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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What to do with results you found but cannot prove(outside your research area)?

Not sure if MathOverflow is still a place to discuss such things, but I'll give it a try. Tell me an alternative site, in case it is wrong here. I translated a representation-theory/combinatorial ...
Mare's user avatar
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16 votes
0 answers
694 views

Decidable open problems

Are there any significant open problems in mathematics which are clearly decidable (in that it is easy to write a clearly corresponding program which will eventually output either Yes or No (or ...
Sridhar Ramesh's user avatar
9 votes
0 answers
295 views

List of modern points of view simplifying or clarifying classical topics

There are many modern mathematical achievements which greatly clarify or (and) simplify classical important topics. I believe a list of such achievements, among other benefits, would be a big help for ...
7 votes
0 answers
579 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 ...
Paul Taylor's user avatar
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7 votes
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Fraction of elements in $\mathbb{Z}_n$ satisfying a certain equation

From a question arising in Game Theory, I want to calculate the sequence $$ a_n = \max_{f_A, f_B : \mathbb{Z}_n \to \mathbb{Z}_n} \frac{\# \left\{ (x,y) | f_A(x) - f_B(y) = xy \mod n \right\}}{n^2} $$...
Michael Mc Gettrick's user avatar
5 votes
0 answers
514 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 ...
5 votes
0 answers
588 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\...
Paul Taylor's user avatar
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5 votes
0 answers
323 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 ...
Sean Eberhard's user avatar
5 votes
0 answers
176 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 ...
Stanley Yao Xiao's user avatar
4 votes
1 answer
539 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 ...
4 votes
0 answers
336 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 ...
J.J. Green's user avatar
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4 votes
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297 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 ...
Bogdan Grechuk's user avatar
4 votes
0 answers
203 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 ...
Reba for Daniel's user avatar
4 votes
0 answers
110 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 ...
tsm's user avatar
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What solutions to useful computational problems could be rewarded through cryptocurrency smart contracts?

What kinds of cryptocurrency smart contracts could be used to reward people for solving specific kinds of useful computational problems? Background In this question, I asked for proposals for useful ...
Joseph Van Name's user avatar
4 votes
0 answers
224 views

Looking for U.K. problem column (?) from 1980s

While digging through some dusty corners of my file cabinet, I found a photocopied sheet of eight (handwritten) problems from 1985 that I recall receiving from my secondary school mathematics teacher ...
Timothy Chow's user avatar
3 votes
0 answers
167 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-...
Mitch's user avatar
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3 votes
0 answers
136 views

What is the name of this substructure/embedding?

I am interested in the following property, be it on an abstract or concrete category: $A$ is a substructure of $B$ such that every automorphism of $A$ extends uniquely to an automorphism of $B$. Or ...
Arnaldo Mandel's user avatar
3 votes
0 answers
133 views

Reference request: name of a transform

Define a transform on polynomials which is linear and acts on each monomial as $$\widehat{z^k} = \frac{(1+z)(2+z)\ldots(k+z)}{k!}.$$ Does anyone know whether this has a name (and therefore has been ...
genneth's user avatar
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2 votes
0 answers
330 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 ...
Riri's user avatar
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2 votes
0 answers
114 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 $2^{< \omega}$ ...
Peter Gerdes's user avatar
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2 votes
0 answers
74 views

How can I prove that the following function is increasing according to x1?

Suppose that $0 \le {X_1} < {X_2} < {X_3}$ . How is it possible to prove the following function is increasing based on ${X_1}$ in the range of $0 \le {X_1} < {X_2}$ ? $f({X_1},{X_2},{X_3})...
Hamed's user avatar
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0 answers
188 views

How to find moment condition for generalized method of moments?

Consider a scalar system with $2K$ outputs and $K+2$ unknowns: $y_{k,1}=x_ka_1+n_{k,1} \quad y_{k,2}=x_ka_2+n_{k,1}$. The variables $n_{k,\ell}$ are zero mean noise variables. To estimate $a_1$ and $...
Jonathan's user avatar
2 votes
0 answers
401 views

Conjectures on fractions where each digit appears once in numerator and denominator

This is a highly redacted version of a question that was asked before. Please see Criteria of considering relevance of the question to the domain of research topics for details. Some numerical ...
Alex's user avatar
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2 votes
0 answers
181 views

Jacobi triple product for multidimensional lattices

The Jacobi triple product identity gives as a special case a product formula for the theta function of a 1-dimensional lattice. Is there a more general product formula for the theta function of an ...
Tom Price's user avatar
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2 votes
0 answers
136 views

Reference needed for: Automatic generation of relevant mathematical exercises (similar to ones written by human) with the help of machine learning

I am doing research on automatic generation of relevant mathematical exercises (similar to ones written by human) with the help of machine learning. I have found several research papers on ...
logicum's user avatar
  • 21
2 votes
0 answers
198 views

Finite topological dimension x local compactness

Of course, the two notions are independent one from the other, but often one of them implies the other under some additional hypotheses. For instance: A topological vector space is finite dimensional ...
Claudio Gorodski's user avatar
1 vote
0 answers
84 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 ...
John117's user avatar
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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 (...
barakugav's user avatar
1 vote
0 answers
170 views

R.H. equivalent statement condition

Is the inequality $\prod \limits_{p \leq \sqrt{x}} (1+\frac{1}{p^2-1}) \prod \limits_{p \leq x} (1+\frac{1}{p}) \leq e^\gamma \ln(\theta(\sqrt{x})+\theta(x))$ where $\theta(x)$ is the Chebyshev's ...
Ahmad's user avatar
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1 vote
0 answers
123 views

repeated addition and square root of fixed number

pick any real number $x$ and integer $k$ and do the following recursive : 1) $x_0 =x $ 2) $x_{n+1} = x_n + \sqrt x_n$ using only $x$ and $k$ how to find the value of $x_k$ without going through ...
Ahmad's user avatar
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1 vote
0 answers
140 views

General procedure to find the determinant of an operator?

I want to learn to find the determinant of an operator. I am given an operator like $\Sigma _{\alpha\beta}=-k^2g_{\alpha\beta}+i\theta\epsilon_{\alpha\beta\gamma} k^\gamma$ $k^2=k^μk_μ$, $g^{αβ}$ ...
Zohaib Aarfi's user avatar
1 vote
0 answers
170 views

Regarding a Feature of Multivariate Real Function

Any real function can be expressed as a function of the sum of two monotonic real functions? More precisely, for real function p(x, y), there exist continuous real functions P(x), h(x,y), g(x) such ...
Wang Tao's user avatar
  • 103
1 vote
1 answer
242 views

Concept of synchronizability

This thread is about the concept of synchronizability. It's a concept I tried to formalize in its most general sense but without success. The goal of this thread is therefore to try to formalize it in ...
sure's user avatar
  • 438
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 ...
Marco Ripà's user avatar
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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 ...
Lili Si's user avatar
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0 answers
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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…
Philosopher's user avatar
0 votes
0 answers
150 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 $E_1$, $...
Memories's user avatar
  • 101
0 votes
0 answers
55 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 ...
Uli Fahrenberg's user avatar
0 votes
0 answers
109 views

Asymptotic inverses of asymptotic functions

The prime number theorem states that two functions are asymptotic. Their inverses (as functions of an integral variable) are also asymptotic. In general, under what conditions are the inverses of ...
user32024's user avatar
0 votes
0 answers
99 views

Repeated function resulting in quadratic time.

Let $$r(f, x) = k$$ such that $$f^k (x) < 2 \hbox{ and } f^{k-1} (x) \geq 2$$ For example $$r(n \rightarrow n-1, 2^n) = 2^n-1; r(n \rightarrow n/2, 2^n) = n.$$ For an arbitrary $c$, we have $$r( ...
Anonymous Coward's user avatar
-1 votes
1 answer
63 views

Idempotent solutions to the implict function theorem other than the identity?

I am interested in the following problem. Assume that an (anti)symmetric function $g:\mathbb{R}^2 \to \mathbb{R}$ satisfies the implicit function theorem. That is, $g(x,y) = \pm g(y,x)$ and $g(x,y)=0$ ...
Charlie's user avatar