I'm studying first order logic, we subset X of the universe is definable if there is some formula φ(v) such that X={x | φ(x)}. We say X is invariant under automorphism if for any automorphism h, h(X)=X.

Definable always implies invariant under automorphism, but I'm wondering about the converse when the universe is finite. I have heard that this is true but I'm having trouble proving it and I can't find anything online about it.

I'm Elliot Glazer, Lead Mathematician of the AI research group Epoch AI. We are working in collaboration with a team of 70+ (and counting!) mathematicians to develop FrontierMath, a benchmark to test AI systems on their ability to solve math problems ranging from undergraduate to research level.

I'm also a regular commenter on this subreddit (under an anonymous account, of course) and know there are many strong mathematicians in this community. If you are eager to prove that human mathematical capabilities still far exceed that of the machines, you can submit a problem on our website!

I'd like to hear your thoughts or concerns on the role and trajectory of AI in the world of mathematics, and would be happy to share my own. AMA!

Relevant links:

FrontierMath website: https://epoch.ai/frontiermath/

Problem submission form: https://epoch.ai/math-problems/submit-problem

Our arXiv announcement paper: https://arxiv.org/abs/2411.04872

Blog post detailing our interviews with famous mathematicians such as Terry Tao and Timothy Gowers: https://epoch.ai/blog/ai-and-math-interviews

Hey! I'm not a mathematician, although I enjoy the subject, and isn't completely inadept. But what I don't know is what resources there are on the internet for easier calculations (I just use my pen, paper and graphing calculator).

I recently started looking over my stocks and funds played around with different exponential functions to see some sort of prognosis of my savings. I just did it by hand with pen and paper and a graphing calculator but it became tedious, especially when I wanted to do alterations of my functions. Is there some app or software that is kind of bare-bones that can run, and save mathematical functions (like f(x))? Derivation and integration would be nice. Otherwise I might have found my first programming project? Any tips would be helpful.

We learn early on that just because a sum converges to zero doesn’t mean its infinite series converges. Case in point, the harmonic series a_n = 1/n. But how often does this actually happen? Shortly after learning this fact, we learn that the infinite series for a_n = 1/n^p converges for all p > 1. So it would appear that for “most” series, if its terms converge to zero then so do the partial sums. Is there some other condition we can impose on a sequence that allows us to say that its partial sums converge when the terms converge to zero?

I'm revisiting topics I learnt in my undergrad and so I'm looking for an overview of it in a concise manner. Say if I want to revisit Real Analysis, I want to see all the important results in one place. Is there some resource you know which does this? Been trying to find something like this for many days but I keep ending up on videos covering the absolute basics.

Let me clarify the question. When you practice art, you get to see your craft get better. When you practice video games, you get to pull off plays you couldn't before.

Let's say I practice and understand set theory, what can I do with it to feel good?

I feel like I'm only practicing maths but not putting it to use, which chips away at my motivation.

I'm sorry if this sounds stupid, I genuinely want to understand what people find in maths, this is curiousity not an angsty teen hating on math.

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.

Matrices can have any number of rows and any number of columns, and how to multiply them is known, if they are compatible for multiplication. But they are still a 2-dimensional array of numbers. The natural question to ask is what if you have higher dimensional arrangement of numbers. Can they be multiplied?
For example extending this to 3-dimensions the numbers would appear in a 3-dimensional arrangement(probably shouldn’t call it an array since that suggests 2-dimensions.) For 3-dimensions, you can interpret as a sequence of matrices extending vertically in the z-directions. Higher dimensions than 3 though would not come so easily for a visual interpretation.
Since tensors have a multiplicity of indices the natural inclination is to think the multiplication can be interpreted as tensor multiplication. Is this valid?

So this "problem" has been on my mind and I feel like the concept probably has been explored, but I am having trouble finding literature on it or the right term to google.

Say you have a set made up of elements with a known order (e.g. 10 integers).

And say that we can define a known order for this set, for example lets call the "base" circular permutation is is just N mod 9 in order so 0->1...->8

We know there are (N-1)! unique circular permutations. Some of them will be "similar" to the base circular permutation. For example, (0,1,2,6,7,8,3,4,5) would be largely "similar" to the base permutation, but has two points where it jumps to other part of the the base cycle. On the other hand, (1, 8, 3, 6, 4, 7, 2, 0, 5) seems completely random form the base permutation.

Is there a known / standard way to quantify / sort all the permutations, based on how similar they are to known ordered "base" circular permutation?

I am at a bit of an impasse. My interest in math bloomed a bit later than it does for most kids, and while I consider that both a good and a bad thing, it has left me in a bit of a pickle.

I am in my Senior year of high school, and became "very good" at math only in my junior year. Now my interests have grown exponentially, and quite beyond the school curriculum.

I have done a lot of self study, currently doing differentials (I'm trying to get a good math basis for quantum physics which is very diff-e and vector heavy). I am considered very advanced by my peers and teachers, but I don't feel like it, and I think I know the reason why.

Because I've been trying to "catch up" to the truly talented kids for the past 6 months, I feel like I have a good grasp on the advanced concepts I've learned myself but lack the rigorous ability to do the basic stuff.

What I mean by this is, when I do the algebra portion of a problem, like let's say an integral was already set up, all the calculus was done, and now it's only 10th grade algebra that's left, there is a 50/50 chance I'll do it right the first time and then another 50/50 chance of me spotting the error (if the solution feels really wrong I usually find it, and it's something insane every time). I feel like the really talented math kids who've been doing math rigorously from like age 6 have this down to muscle memory.

To make a comparison to another hobby of mine, I feel like they are really good at writing prose while I'm very good at structuring a story. The issue is, a good structure with poor prose will give you a story no one will read.

I don't know where to go from here. I feel very concentrated while doing the "hard parts" of a problem and then my head feels full of cotton while I'm doing basic computations. I make errors that are both ludicrous and invisible to me, for example:

Let's say we have a function f(x) = (x^2)/(x^2-x),

This function is defined if x^2-x ≠ 0

After some basic computing we get the penultimate step:

x ≠ 0

x - 1 ≠ 0

And then I'll write

x ≠ 0 x ≠ -1

And I guarantee you I will not spot that error. It will sabotage the entire problem, whatever might come after it, and I won't notice it unless I redo the whole thing again.

I don't know what to do. This is just not good enough, but I don't know how to fix it. I've done probably hundreds and hundreds of practice problems and the same algebra mistakes are ruining me. It has gotten to the point where I never feel confident in my calculations, nor believe I have what it takes to be truly good at math. It's like I lack some mathematical intuition that really talented people have.

I've been feeling rather demotivated because of this, specifically because I can't trust myself and my own ability. If I could at least trust that and only have missing knowledge, I'd be excited to learn. The way things are I just feel useless as a mathematician and I have no plan on how this might be resolved.

I'm thinking along the lines of the pythagorean theorem, pascals triangle, proof of infinitude of primes etc.

Hello, I'm wondering if in mathematics there is a specific name for, or if it's a particular subject of study, the following relationship between functions, or if it’s not really of interest:

Let f(x,y) and g(x,y) be such that f(x,n)=g(f(x,n−1),x).

That is, this describes the relationship between multiplication and addition f(x,y)=x⋅y and g(x,y)=x+y, but it also applies, for instance, to f(x,y)=x/y and g(x, y) = xy/(x + y) and many others…

Thank you

I'm a first year post doc teaching a measure theory and integration course. This is my first time teaching and grading a proof based course. Today the students were filling out their course evaluations and I was also doing some reflecting on how things went. The semester is about to finish and looking back I barely had any time to do research because of how much time this course took! The most time consuming thing was grading homeworks and exams (I'm at a small school so unfortunately there's no teaching assistant to help with grading). I have 24 students and I assign about 10 problems a week with a midterm and final. It takes me forever to go through their logic and determine whether or not their proof is valid. Despite my PhD being in an analysis heavy field I feel like I learned more about measure theory and integration teaching this course than I did while taking a similar course as a grad student!

I began to wonder if grading proofs is normally this time consuming or if I don't understand the material as deeply as I thought and that's why it's taking me hours and hours. I have no idea how you guys manage to produce papers while teaching. Curious to hear what other educators have to say.

i’m wondering if people have a recommendation on an introductory topology textbook to use to self-study with. i’m currently using Armstrong, but i’ve seen others say Munkres is good, and i like the style of it. any thoughts/opinions??

I would like to share a fun exercise that I came up with.

I was thinking about which points on a square grid have integer distance, and the answer is obviously given by Pythagorean triples, which can be generated by the famous Euclid's formula. Then I wondered: What about a triangular (or hexagonal) grid? There, the possible integer distances are given by the triangles with integer sides and an angle of 120°. If the sides are a, b, c, and the 120° angle is between a and b, then they satisfy a² + b² + ab = c² (by the law of cosines).

Then I wondered if I could come up with the analogous of Euclid's formula for this class of triangles, and I found that, if m and n are two integers with m < n, then the triple (a, b, c) where

a = n² - m²

b = 2mn + m²

c = m² + n² + mn

satisfies our condition. Moreover, it is not difficult to prove that if m and n are coprime and not congruent modulo 3, then the triple is primitive (i.e., GCD(a,b,c) = 1). It can also be shown that all primitive triples can be obtained by this formula.

That's it, I just wanted to share this fun little exercise in number theory. I also wonder if this formula already has a name. I am not aware of it, but probably someone knows better than me.

Edit: typos

Title. Let it be reasonably notable, and please mention the size as well.

Also, feel free to bend "structure" and "size" to your wishes.

I’m working through a textbook, and my vector calculus is a bit rusty, so I’m trying to see if my intuition here holds. Any help is appreciated.

I’ll use italics for vectors. Let p(x) be a probability distribution with support on all of R^n. Now, consider a general nxn matrix A. What I’m interested in is the volume integral of div(x_k A x p(x)) (where x_k is the kth element of x) over all of R^n. My intuition is that, due to the divergence theorem, this integral should be the limit of the surface integral of x_k A x p(x) • n over a boundary increasing in size to infinity. My intuition says, since p(x) is a probability distribution, it will decay at infinity, and therefore the integral should be = 0. Is this correct, or are there some conditions on the matrix A for this to be true, or is this just incorrect?

I’ve always been curious about whether mathematics maintains its intrinsic beauty when you reach a high level of specialization. It’s often said that PhD students working in different areas of math, even those that aren’t too far apart, struggle to understand each other’s work. Does this fragmentation take away from the elegance and coherence that many of us love about mathematics?

Does anyone have an idea where I can watch Colors of Math by Ekaterina Eremenko, I can't find it, I appreciate your help 🙏🏼

Like the title says, im looking for some specific examples of where some mathematical models that humans have relied on have failed us with devastating results. Any help is greatly appreciated!

I recently read the following passage on an article online learning regarding the proof that only rational numbers (with the exception of square roots) can be created by straightedge-and-compass constructions:

"In most textbooks, these impossibility proofs are presented only after extensive background in groups, rings, fields, field extensions and Galois theory. Needless to say, very few college-level students, even among those studying majors such as physics or engineering, ever take such coursework. This is typically accepted as an unpleasant but necessary aspect of mathematical pedagogy."

I have a bachelor's degree in Engineering and am quite comfortable with calculus up to differential equations, but as the quote says I'm completely unfamiliar with groups, rings, fields, field extensions and Galois theory. Is there a good resource for somebody with my background to learn these and become familiar with them?

Does anyone how to prove that F(K)≤F(G), where F denotes the Fitting subgroup and K is normal in G?. I think it is true but don't know how to prove it.

Thanks :)

I'm taking a class on probability theory and stochastic processes. So far I've been able to follow all the technical arguments but I've been uncomfortable because I can't build an intuition on any of the results.

For example the Ito integral is an L2 limit of simple processes using a subsequence argument. Since this integral isn't defined pathwise this integral clearly isn't an area of any kind so I have no idea what the random variable that is the Ito integral even represents.

Another example is Ito's lemma. Again the quadratic variation means the "stochastic chain rule" has an extra term (the second derivative) compared to the ordinary chain rule but is there any intuition why it should be 1/2 the second derivative? You can make some sort of argument using Taylor series but we're able to motivate the ordinary chain rule without resorting to any expansions so this seems like a bit of a stretch.

Stochastic differential equations are another example. A non-random ODE or PDE clearly describes the rate of change of different functions. But an SDE isn't even a differential equation and always represents an integral equation. So what exactly are these SDEs modeling?

I'm asking this in r/physics too so to get both perspectives.

Do theoretical and mathematical physicists invent/discover new math in order to explain new emergent phenomena that arises in experimental physics and is therefore used to build theories? Or do physicists also pick up math already invented?
If it's the latter, then there comes another question: are advances in pure mathematics key for developing and understanding theoretical physics?

I'm not talking about rigorous defined frameworks, but new ideas and structures that serve the purpose of explaining specific natural behaviours of matter and energy even though is not defined (at the moment) for general cases.

I know that they’re homology equivalence classes, but I was wondering if there’s something shorter? Can I call them cycles? I know that they’re only cycles in one dimension but is that cool and hip with the topologists to do it casually for all dimensions?

It’s been a while since I studied this in undergrad so apologies for the lack of rigor and specificity.

When I was in Calc 3, I remember learning about curvature, and I believe it was denoted with “k”. We were taught that a small circle was more curvy, than a larger circle with a larger radius.

I understood at the time, why this was defined as such, however I felt like there should also be some notion of curvature that is standardized for the shape of the closed shape.

So that all circles have the same shape regardless of size, but we could still compare the curvature of shapes, i.e. a circle is more curvy than a triangle.

In the same way that Usain Bolt can run really fast for his size, but a Giant 50ft troll running at 30mph would be quite average, if not slow.

Then, I’m sure we could extend this in higher dimensions, where there would probably different ways to quantify curvature when standardized for the shape of the closed N-d shape.

Has anyone come across this line of thinking? I’m sure it’s probably been explored before.

Hi everyone!

I’m currently taking a course in functional analysis, and for the oral exam, I need to present a 20-minute seminar on a topic related to the course. The topic should not overlap with the material already covered but still stay within the realm of functional analysis.

Some of the main areas we’ve covered include:

• Normed and Banach spaces, linear operators, and dual spaces.

• Hilbert spaces and orthogonal projections.

• Hahn-Banach theorem, topologies (weak, weak*), and convexity.

• Spectral theory for bounded operators and compact operators.

• Elements of distribution theory and spaces of sequences.

I’m particularly interested in topics that connect functional analysis to probability theory or ergodic theory, as these are fields I’d like to explore further.

Do you have any suggestions for seminar topics that fit these criteria?

Thanks in advance!

I particularly like popscience books. Of course, they won't teach you the content itself, but they're great for stimulating creativity and imagination, and for arousing the curiosity of the lay public.

I really like Ian Stewart's books, and James Gleick's “Chaos”.

I'm now doing math research on a probability theory question I came up with. Note that I'm an undergraduate, and the problem and my approaches aren't that deep.
First, I googled to see if somebody had already addressed it but found nothing. So I started thinking about it and made some progress. Now I wish to develop the results more and eventually write a paper, but I suddenly began to fear: what if somebody has already written a paper on this?

So my question is, as in the title: how can we know if a certain math problem/research is novel?

If the problem is very deep so that it lies on the frontier of mathematical knowledge, the researcher can easily confirm its novelty by checking recent papers or asking experts in the specific field. However, if the problem isn't that deep and isn't a significant puzzle in the landscape of mathematics, it becomes much harder to determine novelty. Experts in the field might not know about it due to its minority. Googling requires the correct terminology, and since possible terminologies are so broad mainly due to various notations, failing to find anything doesn't guarantee the problem is new. Posting the problem online and asking if anyone knows about it can be one approach (which I actually tried on Stack Exchange and got nothing but a few downvotes). But there’s still the possibility that some random guy in 1940s addressed it and published it in a minor journal.

How can I know my problem and work are novel without having to search through millions of documents?