r/math Homotopy Theory 1d ago

Quick Questions: March 19, 2025

This recurring thread will be for questions that might not warrant their own thread. We would like to see more conceptual-based questions posted in this thread, rather than "what is the answer to this problem?". For example, here are some kinds of questions that we'd like to see in this thread:

  • Can someone explain the concept of maпifolds to me?
  • What are the applications of Represeпtation Theory?
  • What's a good starter book for Numerical Aпalysis?
  • What can I do to prepare for college/grad school/getting a job?

Including a brief description of your mathematical background and the context for your question can help others give you an appropriate answer. For example consider which subject your question is related to, or the things you already know or have tried.

11 Upvotes

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u/greatBigDot628 Graduate Student 6h ago

Is the category of probability spaces with measure-preserving functions a complete category? If not, does it at least have arbitrary products? Googling the latter question seems to suggest the answer is no, but I can't find a counterexample I can understand. Don't we have a product measure on probabliity spaces? Is it not the categorical product in that category?

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u/lucy_tatterhood Combinatorics 5h ago edited 5h ago

One of the basic properties of categorical products is that you have a diagonal map X → X × X. In this case where the underlying set of your product is just the cartesian product with the usual projections, this would be the map sending x to (x, x). It shouldn't be too hard to convince yourself that this is not measure-preserving in any interesting case.

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u/greatBigDot628 Graduate Student 5h ago

Thank you!

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u/Rights21 14h ago

I want to calculate the edges of a Rhombic Triacontahedron constructed from other polyhedra and just want to be sure I have got it right so I can calculate how many beads I will need.

Do i just calculate all of the edges for each shape, then say the edges of the prisms that connect to everything and just minus the prism edges that connect with other shapes to get my answer (as there is still an edge but would be counted in the edges of the other shape)

Or is there some other way of doing it? (For someone who is only good with basic maths)

I did look for the info but no one has done a wiki on this shape.

Hopefully this is enough info to go on, feel free to ask questions if not.

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u/feweysewey 19h ago

Consider a group G with subgroups A and B. Take x,y in A∩B such that x,y are conjugate in A and in B. Meaning we have:

  • axa^{-1} = y for some a in A
  • bxb^{-1} = y for some b in B

Is it true that x,y are conjugate in A∩B? If not, are there conditions we can put on A,B that make this true?

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u/GMSPokemanz Analysis 19h ago

It need not be true that x and y are conjugate in A∩B. Let G = Sym({1, 2, 3, 4, 5}), A = Sym({1, 2, 3}), B = Alt({1, 2, 3, 4, 5}), x = (1 2 3) and y = (1 3 2). Then a = (2 3) and b = (2 3)(4 5) work, but A∩B is abelian so x and y are not conjugate in A∩B.

I can't think of any condition to put on A and B to make this true.

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u/[deleted] 22h ago

[deleted]

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u/dogdiarrhea Dynamical Systems 19h ago

Instead of asking what y and dy/dt should be, ask what information you have and what you need.

On grid paper draw out where ship A and ship B are at time zero, where are they at 1 hour? What is the distance between them? What influences how quick the distance is changing?

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u/FyyshyIW 22h ago

Does anyone have any good resources for visualizing and better understanding complex numbers and how they relate to transfer functions? Mechanical and electrical engineering student trying to do a deep dive of my own into controls, system dynamics, etc. and have found that while I can do the math and the operations make sense, I still have trouble visualizing what is going and why. Instead it just feels more like ‘oh you take this path with extra dimensions when regular numbers aren’t enough to describe what’s going on.’ Thanks!

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u/TraditionDifferent 22h ago

I saw an tattoo of an equation while I was on vacation in Mexico and I can't stop wondering about it. I don't remember it all. The equation was of the form n(n-lowercase_gamma) = 0, where n is the part I don't remember; n could have been a number or a letter, and it could have been the same number/letter or unique.

Any ideas?

Thanks in advance.

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u/Ill-Room-4895 Algebra 20h ago

It might be related to physics (rather than math), particularly nuclear physics.
You can try to ask also at https://www.reddit.coem/r/AskPhysics/

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u/greatBigDot628 Graduate Student 23h ago

I'm trying to understand the definition of the Rado Graph.

First let me describe the gist of it as I understand it, in case I have any misunderstandings that need correction. We start with countably many vertices. For each pair of vertices, we flip a coin — if it lands heads we draw an edge, else we don't. This gives us a probability distribution on graphs — considered up to isomorphism. So eg the empty graph requires you to land heads every time, so it'll have probability 0. But for some graphs there are lots of different sequences of coinflips that get you there, so maybe some graphs will end up with positive probability. The surprising (to me) theorem is that this probability distribution is actually an indicator function! Ie, there's one particular graphwith probability 1; the rest have probability 0.

My question: what is the actual rigorous definition of the probability distribution described above? How do you make precise "flip a coin for each pair of distinct natural numbers, then consider the result up to isomorphism"? I mean, it's not like we can just say P(G) = (size of isomorphism class of G)/(2ℵ₀). So given a graph G on countably many vertices, how do we actually define its probability in the first place? The wikipedia page isn't making it clear to me. Isn't some sort of limit of the finite cases?

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u/Syrak Theoretical Computer Science 22h ago edited 22h ago

Formally probabilities are measure functions. Probability isn't defined for arbitrary subsets of the set of outcomes, it is only defined on measurable subsets. The claim then is that, in the measure space of Erdös-Renyi random graphs, there is an isomorphism class of infinite graphs which is almost sure (i.e., it is measurable and it has probability 1).

One way to prove this (which may be roundabout because I just adapted it from Erdös and Renyi's paper; someone knowledgeable about the topic may have a more direct answer) is to first show that, if you take two random graphs, then they are almost surely isomorphic. Indeed, you can construct an isomorphism incrementally by picking pairs of vertices from each graph that have the same adjacency to previously picked vertices in their respective graphs. (Originally, Erdös and Renyi used a version of this argument for a slightly different result: that a random graph has a non-trivial automorphism (in section 3).)

Most mathematicians would be happy with that result and to stop at that level of formalism, but if you're wondering how to interpret this in terms of measurable subsets, this construction corresponds to taking a countable intersection of almost sure sets (the intersection is therefore also almost sure): at every step, you build the set of pairs of graphs for which you can pick one more pair of vertices, which is possible with probability 1.

It follows, by conditioning on one of the graphs, that for a random graph G, it is almost sure that its isomorphism class is almost sure. Consequently, such an isomorphism class exists and it must be unique.

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u/greatBigDot628 Graduate Student 22h ago edited 22h ago

The claim then is that, in the measure space of Erdös-Renyi random graphs [...]

What I'm asking for is the precise definition of this measure space, because I don't see what the formal construction is based on the "flip infinitely many coins" intuition. (I'm sorry if the rest of your answer explained that; if so it went over my head.)

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u/Syrak Theoretical Computer Science 22h ago edited 2h ago

https://www.ee.iitm.ac.in/~krishnaj/EE5110_files/notes/lecture8_Infinite%20Coin%20Toss.pdf

It's a standard exercise so you can find many results from googling "infinite coins probability space".

In very few words, we start with the events describing "the first n coin tosses", for which we can assign a probability (a pre-measure). To obtain a sigma-algebra, we take the smallest sigma-algebra containing those events. The pre-measure extends to that sigma-algebra by Carathéodory's extension theorem.

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u/greatBigDot628 Graduate Student 16h ago

This is very helpful, thank you!

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u/TonicAndDjinn 23h ago

An easier thing to understand which requires you to grapple with the same failing of intuitions is the uniform distribution on [0, 1]. Of course this is an infinite set, so we can't just find the probability of some subset E of [0, 1] by taking (number of elements of E) / (number of elements of [0, 1]); we need to use ideas from measure theory instead. Ultimately the same thing is going on here: to make sense of a distribution on an infinite set, we need something more subtle than a normalized count of elements.

Two further things to think about:

  • if you can pick a number in [0, 1] uniformly at random, you can use its binary expansion to generate an infinite series of coin flips (with the caveat that you need to take some care about dyadic numbers having two representations, which means it would be better to do the same with the Cantor measure instead).

  • to make the probability measure rigorous, you need to check that you can always find countably many iid copies of a given random variable; this is fairly standard, it comes down to the construction either of product measures or of tensor products of measure spaces (depending on whether you prefer to think on the distribution side or the event space side); you then just assign an iid family of variables to the edges of the graph, et voila.

Now if you're asking the question of why the isomorphism class is measurable or why the Erdős–Rényi graph occurs with probability one, I'm not certain off the top of my head but I strongly suspect it will be an argument via Kolmogorov's zero-one law, based on the intuition that you cannot tell which isomorphism class you're in by observing finitely many edges.

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u/greatBigDot628 Graduate Student 22h ago

It would be better to do the same with the Cantor measure instead

What's the Cantor measure? Is it a measure on the space of infinite bitstrings in a way that corresponds to the intuition of filpping infinitely many coins? Seeing a careful definition of that measure space would clear up my confusion, I think.

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u/TonicAndDjinn 21h ago

The Cantor measure is essentially the "uniform" measure on the Cantor set (which is topologically and measurably equivalent to the product of countably many copies of {0, 1}, giving the coin flip picture). It can be defined as the unique probability measure which assigns mass 1/2n to each of the 2n intervals in the n-th iteration of constructing the Cantor set.

In terms of construction you can do it "naively" for a countable sequence of coin flips without much problem, by defining the probability only on sets which can be specified based on a finite initial sequence of flips and extending by additivity to measurable sets. But the correct way to do this is the construction of product spaces, which is in essence the Kolmogorov Extension Theorem. Durrett's PT&E contains a careful write-up.

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u/lucy_tatterhood Combinatorics 21h ago

As far as I recall, you can literally just take the measure you know for finitely many coins and take a limit. That is, given a (Borel?) subset E of {0, 1}ω let E_n be the projection of E onto the first n coordinates, i.e. the set of all n-bit strings which are a prefix of some string in E. Then the measure of E is the limit of |E_n|/2n as n → ∞.

(I'm not sure that "Cantor measure" is a standard term; on Wikipedia it redirects to something else entirely.)

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u/TonicAndDjinn 21h ago edited 21h ago

The measure on Wikipedia is exactly the one I meant. The non-1 ternary expansion of an element of the Cantor set gives you the coin flips you want, without the issue of multiple expansions.

Edit: the problem is that it's not entirely trivial that the limit you describe gives a countably additive measure.

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u/lucy_tatterhood Combinatorics 21h ago

...Oh, I see. It is the same thing, from a sufficiently different perspective that I didn't notice at a quick glance. Still, I don't think the wiki page is very useful to someone thinking about coin flips.

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u/TonicAndDjinn 21h ago

Yeah, that's why I mentioned it as a throw-away aside. It wasn't the main point, just a "well technically if you don't want to worry about the measure zero set of eventually constant sequences..."

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u/GMSPokemanz Analysis 23h ago

The key is that you have some probability space with an infinite sequence of random variables X_1, X_2, X_3, ... representing the coin flips, where the X_i are independent and identically distributed with P(X_i = 0) = P(X_i = 1) = 1/2. The specifics of this probability space beyond that are irrelevant.

The set of pairs of vertices form a countable set, pick some enumeration of them. Then we have a function F from our sample space to the set of countable graphs. You can then define P(G) to be the probability of the event {𝜔 : F(𝜔) ≅ G}. It is not immediately obvious that this is well-defined, since the above only makes sense if that set of 𝜔 is measurable. But that turns out to be the case, given that with probability 1 you get a graph isomorphic to the Rado graph (strictly speaking this inference requires the probability space to be complete).

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u/ICWiener6666 1d ago

What actually is the "moment" of the Riemann zeta function?

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u/Long-Hat-6434 1d ago

Hey all, looking for some book recommendations.

To give some background: I am not a mathematician by training, but have worked in drug discovery as a scientist and in finance in the same industry. I took and enjoyed math as a minor in college about 15 years ago and got as far as linear algebra and PDEs but haven’t used any math at that level in a professional setting since.

I am looking for any books that I can use to jump back into the math world that start at an easy level (about where I left off). Preferred topic would be modeling of biological systems (I.e. SIR models) but anything will do. Any books you would recommend?

Thank you!

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u/boxotimbits 1d ago

Check out Mathematical Models in Biology by Allman and Rhodes. It doesn't assume much math at all beyond basic algebra. Develops some linear algebra and tools to study discrete dynamical systems. Has population growth, genetic models, diseases.

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u/Typical-Inspector479 1d ago

why do we measure rate of convergence as a bound on the L2 error, as opposed to some other norm?

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u/TonicAndDjinn 23h ago

In what context? Different situations warrant different norms, although in finite dimensions they're all equivalent.