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.
320
questions
12
votes
1
answer
962
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 ...
- 13.4k
12
votes
14
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 ...
51
votes
7
answers
8k
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
229
views
How to solve recurrence relation with 2 variables？
I have the following recurrence relation and boundary condition?
$$
f(n,m) = \frac{\alpha n}{n+m} f(n-1,m) + \frac{\beta m}{n+m} f(n,m-1) + 1
$$
$$
f(n,0) = \frac{1-\alpha^{n+1}}{1-\alpha}, f(0,m) = \...
- 83
21
votes
8
answers
4k
views
Examples of bad notation and its consequences [closed]
An example of bad mathematical notation that comes in my mind and has caused complications throughout history is the notation for imaginary numbers. The original notation used to represent imaginary ...
6
votes
0
answers
460
views
A new and subtle order-theoretic fixed point theorem
Sometimes a very simple argument appears out of the blue and overturns a subject. It is not based on pre-existing theory and heavy involvement in such a theory is actually a handicap in finding such ...
- 7,485
-5
votes
1
answer
367
views
How much would a mathematician cost? [closed]
Recently our department lost one of the best professors who was attracted by a better University. If we were a football club, and he were a leading player, we would receive many millions of ...
55
votes
10
answers
10k
views
What do you do when you're stuck?
I'm pretty sure almost all mathematicians have been in a situation where they found an interesting problem; they thought of many different ideas to tackle the problem, but in all of these ideas, there ...
12
votes
7
answers
3k
views
Books containing new results
In Endless controversy about the correctness of significant papers, Denis Serre writes:
The research community is able to point out incorrect statements, at least among those which have some ...
6
votes
3
answers
597
views
How do I solve the following definite integral (preferably by an asymptotic method)?
$$ \int_{1}^{\infty} \frac{\sin^2 (\mu \sqrt{x^2 -1})}{(x+1)^{\frac{9}{2}} (x-1)^{\frac{3}{2}}} \,dx $$
Note: $\mu$ here is an extremely small constant.
I have tried:
Estimating the integral by ...
- 69
46
votes
2
answers
4k
views
Well known theorems that have not been proved
I believe that there are numerous challenging theorems in mathematics for which only a sketch of a proof exists. To meet the standards of rigor, a complete proof of these theorems has yet to be ...
11
votes
9
answers
1k
views
What are examples of problems we know how to solve for primes (or prime powers), but not for composites?
I am interested in seeing examples of research problems which fall into one of the two following categories:
A problem which is solved in the case of primes (or prime powers), but which remains open ...
-4
votes
1
answer
254
views
Limit of recursion relation
Consider the sequence of functions $\{F_n\}_{n\in \mathbb{N}}$, where each $F_n$ is defined on $\{0,...,n\}$ by recurrence of the following form: $$ F_n(0)=3 \textrm{
and }F_n(k)=\frac{1}{k^2}+\frac{\...
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
391
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
$$ \forall x,y\...
- 7,485
1
vote
0
answers
74
views
Invariance signature in infinite dimension
Let $V$ be an infinite dimensional vector space and suppose we have a smooth family $\{g_t\}_{t\ge 0}$ of symmetric bi-linear forms such that:
$g_0$ is positive-definite
$g_t$ is non-degenerate for ...
- 385
0
votes
0
answers
32
views
Likelihood ratio when true distribution is known
Let $X_{i}, i=1,...,N,$ follow a distribution $p(x)$. Now, we calculate the likelihood ratio between $p$ and a different distribution $q$ using samples $X_i$:
$$
\frac{p(X_i)}{q(X_i)},\ \text{or}\ \...
5
votes
0
answers
307
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 ...
- 8,011
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&...
6
votes
1
answer
247
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 ...
- 91
0
votes
0
answers
52
views
Enumeration of regions between a set of curves
Consider a set of $N$ tuples $\{\alpha_n,\beta_n\}$ where $\alpha_n,\beta_n\in[0,2\pi)$ and functions
$$f_n(x,y)=\cos(\alpha_n-x+y)+\delta\cos(\beta_n-x-y)$$ where $0<\delta<1$ and $x,y\in[0,\pi)...
5
votes
0
answers
171
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 ...
- 23k
3
votes
1
answer
422
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
357
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 ...
- 16k
5
votes
1
answer
321
views
Question on pure mathematics helping climate change research
While I am a pure mathematics tenured professor, still at a relatively young age, and fairly passionate about my area of research, I cannot help but feel that it may be more useful to humanity if I ...
- 2,862
38
votes
5
answers
8k
views
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 ...
- 901
48
votes
14
answers
12k
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 ...
1
vote
1
answer
156
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 ...
- 63
2
votes
2
answers
616
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 ...
0
votes
0
answers
122
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…
10
votes
1
answer
300
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 ...
- 119
9
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 ...
- 525
1
vote
1
answer
144
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 (...
- 11
142
votes
32
answers
12k
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 ...
- 55.4k
9
votes
1
answer
531
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 ...
- 26.5k
2
votes
1
answer
114
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
...
- 16k
4
votes
0
answers
294
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 ...
- 2,447
10
votes
3
answers
815
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 ...
- 4,552
14
votes
4
answers
842
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
263
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 ...
- 59
1
vote
0
answers
95
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 ...
- 11
4
votes
0
answers
285
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 ...
- 3,477
6
votes
0
answers
232
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 ...
- 131
31
votes
2
answers
992
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
110
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 ...
- 21
4
votes
2
answers
465
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... ...
62
votes
27
answers
8k
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
544
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 ...
- 219