All Questions
152,891
questions
0
votes
0
answers
29
views
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 ...
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.
...
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 $\sim$ ...
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 ...
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
$$\displaystyle \mu(n) = \begin{cases} (-1)^{\omega(n)} & \text{if } n \text{ is square-free} \\ 0 & \text{...
8
votes
1
answer
414
views
Every odd integer greater than $1$ is of the form $a+b$ with $a^2+b^2$ being prime
Let $m$ be an odd integer greater than $1$. Is it true that there are positive $a, b$ such that $m=a+b$ and $a^2+b^2$ is a prime number?
It seems that for every odd $m$ there are many $(a,b)\in \...
5
votes
1
answer
385
views
surface with rational curve in the double locus
I am interested in the existence of a surface $X$ over $\mathbb{C}$ with the following properties (or a reason for why one cannot exist):
$X$ is slc (and not-normal)
There is rational curve $C \...
1
vote
0
answers
102
views
What happens if in Randers metric the norm of the wind is not less than 1
One way to define the Randers metric is using the data $(h,W)$ associated to the Zermelo problem. Here $h$ is the Riemannian metric and $W$ is the wind. In order to define the Randers metric we must ...
7
votes
1
answer
225
views
Discriminant of numerator of inverse logarithmic derivative operator iteration
Let $T:\mathbb Q(x)\to \mathbb Q(x)$ be the operator of inverse logarithmic derivative, i.e. $$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 ...
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 ...
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 ...
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 ...
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 ...
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. (...
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 ...
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 ...
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 ...
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 ...
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 ...
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 ...
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$...
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 ...
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\...
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 ...
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 ...
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 ...
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.
...
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&...
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 ...
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?
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{...
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\...
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?
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 ...
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 ...
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 ...
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 ...
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$^*$...
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 $P(i \in S) = k p_i$ ...
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 $\...
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 ...
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 ...
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. ...
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 ...
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\...
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 $|z|\le 1$ (...
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 ...
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 $\...
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 ...