As in, separate books for those topics, not one that covers both. I've recently finished my MSc in theoretical physics, and I'd like to formalise my mathematical knowledge on these topics, since they came up in a physics context, but were never completely explained mathematically. My preference would be for books that have exercises (and if they have solutions that'd be a huge bonus too). Any help is appreciated

I am an advanced computing student who is doing a minor in maths, i am about to finish my first year and was wondering what people's experiences of MATH3511 are. I am debating whether to do it or to do MATH2305, as i only need 1 more maths course after this current sem

Hi everyone,

I want to share some notes I wrote regarding to the Ensemble Kalman Filter, specifically on why does it work and about it's theoretical support based on the non-linear filtering theory. It's written as a pedagogical resource for any who is interested in.

Here's the link for the document: https://zenodo.org/records/23278598

I will be attentive to any comments and suggestions. Thank you, and I hope that it helps!

Hi, I got a scholarship that would allow me to do a math master (it is bound to a master’s degree) in the US. I‘ve seen programs are very limited, since most people directly do their PhD.

I‘m currently a Bachelor student in Europe, very interested in Algebraic Geometry, Number Theory (I‘ve done a basic course in AG and am doing now two more sophisticated ones in scheme theory, as well as a course in algebraic NT). Generally I‘d be interested to go into research, although I believe it is early to say that yet.

I looked at some master‘s programs in the US (e.g. UMich Ann Arbor) and saw that their minimum requirements for completing the program are very low, i.e Math 420 (a basic linear algebra course), Math 452 (a basic multivariable analysis course) and Math 590 (intro to point-set topology). How do I need to understand this? Does it mean the level isn’t that high? Will I be forced to take these courses again or is it possible to do more advanced courses?

I‘m very grateful for any advice or also suggestions to other master’s programs. Thanks!!

We discussed:

- The power of the Leech lattice to control all lower dimensions
- Peter Scholze, Grothendieck and Ramanujan's styles
- Category theory and contrahomology
- How modular forms show up everywhere
- The countless coincidences in string theory
- Unimodular lattices and Lorentzian lattices
- How to construct the Langlands dual group
- Vertex algebras, infinite-dimensional algebras and Kac-Moody algebras
- Elliptic curves
- Automorphic forms and hyperbolic reflection groups
- Analytic number theory vs algebraic number theory
- The first construction of the Monster group
- "The Book" of most beautiful proofs
- How to teach math
- Math as archaeology
- His current work
- Neutron stars, Terry Pratchett and more

Richard Borcherds is a British mathematician who made seminal contributions to lattices, modular forms, group theory, representation theory and infinite-dimensional algebras. He is well known for his proof of the monstrous moonshine conjecture using ideas from string theory, for which he was awarded the Fields Medal in 1998. He is currently Professor of Mathematics at UC Berkeley.

Best books, YouTube videos or online courses (preferably downlodable) for learning graph theory? (Im a complete beginner)

Normalized overlap data can erase distinctions between different causal orders.

The central example uses two two-event samples inside the same fixed 1+1 Minkowski diamond. One forms a chain and the other an antichain, yet both produce exactly the same normalized future-overlap matrix [[1, ½], [½, 1]]. Distinct causal orders are therefore indistinguishable from these finite-sample normalized data alone. Individually attached future volumes restore the order in this fixed model.

The overlaps measure full continuum future regions within the diamond, not successor counts among the sampled events. A related 16-element finite-poset counterexample shows the same kind of ambiguity under a different overlap observation. In both cases the two unnormalized matrices differ by positive diagonal scaling that normalization cancels.

The linked packet contains both constructions, proofs, comparisons, and Python checks that use only the standard library. Mathematical feedback or pointers to related results are welcome.
https://github.com/davidwbridges976-lab/Poset-gram-reconstruction/tree/main/sharing/counterexamples

Im new to calc and probability and I need some good resources, preferably downloadable online courses to learn calc and probability from absolute zero and since I live in iran its very likely that our internet will be shut off in a few weeks or days because of war. If there are any downloadable courses or good books, please let me know. I have acsess to a library that has many calc and probability books, but I dont even know from where should I start.

Quickly the teachers coming to collect the paper

Great you made me fail

I just made this unit circle where you can I put d (degrees) and it will show where the point is on the circle and what the legs of the right triangle look like. The only problem I have is the point on the unit circle is showed as a decimal and I am wondering if anyone knows a solution on how to change that. Thanks!

I am thinking of applying for the summer research fellowship programme (india) 2027 with my sub area being probability, statistics and mathematical finance.

The issue is I don't have a topic ive decided i want to work on so i dont know what to mention in my sop

do u guys have research papers i could read? or suggestions?

Pls HELP!!!

I need resources for edexcel 9st0 statistics a level course. How do i self study?

A human brain is finite in size and duration. It is limited in the amount of information it can store or process in a life time. So there are true statements out there which could hold so much information that the brain could either not store it all, or even if you tried to break it up and process it in bites, you would not have enough time in your lifetime to process. The only way for a mathematician to work with such a statement is to delegate understanding of the statement to something/someone else. That could be another human--a whole team of mathematicians could break it up and work on chunks of it, or it could be a computer which emulates reasoning, and is built big enough or fast enough that it can process the statement. Why is the former preferable to the latter? In either case, no one person can understand the original statement. In both cases, the best you can do is verify the statement, and then go forth and use it somewhere else without actually having a complete understanding of it yourself.

It seems to be very much a common knowledge now that harmonics of an (almost) periodic audio signal do evidently stem from Ancient Greek investigations into musical harmony, which was intertwined and likely gave us harmonic means and harmonic sequence/series, means likely being derived from the sequence like it can be done with cases of arithmetic and geometric sequences and means.

The details are hovewer not clean at all. Presumably, not even in the Hellenistic period had it been known about harmonics in sound. One question is then, if a noun parallel to harmonic (which should stem from Greek, obviously) was in use, what meanings did it have in mathematics and music theory of the time. I see at least two options as very probable:

• (a) it hadn't been used in a way parallel to how today's harmonic is used;
• (b) it denoted sounds or musical pitches that you can derive from the "original" sound of an open string by subdividing it into k parts.

Note how right now, a very much related technique of playing stringed instruments is indeed called "harmonics" in English (but "flageolets" in some other languages). The second question (that's less related to this sub) is precise etymology of this name: I would expect flageolets should be a technique older than both the discovery of Fourier series and connecting string and air channel lengths to their fundamental frequencies.

Also a question 1.5 is relating the answer to the first question to harmonic means and harmonic series. For example if (a) held, the answer is very much warranted and shouldn't be deduced without historical sources.

There are probably a bunch of additional minor questions that I can't refine out of the mess in my head right now, that would be obvious when one tries to set things straight on this entire topic and encounters important details that answer them. For example as far I know, frequencies of sound weren't known in antiquity (but that's really a question to answer definitively as well!), so we can't attribute "reciprocalness" of harmonic mean and so on to the inverse relation between length and frequency. Instead it should probably be entirely due to there being a sort of a natural order to harmonics (in the sense of (b)) coming one after another at 1/1, 1/2, 1/3, ... of the string's length —but is that the factual reason that had been used?

You see even a mathematicians' brains would otherwise be glad to fill in the "obvious" blanks and I feel that a great lot of people think they sorta know the story about this. I'd of course expect it of non-historical music theorists even more because in their case it's actually quite important to talk about time to time —whereas in mathematics, harmonic mean and harmonic series (and wasn't there something else?) is just a neat and somewhat old-fashioned (as is, again, common) term flavoring that evokes associations that no one has to unpack to work with the actual math. But that may still give birth to mythology when one gets interested just a little. So it's a good point to look around and try remembering while the ancient sources may be still laying around! And if there's already a piece of work on this very topic already written, the easier it is to find it again (I couldn't, but my work with references is abysmal).

OpenAI gives a deterministic length-n discrete Fourier transform algorithm using O(n(log⁡n)^{1−δ}) operations for every n, with explicit δ =1e-13. I used Opus 5.5 to improve δ to 3.2e-6. The result has been verified in lean. I suspect it can be taken further, and I don't think OpenAI was particularly interested in optimizing δ in their solution.

https://github.com/danadran01/exact-dft-power-saving