# Questions tagged [gm.general-mathematics]

Questions about mathematics which don't fall into the other arXiv categories. If you have a general question about mathematics but it is not research level, it's off-topic but it might be welcomed on Mathematics Stack Exchange.

320
questions

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### Smale's view of mathematical artificial intelligence

This snippet is from Smale's paper Smale, Steve (1999). "Mathematical problems for the next century". In Arnold, V. I.; Atiyah, M.; Lax, P.; Mazur, B. (eds.). Mathematics: frontiers and ...

12
votes

14
answers

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### Oddities of evenness

Being initially a little bit perplexed by the observation that the possibility of calculating vertex potentials $\lbrace\pi_1,\dots,\pi_n\rbrace$ for weighted cycle graphs $C_n,\,2\lt n$ such that the ...

51
votes

7
answers

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### Are there any fields of academic mathematics whose epistemic status as math is controversial within the academic community?

String theory (and related areas of purely theoretical quantum gravity, like loop quantum gravity) has a unique position within the academic physics community. Many academic physicists don't really ...

6
votes

1
answer

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### How to solve recurrence relation with 2 variables？

I have the following recurrence relation and boundary condition?
$$
f(n,m) = \frac{\alpha n}{n+m} f(n-1,m) + \frac{\beta m}{n+m} f(n,m-1) + 1
$$
$$
f(n,0) = \frac{1-\alpha^{n+1}}{1-\alpha}, f(0,m) = \...

21
votes

8
answers

4k
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### Examples of bad notation and its consequences [closed]

An example of bad mathematical notation that comes in my mind and has caused complications throughout history is the notation for imaginary numbers. The original notation used to represent imaginary ...

6
votes

0
answers

460
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### A new and subtle order-theoretic fixed point theorem

Sometimes a very simple argument appears out of the blue and overturns a subject. It is not based on pre-existing theory and heavy involvement in such a theory is actually a handicap in finding such ...

-5
votes

1
answer

367
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### How much would a mathematician cost? [closed]

Recently our department lost one of the best professors who was attracted by a better University. If we were a football club, and he were a leading player, we would receive many millions of ...

55
votes

10
answers

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### What do you do when you're stuck?

I'm pretty sure almost all mathematicians have been in a situation where they found an interesting problem; they thought of many different ideas to tackle the problem, but in all of these ideas, there ...

12
votes

7
answers

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### Books containing new results

In Endless controversy about the correctness of significant papers, Denis Serre writes:
The research community is able to point out incorrect statements, at least among those which have some ...

6
votes

3
answers

597
views

### How do I solve the following definite integral (preferably by an asymptotic method)?

$$ \int_{1}^{\infty} \frac{\sin^2 (\mu \sqrt{x^2 -1})}{(x+1)^{\frac{9}{2}} (x-1)^{\frac{3}{2}}} \,dx $$
Note: $\mu$ here is an extremely small constant.
I have tried:
Estimating the integral by ...

46
votes

2
answers

4k
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### Well known theorems that have not been proved

I believe that there are numerous challenging theorems in mathematics for which only a sketch of a proof exists. To meet the standards of rigor, a complete proof of these theorems has yet to be ...

11
votes

9
answers

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### What are examples of problems we know how to solve for primes (or prime powers), but not for composites?

I am interested in seeing examples of research problems which fall into one of the two following categories:
A problem which is solved in the case of primes (or prime powers), but which remains open ...

-4
votes

1
answer

254
views

### Limit of recursion relation

Consider the sequence of functions $\{F_n\}_{n\in \mathbb{N}}$, where each $F_n$ is defined on $\{0,...,n\}$ by recurrence of the following form: $$ F_n(0)=3 \textrm{
and }F_n(k)=\frac{1}{k^2}+\frac{\...

41
votes

11
answers

4k
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### Topology in non-mathematical literature

A great piece of knowledge that I heard from a talk of Robert Ghrist, is that one of the earliest instances of non-trivial manifolds (i.e. of dimension higher than 2) appears in Dante's Paradise, ...

5
votes

0
answers

391
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### Bourbaki-Witt in a textbook, other than in logic?

The Bourbaki-Witt theorem states that, in a chain-complete poset, the subset $X$ generated by an inflationary monotone function $s$ from the least element and joins of chains satisfies
$$ \forall x,y\...

1
vote

0
answers

74
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### Invariance signature in infinite dimension

Let $V$ be an infinite dimensional vector space and suppose we have a smooth family $\{g_t\}_{t\ge 0}$ of symmetric bi-linear forms such that:
$g_0$ is positive-definite
$g_t$ is non-degenerate for ...

0
votes

0
answers

32
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### Likelihood ratio when true distribution is known

Let $X_{i}, i=1,...,N,$ follow a distribution $p(x)$. Now, we calculate the likelihood ratio between $p$ and a different distribution $q$ using samples $X_i$:
$$
\frac{p(X_i)}{q(X_i)},\ \text{or}\ \...

5
votes

0
answers

307
views

### What's with the speaker's initial thing?

If you've ever been to a math talk (at least in pure maths in the UK) you've probably seen something like this written:
Theorem (E.--Johnson--Smith, 2022+) The XYZ conjecture is true.
At some point ...

1
vote

0
answers

190
views

### On the definition of "natural" in Mathematics [duplicate]

Often in mathematics we found objects which are qualified as "being natural". The first example appears in the vector space $\mathbb R^n$, where we say that we have the "natural basis&...

6
votes

1
answer

247
views

### Classification results

A typical classification result for a class $C$ of objects looks like that:
Theorem. Each object of $C$ is isomorphic to one object of the following list: [insert list here].
Examples are the ...

0
votes

0
answers

52
views

### Enumeration of regions between a set of curves

Consider a set of $N$ tuples $\{\alpha_n,\beta_n\}$ where $\alpha_n,\beta_n\in[0,2\pi)$ and functions
$$f_n(x,y)=\cos(\alpha_n-x+y)+\delta\cos(\beta_n-x-y)$$ where $0<\delta<1$ and $x,y\in[0,\pi)...

5
votes

0
answers

171
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### Generalization of IMO5 from 1987

The following question appeared as question 5 on the IMO in 1987:
Prove that for all $n \geq 3$ one can find $n$ distinct points on the Euclidean plane with the property that the distance between any ...

3
votes

1
answer

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### Novel examples, proofs or results in mathematics from arithmetic billiards

The goal of the post is get a repository of mathematical results, proofs or examples by users of the site, arising from arithmetic billiards in number theory, analysis, geometry,….
Wikipedia has an ...

9
votes

1
answer

357
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### Why are discreteness and smoothness in physics inversed with respect to geometry?

In a closed (say differentiable) Riemannian manifold you see only continuous features when looking at small neighbourhoods of points. From afar,
discrete features appear ((co)homology, closed ...

5
votes

1
answer

321
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### Question on pure mathematics helping climate change research

While I am a pure mathematics tenured professor, still at a relatively young age, and fairly passionate about my area of research, I cannot help but feel that it may be more useful to humanity if I ...

38
votes

5
answers

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### Is spherical trigonometry a dead research area?

When I was an undergrad, the field of spherical trigonometry was cited as a once-popular area of math that has since died. Is this true? Are the results from spherical trigonometry relevant for ...

48
votes

14
answers

12k
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### Is amateur research in mathematics viable?

After a long reflection, I've decided I won't go to graduate school and do a thesis, among other things. I personally can't cope with the pressure and uncertainty of an academic job.
I will therefore ...

1
vote

1
answer

156
views

### Computation involving Gauss integer function

I have used mathematica to test following equation is true
\begin{equation}
\sum_{a,b=0}^{m-1} \left[\frac{a+b n}{m}\right] = \frac{n m^2}{2} -\frac{nm}{2}
\end{equation}
where $[x]$ is the floor ...

2
votes

2
answers

616
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### What is the most "informative" Yes/No math question you know? [closed]

Imagine that alien civilization contacted you and offered to answer one math question. This should be a Yes/No question (so, you cannot ask for a million-digit binary string encoding the answers to a ...

0
votes

0
answers

122
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### What is the meaning of PUP in remarks?

I see the remarks in Gowers’s The Princeton Companion to Mathematics. What is the meaning of the abbreviation ‘PUP’? Never seen before…

10
votes

1
answer

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### Can you get any natural number from 4 by performing given operations?

You can perform the following operations on numbers:
divide the number by 2,
add 0 or 4 at the end of the number.
Can you get any natural number from 4 by performing only these operations?
So far I ...

9
votes

6
answers

1k
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### Surprising applications of the theory of games?

I am currently studying the applications of games in quantum information theory and related fields and I am aware of its uses in places like model theory and set theory. So I was curious, what are ...

1
vote

1
answer

144
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### How to eliminate angle in a Glissette equation of carried point of a line sliding along two lines not at right angles

Glissettes are the curves trances out by a point carried by a curve, which is made to slide between given points or curves. My problem specifically include a line which slides between two fixed lines (...

142
votes

32
answers

12k
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### Conceptual reason why the sign of a permutation is well-defined?

Teaching group theory this semester, I found myself laboring through a proof that the sign of a permutation is a well-defined homomorphism $\operatorname{sgn} : \Sigma_n \to \Sigma_2$. An insightful ...

9
votes

1
answer

531
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### How far away can we get by multiple rounding and unit change?

This question is inspired by xkcd #2585 (Rounding):
Let $u_0,\ldots,u_n$ be positive real numbers (we can assume w.l.o.g. that $u_0=1$) or “units”.
Consider the following directed graph: its vertices ...

2
votes

1
answer

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### Algorithm for compact polynomial expressions

Sometimes an ugly polynomial (perhaps in several variables) can be expressed as a small sum of much simpler polynomials. Can this be done algorithmically? More precisely:
Is there a reasonable
...

4
votes

0
answers

294
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### Récoltes et Semailles for the non (algebraic-) geometer

With the recent publication of Grothendieck's Récoltes et Semailles, I've been umm-ing and ah-ing about whether to get a copy, if only for the "soaking nuts" story. My French reading is ...

10
votes

3
answers

815
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### Progress in robustifying mathematics - i.e. making mathematical theorems robust to small changes in hypotheses

The idea of making a mathematical theorem robust to small changes in its hypotheses has been known for some time. In areas such as group theory reasonable progress has been made leading to the theory ...

14
votes

4
answers

842
views

### What are some examples of understanding a space by studying the functions on this space?

In Quantum theory, groups and representations, Peter Woit writes:
A fundamental principle of modern mathematics is that the way to
understand a space $M$, given as some set of points, is to look at $...

5
votes

1
answer

263
views

### How the solve the equation $\frac{(a+b\ln(x))^2}{x}=c$ [closed]

I need to solve the equation
$$\frac{(a+b\ln(x))^2}{x}=c$$
where $a$, $b$, and $c$ are given. It is known that $a$ and $b$ are fixed and satisfy some condition such that the left hand side is ...

1
vote

0
answers

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views

### Drawing a 3D object in a 3D environment, and converting to math [closed]

So I have been granted a free time and I want to work on a project but first I had to research.
As we know, lines have infinite points, and with lines, we can create infinite shapes. I want to let ...

4
votes

0
answers

285
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### What does it mean to solve an equation?

Assume that we want to find all integer (or rational) solutions to the polynomial Diophantine equation
$$
P(x_1,\dots,x_n) = 0
$$
where $P$ is a polynomial with integer coefficients. Do we have a ...

6
votes

0
answers

232
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### Mathematical questions or areas amenable to AI [duplicate]

This question regards the new paper "Advancing mathematics by guiding human intuition with AI" by Davies et al. (Nature, 2021) (DOI link in open access) in which researchers at Deepmind ...

12
votes

2
answers

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### Is it ever unnecessary to mathematically formalize a concept?

From my understanding, mathematics sometimes gives rise to new physical/tangible laws and the converse is also true. In particular, physical phenomena give rise to new mathematics.
In all of the cases ...

31
votes

2
answers

992
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### Math videos containing real time rough thinking

I find this vlog experiment of Gowers very brave, and I think his idea of having examples of real-time mathematical thinking by experts can be very encouraging for young mathematicians, who imagine ...

41
votes

18
answers

4k
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### Results in linear algebra that depend on the choice of field

Linear algebra as we learn it as undergraduates usually holds for any field (even though we usually learn it for the complex, or real, numbers).
I am looking for a list of concepts, and results, in ...

1
vote

0
answers

110
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### Closed form for $\sum _{k=1} ^n \frac k 2 \,\operatorname{sgn} \left( \frac 1 {k^2} + \cos \frac {2\pi n} k-1 \right)$ [closed]

I am looking for the closed form of
$$\sum _{k=1} ^n \frac k 2 \,\operatorname{sgn} \left( \frac 1 {k^2} + \cos \frac {2\pi n} k-1 \right) \ .$$
Wolfram Alpha cannot do this for me, so I am forced to ...

4
votes

2
answers

465
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### Theorems with finite sets of exceptions

Exceptions are interesting. Sometimes, they're also important. If a theorem with exceptions is important for a subject, there are liable to be many corollaries of the form "either this is true... ...

62
votes

27
answers

8k
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### Golden ratio in contemporary mathematics

A (non-mathematical) friend recently asked me the following question:
Does the golden ratio play any role in contemporary mathematics?
I immediately replied that I never come across any mention of ...

11
votes

3
answers

544
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### Any hints on how to prove that the function $\lvert\alpha\;\sin(A)+\sin(A+B)\rvert - \lvert\sin(B)\rvert$ is negative over the half of the total area?

I have this inequality with $0<A,B<\pi$ and a real $\lvert\alpha\rvert<1$:
$$ f(A,B):=\bigl|\alpha\;\sin(A)+\sin(A+B)\bigr| - \bigl| \sin(B)\bigr| < 0$$
Numerically, I see that ...