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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4 answers
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Is spherical trigonometry a dead research area?

When I was an undergrad, the field of spherical trigonometry was cited as a once-popular area of math that has since died. Is this true? Are the results from spherical trigonometry relevant for ...
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38 votes
12 answers
9k views

Is amateur research in mathematics viable?

After a long reflection, I've decided I won't go to graduate school and do a thesis, among other things. I personally can't cope with the pressure and uncertainty of an academic job. I will therefore ...
0 votes
1 answer
59 views

Computation involving Gauss integer function

I have used mathematica to test following equation is true \begin{equation} \sum_{a,b=0}^{m-1} \left[\frac{a+b n}{m}\right] = \frac{n m^2}{2} -\frac{nm}{2} \end{equation} where $[x]$ is the floor ...
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2 votes
2 answers
536 views

What is the most "informative" Yes/No math question you know? [closed]

Imagine that alien civilization contacted you and offered to answer one math question. This should be a Yes/No question (so, you cannot ask for a million-digit binary string encoding the answers to a ...
1 vote
0 answers
38 views

The exact distribution of the conditional distribution of the OLS estimator

This is the problem that I have tried figuring it out for a while, and I still need some advice because there is no explicit derivation in the textbook that I have seen so far. The problem looks easy ...
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0 votes
0 answers
99 views

What is the meaning of PUP in remarks?

I see the remarks in Gowers’s The Princeton Companion to Mathematics. What is the meaning of the abbreviation ‘PUP’? Never seen before…
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10 votes
1 answer
277 views

Can you get any natural number from 4 by performing given operations?

You can perform the following operations on numbers: divide the number by 2, add 0 or 4 at the end of the number. Can you get any natural number from 4 by performing only these operations? So far I ...
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  • 119
8 votes
6 answers
1k views

Surprising applications of the theory of games?

I am currently studying the applications of games in quantum information theory and related fields and I am aware of its uses in places like model theory and set theory. So I was curious, what are ...
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0 votes
0 answers
67 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 (...
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133 votes
31 answers
10k 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 ...
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9 votes
1 answer
488 views

How far away can we get by multiple rounding and unit change?

This question is inspired by xkcd #2585 (Rounding): Let $u_0,\ldots,u_n$ be positive real numbers (we can assume w.l.o.g. that $u_0=1$) or “units”. Consider the following directed graph: its vertices ...
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0 votes
0 answers
47 views

How can C2 double-helices, produce C3 rotational symmetry?

I am trying to solve the structure of a helical protein complex using a software for maximum a posteriori refinement of (multiple) 3D reconstructions (Relion). When searching for helical symmetry, the ...
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2 votes
1 answer
99 views

Algorithm for compact polynomial expressions

Sometimes an ugly polynomial (perhaps in several variables) can be expressed as a small sum of much simpler polynomials. Can this be done algorithmically? More precisely: Is there a reasonable ...
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3 votes
0 answers
218 views

Récoltes et Semailles for the non (algebraic-) geometer

With the recent publication of Grothendieck's Récoltes et Semailles, I've been umm-ing and ah-ing about whether to get a copy, if only for the "soaking nuts" story. My French reading is ...
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10 votes
3 answers
801 views

Progress in robustifying mathematics - i.e. making mathematical theorems robust to small changes in hypotheses

The idea of making a mathematical theorem robust to small changes in its hypotheses has been known for some time. In areas such as group theory reasonable progress has been made leading to the theory ...
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12 votes
4 answers
663 views

What are some examples of understanding a space by studying the functions on this space?

In Quantum theory, groups and representations, Peter Woit writes: A fundamental principle of modern mathematics is that the way to understand a space $M$, given as some set of points, is to look at $...
5 votes
1 answer
258 views

How the solve the equation $\frac{(a+b\ln(x))^2}{x}=c$ [closed]

I need to solve the equation $$\frac{(a+b\ln(x))^2}{x}=c$$ where $a$, $b$, and $c$ are given. It is known that $a$ and $b$ are fixed and satisfy some condition such that the left hand side is ...
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1 vote
0 answers
72 views

Drawing a 3D object in a 3D environment, and converting to math [closed]

So I have been granted a free time and I want to work on a project but first I had to research. As we know, lines have infinite points, and with lines, we can create infinite shapes. I want to let ...
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3 votes
0 answers
232 views

What does it mean to solve an equation?

Assume that we want to find all integer (or rational) solutions to the polynomial Diophantine equation $$ P(x_1,\dots,x_n) = 0 $$ where $P$ is a polynomial with integer coefficients. Do we have a ...
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6 votes
0 answers
214 views

Mathematical questions or areas amenable to AI [duplicate]

This question regards the new paper "Advancing mathematics by guiding human intuition with AI" by Davies et al. (Nature, 2021) (DOI link in open access) in which researchers at Deepmind ...
12 votes
2 answers
2k views

Is it ever unnecessary to mathematically formalize a concept?

From my understanding, mathematics sometimes gives rise to new physical/tangible laws and the converse is also true. In particular, physical phenomena give rise to new mathematics. In all of the cases ...
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31 votes
2 answers
907 views

Math videos containing real time rough thinking

I find this vlog experiment of Gowers very brave, and I think his idea of having examples of real-time mathematical thinking by experts can be very encouraging for young mathematicians, who imagine ...
41 votes
18 answers
4k views

Results in linear algebra that depend on the choice of field

Linear algebra as we learn it as undergraduates usually holds for any field (even though we usually learn it for the complex, or real, numbers). I am looking for a list of concepts, and results, in ...
1 vote
0 answers
103 views

Closed form for $\sum _{k=1} ^n \frac k 2 \,\operatorname{sgn} \left( \frac 1 {k^2} + \cos \frac {2\pi n} k-1 \right)$ [closed]

I am looking for the closed form of $$\sum _{k=1} ^n \frac k 2 \,\operatorname{sgn} \left( \frac 1 {k^2} + \cos \frac {2\pi n} k-1 \right) \ .$$ Wolfram Alpha cannot do this for me, so I am forced to ...
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4 votes
2 answers
395 views

Theorems with finite sets of exceptions

Exceptions are interesting. Sometimes, they're also important. If a theorem with exceptions is important for a subject, there are liable to be many corollaries of the form "either this is true... ...
55 votes
23 answers
7k views

Golden ratio in contemporary mathematics

A (non-mathematical) friend recently asked me the following question: Does the golden ratio play any role in contemporary mathematics? I immediately replied that I never come across any mention of ...
11 votes
3 answers
529 views

Any hints on how to prove that the function $\lvert\alpha\;\sin(A)+\sin(A+B)\rvert - \lvert\sin(B)\rvert$ is negative over the half of the total area?

I have this inequality with $0<A,B<\pi$ and a real $\lvert\alpha\rvert<1$: $$ f(A,B):=\bigl|\alpha\;\sin(A)+\sin(A+B)\bigr| - \bigl| \sin(B)\bigr| < 0$$ Numerically, I see that ...
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36 votes
22 answers
6k 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 ...
62 votes
20 answers
8k views

Situations where “naturally occurring” mathematical objects behave very differently from “typical” ones

I am looking for examples of the following situation in mathematics: every object of type $X$ encountered in the mathematical literature, except when specifically attempting to construct ...
18 votes
3 answers
2k views

Heat map of current mathematics

The recent article on Quanta (by Natalie Wolchover) concerning $\aleph_1$ vs. $\aleph_2$ suggests that there is excitement within that community: Juliette Kennedy: "It’s one of the most ...
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20 votes
2 answers
1k views

sci.math.research archive?

Does there exist an archive somewhere of posts to the USENET newsgroup sci.math.research? The best approximation I'm aware of is Google Groups. However, despite ...
51 votes
10 answers
6k views

Changes forced by the pandemic

The Covid-19 pandemic has changed our work-lives in ways few of us could have anticipated. These exceptional circumstances have forced each one of us and each one of our institutions to adapt, ...
0 votes
1 answer
130 views

What are examples of mathematical objects that are 'constructed out of' a range of other objects but fall out of them? [closed]

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: $\...
33 votes
14 answers
6k views

Is pure mathematics useful outside of mathematics itself? [closed]

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 ...
5 votes
0 answers
196 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 ...
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1 vote
2 answers
193 views

The sum of triangular functions

I have a sum of $N-1$ triangular functions 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\...
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2 votes
1 answer
913 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 ...
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4 votes
0 answers
102 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 ...
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74 votes
10 answers
9k 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
0 answers
183 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 $$ a_n = \max_{f_A, f_B : \mathbb{Z}_n \to \mathbb{Z}_n} \frac{\# \left\{ (x,y) | f_A(x) - f_B(y) = xy \mod n \right\}}{n^2} $$...
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5 votes
1 answer
238 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: ...
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1 vote
1 answer
260 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(\...
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  • 153
3 votes
6 answers
439 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 ...
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0 votes
2 answers
94 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} ...
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  • 299
5 votes
1 answer
269 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(\...
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  • 153
3 votes
0 answers
150 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-...
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  • 153
3 votes
1 answer
124 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,\...
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2 votes
0 answers
82 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 $2^{< \omega}$ ...
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  • 1,515
0 votes
0 answers
111 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 $E_1$, $...
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44 votes
4 answers
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 ...

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