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.
265
questions
0
votes
0answers
16 views
What are examples of mathematical objects that are 'constructed out of' a range of other objects but fall out of them?
What are examples of mathematical objects that are somehow 'constructed out of' a whole range of other objects but fall out of them? One example that comes to my mind is that of ordinal numbers: $\...
27
votes
16answers
4k views
Is pure mathematics useful outside of mathematics itself?
From time to time Mathoverflow allows soft questions because they are arguably best answered by active mathematicians and they can benefit other mathematicians/PhD students/math undergraduates. I ...
4
votes
0answers
178 views
Is my equation for finding the length of a curve correct?
My son has been working on equations for finding length of the length of a curve without any resources (computers, calculators...) while in prison. He is wondering if he is on the right path. Sadly my ...
1
vote
1answer
59 views
the sum of triangular functions
I have a sum of $N-1$ triangular numbers to calculate and it is $$s = \sum_{q=0}^{N-1} \frac{W_q \left(\cos\left(\dfrac{n\pi q}{N}\right) + \cos\left(\dfrac{(n-1)\pi q}{N}\right)\right) }{ 1 + \cos\...
2
votes
1answer
718 views
Examples of reputable journals in mathematics without impact factor? And is it good to publish in them?
I came across a journal which is reputable in terms of not being listed in Beall's list, but which according to scimagojr and clarivate does not have an impact factor. The journal is Journal of ...
4
votes
0answers
85 views
What do you call such a relation between subsets in a poset
Consider a poset $(X, \geq)$. Let's define a new relation $\succsim$ on subsets of $X$: for $A, B\subseteq X$, say $A\succsim B$ if for any $a\in A$ and any $b\in B$, we have $a\geq b$.
Does such a ...
67
votes
9answers
6k views
What are possible applications of deep learning to research mathematics?
With no doubt everyone here has heard of deep learning, even if they don't know what it is or what it is good for. I myself am a former mathematician turned data scientist who is quite interested in ...
7
votes
0answers
171 views
Fraction of elements in $\mathbb{Z}_n$ satisfying a certain equation
From a question arising in Game Theory, I want to calculate the sequence
$$
a_n =
\max_{f_A, f_B : \mathbb{Z}_n \to \mathbb{Z}_n}
\frac{\#
\left\{
(x,y) | f_A(x) - f_B(y) = xy \mod n
\right\}}{n^2}
$$...
2
votes
0answers
165 views
Integral representation of $f(x)=0^x$ [closed]
Recently I had an argument with Luboš Motl on Quora, where he had argued that $0^0$ should be left undefined in computer algebra systems, because $x^y$ has no limit at $(0,0)$ and $0^x=0$ at all $x>...
5
votes
1answer
211 views
Get the commands history from GAP system
I am not sure whether this was asked before, but I didn't find a reference in GAP system documentation on how to print the history of the command line (Ubuntu installation).
For instance:
...
1
vote
1answer
246 views
Can the solutions to this beautiful equation always be expressed in terms of algebraic numbers?
For $n\geq1$, the largest solution to this lovely equation is a local extremum on a function related to the Fibonacci sequence:
$$\sum_{k=1}^{n} k{(-1)^{k}} \cdot \frac{\sin(\frac{k\pi}{x} )}{3+2\cos(\...
3
votes
6answers
390 views
how to prove this high degree inequality
Let $x$,$y$,$z$ be positive real numbers which satisfy $xyz=1$. Prove that:
$(x^{10}+y^{10}+z^{10})^2 \geq 3(x^{13}+y^{13}+z^{13})$.
And there is a similar question: Let $x$,$y$,$z$ be positive real ...
0
votes
2answers
88 views
Existence of functions satisfying a homogeneity condition
Do there exist a (non-trivial) globally Lipschitz continuous function $g:\mathbf{R}\to\mathbf{R}$ and a non-decreasing function $f:\mathbf{R}_+\to\mathbf{R}_+$ such that the identity
\begin{equation}
...
5
votes
1answer
237 views
Is the solution to this trig function known to be algebraic or transcendental?
This largest solution to this gorgeous equation is the first local extremum on a function related to the Fibonacci sequence:
$$x^2 \cdot \sin \left(\frac{2\pi}{x+1} \right) \cdot \left(3+2 \cos \left(\...
3
votes
0answers
140 views
Does this trig equation have a closed-form solution?
This equation came up when I was looking at the Fibonacci sequence; I adore its symmetry:
$$x^2 \cdot \sin \left(\frac{2\pi}{x-1} \right) \cdot \left(3+2 \cos \left(\frac{2\pi}{x} \right) \right) = (x-...
3
votes
1answer
112 views
What is the ideal form of an h-curve?
This question concerns mathematical modelling of the citation curve, well-known in the sciencemetry.
The citation curve (or else the $h$-curve) of an individual researcher is the vector $(c_1,c_2,\...
2
votes
0answers
70 views
Good notation for finite partial functions from $\omega$ to 2
I'm working in computability theory and need to use partial functions with finite domain from $\omega$ to 2 as approximations in my current paper. Normally this is simply done using $2^{< \omega}$ ...
0
votes
0answers
100 views
Formula from a form having one dirac delta to a form having two dirac delta
I'd like to ask at which condition I can transform a formula with one delta function to a form with two delta function.
Suppose a physical system has a quasi-continuous energy-level spectrum $E_1$, $...
42
votes
4answers
4k views
How to invoke constants badly
In a nice and witty lecture titled "how to write mathematics badly" (available on YouTube at https://www.youtube.com/watch?v=ECQyFzzBHlo&t=23s), Jean-Pierre Serre describes various ways ...
0
votes
1answer
68 views
Closed-form for recursive “geometric-like” recursion
I asked this question of MSE, but to no avail; alas, here I am.
Let $k>0$, $C\geq 1$, $\alpha \in (0,1]$, and let $(x_n)_{n\geq 1}$, be a sequence of real numbers given by the recursion
$$
x_{n+1} =...
3
votes
2answers
276 views
Are gyrogroups useful for anything else other than the Einstein velocity addition rule?
Gyrogroups were discovered by Ungar in modelling the Einstein velocity addition rule in relativity. Have they been shown to be useful elsewhere in mathematics (or mathematical physics)?
53
votes
9answers
4k views
Examples of back of envelope calculations leading to good intuition?
Some time ago, I read about an "approximate approach" to the Stirling's formula in M.Sanjoy's Street Fighting Mathematics. In summary, the book used a integral estimation heuristic from ...
1
vote
1answer
112 views
An ambitiouser binomial coefficients sum
I asked how to calculate $$\sum_{i = 0}^b(-1)^i\binom{b}{i}\binom{a+b-i-1}{a-i}$$ and got amazing answers. A bit later, however, I figured I needed something rather more complicated: I need to find ...
-1
votes
3answers
190 views
Binomial Coefficients sum [closed]
Any idea on whether or not $$\sum_{i = 0}^b(-1)^i\binom{b}{i}\binom{a+b-i-1}{a-i}$$
has a closed formula on $a$ and $b$ (and on what it is, in case it does)?
It is supposed that $b \le a$.
20
votes
14answers
4k views
Math talk for all ages
I've been asked to give a talk to the winners of a recent math competition. The talk can be entirely congratulatory, or it can contain a bit of actual mathematics. I'd prefer the latter. I'd also ...
7
votes
1answer
759 views
Do mathematicians use notebooks to keep their results these days? [closed]
Mathematicians work a lot and are usually inspired by many things. In their lifetimes they get to publish only portions of their results. There have been stories of how Gauss, Euler, Ramanujan, ...
26
votes
5answers
5k views
Has any open/difficult problem in ordinary mathematics been solved only/mostly by appeal to set theory?
We know that many (if not all) mathematical notions can be reduced to the talk of sets and set-membership. But it nevertheless sounds like a grueling task (if at all possible) to actually get advanced ...
7
votes
1answer
381 views
Abandoned notions in mathematics? [duplicate]
I'm looking for examples of abandoned or demised notions/concepts in mathematics, preferably (but not necessarily) after the age of foundations. To be clear: I'm not looking for abandoned ideas or ...
1
vote
0answers
174 views
Online courses for mathematics [closed]
I'm sorry if I'm posting this in the wrong forum. My background is in biology and medicine. I am looking to re-learn undergraduate-level mathematics, in particular discrete mathematics, calculus, and ...
37
votes
5answers
6k views
How to deal with an advisor that offers you nearly no advising at all?
I am a young PhD student (24) at a Germany university and I am not sure whether this is the right place to ask this kind of question. If not feel free to move it elsewhere or delete it completely.
...
14
votes
9answers
2k views
Tools from other disciplines useful to mathematics research?
Obviously, mathematics provides essential tools for physicists, biologists, economists, engineers and many others to use in their research. Equally obviously, physics, biology, economy and engineering ...
63
votes
22answers
8k views
Small ideas that became big
I am looking for ideas that began as small and maybe naïve or weak in some obscure and not very known paper, school or book but at some point in history turned into big powerful tools in research ...
1
vote
1answer
174 views
Examples of conjectures whose direct falsity implies different consequences than indirect falsity
Mathematics several times has statements of form
$$\mathsf{Statement A}\implies\mathsf{Statement B}$$
where $\mathsf{Statement A}$ and $\mathsf{Statement B}$ are conjectures while the implication is ...
38
votes
8answers
4k views
Counterexamples against all odds
What are some examples of conjectures proved to be true generically (i.e. there is a dense $G_{\delta}$ of objects that affirm the conjecture) but are nevertheless false?
Also, it would be cool to see ...
2
votes
0answers
134 views
Dreaming of mathematics [duplicate]
Ramanujan said that his mathematical inspirations came in dreams (https://en.wikipedia.org/wiki/Namagiri_Thayar, https://www.foxnews.com/science/100-year-old-deathbed-dreams-of-mathematician-proved-...
66
votes
14answers
10k 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 ...
147
votes
46answers
25k 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 ...
5
votes
0answers
276 views
Algebraic applications of an order-theoretic idiom of recursion
Many algebraic constructions must surely use the following observation, probably disguised as one of its proofs:
Lemma Let $s:X\to X$ be an endofunction of a poset such that
$X$ has a least element $...
0
votes
0answers
49 views
“Anti-Leibniz order”
It seems that some people use the term "anti-Leibniz order" for what I'd call the "diagrammatic order" of composition: writing $f;g$ for the composition of $f$ and $g$ instead of $g\circ f$.
(I have ...
21
votes
7answers
1k views
Examples of improved notation that impacted research?
The intention of this question is to find practical examples of improved mathematical notation that enabled actual progress in someone's research work.
I am aware that there is a related post ...
2
votes
0answers
144 views
Studying the vast world of Number Theory [closed]
I'm a high school student, interested in mathematics, especially in number theory.
While preparing for the IMO test, and thinking about generalizations or the root of many olympiad problems led me to ...
33
votes
5answers
3k views
The origin(s) of the word “elliptic”
The word elliptic appears quite often in mathematics; I will list a few occurrences below. For some of these, it is clear to me how they are related; for instance, elliptic functions (named after ...
13
votes
2answers
452 views
Number triangle
This question arose just out of curiosity. Note the triangle of 0-1's below, whose construction is as follows. Choose any number, say 53 as done here. The first line of the triangle is the binary ...
8
votes
1answer
686 views
Recreational mathematical papers [closed]
Sometimes it is nice to get a less technical paper on mathematics to read and learn something different for a change. These papers often make us discover some new curiosity, to think about the process ...
2
votes
0answers
239 views
Advice for graphic tablet for math [closed]
With the current Coronavirus disease (COVID-19), many of us had to switch all our activity to full online mode.
I am wondering whether some of you had the chance to use graphic tablets. I am looking ...
-4
votes
1answer
291 views
PCP theorem to check hard proofs [closed]
Is it technically possible to check formidable proofs like Mochizuki's using PCP theorem before mathematicians spend time in understanding the mechanics of the proof? If so why have mathematicians not ...
0
votes
1answer
51 views
Rule to determine rotationally invariant orders of the points of arbitrary 2d splines
I would like to find a rule to determine the order of the points of arbitrary 2d splines, which should be invariant with respect to rotation (as far as possible).
To illustrate the problem, let us ...
3
votes
1answer
325 views
Journey into a strange wilderness [closed]
W. S. Anglin wrote
Mathematics is not a careful march down a well-cleared highway, but a journey into a strange wilderness, where the explorers often get lost. Rigour should be a signal to the ...
11
votes
3answers
1k views
How can I simplify this sum any further?
Recently I was playing around with some numbers and I stumbled across the following formal power series:
$$\sum_{k=0}^\infty\frac{x^{ak}}{(ak)!}\biggl(\sum_{l=0}^k\binom{ak}{al}\biggr)$$
I was able ...
76
votes
6answers
10k views
Is data science mathematically interesting?
I have seen a plethora of job advertisements in the last few years on mathjobs.org for academic positions in data science. Now I understand why economic pressures would cause this to happen, but from ...
