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.

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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 ...
Anthony's user avatar
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-1 votes
0 answers
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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 ...
Leon Archinger's user avatar
2 votes
0 answers
140 views

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?
Pace Nielsen's user avatar
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4 votes
2 answers
388 views

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 views

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 views

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 ...
ofw2jopfpo2's user avatar
49 votes
1 answer
8k views

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 ...
Ryan Hendricks's user avatar
0 votes
1 answer
104 views

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 ...
Tom Lechner's user avatar
0 votes
0 answers
52 views

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 ...
Marco Ripà's user avatar
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4 votes
2 answers
513 views

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....
xFioraMstr18's user avatar
8 votes
1 answer
799 views

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 views

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')\...
Math_Y's user avatar
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0 votes
1 answer
103 views

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 ...
li ang Duan's user avatar
2 votes
1 answer
221 views

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 ...
Jiacai Liu's user avatar
1 vote
1 answer
339 views

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 views

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!
li ang Duan's user avatar
2 votes
1 answer
112 views

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&...
li ang Duan's user avatar
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 ...
Jiacai Liu's user avatar
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 ...
Riri's user avatar
  • 31
0 votes
1 answer
39 views

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 ...
li ang Duan's user avatar
26 votes
2 answers
3k views

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 ...
Sofia Tirabassi's user avatar
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)\...
li ang Duan's user avatar
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 $$ \...
Jiacai Liu's user avatar
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{\...
Jiacai Liu's user avatar
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 ...
Joe's user avatar
  • 361
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 ...
Lili Si's user avatar
  • 95
-4 votes
1 answer
432 views

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 (...
SARTHAK GUPTA's user avatar
11 votes
1 answer
1k views

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 ...
Turbo's user avatar
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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 views

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) = \...
Lili Si's user avatar
  • 95
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 ...
Paul Taylor's user avatar
  • 8,006
-5 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 views

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 ...
Abdullah's user avatar
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 views

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{\...
José María Grau Ribas's user avatar
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\...
Paul Taylor's user avatar
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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 ...
John117's user avatar
  • 395
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 ...
Sean Eberhard's user avatar
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&...
A. J. Pan-Collantes's user avatar
6 votes
1 answer
262 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 ...
user493267's user avatar
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 ...
Stanley Yao Xiao's user avatar
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 ...
Roland Bacher's user avatar

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