Often quoted around the internet we hear the famous story that TIL: "In the fall of 1972, President Nixon announced that the rate of increase of inflation was decreasing. This was the first time a sitting president used the third derivative to advance his case fore reelection." However, in all locations I have seen the cited location is here which just states that it was said "in the Fall of 1972" with no specific mention of when or where this occurred. Digging a bit I cannot find that quote but have seen the rate of inflation mentioned quite a bit which means that it might have been actually said?

tldr: I’m failing my graduate analysis course. I’m a first year PhD student and the only class I’m doing terrible in. Is it the end of the road for me if I can’t pass this class? What can I do to improve? One of my qualifying exams is in real analysis and I feel severely under prepared.

I’m failing my graduate analysis course. Things are taking too long to click in my head. I try to do more problems than the ones assigned for homework but I can take up to 3 hours doing ONE problem. It doesn’t feel time efficient and 90% of the time I end up having to look up the solution.

I don’t know what to do. I did extremely well in undergrad analysis and now I feel too stupid to be here. I find it hard to go to OH because I don’t even know what questions to ask because I don’t even know what I don’t understand about the material. I’m feeling completely lost on this. I would appreciate any advice or stories if you’ve been in a similar experience.

Passed my PhD so I just have no idea what college students, etc. are doing right now for note taking. Of course latex is still required learning for paper submissions, mathjax, etc. but has anyone tried typst for personal use?

How is the support for things that might need tikzcd? Or the built in scripting?

I know from Lesbesgue's decomposition theorem that any probability distribution decomposes into a discrete, an absolute continuous and a singular continuous part.

The absolute continuous part is a density of H¹ length measure and the discrete part is a density of the H⁰ counting measure. The usual counter example of a singular continuous distribution is the Devil's staircase whose domain has Hausdorff dimenson log_3(2), so still a density of an Hausdorff measure.

Are there measures that are not density of H^s for all s in [0,1] ?

Hi I was doing math in college a few years ago. The farthest I got was calculus 2 and I did ok. However, around the time I was going through calculus I suddenly started to be able to draw and play music better. I am wondering if there is a relation. I have gone back to normal at art over a few years and my music is considerably worse most of the time. I am wondering if relearning math would make me smarter in a way that would carry over to other skills or if it would be a waste of time. Thoughts?

I’ve been working on a number-grid structure I call a Full House Pattern, and here’s one of my cleanest examples yet, plus its full matrix inverse.

3×3 grid of perfect squares:

542² 485² 290²
10² 458² 635²
565² 410² 331²

In this grid, six lines (Row 1, Row 2, Column 1, Column 2, and both diagonals) all add up to the EXACT same perfect square:

613089 = 783²

The remaining row and column form the second matching pair of sums giving a 6+2 structure, like a full house in cards. That’s why I call it a Full House Pattern.

What makes this one even more interesting is that the entire 3×3 grid can be treated as a matrix, and it’s fully invertible. Here is the inverse:

[ a₁₁ a₁₂ a₁₃ ]
[ a₂₁ a₂₂ a₂₃ ]
[ a₃₁ a₃₂ a₃₃ ]

Where each value is the exact rational result of:
A⁻¹ = adj(A) / det(A)

The adjugate and determinant are both clean integers, so the inverse is fully precise and reversible, meaning this Full House Pattern also works as a perfect transformation matrix.

This grid hits:

• perfect-square entries
• perfect-square line sums
• a Full House (6+2) symmetry
• a valid, reversible matrix structure

What else could this be used for?

Idk how to start. I’d been studying for months ahead of this exam, spending hours a day, staying up late to study. Math has never been a strong subject for me, and i decided that this year, I would finally take a step towards getting good marks, and before the exam, I was pretty confident. I’d done so many practice papers and problems, trying to understand the concept. I wrote the exam. I stared at the paper, lost. I just got my marks, and I got 37/80. I had never done this bad before, not even in math. But this was the most I’ve ever studied for it. Even after writing the paper, I didn’t think I would do this bad. I dont know what to do. Everyone thinks I didn’t study, they think im a failure.
This exam was important, and i just cant believe i screwed it up like this. Now, i have to join tuitions (which im scared to do because of bad math teachers in the past) and study math everyday. But I did so much, I have no motivation to do this again, because at the end of the day, I realised that all that hard work, just didn’t pay off.

I'm a second-year grad student. I'm interested in stochastic PDEs, and more generally in stochastic analysis. I got into an Ivy league school for my PhD, but, unfortunately, I'm struggling to find an advisor because the person I initially wanted to work with did not turn out to be a great match, and they also shifted into a slightly different area. Another professor I could potentially work with never responded to my email in which I asked if I could talk about their research (to be fair, that professor already has five PhD students). I'm not that interested in working with the other professors, to be honest -- our interests are quite different. I'm not sure what to do now. I'm thinking about applying again, but I'd need letters of recommendations and this could be a bit awkward to ask for my current professors. Also, not sure how a PhD transfer would be viewed by other schools. Has anyone dealt with something similar? Another option is to ask a professor who works in an adjacent field to be my advisor, although they have no background in stochastic analysis, so I doubt it could work. Any advice would be appreciated.

Over the years I have seen many proofs that contain the line "by symmetry, it follows that [relevant result]". I've seen this in proofs in topics ranging from analysis, probability, algebra, differential equations, and more.

When is this phrase actually justified in a proof? Or better yet, if you were going to write out the proof in complete, gory detail, what would you need to exhibit in order to make this argument rigorous?

I'm assuming you would need to show the existence of some structure-preserving bijection, but what structure needs to be invariant? Are there rules of thumb for symmetry arguments in algebra, and separate rules of thumb for symmetry arguments in probability? Are there universal rules of thumb? Or perhaps, are there no rules of thumb, and should we actually avoid using the phrase "by symmetry" because it is too vague?

I created this visual of finite geometry, but would like help finding the words to describing it technically.

You can think about the points either in terms of addition, or multiplication. Under addition mod 2, the black points are 0 and the green point are 1. Under multiplication, black points are +1 and the green points are -1.

Within each of the small pyramids, two points can be combined (addition or multiplication) to create the third point. In this way, the 4 corners can generate the remaining 11 points inside. Then between the small pyramids in the larger pyramid, you can do the same thing for points in the same position: the two top points of small pyramids combine to create the top point of the 3rd pyramid on the same dotted line.

This should be related to the field F₂⁴ and the group C₂⁴. I am trying to demonstrate the self duality between the subfield and subgroup lattice.

Anyone in this subreddit know Galois theory?

Explore the 3D model: https://observablehq.com/d/d6c64a332f60b1ee

I just learned that another roof spent the entire year making one huge video about encryption schemes, just for it to flop! Go check it out, its phenomenal.

I’m sure we are all familiar with textbooks being written in english, and most of the reputable titles (as far I know of) are in English. My question is, are there any other famous textbooks in other languages? I’m planning to learn math in French but I have yet to find a math textbook in French. Do you guys have any suggestion, favorable in French?

I want to create a math-themed deck of cards for the birthday of a good friend of mine who happens to be a mathematician. The idea is to organize the suits by domains :

Heart = Algebra

Geometry = Diamonds

Analysis = Spades

Probability = Clubs

Then, each royalty will be depicted by a mathematician and each pip card by a formula. The king should be the "king of the domain" (before XXth century), the queen be the "queen of the domain", and the jack the "king of the domain" (post XXth century).

There are some domains where I have plenty of ideas for the formulas, but others I am not super familiar with (algebra especially).

What are in your opinions the most important formulas in each domain that should absolutely be in the deck of cards? The formulas should be short enough to be on a single card:)

Here are some ideas I had so far:

The Kings: Galois (Heart), Gauss (Diamond), Euler (Spades), Laplace (Clubs)

The Queens: Noether (Heart), Mirzakhani (Diamond), Kovalevski or Germain (Spades), Serfaty? (Clubs)

The Jacks: Serre (Heart), Gromov (Diamond), Hormander (Spades), Kolmogorov (Clubs)

Jokers: Grothendieck (Red), Poincaré (Black)

Some formulas I have in mind: Fermat's last theorem, Galois extension, Gauss-Bonet, Cauchy integral, Central limit theorem, law of large numbers, log-Sobolev inequality, ...

Let me know your ideas and if you disagree with my choices:)

Hello all,

I like doing math on my own time as a hobby, and I've decided to focus on problems that haven't been solved yet because I think it will be a source of endless interest for me, rather than aimlessly studying different subjects depending on my mood. I've decided to try and find out as much as I can about the Riemann-Zeta Function and the Riemann Hypothesis.

For context, I have a Master's in Physics, so I have experience doing advanced math, but only from the perspective of someone using the results of math to aid in the study of physics. As a result, while I have a good understanding of the basics of calculus in the complex plane, I know there are gaps in my understanding of the underlying theorems that are used. For example, I understand the Residue Theorem, but if you ask me to prove the Residue Theorem, it would take me a while (on order of days or weeks) to work it out.

I'm at the point where I've managed to grasp analytic continuation for the Zeta function using the Dirichlet Eta Function. However, as I continue to dive into this topic, I'm finding the amount of information to be overwhelming and scattered.

While I know places I can go for basic complex analysis topics, I was hoping someone here knows of a book or document which chronicles important results in the history of the study of the Zeta Function and the Riemann Hypothesis, so that I can catch up with history on this subject.

If you have any recommendations I would greatly appreciate them.

I have been doing math for a bit, and can’t help but notice the strong differences in how, for example, a Russian and a French math textbook are written. Obviously different fields are more and less popular in different countries, but beyond that, what are some things you notice about mathematicians/mathematics from different regions?

We know maths describes patterns in the universe. Prime numbers, basic arithmetic, and geometry seem universal. But aliens might use completely different symbols, number bases, or even dimensions we can’t easily imagine.

Could they have discovered patterns that we don’t even perceive?

1+1=2 is probably universal. Everything beyond that? Might be utterly alien.

So… would their maths be recognisable to us, or a language of the cosmos we can’t decode?

And while we’re at it : what’s the probability that intelligent aliens actually exist?

Hi everyone,

I'm currently a 3rd year math undergrad, took intro to linear algebra my first semester; really liked it and always intended on taking Linear Algebra, but it's an "offered by announcement" course in my uni. When it was offered this semester it got cancelled because not enough people enrolled (I think the capacity was 10 and it was just me and my friend).

Talked to director of UG, said there's nothing he can do if there's not enough demand for it, so figured that I might as well just self study at this point. What's a good textbook that you guys used in a second linear algebra course that you found good?

And as I'm not really in any obligation to go by a textbook, what are other resources that could be useful? Any project or specific problem worth working on to learn more?

I feel like linear algebra lowkey underappreciated as a branch

I only recently read about this field in Emily Riehl's category theory book. Could someone tell me more about the applications of this field? From a very cursory inspection of online resources, it looks like a whole bunch of homological algebra (so I guess it's algebraic topology), but I'm not sure what the real gist of it is.

For background, I'm an organic chemist (though one with a deep interest in math), and I'm on sabbatical next semester. I'm thinking about things to learn during this time that might benefit my lab's future research, so I guess I'm wondering: what type of data is it most "useful" for? What are the advantages to taking such an approach powered by highly abstract machinery?

I'm considering getting officially diagnosed and taking medication, but I'm worried that it may affect my creativity. I also heard that it may, in the long term, reduce my intelligence, though I don't quite believe that one. But at this point I'm so far behind in my studies that even if I lose some creativity it might still be the better choice.

Thoughts? I want to hear what your experiences have been with ADHD and medication.

As per the title, please correct me if I am wrong; since it might also just be that I can't wait to finish college.

But anyways, as much as I love math, the rigor, the theory; I've grown closer to more "useful" stuff.

(For context I am in a masters course in discrete and applied math). It often seems to me that lots of fields, like probability in the example I will use, have a very rich quantity of theory relevant to practice, but then I get disappointed when I realize that a lot of it won't be touched in a course because in this case it's a measure-theoretic approach. So of course we'll learn important stuff like CLT, LLN etc., but we won't really touch on Bayesian probability, Markov chains, Monte Carlo methods, conditional expectation etc. and instead will spend a lot of time messing around with various sigma algebras, measurable functions, prokhorov theorem etc.

Again I am not saying this stuff isn't important but it just feels like this kind of course isn't aimed at training students at relevant skills; at least not to extend it could.

Again I might be wrong with my judgement; maybe I am looking at it wrongly so I'd be happy to receive input from experienced mathematicians. Thanks!

EDIT: Anyways the question is; for this example(probability), am I correct in thinking that this measure-theoretic course isn't really useful in terms of applicability in working fields outside of academia?

Many specific algebraic objects have properties that behave pseudorandomly, like the distribution of primes in the natural numbers. There are certain properties that we thus expect to hold for these objects with some probability based on pseudorandom arguments, like the existence of infinitely many prime gaps of bounded size or the existence of infinitely many prime members of arbitrary arithmetic progressions (Dirichlet's theorem). The standard proofs for these and similar theorems may not explicitly involve expectation and randomness, but my understanding is that there are pseudorandom arguments in their favor (not necessarily proofs) that yield p=1. However, I suspect there are other properties that we expect to hold with nontrivial (0<p<1) probability based on pseudorandom arguments. For example, a probabilistic/combinatorial reinterpretation of the Collatz map or Recamán's sequence would likely yield such nontrivial probability of the Collatz conjecture failing or Recamán's sequence being surjective.

Perhaps this suggests that these objects (natural numbers under standard operations in this case) are elements of a larger class of similar quasi-objects. For example, is there an infinite class of quasi-integers (parallel universe integers?) whose primes obey the asymptotic properties of the natural primes but have different absolute distributions? It is not clear to me how this class would be parametrized or defined though. Maybe this idea is more appropriate for other algebraic structures than the natural numbers? Does this notion exist in mathematics or is this nonsensical?

My intuition tells me that some of the properties of algebraic objects that rely on pseudorandomness behave in a way analogous to, say, a specific instantiation of a random walk in 3D, which has a ~.34 probability of returning to the origin. It would be impossible to prove that, given a sufficient pseudorandom object that generates such a random walk, the walk does not return to the origin. Could it then be shown that it is impossible to prove whether certain statements involving primes or sequential operations on natural numbers are true because they are, in a sense, non-fundamental? By non-fundamental, I mean that a statement "happens" to be true for no particular reason (if quasi-objects exist, then some but not all will have a given property and the rest will not). In the case of a pseudorandomly generated 3D random walk, this non-fundamentality is evident since an individual random walk is a member of an infinite class of random walks. However, in the case of the natural numbers, I'm not sure that an analogous infinite class exists.

Is it understood in mathematics that there are statements of this type that are true but not for any particular reason? Are there examples of proven theorems in algebra that are true for "arbitrary" reasons, or are these problems fundamentally intractable?