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Infinite number of points reflecting on the circle, must some two (or more) ever meet?

I just created a following problem. Suppose that we have an infinite number of points on the circle and that they at the same time start to travel (all with the same speed) from the point where they ...
Shalom's user avatar
  • 513
2 votes
1 answer
280 views

Properties of harmonic maps into spheres

Let $(M, g)$ be a complete, noncompact Riemannian $n$-dimensional manifold and let $\phi \colon M \to \mathbb S^n$ be an harmonic map, where $\mathbb S^n$ is the euclidean $n$-dimensional sphere. ...
Onil90's user avatar
  • 823
4 votes
1 answer
359 views

Definition of quotient manifolds, and $\Gamma \backslash \mathscr H$ as a quotient manifold

I have just encountered some subtlety with quotient manifolds and now I don't think I understand some things as well as I thought I did. Let $X$ be a real or complex analytic manifold, and $\sim$ ...
D_S's user avatar
  • 6,100
1 vote
0 answers
57 views

Bound of analytic torsion for a line bundle

Let $E\rightarrow X$ be a line bundle over a compact Riemann surface. Let $\Delta_{E}$ be the Dolbeault Laplacian $\Delta_{\overline{\partial}}$ extended to sections of $E$. Let $g_1, g_2$ be two ...
Bombyx mori's user avatar
  • 6,141
0 votes
0 answers
252 views

Question on sums of multiplicative functions twisted by the Mobius function

Let $\mu(\cdot)$ be the Mobius function, defined on the natural numbers by $$\displaystyle \mu(n) = \begin{cases} (-1)^{\omega(n)} & \text{if } n \text{ is square-free} \\ 0 & \text{...
Stanley Yao Xiao's user avatar
8 votes
1 answer
414 views

Every odd integer greater than $1$ is of the form $a+b$ with $a^2+b^2$ being prime

Let $m$ be an odd integer greater than $1$. Is it true that there are positive $a, b$ such that $m=a+b$ and $a^2+b^2$ is a prime number? It seems that for every odd $m$ there are many $(a,b)\in \...
Konstantinos Gaitanas's user avatar
5 votes
1 answer
385 views

surface with rational curve in the double locus

I am interested in the existence of a surface $X$ over $\mathbb{C}$ with the following properties (or a reason for why one cannot exist): $X$ is slc (and not-normal) There is rational curve $C \...
Srks's user avatar
  • 379
1 vote
0 answers
102 views

What happens if in Randers metric the norm of the wind is not less than 1

One way to define the Randers metric is using the data $(h,W)$ associated to the Zermelo problem. Here $h$ is the Riemannian metric and $W$ is the wind. In order to define the Randers metric we must ...
Majid's user avatar
  • 227
7 votes
1 answer
225 views

Discriminant of numerator of inverse logarithmic derivative operator iteration

Let $T:\mathbb Q(x)\to \mathbb Q(x)$ be the operator of inverse logarithmic derivative, i.e. $$Tf=\frac{f}{f'}.$$ Define $$p_n(x)=T^n\left(x-\frac{x^2}{2}\right).$$ Let $f_n(x) \in \mathbb Z[x]$ be ...
Alexander Kalmynin's user avatar
0 votes
1 answer
201 views

Field of constants of a Galois extension of function fields

Let $F$ be a finite Galois extension of the rational function field $\mathbb Q(x)$. Let $k$ be the field of constants of $F$, i.e., the algebraic closure of $\mathbb Q$ in $F$. Is $k$ necessarily a ...
352506's user avatar
  • 991
2 votes
1 answer
139 views

Convergence of sequence of images of Schur multipliers

Let $\eta$ be a continuous bounded function on $(0, \infty)^{2}$ so that $\eta(0,0)=1$. Let $A$ be a bounded operator on $\ell^{2}(\mathbb{Z}_{\geq 0})=\ell^{2}$ (by bounded operator I will always ...
Raphael's user avatar
  • 31
1 vote
0 answers
254 views

When is an algebra a container? Can we compute the container for an algebra?

Containers capture data structures like lists, bags and trees. Containers can be monads and comonads as we see here. There is a fascinating link between containers and algebras. For instance, take ...
Ben Sprott's user avatar
  • 1,281
191 votes
12 answers
30k views

Do you know important theorems that remain unknown?

Do you know of any very important theorems that remain unknown? I mean results that could easily make into textbooks or research monographs, but almost nobody knows about them. If you provide an ...
1 vote
1 answer
628 views

Inception of modern view of Sheaf Cohomology in Mathematical Literature

From wikipedia entry on Sheaf Cohomology I have found the intriguing passage: 'The essential point is to fix a topological space X and think of cohomology as a functor from sheaves of abelian groups ...
user122465's user avatar
0 votes
1 answer
166 views

How to play the following game?

Let $n,k\in\mathbb N$, $x\in(0,1/2)$. You start $n$ empty bins; each can accommodate at most $k$ balls. At each iteration, you choose an $x$ fraction of the non-full bins and add one ball to each. (...
R B's user avatar
  • 608
1 vote
0 answers
190 views

Is this the only integer point of this curve?

$\textbf{How to attempt proving that $4x^3-3z^2=1$ has no positive integer solutions?} $ I want to show there is no positive integer solution to this equation other than $(1,1)$. How to do that? Is ...
diffusiondiver11's user avatar
0 votes
2 answers
145 views

coloring infinite vertex transitive graph without large cliques

Let $G$ be an infinite vertex-transitive graph (this means that for every $u,w \in V(G)$ there exists an automorphism $\tau$ of $G$ such that $\tau(u) = v$). We assume that $G$ is undirected, and does ...
Pablo's user avatar
  • 11.2k
10 votes
0 answers
313 views

Is there a spectral sequence of Atiyah's topological KR-theory that can be used to compute basic examples?

For Segal's complex $G$-equivariant $K$-theory, it is well-known that there is an Atiyah-Hirzebruch spectral sequence. If say $G$ a finite group and $X$ a finite CW-complex, the second page of this ...
Luuk Stehouwer's user avatar
0 votes
0 answers
213 views

Upper bound on matrix perturbation such that all eigenvalues lie within the unit circle

Consider the matrix $$N=\left[\matrix{\mathbb{I}_n-\epsilon L & X\\ \epsilon Y & Z}\right]$$ where $\epsilon>0$ is a small positive parameter and $Z$ is a square $m\times m$ matrix with ...
CTNT's user avatar
  • 101
5 votes
1 answer
175 views

Fields of rationality as a notion of automorphic size

I want to interpret the degree of the field of rationality of an automorphic form as a notion of size, analogously to the conductor, and this question is about the possible obstructions to do so. The ...
Desiderius Severus's user avatar
4 votes
1 answer
512 views

Maximal subgroups not containing a specific element

Given a non-trivial group $G$ and $g\in G\setminus \{e_G\}$ where $e_G$ is the neutral element, it is easy to show using Zorn's Lemma, that there is a subgroup not containing $g$ that is maximal ...
Dominic van der Zypen's user avatar
1 vote
1 answer
139 views

Topological full groups and minimal orbit closures

Let $X$ be the Cantor set, and let $g$ be a minimal homeomorphism of $X$. Let $h$ be a homeomorphism in the topological full group of $g$, that is, for every $x \in X$, there is a neighbourhood of $x$...
Colin Reid's user avatar
  • 4,678
1 vote
0 answers
189 views

Search strategy for Babson task in chess

I asked this on a computer chess forum (programmers hang out there, etc.) and got no substantive answers, which makes me think it's a research question. Whether it's sufficiently mathematical is ...
anonymous's user avatar
1 vote
0 answers
153 views

Compactifying morphisms and ample line bundles

Let $f:X\to Y$ be a projective morphism between two normal quasi-projective varieties, and $L$ a $f$-ample line bundle on $Y$. Then the claim is: There is a compactication $\bar{f}:\overline{X}\to\...
Omprokash Das's user avatar
3 votes
0 answers
155 views

Must a counterexample $f$ to the $n$-dimensional JC satisfy $\cap f^i(k[x_1,\ldots,x_n])=k$?

There is a known result concerning the two-dimensional Jacobian Conjecture which says the following: Let $k$ be a field of characteristic zero. If $f:k[x,y] \to k[x,y]$ has an invertible Jacobian and ...
user237522's user avatar
  • 2,783
7 votes
0 answers
233 views

Reconstitution from reduction and tropicalization for $p$-adic varieties

For a variety $X$ over a $p$-adic field $k$, its reduction to a variety $X_p$ over residue field $k_p$ of $k$ and its tropicalization to $X_{tr}$ seem to me to be somewhat orthogonal processes. Taken ...
user avatar
3 votes
0 answers
805 views

A "surjective implies injective" property for endomorphism rings of modules

Fix a unital commutative ring $R$ and consider a left $R$-module $M$. $\newcommand{\End}{{\rm End}}$ (For the indirect application I have in mind, which would require another post, $\End_R(M)$ will ...
Yemon Choi's user avatar
  • 25.5k
21 votes
1 answer
1k views

Tropicalization of perfectoid spaces and their tilts

Does tropicalization exist in the world of perfectoid spaces? Since it does for Huber's adic spaces, I thought it might for perfectoid spaces too, yet I can't find any explicit references so far. ...
user avatar
5 votes
1 answer
601 views

Eulerian ordering of the integers modulo n

Let $n>1$ be an integer. Consider the set $C_n := \{0,1, \dots , n-1\}$. An Eulerian ordering of $C_n$ is an ordering $r_1, \dots, r_n$ of its elements such that: $$\forall i \le n \ \forall j&...
Sebastien Palcoux's user avatar
7 votes
1 answer
304 views

$p$-adic lifts of tropical varieties

What is currently known about lifts of tropical varieties to varieties over $\mathbb{Q}_p$ or its extensions? Starting with an appropriate rational polyhedral cone complex what are the obstructions ...
user avatar
15 votes
1 answer
1k views

Ramsey Number R(3,3,4)

How much is known about the Ramsey number R(3,3,4)? There is a trivial upper bound of 34, but are any tighter bounds known?
Thomas's user avatar
  • 2,691
2 votes
2 answers
687 views

"Minkowski Multiplication" of Convex Sets?

Apologies if this question might be trivial or has been asked already (haven't found an equivalent post), but I am trying to figure out whether the following is true: Given two convex sets $\mathcal{...
MrRed's user avatar
  • 123
1 vote
0 answers
66 views

Showing that $b$ is a inner point of $\mathcal{G}$ where $\mathcal{G}$ is a subset of $\mathbb{R}^{N+3}$ determined by $\mathcal{M}^{+}$

Let $(\Xi,\mathscr{E})$ be a measurable space, $(\mathbb{R_{+}},\mathfrak{B})$ other measurable space where $\mathfrak{B}$ a $\sigma$-algebra. We consider the measurable space $(\Xi\times\Xi\times\...
PepitoPerez's user avatar
9 votes
1 answer
235 views

Liftable rational varieties

Is there an example of a rational smooth projective variety over a perfect field of characteristic $p$, that is not liftable to characteristic zero?
user avatar
4 votes
0 answers
250 views

$p$-adic lifts of varieties over finite fields

Fix a finite field $k_p$ of characteristic $p$, as well as a $p$-adic lift of it, $k_0$. In other words $k_0$ is a $p$-adic field and $k_p$ its residue field. Let $X_p$ be a non-singular variety over ...
user avatar
6 votes
1 answer
208 views

Jones-Wenzl-type projectors for Brauer algebras

Jones-Wenzl projectors are an explicit combinatorial description of the central projectors of the Temperley-Lieb algebra. They also describe very explicitly the failure of certain representations to ...
Calvin McPhail-Snyder's user avatar
3 votes
0 answers
312 views

Philosophical question on the role of motivic cohomology

As a physicist who has read almost nothing on the beautiful theories of motives and motivic cohomology, I have some philosophical questions on motivic cohomology. I apologize first for my naive ...
Wenzhe's user avatar
  • 2,961
3 votes
0 answers
234 views

Use GAP program to obtain explicit cocycles in group cohomology

I'm trying to compute group cohomology $H^n(G,\mathbb{Z})$ of some crystal groups $G$ which are infinite but finitely generated groups. I succeed in obtaining cohomology groups using projective ...
Xu Yang's user avatar
  • 123
12 votes
0 answers
336 views

Does Thompson's group $V$ have property AP?

Property AP: A discrete group $\Gamma$ has property AP (Approximation Property) if there exists a net $(\phi_i)_{i \in I}$ of finitely supported functions on $\Gamma$ such that $\phi_i \to 1 $ weak$^*$...
tattwamasi amrutam's user avatar
2 votes
1 answer
72 views

Sampling with non-uniform probabilities

Let $p_1,p_2,...,p_n$ are given probabilities. ($\sum_{i=1}^n p_i =1, p_i \geq 0 $). Is there any distribution, which picks $k\leq n$ distinct elements from $1,2,...,n$ such that $P(i \in S) = k p_i$ ...
Ethan's user avatar
  • 145
6 votes
1 answer
191 views

Is there a well-defined notion of "pitch shift" without time dilation?

Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a continuous periodic function. If we replace $f$ by $$g(t) := f(\frac{1}{\lambda}t)$$ for some $\lambda > 0$, this new function $g$ has period $\...
James's user avatar
  • 1,498
2 votes
0 answers
210 views

Expected length of a random path in a graph

Let $G$ be a graph and $v$ one of its vertices. Are there any known formulas or fast algorithms for calculating the expected length of a random path in $G$ starting in $v$? In each step we choose a ...
Gorka's user avatar
  • 1,825
2 votes
0 answers
76 views

Curvature on bad 2-orbifold

It may a stupid question. As stated in Lang-Fang Wu' paper (JDG 33(1991)) “Bad orbifolds do not admit metrics of constant curvature. ” Can anyone give a proof or a reference about this statement. How ...
AlexJChow's user avatar
1 vote
0 answers
50 views

Extension of a function from a subset of a manifold to the unit sphere

There is a line in this following paper (page no- 221, the paragraph before the Lemma 2.2) Otsu, Yukio; Shiohama, Katsuhiro; Yamaguchi, Takao, A new version of differentiable sphere theorem, Invent. ...
MAS's user avatar
  • 872
4 votes
0 answers
113 views

Determining whether a morphism is the induced morphism?

Let $F\colon \mathcal A \to \mathcal B$ be a left exact functor between Grothendieck abelian categories. Given a morphism $f\colon A\to B$ in $\mathcal A$ and a morphism $g\colon RF(A)\to RF(B)$ in ...
Avi Steiner's user avatar
  • 3,031
2 votes
0 answers
149 views

Kaehler differentials of tensor products over another ring

Is this result well-known (or even true)? Let \begin{align*} f:A & \to B\\ g:A & \to C \end{align*} be homomorphisms of finitely generate $k$-algebras and let $R=B\otimes_{k}C$ and $S=B\...
Justin Smith's user avatar
0 votes
1 answer
1k views

Bounding the derivative of a holomorphic function on a disk by its absolute value

Let $f(z)$ be a holomorphic function defined on the disk $|z|\le 2$. Suppose $|f(z)|<1$ for $|z|\le 2$. It looks like there is a constant $c>0$ such that $|f(z)'|<c$ on the disk $|z|\le 1$ (...
aglearner's user avatar
  • 14k
3 votes
1 answer
354 views

Is it known that MLC is sufficient to prove the density of hyperbolic conjecture of rational maps (or not)

Is it known that local connectivity of the Mandelbrot set (MLC) is sufficient prove the density of hyperbolic conjecture of qudratic family. I wondered is it known that the MLC is not enough (or ...
yaoxiao's user avatar
  • 1,664
9 votes
2 answers
489 views

PCF theory and good points in scales

If $\kappa$ is a singular cardinal, a scale for $\kappa$ consists of an increasing sequence $\langle \kappa_i : i < \mathrm{cf}(\kappa) \rangle$ converging to $\kappa$ and a sequence of functions $\...
Monroe Eskew's user avatar
  • 18.1k
3 votes
1 answer
119 views

an inverse semigroup (and perhaps a $C^*\!$-algebra) associated with a directed graph

The following inverse semigroup associated to a directed graph came up in my research. I've read that from an inverse semigroup one may derive a $C^*\!$-algebra whose generators are partial isometries ...
David Hillman's user avatar

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