/r/math

This subreddit is for discussion of mathematics. All posts and comments should be directly related to mathematics, including topics related to the practice, profession and community of mathematics.

This subreddit is for discussion of mathematics. All posts and comments should be directly related to mathematics, including topics related to the practice, profession and community of mathematics.

**Please read the FAQ before posting.**

**Rule 1: Stay on-topic**

All posts and comments should be directly related to mathematics, including topics related to the practice, profession and community of mathematics.

In particular, any political discussion on /r/math should be directly related to mathematics - all threads and comments should be about concrete events and how they affect mathematics. Please avoid derailing such discussions into *general political discussion*, and report any comments that do so.

**Rule 2: Questions should spark discussion**

Questions on /r/math should spark discussion. For example, if you think your question can be answered quickly, you should instead post it in the Quick Questions thread.

Requests for calculation or estimation of real-world problems and values are best suited for the Quick Questions thread, /r/askmath or /r/theydidthemath.

If you're asking for help learning/understanding something mathematical, post in the Quick Questions thread or /r/learnmath. This includes reference requests - also see our list of free online resources and recommended books.

**Rule 3: No homework problems**

Homework problems, practice problems, and similar questions should be directed to /r/learnmath, /r/homeworkhelp or /r/cheatatmathhomework. Do not ask or answer this type of question in /r/math. If you ask for help cheating, you will be banned.

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If you are asking for advice on choosing classes or career prospects, please post in the stickied Career & Education Questions thread.

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**Recurring Threads and Resources**

*What Are You Working On?* - every Monday

*Discussing Living Proof* - every Tuesday

*Quick Questions* - every Wednesday

*Career and Education Questions* - every Thursday

*This Week I Learned* - every Friday

*A Compilation of Free, Online Math Resources*.

*Click here to chat with us on IRC!*

**Using LaTeX**

To view LaTeX on reddit, install *one* of the following:

MathJax userscript (userscripts need Greasemonkey, Tampermonkey or similar)

TeX all the things Chrome extension (configure inline math to use [*;* *;*] delimiters)

`[; e^{\pi i} + 1 = 0 ;]`

Post the equation above like this:

`[*;* e^{\pi i}+1=0 *;*]`

**Using Superscripts and Subscripts**

x*_sub_* makes x*sub*

x*`sup`* and x^(sup) both make x^{sup}

x*_sub_`sup`* makes x*sub*`sup`

x*`sup`_sub_* makes x`sup`

*sub*

**Useful Symbols**

Basic Math Symbols

≠ ± ∓ ÷ × ∙ – √ ‰ ⊗ ⊕ ⊖ ⊘ ⊙ ≤ ≥ ≦ ≧ ≨ ≩ ≺ ≻ ≼ ≽ ⊏ ⊐ ⊑ ⊒ ² ³ °

Geometry Symbols

∠ ∟ ° ≅ ~ ‖ ⟂ ⫛

Algebra Symbols

≡ ≜ ≈ ∝ ∞ ≪ ≫ ⌊⌋ ⌈⌉ ∘∏ ∐ ∑ ⋀ ⋁ ⋂ ⋃ ⨀ ⨁ ⨂ 𝖕 𝖖 𝖗 ⊲ ⊳

Set Theory Symbols

∅ ∖ ∁ ↦ ↣ ∩ ∪ ⊆ ⊂ ⊄ ⊊ ⊇ ⊃ ⊅ ⊋ ⊖ ∈ ∉ ∋ ∌ ℕ ℤ ℚ ℝ ℂ ℵ ℶ ℷ ℸ 𝓟

Logic Symbols

¬ ∨ ∧ ⊕ → ← ⇒ ⇐ ↔ ⇔ ∀ ∃ ∄ ∴ ∵ ⊤ ⊥ ⊢ ⊨ ⫤ ⊣

Calculus and Analysis Symbols

∫ ∬ ∭ ∮ ∯ ∰ ∇ ∆ δ ∂ ℱ ℒ ℓ

Greek Letters

**Other Subreddits**

**Math**

- /r/learnmath
- /r/mathbooks
- /r/cheatatmathhomework
- /r/matheducation
- /r/casualmath
- /r/puremathematics
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- /r/mathmemes

**Tools**

**Related fields**

/r/math

125

On MSE and other forums I've heard apocrypha of PhD students that discover some amazing structure as a part of their thesis, but under scrutiny it turns out the amazing structure they have was the empty set and all their work was trivial.

Has there ever been a real instance of something like this? Where a mathematician or student has had their time wasted chasing a dead-end for longer than they should have?

41 Comments

2024/02/12

11:50 UTC

11:50 UTC

2

2 Comments

2024/02/11

22:03 UTC

22:03 UTC

1

Is there anyone that can explain to me rhe difference between a paravector and a multivector? Or even more generally, what is the motivation behind the algebra of physical space? If it's isomorphic to the even subalgebra of Cl(3,1), why not just use Cl(3,1)?

2 Comments

2024/02/12

07:10 UTC

07:10 UTC

1

I am looking for topics to research for my thesis, I have taken calc 1,2&3, linear algebra, probability and stats, and abstract algebra. I'm currently taking Intro to Analysis and Differential equations but I'm mostly interested in topics surrounding linear algebra but will take any suggestions! Appreciate it!

2 Comments

2024/02/12

03:31 UTC

03:31 UTC

1

Hello, I've been reading Mizenmacher and Upfal's Probability and Computing for a randomized algo class. I think I have pretty good fundamentals in stats and combinatorics, but this book has been particularly difficult for me to work through. It took me about a total of 8 hours just to read and understand the 3rd chapter in the book (doesn't include time to do the exercises). I don't know if it's too hard for me, as I'm taking very long to finish reading a chapter or If I'm being too inefficient. Every time the author introduces a lemma/theorem I try to prove it myself and look at the solution. The thing is I understand how the pieces in the solution fit together but can't seem to build an intuition to start solving the problems at all. If anyone's read the book, then can you give me some tips to get better?

0 Comments

2024/02/11

12:27 UTC

12:27 UTC

0

I was just wondering what would be the effect of solving Traveling Salesman Problem/Hamiltonian Graph problem in optimal time and by extension P vs NP? Specifically with regards to any militarization or advancement of that it could be used for.

6 Comments

2024/02/11

16:55 UTC

16:55 UTC

1

I am an undergrad doing a double major, and am just curious where most people end up.

4 Comments

2024/02/12

03:24 UTC

03:24 UTC

3

In physics there is a concept called information.

I've been increasingly interested in it since learning about the black hole paradox and the relational theory of quantum mechanics.

I wanted to better study the physics and science of information so I looked into information theory.

I'm not sure if it is what I am looking for though. It is an interesting field in and of itself, but it seems mainly to relate to the theory of computer science? I'm not sure.

Is information theory relates to the physics of information?

3 Comments

2024/02/12

00:25 UTC

00:25 UTC

30

I don't know if this is true but I feel like mathematicians have a smaller research output and citations compared to physicists, biologists etc. Actually this is understandable but does that make you feel unproductive?

14 Comments

2024/02/11

20:38 UTC

20:38 UTC

8

Hey everyone!

As a high school kid in the early 2000s I enjoyed the way math was portrayed in the show Numb3rs and it got me interested. I didn't pursue math professionally, but I ended up in a field that uses set theory a whole lot.

Recently, I have been rewatching the series and I've found some mathematical inconsistencies (expected because it's a show first) but that got me thinking about what the math community thinks about Numb3rs and if there are any other math heavy shows that you guys would recommend

16 Comments

2024/02/11

19:16 UTC

19:16 UTC

101

So I've been noticing that when mathematicians do a presentation they typically use this distinct (kind of ugly) presentation sofware (link to a picture of it being used: https://imgur.com/a/gfXIeqs). What is it called and why do they do it?

62 Comments

2024/02/11

19:11 UTC

19:11 UTC

8

I am following some material on the SVD derivation, and I am struggling to follow the logic that I keep seeing about the "dyadic form" of SVD.

The dyadic form represents SVD as the sum of rank 1 matrices, formed from the outer products of the left & right singular vectors.

However, each derivation I can find seems to take a liberal step in getting from the matrix product of UDV to the sum of outer products u_i * v_i'.

Examples:

Ignoring the sigmas for now, I cannot follow how the matrix product of [u_i ] [—v_i'—] (i.e. columns of U, rows of V') leads to the sum of outer products of (u_i v_i). I am used to thinking of matrix product UV' as rows of U * columns of V', so I presume I am stuck on a reversal of this notation more than anything.

If anybody could clear this up or explain the matrix product in terms of the outer product, I would appreciate it. Thank you.

5 Comments

2024/02/11

18:36 UTC

18:36 UTC

9

Hello,

I have done a fair amount of algebraic topology, and this semester doing my bachelor thesis in algebraic K-theory. I have heard at several opportunities of instances where geometric and differential topology are very close to algebraic topology. For example cobordisms as a generalised (co)homology theory and Wall's finiteness obstruction for example.

I am wondering, if these are one of examples, or if it possible to do a lot of interesting stuff in differential/geometric topology using mostly the tools of algebraic topology (i.e abstract algebraic machinery capturing properties of the space)?

If so, does anybody have any book recommendations for these topics, which would come after a good amount of knowledge in algebraic topology. All the books I have found so far dont assume in depth knowledge of algebraic topology, and so don't scratch this itch of algebraic topology used in very geometric/differential setting.

Any opinion on the matter is welcome, big picture stuff, specific examples, etc.

12 Comments

2024/02/11

17:40 UTC

17:40 UTC

127

I realized this morning that I implicitly assumed that contradiction proofs always imply a direct proof, but I’m not sure if it must be the case logically. The toy example I had in mind was “If an unstoppable force meets an immovable object, then it will shift it.” I’m not sure if proving the negation is false makes it possible to prove the statement is true.

46 Comments

2024/02/11

17:22 UTC

17:22 UTC

13

11 Comments

2024/02/11

16:26 UTC

16:26 UTC

155

At my university it seems par for the course that none of the PhD students focusing on pure math and theory use any programming at all. Most of the work seems pen and paper. How true is that across the board, and what has been your experience?

77 Comments

2024/02/11

13:18 UTC

13:18 UTC

42

One of my exams consists of a 20 minute oral discussion (10 minutes presentation, 10 minutes questions) answering a question related to math, physics or both. I have 20 minutes before the exam to write down any formulae, draw any graphs, or write down notes, but I can prepare the question I'll be answering in advance.

For example, a possible question could be: "How can logarithms be useful for modelling seismic intensity?" or "What phenomena can we really represent using the Normal Law?" (Though I personally don't find these two very interesting).

Moreover, the question needs to be understandable enough to present it to a member of the jury which has no mathematics/physics background.

I am having some trouble finding interesting and unique questions which could grab the attention of a non-STEM person, and which I could feasibly answer in the duration of my presentation. Any ideas? Thanks.

13 Comments

2024/02/11

12:59 UTC

12:59 UTC

4

I'm a second-year maths student in Australia going on exchange to Europe in August. I'm travelling around Europe with some mates in July and then have to kill time before uni starts in late August. I came across the IMC - it sounds interesting and it fits perfectly in my schedule.

I don't have much experience with maths competitions, but I've taken courses in all the fields covered by the competition, aside from complex analysis (even if there is a complex analysis question on the exam, that will be the least of my problems). If I were to go, I'd obviously practice so I don't completely flop, although that'll probably happen anyway.

Has anyone done the in-person competition, and if so would you recommend it? Also do you need some prior results in maths competitions to compete in-person?

3 Comments

2024/02/11

11:28 UTC

11:28 UTC

3

Occasionally someone asks me what do people like of mathematics, why do they find it enjoyable or even funny. Usually I try to introduce them to recreational mathematics and/or some classical problems, and to do this I let them explore the cut-the-knot website. It's a great resource, I have seen people falling in love with it more than once (and I love it first). Sadly, the website has not been updated since its creator and owner passed away in July 2018.

Are there any other online resources similar to it that I can use? Finding other cool stuff to have fun and learn with would be awesome both for me and for other people. The longer the list can become, the best! Thank you!

1 Comment

2024/02/10

13:23 UTC

13:23 UTC

11

Sorry if this is a really random question. Are there any instances of where calculus (that too, basic calculus, like differentiation/definite integration) was/is being used for some sort of modern-day problem or just in a way you wouldn't expect. I'm defining 'expected' uses as stats/finance/economics things and also kinematics.

17 Comments

2024/02/11

02:27 UTC

02:27 UTC

8

2 Comments

2024/02/10

12:47 UTC

12:47 UTC

107

For context, I'm a theoretical/computational physicist and I had a conversation a couple of days ago about how physical 'intuition' and problem solving is just a matter of constructing a series of mental maps over time as you work on/solve hard problems. For instance, if I had to model a crazy physical system, my first line of attack would be guessing a Lagrangian from things I knew about it's dynamics, or look for symmetries or try constraining it with energy conservation.

I personally have a undegraduate degree in math and most of the core classes of a math PhD, though I mainly did analysis and diffeq -- so I can see the beginnings of constructing similar attack strategies for theorems in those. But number theory completely defeats me, i.e I can follow a proof if I work through it but I have no intuition whatsoever on how you would start formulating a proof that some crazy series ends up being pi/2 or similar. For instance, take a series like this (I know it's one of Ramanujan's results and his intuition is famously inscrutable), how would you start attacking it? And, how do you think someone like Ramanujan would've come up with it in the first place? What I'm asking is the sequence of ideas/breakthroughs that might've led to the result. For instance, in relativity, I can explain how something like the equivalence principle made sense to Einstein and how he might have come up it (in fact, that's how we teach general relativity to grad students!).

PS: This is my first post on r/math, so if something like this is against the sub's rules, I'm happy to delete.

11 Comments

2024/02/11

01:03 UTC

01:03 UTC

21

In the past couple of days, I have seen a few social media mathematicians talk about Parabolic Trig Functions. Michael Penn has this video which follows this paper. I've also seen a number theorist on TikTok try to come up with something similar. While obviously good approaches, I don't really think these parameterizations should be considered "trig functions". And this is for much the same reason that we parameterize a hyperbola by area rather than arclength - the functions parameterized by arclength simply don't have nice properties.

The thing to note is that both circular and hyperbolic trig functions are basically homomorphisms from the additive reals to the conic. The circular trig functions are a homomorphism to planar rotations, and the hyperbolic trig functions are a homomorphism to real valued multiplication (this can be seen most clearly when we look at (e^(u),e^(-u)) as the "trig functions" for the hyperbola xy=1). When viewed over complex numbers, planar rotations get absorbed in the arithmetic of multiplication which is why Euler's formula relates something clearly multiplicative (e^(x)) with the circular trig functions. Circles are just larval hyperbolas afterall.

Consequently, a "true" trig function for the parabola should act as a homomorphism from the additive reals to some other arithmetic natural to the parabola. This natural arithmetic on parabolas is actually addition. In fact, if (a,b) and (c,d) are points on y=x^(2), then we can add them through the formula

- (a+b) + (c+d) = (a+b, b+d+2ac)

That this is "natural" follows from the fact that there is a geometric way to reproduce both the circular and hyperbolic arithmetic and this is what you get when you apply it to the parabola (eg). There is then a very clear homomorphism from the reals to the parabola given by x -> (x,x^(2)). This means that we get a very disappointing result in that if cosp(x) and sinp(x) are the parabolic trig functions, then we have cosp(x)=x and sinp(x)=x^(2). These obey they additive relationship

- cosp(x+y) = cosp(x) + cosp(y)
- sinp(x+y) = sinp(x)+sinp(y) + 2cosp(x)cosp(y)

If you know Lie Groups, then this geometric construction makes the parabola a Lie Group and this map *is* the exponential map from the Lie Algebra, just as it works with the other circular/hyperbolic trig functions. So, not a super fun result and ultimately probably disappointing, but I think it fits the parameters more naturally than these other constructions which start by defining the parameter first rather than finding the functions with certain additive properties. I haven't found a satisfying elementary geometric interpretation for this "parabolic angle" like the area under a curve. The only real things are the slope of the line from (0,0) to the point on the parabola, or just the distance from the point to the x-axis both of which seem pretty meh. (A disadvantage the other approaches do not have.)

6 Comments

2024/02/10

22:48 UTC

22:48 UTC

3

Hi, I’m trying to learn about Thompsons Groups F T and V, I have found loads of introductory material on F but haven’t stumbled across similar for T. Does anybody know of any good papers/articles/books that i could read? (I should say I have read Cannon-Parry-Floyd but i couldn’t get on with it very well)

1 Comment

2024/02/10

14:44 UTC

14:44 UTC

143

I'm talking about the linear algebra that could be encountered at an undergraduate level. I know that "difficult" is subjective, but what is the topic that you found most challenging to understand/to do exercise of? These days I have read about (not studied seriously yet, I will within two weeks) scalar products and stuff about orthogonal/symmetric matrices, and it looks really confusing and intimidating at first sight, the exercises particularly. I was just curious to know if you had similar experiences and what you found most challenging.

177 Comments

2024/02/10

19:59 UTC

19:59 UTC

343

That's me whenever I give a quick read through my thesis.

52 Comments

2024/02/10

16:48 UTC

16:48 UTC

30

There is no area of math that will ever be "complete". After a few millenia of work in euclidean geometry, what are the problems that mathematicians are concerned right now?

I'm really talking about euclidean geometry here. Not topology, not non-euclidean geometry. Just the good old ruler and compass math.

The last i saw something on this area was on a Galois Theory book showing how we can't make a regular heptagon using just a ruler and compass. Also some other things like how we can't trisect an angle or square the circle.

So this made me wonder, what else are we doing in this area with our modern math?

24 Comments

2024/02/10

14:59 UTC

14:59 UTC

3

I am currently coding up a problem, and part of it involves sampling indices 0, ..., n-1 uniformly at random. For powers of U=2^(32) this is usually available from PRNG sources. However, for other n you need to transform.

A quick way to do it is to take the input u mod n, but that introduces bias when n does not divide 2^(32) evenly, and it gets bad when n is big. With rejection sampling, you can get back an unbiased version by throwing out values in the last incomplete block, and repeating until you get a good value. However, you're throwing away a lot of entropy when you do this, along with what you already lose by condensing from 2^(32) states down to n.

I came up with a little algorithm (which probably already exists) to preserve the extra entropy and pass it into subsequent draws.

For a given n, k = U - U % n can be used as a break point to divide the sampling into two cases. Conditional on the event u < k, u % n is uniform on 0, ..., n-1, with u / n being independently uniform on 0, ..., U / n - 1. Likewise, conditional on the event u >= k, u-k is uniform on 0, ..., U-k-1. In either case, we have an auxiliary random draw b from 0, ..., B-1 where either B = U / n or B = U-k. If B >= U, we can use that draw to replace one PRNG call. If B < U we can take that extra draw and incorporate it to a new draw u', v = u' * B + b will then be uniform on 0, ..., U * B-1, and the procedure repeats from the start until you get a draw which does not reject.

I'm pretty sure this will, on average, achieve the entropy limit, but I haven't proven it. Care has to be taken when you're trying to get this to work with capped integers, to avoid overflows. It can still be done.

3 Comments

2024/02/10

11:03 UTC

11:03 UTC

18

If I have experience with analysis and pde theory/functional analysis, what is a good way to start learning numerical analysis and finite element method for pde’s and being able to implement it? What kind of software is used? (I only know basic Java and python) Also, does experience with pure analytic pde theory help with learning computational pde?

9 Comments

2024/02/10

03:45 UTC

03:45 UTC