see myself being content, let alone happy.

Ben Steffan

I think I've pretty much exhausted my toolkit

Shaun

@XanderHenderson It's a beautiful place :)

Xander Henderson

@Shaun Yeah, and they keep all the good whisky there.

Shaun

Often, it gets voted as the most beautiful beach in the UK.

Thorgott

@BenSteffan yeah, just saw the edit, it's still nice progress, but the Prüfer groups seem hard...

one (probably stupid) detail I'm getting caught up again: what do you mean when you conclude that $H_3(M)=0$ "by Hurewicz"?

Ben Steffan

16:27

@Thorgott there's a stronger form of the hurewicz theorem that's not so well known: If $M$ is $n$-connected, $n \geq 1$, then the Hurewicz map $\pi_* M \to H_*(M)$ is an iso. in degree $n + 1$ and surjective in degree $n + 2$

Thorgott

oh of course

I did too much category theory the last year, my alg top is getting rusty

Xander Henderson

@Thorgott So you spent in excess of 30 seconds thinking about category theory last year? FOR SHAME!

Thorgott

this is the trick where you build a $K(G,n)$ from an $M(G,n)$ by attaching cells in dimension $\ge n+2$

Ben Steffan

@Thorgott yeah

you can also prove it using the Serre specseq if you know the weaker version I think

much less elegant but a good exercise

hm, I wonder if you can also prove it for the map of spectra directly

Thorgott

wait

ok either I'm missing something stupid or I have an answer (probably the former)

Ben Steffan

16:37

you have my attention

actually you don't even need the weaker version as input I think. I've forgotten how that proof works in detail

well you'd need the $n = 1$ case as input

Thorgott

ok nvm, I did not have something

just observed that if we fix a $3$-dimensional CW-model without $1$-cells, its cellular chain complex presents a length $1$ free resolution of $G$ in the degrees $2$ and $3$

Ben Steffan

maybe it's possible to do something with that

but then why can't the map be injective...

Thorgott

hmm, does it imply our $K(G,2)$ is the suspension of a $K(G,1)$?

Ben Steffan

not sure, but doesn't it imply that $H_*(K(G, 2)) \cong H_*(G, \mathbb{Z})$ where the rhs is group homology?

which I guess is the same thing in green

Thorgott

16:53

you can attach $1$- and $2$-cells to a point according to the same presentation to get the $2$-skeleton of a $K(G,1)$

so $K(G,2)$ is the suspension of the $2$-skeleton of a $K(G,1)$

this implies $K(G,2)$ is the $3$-skeleton of $\Sigma K(G,1)$, in particular they have the same $\pi_2$, but that's actually always true rather than a contradiction

Ben Steffan

yeah, I don't immediately see how this helps

Thorgott

also, by the way, can you sanity check me on whether such a CW-model exists? obviously, there is a $3$-dimensional one since $M$ is an open $4$-manifold and there is one with trivial $1$-skeleton cause $M$ is simply connected, but can both be achieved at once?

oh wait, we're simply connected, so this follow by general minimal cell structure stuff

nevermind, the version I recall from Hatcher only applies to f.g. groups

Ben Steffan

yeah

I have no idea, to be honest

oh this is false, no?

if there are no 1-cells than $M$ cannot be locally finite

and then you have no chance of being a manifold

what with infinitely many 2-cells now all attached to a single 0-cell

Xander Henderson

Oh, sh*t... David Lynch died. :(

Ben Steffan

@XanderHenderson yeah :(

Xander Henderson

17:04

Man... that makes an already annoying day worse.

(It looks like I am going to have to drop a significant percentage of Kelly BlueBook on my car today in order to keep it running... almost tempted to just buy a new car, but I don't want another bill to pay each month.)

Thorgott

@BenSteffan well, these models are all only up to equivalence, as far as I'm concerned

here's a different idea: analyze the Serre SS for $\Omega M\rightarrow\ast\rightarrow M$. since $M$ is a $M(G,2)$, the second page lives only in the $0$-th and $2$-nd column and we converge to something acyclic, so this yields isomorphisms $H_{n+1}(\Omega M)=H_2(M;H_n(\Omega M))=G\otimes H_n(\Omega M)$, but $\Omega M$ is a $K(G,1)$

surely the group homology of $\mathbb{Z}[p^{\infty}]$ is computable?

and then this might yield a contradiction (or a tautology)?

Ben Steffan

I don't quite follow. We know that $H_*(K(\mathbb{Z}[p^\infty], 2))$ is concentrated in degree 2, by the answer you linked me a while ago.

You cannot get a contradiction using co/homology alone.

Thorgott

I suppose that means the second option (tautology)

Ben Steffan

yeah

Thorgott

which means the group homology of $\mathbb{Z}[p^{\infty}]$ is the free tensor algebra on $\mathbb{Z}[p^{\infty}]$??

Ben Steffan

17:16

I, uh, guess?

$H^*(\mathbb{Z}[p^\infty], \mathbb{F}_p)$ is polynomial if I read that answer correctly

Actually $\mathbb{Z}[p^\infty] \otimes \mathbb{Z}[p^\infty] = 0$

Thorgott

oh yeah...

what an annoying problem

Ben Steffan

yeah :))

part of me is waiting for someone to come around and tell me that there is an example

well I'll let it sit until the bounty runs out and if I get now answer I'll take it to MO

hbghlyj

I have a question about asymptotics: Within $(\pi-\varepsilon,\pi+\varepsilon)$, what is the growth rate of the denominator of the rational number with the smallest denominator?
For $\varepsilon=\frac1{700}$, we get $\dfrac{22}7$, with a denominator of $7$
For $\varepsilon=\frac1{10^6}$, we get $\displaystyle \frac{355}{113}$, with a denominator of $113$
For $\varepsilon=\frac1{10^{10}}$, we get $\displaystyle \frac{312689}{99532}$, with a denominator of $99532$

pie

18:00

which one do you recommend more for first complex analysis course? " Marsden and Hoffman, Basic Complex Analysis, Freeman" or "Complex Analysis
Book by Lars Ahlfors" or "Complex Analysis
Elias M. Stein and Rami Shakarchi"?

leslie townes

20:16

pie for self study i would maybe de-recommend ahlfors, it is written in an older style that a lot of people used to more modern books will struggle with, particularly without an instructor to guide them. other than that i think the most important thing would be just picking one book (either of those, or some other) and sticking to it as opposed to mixing and matching the treatments of different books.

complex analysis maybe more than other subjects is something where a lot of the fundamental material that an intro course would develop, can be brought out in many different inequivalent logical orderings, to the point of the definitions in one book might be theorems in another, and without previous background in the subject a self studier is likely to confuse themselves if they work with a pile of books that are putting things in a different logical framework and order

this is a general hazard of self study with math, that we are basically always dealing with in here and on main, but i think it is particularly acute in complex analysis

pie

20:29

How can I effectively address the topics I've rushed through? For instance, I've never studied multivariable calculus thoroughly; I've rushed through it multiple times due to time constraints or impatience. This has significantly impacted my understanding of certain areas in complex analysis, particularly sections involving implicit differentiation. I struggle because I neither remember the rules nor studied them properly in the past.

To tackle this, I tried revisiting these concepts through Rudin's book, focusing on Chapter 9. However, that chapter heavily relies on earlier sections, whic…

That is one of the reasons that made me reread rudin multiple times, before I can go to complex I think I need ton finish this part that I never did, but how?n should I pick a multi variable analysis book? read rudin chapter 9 and 10? or what do you think I should do?

leslie townes

i dunno, maybe find a book that doesn't assume a lot of multivariable calculus? i don't actually think a lot of them do? anyway i don't recall that stein and shakarchi or marsden do. you can do a fine intro to subject using only tools you would have met in single variable real analysis and tools you develop on the way

C is just R^2, there are only two directions :)

the fact that you can write slick proofs of things or understand how it generalizes better if you know more shouldn't stop you from just diving in

it seems less daunting if you just pick one book and stick with it. the book itself will give you most of what you need to know, or very specific indications of what they are using (not at the level of "go reread rudin," but "go understand why a limit like this will exist under the hypothesis that f is continuous at c")

i might again de-recommend ahlfors on this basis, when authors write in an older and more classical style, as he does, they fairly commonly make reference to all sorts of outside things for the "benefit" of the reader who might recognize them, but in a way that might not be fully logically supported or referenced in the text

which is like quicksand for a lot of self studiers

i don't know if you are having this problem, but the common self study thing you see on main, and here, is people will be reading one book, get confused at some minor technical or definitional point, go to wikipedia, page down 6 times in a long exhaustive discussion of something elementary, and 5 clicks through wikipedia later they've convinced themselves they need to understand complex differential forms before even touching the book they were reading again

its less daunting and more helpful to just fix one single logical path through the subject, it basically doesn't matter which one, and then see how much of that path you can understand. books are actually written expecting readers to do this (or at least were until maybe 10 years ago)

the other self study twilight zone is "if i can't understand this one minor point i cannot read a single page more of the book, i have to go back to page 1" which is usually wrong. one of the most useful skills that anyone can develop is being able to cabin off something they don't yet understand in a box, not to be able to "prove it rigorously," but just to recognize where it's being used and where it isn't in the future, and then trying to see what they can do with that thing as a black box

but the short answer to that is marsden or stein (but not both), and not ahlfors :)

pie

@leslietownes ِ "i don't know if you are having this problem," yeah it happened a lot..

leslie townes

20:45

it's often basically what causes people to show up here :)

pie

@leslietownes BTW, MSE is an absolute treasure! It has helped me immensely with so many things. I never would have expected there to be a place where math enthusiasts can ask and answer questions like this. I'm truly grateful to all of you.

leslie townes

21:05

it has both good days and bad days :D

pie

@leslietownes why?

leslie townes

well, some days are worse than others for the site being flooded with low effort questions, AI/useless/point-chasing answers, etc. if by "why" you mean, do i know what makes that worse on some days than others, i dunno

psie

21:44

I'm reading a chapter on the change of variables formula. It has not been established yet that Lebesgue measure is invariant under rotations in $\mathbb R^d$, but it has been proved that Lebesgue measure is invariant under translations (and that the pushforward of Lebesgue measure under dilations, i.e. $x\mapsto ax$ where $a\in\mathbb R\setminus\{0\}$, is Lebesgue measure times $|a|^{-d}$). I'm wondering, is it possible to show a hyperplane has Lebesgue measure $0$ with these given facts?

If the hyperplane is parallel to one of the coordinate planes (i.e. the $xy$-plane or the $yz$-plane), then it is simply a translation of that plane. And the coordinate plane has Lebesgue measure $0$ since it is a Cartesian product with one set being equal to $\{0\}$, so by the definition of a product measure (and $\{0\}$ having Lebesgue measure $0$), the measure is $0$. But how can I reason that the plane $y=x$ in $\mathbb R^3$ has Lebesgue measure $0$, for instance?

Ben Steffan

You can probably show that directly from the definition

Xander Henderson

@BenSteffan I was just typing the same thing!

Ben Steffan

but observe that your scaling formula implies that the measure of a linear subspace must be either 0 or $\infty$, so

you can get away with just showing it must be finite... so find a sequence of cubes, etc.

@XanderHenderson beatcha :)

Xander Henderson

@BenSteffan Congrats.

Ben Steffan

thanks

hbghlyj

21:52

Given a differentiable mapping $f:\mathbb{R}^2\to\mathbb{R}^2$, how to find all points such that $f$ maps all local angles $\alpha$ at that point to an angle of $2\alpha$? For example, $f(x,y)=(x^2-y^2,2xy)$ satisfies this condition at $(0,0)$, but how to write down an equation for this condition in general? I think this is related to the Jacobian matrix of $f$, but I am not sure how to proceed.

hbghlyj

22:07

Let $\gamma_1$ be a line parallel to $x$ axis and $\gamma_2$ be a line parallel to $y$ axis. They intersect at a point $p$. Then, the angle between $\gamma_1$ and $\gamma_2$ is $\frac{\pi}{2}$. Let $f$ be a differentiable mapping. Then, $f$ maps $\gamma_1$ to a curve $f\circ\gamma_1$ and $\gamma_2$ to a curve $f\circ\gamma_2$. They intersect at a point $f(p)$. If $f$ satisfies the condition at $p$, then the angle between $f\circ\gamma_1$ and $f\circ\gamma_2$ is $2\alpha=\pi$.

But $(f\circ\gamma_1)'$ and $(f\circ\gamma_2)'$ are $\frac{\partial f}{\partial x}$ and $\frac{\partial f}{\partial y}$, respectively. So, the condition implies $\frac{\partial f}{\partial x}=-k\frac{\partial f}{\partial y}$ for some $k\in\mathbb{R}^+$. So $p$ is a critical point of $f$.

psie

ah yes, if Lebesgue measure $\lambda_d$ in $\mathbb R^d$ is just $\lambda_d=\lambda\otimes\cdots\otimes\lambda$, where $\lambda$ is $1$-dimensional Lebesgue measure, then indeed $\mathbb R^{d-1}\times\{0\}$ has $\lambda_{d}$ measure $0$. Thanks!

Ben Steffan

but you knew this already, and the question was whether you can apply this to hyperplanes not of this form?

psie

hmm, right...

@BenSteffan but what did you mean by showing this from definition then?

Ben Steffan

well evidently the text you're working off is fancy

psie

yes :)

Ben Steffan

22:16

I mean that you take the definition of the lebesgue measure as infimum over a sum of volumes of cubes covering your set

surely you can also do it from the product definition

psie

ok 👍

Ben Steffan

but then why do you care, if you're going to prove the transformation formula anyways

psie

well, the section starts with the following proposition, and before it is proved, there's this remark which led to me ask my question in the very beginning

> Proposition: Let $b\in\mathbb R^d$ and let $M$ be a $d\times d$ invertible matrix with real coefficients. Define a function $f:\mathbb R^d\to\mathbb R^d$ by $f(x)=Mx+b$. Then, for every Borel subset $A$ of $\mathbb R^d$, $$\lambda_d(f(A))=|\det(M)|\lambda_d(A),$$where $\det(M)$ is the determinant of $M$.

> Remark: If $M$ is not invertible, $f(A)\subset f(\mathbb R^d)$ is contained in a hyperplane, which has zero Lebesgue measure.

Ben Steffan

well the proposition clearly implies the remark

psie

indeed, it does, but the remark comes before the proof :)

Ben Steffan

22:23

that doesn't have to mean much

putting a remark between statement and proof is not uncommon

psie

yeah, true

hbghlyj

I have a calculus question:
Given a differentiable mapping $f:\mathbb{R}^2\to\mathbb{R}^2$, if $f$ maps all local angles $\alpha$ at a point $p$ to an angle of $k\alpha$ for some $k\ne\pm1$, then the Jacobian matrix of $f$ at $p$ is $\begin{pmatrix}0&0\\0&0\end{pmatrix}$. Intuitively, I think that no linear mapping can satisfy this condition except the zero mapping, so the Jacobian matrix must be zero.

also $k\ne0$

Shaun

23:04

If $G$ is a one-headed group, then is $G\rtimes\Bbb Z_2$ also one-headed?

I believe so.

The unique maximal normal subgroup would be $G\times\{1\}$, right?

I'll ask on the main site.

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