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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2 votes
0 answers
130 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?
13 votes
2 answers
985 views

Are there any organized websites for seminar/conference videos?

These days, there are many conference centers and universities recording seminars and conference talks and make them available on the web. Some examples: http://www.fields.utoronto.ca/video-archive ...
4 votes
2 answers
382 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
193 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 ...
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 ...
2 votes
1 answer
260 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 ...
106 votes
32 answers
14k views

Special rational numbers that appear as answers to natural questions

Motivation: Many interesting irrational numbers (or numbers believed to be irrational) appear as answers to natural questions in mathematics. Famous examples are $e$, $\pi$, $\log 2$, $\zeta(3)$ etc. ...
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 ...
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 ...
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 ...
4 votes
2 answers
509 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....
90 votes
8 answers
12k views

Books on music theory intended for mathematicians

Some time ago I attended a colloquium given by Princeton music theorist Dmitri Tymoczko, where he gave a fascinating talk on the connection between music composition and certain geometric objects (as ...
93 votes
20 answers
10k views

Short papers for undergraduate course on reading scholarly math

(I know this is perhaps only tangentially related to mathematics research, but I'm hoping it is worthy of consideration as a community wiki question.) Today, I was reminded of the existence of this ...
77 votes
15 answers
13k views

Each mathematician has only a few tricks

The question "Every mathematician has only a few tricks" originally had approximately the title of my question here, but originally admitted an interpretation asking for a small collection ...
163 votes
46 answers
31k views

Every mathematician has only a few tricks

In Gian-Carlo Rota's "Ten lessons I wish I had been taught" he has a section, "Every mathematician has only a few tricks", where he asserts that even mathematicians like Hilbert ...
1 vote
1 answer
174 views

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 (...
160 votes
37 answers
15k views

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 ...
7 votes
6 answers
1k views

Another chicken or egg: sequence or series

This is a side question which is more motivated by teaching than research. First, I am trying to convince myself that sequences appear before series (as numerical approximations to "interesting" ...
8 votes
1 answer
793 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
367 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')\...
78 votes
49 answers
43k views

Examples of interesting false proofs

According to Wikipedia False proof For example the reason validity fails may be a division by zero that is hidden by algebraic notation. There is a striking quality of the mathematical fallacy: as ...
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 ...
2 votes
1 answer
215 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 ...
1 vote
1 answer
329 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
158 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!
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 ...
18 votes
8 answers
2k views

Computationally challenging integer sequences

I wonder what are the examples of integer sequences, where only few elements are known and the researchers are still actively looking for the new terms. I think this discussion might be a good ...
90 votes
11 answers
13k views

What are possible applications of deep learning to research mathematics?

With no doubt everyone here has heard of deep learning, even if they don't know what it is or what it is good for. I myself am a former mathematician turned data scientist who is quite interested in ...
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&...
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 ...
-1 votes
1 answer
120 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)\...
2 votes
0 answers
334 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 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 ...
5 votes
0 answers
515 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 ...
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
253 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{\...
14 votes
6 answers
4k views

How to refer to a “theorem” that you have shown to be wrong

I am writing a paper about a flaw that I found in a published paper. There, the statement is called “Theorem 2”. In my paper, I am reproducing the other paper’s definitions, and steps leading towards ...
41 votes
23 answers
8k views

Theorems with many distinct proofs

I was told that whenever one learns a new technique, it is a good idea to see if one can prove a well-known theorem using the new technique as an exercise. I am hoping to build a list of such theorems ...
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 ...
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 ...
-4 votes
1 answer
425 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 (...
5 votes
0 answers
591 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\...
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 ...
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 ...
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 ...
163 votes
9 answers
28k views

Endless controversy about the correctness of significant papers

In principle, a mathematical paper should be complete and correct. New statements should be supported by appropriate proofs. But this is only theory. Because we often cannot enter into the smallest ...
6 votes
1 answer
426 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) = \...
7 votes
8 answers
4k views

Beautiful theorems with short proof [closed]

I like to ask for beautiful mathematical theorems with short proof. A proof is short in my sense if it is at most one page assuming basic notations and very basic results a second year student will ...
78 votes
21 answers
17k views

Is rigour just a ritual that most mathematicians wish to get rid of if they could?

"No". That was my answer till this afternoon! "Mathematics without proofs isn't really mathematics at all" probably was my longer answer. Yet, I am a mathematics educator who was ...

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