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.
351 questions
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When is 4 qualitatively different than $n\leq 3$?
Inspired by When is 2 qualitatively different from 3?
Also similar to Are there mathematical concepts that exist in dimension 4, but not in dimension 3? (Math SE), but with the restriction of being ...
61
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71
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When is 2 qualitatively different from 3?
I'd like to get a list of instances in mathematics where a problem with two parameters (or some parameter set to $2$) is qualitatively different from the instance of that problem with the value set to ...
14
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6
answers
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What does keep you "doing what you do"? [closed]
I am towards the end of my Phd (with some difficultues to overcome, I can say I am really satisfied about it) and I was wondering about what to do next. There are basically two paths: academia or ...
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What are some (popular) references on variants of the classical gambler's ruin problem that exists in literature?
It is fascinating that the gambler's ruin problem which is so ubiquitous in modern probability theory (cf. the Levin-Peres text on Markov chain and Mixing Times) actually dates back to a letter from ...
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1
answer
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Specific distance between sets of points
Let us have closed curve without self-intersections,
initial point $O$ and curve parameter $t$, $0 \leq t \leq t_{\max}$ so $t(O) = 0 = t_{\max}$.
There are two sets of points on the curve, which are ...
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vote
0
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From a constraint satisfaction problem (CSP) to a sudoku grid [closed]
one of the existing methods of solvin a sudoku grid is via constraints satisfaction (CSP), but can we do the inverse ie convert a CSP problem into a sudoku grid and then solve it ?
51
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9
answers
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Examples of theorems where numerical bounds on $\pi$ played a role
This is a whimsical question, motivated purely by curiosity rather than for any application.
We are all familiar with countless mathematical results which use Archimedes' constant $\pi$ either in ...
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2
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1
answer
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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 ...
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0
answers
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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?
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4
votes
2
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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
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1
answer
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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
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1
answer
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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 ...
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53
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1
answer
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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 ...
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1
answer
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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 ...
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0
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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 ...
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2
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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....
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votes
1
answer
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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
426
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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')\...
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1
answer
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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 ...
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1
answer
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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 ...
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1
answer
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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
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1
answer
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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!
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1
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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&...
- 109
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votes
1
answer
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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 ...
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2
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0
answers
381
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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 ...
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1
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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 ...
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2
answers
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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 ...
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0
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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
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1
answer
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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)\...
- 109
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1
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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
$$
\...
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2
votes
1
answer
275
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Does this KL divergence inequality hold?
Suppose $p$ and $q$ are two discrete distributions. Given a positive constant $\beta\in(0,1)$, we create a new discrete distribution $y$ such that
$$
\frac{y\left( x \right)}{p\left( x \right)}=\frac{\...
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7
answers
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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 ...
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0
answers
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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 ...
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1
answer
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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 (...
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votes
1
answer
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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 ...
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16
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 ...
53
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?
$$
f(n,m) = \frac{\alpha n}{n+m} f(n-1,m) + \frac{\beta m}{n+m} f(n,m-1) + 1
$$
$$
f(n,0) = \frac{1-\alpha^{n+1}}{1-\alpha}, f(0,m) = \...
- 105
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8
answers
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Examples of bad notation and its consequences [closed]
An example of bad mathematical notation that comes in my mind and has caused complications throughout history is the notation for imaginary numbers. The original notation used to represent imaginary ...
8
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1
answer
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A new and subtle order-theoretic fixed point theorem
Sometimes a very simple argument appears out of the blue and overturns a subject. It is not based on pre-existing theory and heavy involvement in such a theory is actually a handicap in finding such ...
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votes
10
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What do you do when you're stuck?
I'm pretty sure almost all mathematicians have been in a situation where they found an interesting problem; they thought of many different ideas to tackle the problem, but in all of these ideas, there ...
12
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7
answers
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Books containing new results
In Endless controversy about the correctness of significant papers, Denis Serre writes:
The research community is able to point out incorrect statements, at least among those which have some ...
6
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3
answers
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How do I solve the following definite integral (preferably by an asymptotic method)?
$$ \int_{1}^{\infty} \frac{\sin^2 (\mu \sqrt{x^2 -1})}{(x+1)^{\frac{9}{2}} (x-1)^{\frac{3}{2}}} \,dx $$
Note: $\mu$ here is an extremely small constant.
I have tried:
Estimating the integral by ...
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49
votes
2
answers
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Well known theorems that have not been proved
I believe that there are numerous challenging theorems in mathematics for which only a sketch of a proof exists. To meet the standards of rigor, a complete proof of these theorems has yet to be ...
11
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9
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What are examples of problems we know how to solve for primes (or prime powers), but not for composites?
I am interested in seeing examples of research problems which fall into one of the two following categories:
A problem which is solved in the case of primes (or prime powers), but which remains open ...
-4
votes
1
answer
310
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Limit of recursion relation
Consider the sequence of functions $\{F_n\}_{n\in \mathbb{N}}$, where each $F_n$ is defined on $\{0,...,n\}$ by recurrence of the following form: $$ F_n(0)=3 \textrm{
and }F_n(k)=\frac{1}{k^2}+\frac{\...
41
votes
11
answers
5k
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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
651
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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\...
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1
vote
0
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94
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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 ...
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5
votes
0
answers
357
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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 ...
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