All Questions
152,891
questions
6
votes
0
answers
137
views
Existence of $G$-map between finite $G$-simplicial complex
Let $X, Y$ be finite free $G$- simplicial complex. What kind of properties are necessary for existence a $G$-map,i.e, a continuous map which preserves $G$-action, from $X$ to $Y$? Does existence of ...
4
votes
2
answers
424
views
A homeomorphism between the unit interval $[0,1]$ and a linearly independent subset of a Hilbert space
Let $H$ be a infinite dimensional, separable Hilbert space over $\mathbb{C}$
Let $B$ a subset of $H$ such that $B$ is linearly independent and such that exists a homeomorphism $f : [0,1] \to B$ ...
1
vote
1
answer
155
views
Are there polygonal tilings with infinitely many positions, each (or at least one) occurring infinitely often?
My recent question about polygonal tilings where tiles can occur in infinitely many positions has been answered by two nice constructions (besides Jan Kyncl's answer, there is the Conway tessellation ...
2
votes
0
answers
228
views
Question about the term $\sum_{ \rho} \frac{X^{\rho}}{\rho}$ in the explicit formula of $\sum_{n \leq X} \Lambda(n) \chi(n)$
Let $\Lambda$ be the von Mangoldt function and $\chi$ a primitive character mod $q$, then we have the explicit formula
$$
\sum_{n \leq X} \Lambda(n) \chi(n) = \delta_{\chi} X - \sum_{ |Im \ \rho| \leq ...
1
vote
1
answer
221
views
A double sequence in a Banach space
Let $V$ be a infinite dimensional Banach space over $\mathbb{C}$
Let $\{a_{m,n} \cdot v_{m,n}\}_{m,n \in \mathbb{N}}$ be a double sequence with $a_{m,n} \in \mathbb{C}$ and $v_{m,n} \in V$ such that:
...
6
votes
0
answers
210
views
Reconstruct orthogonal from an orthostochastic matrix
Given an $n \times n$ orthostochastic matrix $\mathbf{A}$, i.e., there exists an orthogonal matrix $\mathbf{O}$ with $A_{ij} = O_{ij}^2$ for all $1\leq i,j \leq n$. What is the fastest way to find $\...
2
votes
2
answers
223
views
A Characterization of the traces of functions in $W^{1,2}$
I have a question about the traces of functions in $W^{1,2}$.
Let $D$ be a connected open subset of $\mathbb{R}^d$.We denote $W^{1,2}(D)$ by
\begin{align*}
W^{1,2}(D)=\{f \in L^{2}(D,dx) \mid \...
-2
votes
1
answer
74
views
Matching and minimal degree
Let $n\in\mathbb{N}$ be a positive integer and let $G =(V,E)$ be a connected simple undirected graph with $|V| = 2n$. Is it true that if for the minimal degree $\delta(G)$ we have $\delta(G) \geq n$, ...
6
votes
1
answer
316
views
The Euclidean norm and $k$ largest elements
This is not a homework problem, although I fear it may turn out to be at that level. For any nonnegative $x\in\mathbb{R}^n$, let $f_k(x)$ be the sum of the $k$ largest values in $x$, and define $$f(x)...
9
votes
1
answer
279
views
Essential maps of spectra which are null when localized at any prime
There are maps of spaces which are not null-homotopic, but when localized at any prime become null. I don't know explicit constructions of any, but an example is given in Section 6 of Chapter 25 of ...
12
votes
0
answers
802
views
Number field analog of Artin-Tate $\Rightarrow$ BSD?
What is the difference between the alternating product of the Hasse-Weil $L$-functions of the generic fiber of an arithmetic scheme $X\to\text{Spec}(\mathbf{Z})$ and the zeta function of $X$? (each ...
14
votes
2
answers
825
views
Automorphism of $\mathbb{P}_A^n$
Let $k$ be a field.
The proof that $Aut(\mathbb{P}_k^n)=PGL(n+1:k)$ relies on the fact for any automorphism $\alpha$ of $\mathbb{P}_k^n$, $\alpha^*(\mathcal{O}_{\mathbb{P}_k^n}(1)) = \mathcal{O}_{\...
9
votes
0
answers
617
views
Hartshorne's Conjectures about Algebraic Bundles?
In 1978 Hartshorne published a list of 26 open problems about algebraic bundles on projective spaces [Hart], proceeding from an Oxford conference organized by Atiyah.
I understand that many of these ...
4
votes
1
answer
444
views
Is there a measure on the sphere with positive Fourier transform?
Is it possible to have an even probability measure $\mu$ (that is $\mu(A)=\mu(-A)$ for any set $A\subset \mathbb{R}^d$) supported on the unit sphere $S^{d-1}$ such that its Fourier Transform
$$
\...
2
votes
1
answer
97
views
Reflexive modules up to multiplicity
Call an indecomposable module $M$ over a ring $A$ (restrict to finite dimensional algebras if you like or if it helps) $n$-almost reflexive in case $M^{**} \cong nM$, when $(-)^{*}=Hom_A(-,A)$ and $nM$...
1
vote
0
answers
51
views
A Pathwise Optimization Problem
Suppose $X$ is a nonnegative semimartingale that is progressively measurable with respect to a given filtration $\{\mathcal F_t\}_{t\ge 0}$. Let $c:\mathbb R_+ \mapsto \mathbb R_+$ be a strictly ...
10
votes
1
answer
374
views
How many positions of a tiling polygon can occur simultaneousy?
Let $T$ be a polygon which tiles the plane. For an instance of $T$ (mirrored or not), call the set of its translates a position of $T$.
My question:
How many different positions can occur in ...
5
votes
0
answers
478
views
Poincaré duality for motivic cohomology
Is Poincaré duality for étale motivic cohomology known for projective regular (not necessarily smooth, just regular) schemes over Dedekind rings?
More precisely, two questions. Let $f: \mathcal{X}\to\...
1
vote
0
answers
18
views
Empirical approaches to validate observational bounds on minimum gap between least eigenvalues of $n \times n$ correlation matrix and its submatrices
Let
$\Sigma$ be an $n \times n$ correlation matrix whose least eigenvalue is denoted by $\lambda$.
$\Sigma_i'$ be an $(n-1) \times (n-1)$ submatrix of $\Sigma$ obtained by eliminating the $i$-th row ...
8
votes
0
answers
258
views
Shift on trivalent directed tree, operator and von Neumann algebra
Let $\mathcal{T}$ be the trivalent directed tree, with two parents and one child for each vertex (see below). Let $\mathcal{V}$ be the set of vertices of $\mathcal{T}$ and $H$ be the Hilbert space $\...
2
votes
0
answers
87
views
Functional equation involving integrals and exponential
Can we find on $\mathbb{R}^+$ a real positive function $f(x)$ (in $C^{\infty}$) such that:
$$\int_0^{\infty} f(x) e^{\lambda \int_1^{x} f(t)^2 dt} dx=0$$
where $\lambda$ is a complex number (with $0&...
1
vote
2
answers
194
views
Isolated periodic trajectories of Hamiltonian systems
Is there any example of an autonomous Hamiltonian system with a periodic trajectory isolated in the whole phase space? The Poincar\'e map of such a trajectory within its energy level should be very ...
0
votes
0
answers
219
views
Angle between two vectors in a Minkowski (Finsler) space
Given a Minkowski (or Finsler) space $(V,F)$, I am wondering how to define the angle between two vectors $w$ and $v$. I first thought it must be as $$\cos\theta(w,v)=\frac{g_w(w,v)}{\sqrt{g_w(w,w)g_w(...
7
votes
1
answer
235
views
Independent/Easy fraction of sentences over PA
Let $S(n)$ be the set of all sentences over PA of length at most $n$ (counting the quantifier symbols, boolean connectives, arithmetic operations and constants, and counting each variable as length $1$...
2
votes
0
answers
125
views
Computing the $K$-theory of the free inverse semigroup $C^*$-algebra
A typical example where I know the $K$-theory of the quotient, and wonder if this could help to compute the $K$-theory of the extension. (Or other way round: knowing one part out of three ones.)
I ...
5
votes
2
answers
374
views
Can the relative degree and ramification index can be read off the characteristic polynomial?
Let $L/K$ be a Galois extension of global fields with Galois group $G$. Assume that for a prime $p$ of $K$ we are given the datum $(L_{p}(s,\rho))_{\rho}$, where $\rho$ varies over the irreducible ...
2
votes
0
answers
70
views
Convergence of empirical measure in case of proliferation
I am currently working on the theory of mean field limits of interacting particles. Here are two slides of a talk from an Italian researcher:
I don't understand why he calls $u(t,x)$ a time dependent ...
5
votes
1
answer
154
views
A regular first countable space of cellularity at most $2^\omega$
Let $X$ be a regular first countable space of cellularity at most $2^\omega$.
Is it true that the cardinality of $X$ is at most $2^\omega$?
A cellular family is a family of pairwise disjoint non-...
7
votes
1
answer
330
views
Crafting Suspension Spectra
There is a theorem by Hopkins and Smith which states that for every $n > 0$ there is an ideal $I_n = (v_0^{k_0}, \dots, v_n^{k_n})$ such that there exist a spectrum $X_n$ with the following ...
3
votes
0
answers
164
views
Characterisation of reflexive modules for general rings
A module $M$ over a general ring A is called reflexive in case the canonical evaluation map $e_M: M \rightarrow M^{**}$ is an isomorphism, when $(-)^{*}:=Hom_A(-,A)$. Is $M$ reflexive iff $M \cong M^{*...
2
votes
0
answers
142
views
Any proved connection between Roth theorem and hartmanis stearns conjecture?
Roth theorem classifies numbers into two classes, one is rational and transcendental, another is irrational algebraic numbers, by the number of solutions to the inequality (finite or infinite), and ...
9
votes
3
answers
1k
views
Does there exist a notion of discrete riemannian metric on graph?
I would like to know if there is any notion of a discrete Riemannian metric on graphs. C. Mercat has worked on discrete Riemann Surfaces, but that's not exactly what I am working on.
To be more ...
4
votes
1
answer
320
views
Decompostion of hook schur function in terms of cauchy product of holonomic functions
Let $s_{\lambda}$ denote the schur function and $\lambda$ is the partion of an integer. The schur function written in power sum symmetric basis apper as following. $\chi$ denote the character.
\begin{...
-1
votes
1
answer
117
views
Hierarchies of Operator Norms [closed]
Given some linear operator $T: V \mapsto W$, we can talk about the operator norm between the spaces V and W, i.e.
$$
\| T \|_{V \mapsto W} \ = \ \sup_{g} \| Tg \|_W \ , \quad \mbox{ with } \| g \|_V \...
3
votes
1
answer
1k
views
Completing the square of a matrix expression
Let $A,C\in\mathbb{R}^{m\times n}$, $n\ge m$, $B\in\mathbb{R}^{n\times m}$, and $P$ be a real positive definite $m\times m$ matrix. Denote by $\mathcal{S}^n$ the space of $n\times n$ real symmetric ...
6
votes
0
answers
328
views
Current state of Serre's Motives conjectures in Seattle
It would be worth if we have a current state of the conjectures of
Propriétés conjecturales des groupes de Galois motiviques et des représentations l-adiques. J P Serre. In Motives, Seattle
And ...
10
votes
0
answers
193
views
A theorem of Gugenheim on twisted tensor products
Suppose $A$ is a DGA algebra and $C$ a DGA coalgebra. An $(A,C)$-bimodule is an object $M$ that is both a right $A$-module and a left $C$-comodule in the evident compatible way. An $(A,C)$-bundle is ...
3
votes
0
answers
128
views
Improving prime number generation probability?
Deterministic generation of primes in polynomial time is unknown.
Is there a way to probablistically in $O(n^c)$ time bound for some $c>0$ generate polynomially $\Omega(n^c)$ many integers in $[0,...
5
votes
1
answer
273
views
Factorization of Gabriel-Zisman localization construction?
My question concerns whether the Gabriel-Zisman localization construction $S^{-1}$ for categories admits a known factorization into a pair of commuting constructions $S^l$ and $S^r$.
The localization ...
4
votes
0
answers
165
views
A forcing which can build weird models of $\neg$ADS
There is a class of forcing notions I've been playing around with recently. They have a couple nice properties, and all have the same theme, but I've found them difficult to analyze beyond the basics. ...
3
votes
1
answer
192
views
Why are supplemented line bundles the correct generalization of "lines"?
Originally I had asked this on mathstack exchange but seems possibly appropriate for overflow as well.
Let $k$ be a field. For the vector space $k^d$, a line in $k^d$ is a one dimensional subspace ...
2
votes
1
answer
512
views
Characterisation of reflexive modules
Let $A$ be a semiperfect noetherian ring.
A module $M$ is called reflexive in case the canonical map $f_M: M^{**} \cong M$ is an isomorphism, when $(-)^{*}:=Hom_A(-,A)$. This is equivalent to say that ...
6
votes
0
answers
365
views
K-theory of the infinite dimensional projective space
What is the $K$-theory of the category of coherent sheaves on the infinite (countable) dimensional projective space over a field? As far as I know, $K$-theory is oriented; hence this theory should be "...
5
votes
0
answers
125
views
Volumes of Hecke operators
Let $G=GL(2, F)$ and $K$ a maximal compact subgroup. Unramified Hecke operators are defined by the action of the double cosets
$$T(n) = \bigcup_{\substack{ad=n, a>0 \\ a|d}} K \left( \begin{array}{...
3
votes
2
answers
206
views
(How) do Better TSP Heuristics help in Answering the $NP=P$ Question?
This question is motivated by my impression, that finding better heuristics for the TSP problem (or any other $NP$-complete problem) is "only" of practical interest, but doesn't provide any progress ...
11
votes
0
answers
244
views
Poset of nonvanishing minors of a matrix
This question was posed on MSE here three days ago, but hasn't gotten any answers or suggestions. I hope it's okay to ask it on MO, but if I should wait a little longer, please just let me know.
Say $...
3
votes
0
answers
435
views
"Frobenius Descent"
The following proposition is there in Pink's lecture notes on finite group schemes.
Let $k$ be an algebraically closed field of characteristic $p$. The category of finite length $W(k)$-modules $N$ ...
4
votes
0
answers
105
views
Valued fields with quantifier elimination in the Macintyre language
For which fields $k$ of characteristic $p$ does the Witt construction of a discretely valued field $W(k)$ of characteristic $0$ with residue field $k$ eliminate quantifiers in the language of rings ...
1
vote
1
answer
351
views
Are $\| \Delta u \|_{L^p(\Bbb R^d)} + \| u \|_{L^p(\Bbb R^d)}$ and $\| u \|_{W^{2,p}(\Bbb R^d)}$ equivalent norms?
Is it true that $$\| \Delta u \|_{L^p(\Bbb R^d)} + \| u \|_{L^p(\Bbb R^d)}\quad\text{and}\quad\| u \|_{W^{2,p}(\Bbb R^d)}$$ are equivalent norms?
This results is pretty easy and straightforward for $...
5
votes
0
answers
244
views
A conjecture in rate distortion theory and stochastic filtering
Let $(X_t)_{t\in T}$ be a stationary random process with known and fixed law $P_X$ describing a dynamic source.
This source is to be encoded real-time by an encoder $e$ into an encoded message $E_t$ ...