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.
42
questions with no upvoted or accepted answers
17
votes
0
answers
971
views
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 ...
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 ...
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 ...
7
votes
0
answers
198
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
$$
a_n =
\max_{f_A, f_B : \mathbb{Z}_n \to \mathbb{Z}_n}
\frac{\#
\left\{
(x,y) | f_A(x) - f_B(y) = xy \mod n
\right\}}{n^2}
$$...
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
$$ \forall x,y\...
5
votes
0
answers
323
views
What's with the speaker's initial thing?
If you've ever been to a math talk (at least in pure maths in the UK) you've probably seen something like this written:
Theorem (E.--Johnson--Smith, 2022+) The XYZ conjecture is true.
At some point ...
5
votes
0
answers
176
views
Generalization of IMO5 from 1987
The following question appeared as question 5 on the IMO in 1987:
Prove that for all $n \geq 3$ one can find $n$ distinct points on the Euclidean plane with the property that the distance between any ...
4
votes
1
answer
539
views
Novel examples, proofs or results in mathematics from arithmetic billiards
The goal of the post is get a repository of mathematical results, proofs or examples by users of the site, arising from arithmetic billiards in number theory, analysis, geometry,….
Wikipedia has an ...
4
votes
0
answers
336
views
Récoltes et Semailles for the non (algebraic-) geometer
With the recent publication of Grothendieck's Récoltes et Semailles, I've been umm-ing and ah-ing about whether to get a copy, if only for the "soaking nuts" story. My French reading is ...
4
votes
0
answers
297
views
What does it mean to solve an equation?
Assume that we want to find all integer (or rational) solutions to the polynomial Diophantine equation
$$
P(x_1,\dots,x_n) = 0
$$
where $P$ is a polynomial with integer coefficients. Do we have a ...
4
votes
0
answers
203
views
Is my equation for finding the length of a curve correct?
My son has been working on equations for finding length of the length of a curve without any resources (computers, calculators...) while in prison. He is wondering if he is on the right path. Sadly my ...
4
votes
0
answers
110
views
What do you call such a relation between subsets in a poset
Consider a poset $(X, \geq)$. Let's define a new relation $\succsim$ on subsets of $X$: for $A, B\subseteq X$, say $A\succsim B$ if for any $a\in A$ and any $b\in B$, we have $a\geq b$.
Does such a ...
4
votes
0
answers
588
views
What solutions to useful computational problems could be rewarded through cryptocurrency smart contracts?
What kinds of cryptocurrency smart contracts could be used to reward people for solving specific kinds of useful computational problems?
Background
In this question, I asked for proposals for useful ...
4
votes
0
answers
224
views
Looking for U.K. problem column (?) from 1980s
While digging through some dusty corners of my file cabinet, I found a photocopied sheet of eight (handwritten) problems from 1985 that I recall receiving from my secondary school mathematics teacher ...
3
votes
0
answers
167
views
Does this trig equation have a closed-form solution?
This equation came up when I was looking at the Fibonacci sequence; I adore its symmetry:
$$x^2 \cdot \sin \left(\frac{2\pi}{x-1} \right) \cdot \left(3+2 \cos \left(\frac{2\pi}{x} \right) \right) = (x-...
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 ...
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 ...
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 ...
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 $2^{< \omega}$ ...
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})...
2
votes
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 $...
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 ...
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 ...
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 ...
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 ...
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 ...
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 (...
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 ...
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 ...
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
$\Sigma _{\alpha\beta}=-k^2g_{\alpha\beta}+i\theta\epsilon_{\alpha\beta\gamma} k^\gamma$
$k^2=k^μk_μ$, $g^{αβ}$ ...
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 ...
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 ...
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 ...
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 ...
0
votes
0
answers
149
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…
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 $E_1$, $...
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 ...
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 ...
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( ...
-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 $g(x,y)=0$ ...