# 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.

346
questions

2
votes

1
answer

61
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### Subspaces of $C_0$ on which $p$-norm are equivalent?

I have a question concerning the generalization of the following fact.
Let $E = C^0([0,1],\mathbb{R})$ endowed with the $\|.\|_\infty$ norm. One can show that if $F$ is a subspace of $E$ for which ...

-1
votes

0
answers

34
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### Parametric form of ellipsoid

I need help solving a problem I encountered while simulating ellipsoids in space.
I have a running implementation of generating rotational ellipsoids.
The problem is that I generate a grid of points ...

2
votes

0
answers

140
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### Search publications for LaTeX code

Is there a way to search the literature for specific instances of formulas (and variants), perhaps using $\rm\LaTeX$ code?

4
votes

2
answers

388
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### Are there any undergraduate-friendly research areas in algebra? [closed]

I don't know if this question is more appropriate for the academia stack exchange, but I'm posting it here because it's more closely related to math itself.
I'm not actually an undergraduate, I'm a ...

6
votes

1
answer

198
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### Results with a flavor “every automorphism of automorphisms is inner”

It seems that there are a number of results which take more or less the following form: let $X$ be some (specific) kind of structure, let $Y$ be the group of automorphisms of $X$ or perhaps ring of ...

2
votes

1
answer

287
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### Where can I access American Mathematical Monthly problems given an index?

I don't know if this is the appropriate website to ask, so I understand if this post gets closed. I want to explore (and maybe solve) some of the currently-unsolved problems submitted by readers on ...

49
votes

1
answer

8k
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### What mathematical problems can be attacked using DeepMind's recent mathematical breakthroughs?

I am a research mathematician at a university in the United States. My training is in pure mathematics (geometry). However, for the past couple of months, I have been supervising some computer science ...

0
votes

1
answer

104
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### 4 triangular faces 6 vertices not tetrahedron [closed]

I have made a solid and would like to know its' name, volume and related formulas. It is made using a flat potato chip bag. The end opposite the factory seal is sealed perpendicular to the factory ...

0
votes

0
answers

52
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### Showing that the congruence speed of any integer exponentiation $a^b$ is constant and $\geq 1$ iff $a>1$ is a multiple of $10$

Years ago, I defined the "congruence speed" (radix-$10$) of the integer tetration $^{b}a$ as $V(a,b)$, which is the number of the new(!) rightmost digits that freeze when we move from $b \in ...

4
votes

2
answers

513
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### Zentralblatt MATH volume numbering

Recently, I learned how to read some of codes that appear on specific pages in zbMATH Open, formerly known as Zentralblatt MATH.
For example, in the above review, the circled code "Zbl 1218....

8
votes

1
answer

799
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### Examples of ZBMath reviews that motivated you to read the paper

This is community wiki question.
I will be writing my first review for ZBMath. I would like to take some suggestion through examples.
In general, abstract is too small and introduction is too lengthy ...

6
votes

1
answer

368
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### When are the chirp signals orthogonal?

Assume that we have two bounded-time chirp signals,
\begin{align}
x(t)&=\exp\Big(j\pi(\alpha t^2+\beta t+\gamma)\Big),\quad 0\leq t\leq T,\\
y(t)&=\exp\Big(j\pi(\alpha' t^2+\beta' t+\gamma')\...

0
votes

1
answer

103
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### Can orientation preserving diffeomorphism in $\mathbb{R}^d$ be presented by flowmap of dynamical systems? (time-varying case)

Because flowmaps are homeomorphic maps on a compact domain $\Omega$, I was wondering if there is any literature that proves that diffeomorphism $\Phi(x)$ can be expressed as a flowmap of a certain ...

2
votes

1
answer

221
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### What's the lower bound of the correlation coefficient?

Suppose a random variable $X \in \mathbb{R}$ follows a discrete distribution $p$ and takes $n$ values. We assume $E[X]=0$ and $|X|\le M$, where $M$ is a constant. Given a smooth and monotonic ...

1
vote

1
answer

339
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### Book on analysis and algebra at the undergraduate level [closed]

I am writing this post because I would like to know what are your references concerning math book showing the interplay between analysis and algebra at an undergraduate-advanced undergraduate level.
...

-2
votes

1
answer

164
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### If a continuous function is differentiable at a point, is it differentiable in some neighborhood around that point? [closed]

This seems like it should be true but I was wondering if anyone could prove it. Thanks!

2
votes

1
answer

112
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### Any theorem shows that flowmap $\phi_{\sum_{i=1}^n a_i f_i(x)}^\tau$ can be approximated by $\phi_{f_{\theta(t)}(x)}^{\tau'}$?

Given a control family $F:=\{f_1,\dotsc,f_n\}$, and $\phi_f^\tau(x)$ is the flowmap of the dynamical system
$$
\begin{cases}
z'(t)=f(z),\\
z(0)=x,
\end{cases}
$$ at end time point $\tau$.
Suppose $a_i&...

0
votes

1
answer

101
views

### What's the lower bound for this quantity?

Suppose $p$ is a discrete distribution with $n$ values and the random variable $x$ satisfies $\mathbb{E}_p[x] = 0$ and $|x| < \infty$. Given $\alpha \in (0,1)$, does there exist a lower bound for ...

2
votes

0
answers

338
views

### How to handle a research identity crisis

I have studied applied math and got a PhD (3yrs) in that field with applications in fluid dynamics. Then in my first postdoc (1.5yrs) I did again a postdoc in applied math but studied applications in ...

0
votes

1
answer

39
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### Is the right-hand term of the autonomous dynamic system equivalent to the original system after being multiplied by a constant?

Given two dynamical systems where $f$ is lipschitz for $x$ : $\begin{cases} x'(t)=af(x),\\ x(0)=x_0,\end{cases} t\in[0,\tau]$ and $\begin{cases} z'(t)=f(z),\\ z(0)=x_0,\end{cases} t\in[0,\tau']$, and ...

26
votes

2
answers

3k
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### History of right hand rule

I am not sure if this is the right place to ask, but many mathematicians are knowledgeable and interested also in history of math, so here I am.
I am curious to know when the right-hand-rule for ...

5
votes

0
answers

517
views

### What sets are known to have cardinality equal to $\mathbb{N}$ or $\mathbb{R}$ but open as to which?

A long time ago a similar question was asked on math.stackexchange.
There are many sets which we know to be either finite or infinitely countable but do not know which cardinality specifically.
An ...

-1
votes

1
answer

123
views

### Can we use linear map to approximate lipschitz continuous function $f$ in a compact domain after some linear transform?

Suppose $f : \mathbb{R}\to \mathbb{R}$ is lipschitz continuous function , $K$ is a compact domain, for any $\varepsilon>0$, can we find $d,a\neq 0,c,w,b \in \mathbb{R}$ such that $\|df(ax+c)-(wx+b)\...

0
votes

1
answer

127
views

### Does this inequality hold for the cumulant generating function?

Suppose a random variable $X$ is zero-mean and the cumulant generating function is
$$
K\left( t \right) =\log \mathbb{E}[e^{tX}].
$$
Given any positive constant $\tau > 0$, does this inequality
$$
\...

2
votes

1
answer

256
views

### Does this KL divergence inequality hold?

Suppose $p$ and $q$ are two discrete distributions. Given a positive constant $\beta\in(0,1)$, we create a new discrete distribution $y$ such that
$$
\frac{y\left( x \right)}{p\left( x \right)}=\frac{\...

18
votes

7
answers

4k
views

### Why do infinite-dimensional vector spaces usually have additional structure?

On Mathematics Stack Exchange, I asked the following question: Why are infinite-dimensional vector spaces usually equipped with additional structure? Although it received one good answer, I feel that ...

0
votes

0
answers

59
views

### A recurrence relation with two variables

How to solve the following recurrence relation?
$$f(i,j) = 2 f(i,j-1) + (\alpha^j+\beta^j) f(i-1,j), 0<\alpha,\beta < 1$$
With the boundary condition
$$ f(0,0) = f(1,0) = f(0,1) = 1 $$
A special ...

-4
votes

1
answer

432
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### Amount of mathematical knowledge required for starting Ph.D. in pure mathematics [closed]

How much mathematics should one know before starting a Ph.D. program in pure mathematics? For example what topics one must understand well to pursue a Ph.D. in US University in Number Theory (...

11
votes

1
answer

1k
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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 ...

13
votes

16
answers

3k
views

### 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 ...

54
votes

7
answers

9k
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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

432
views

### 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
views

### 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 ...

7
votes

0
answers

588
views

### 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 ...

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votes

1
answer

467
views

### 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 ...

60
votes

10
answers

12k
views

### 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

3k
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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

699
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 ...

47
votes

2
answers

5k
views

### 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

1k
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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

289
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
views

### 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

595
views

### 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

85
views

### 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 ...

5
votes

0
answers

326
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
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### 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

262
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### 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 ...

5
votes

0
answers

176
views

### 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 ...

4
votes

1
answer

542
views

### 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

374
views

### 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 ...