0
votes
0answers
18 views

Criteria for existence of stable principal submatrices of a stable matrix?

Let $A$ be an $n\times n$ real matrix. Suppose $A$ is stable, that is, all the eigenvalues of $A$ have strictly negative real part. Question: What are some results about existence of stable ...
3
votes
1answer
128 views

Relation between conjugacy class, quotient isomorphism class, and signature of Fuchsian groups

Let $\Gamma\le SL(2,\mathbb{Z})$ be a finite index subgroup, not necessarily "congruence". Let $c_4,c_6$ be the number of conjugacy classes of elements of order 4 and 6 respectively, let $c_{-1}$ be ...
0
votes
0answers
27 views

Does a polynomial system with precisely e solutions have a Groebner basis of degree bounded by e?

Let $k$ be a field and let $R=k[X_1,...,X_n]$ be a polynomial ring. Let $F \subset R$ be a finite subset generating a radical ideal $I$ with precisely $e$ solutions over an algebraic closure of $k$. ...
1
vote
1answer
63 views

Every $W^{1,p}$ has a representative in ACL

Let $\Omega:=(0,1)^n$ and define $ACL_i(\Omega)$ as the set of all Borel functions $u:\Omega\to\mathbb{R}$ such that $$ t\mapsto u(x_1,\dots,x_{i-1},t,x_{i+1},\dots,x_n) $$ is $AC$ for a.e. $(x_1,\...
1
vote
0answers
114 views

Trivial cohomology for fibers implies isomorphism on cohomology

Let $f: Y \rightarrow X$ be a map of topological spaces such that for any $x \in X, f^{-1}(x)$ has trivial cohomology for some cohomology theory (in my case, cohomology with rational coefficients is ...
2
votes
0answers
73 views

Compatibility of formal completion and rigid analytic generic fiber

Let $R$ be a complete valuation ring of rank $1$ (e.g., a complete discrete valuation ring) and let $K$ be its field of fractions. Consider a proper $R$-scheme $X$ that is, say, normal (if needed). ...
0
votes
0answers
119 views

self-intersection number of rational curves in smooth projective surfaces

Given a smooth projective surface $X/\mathbb{C}$, denote the $Hom_1(\mathbb{P}^1, X)$ to be the set of all degree 1 morphisms from $\mathbb{P}^1$ to $X$. We know that we can regard $Hom_1(\mathbb{P}^1,...
6
votes
0answers
118 views

Is the theory of weak $n$-categories a cofibrant replacement of the theory of strict ones?

I have algebraic models of $n$-categories in mind. By "theory of (weak) $n$-categories", I mean "[monad / operad / whatever] whose algebras are (weak) $n$-categories". To be more precise: fix an ...
2
votes
0answers
68 views

relative quantization on fibration

Let $\pi:X\to B$ be a holomorphic submerssion of two Kaehler varieties $X,$ $B$ and $(B,\omega)$ be quantizable and fibres $X_s$ also are quantizable, then $X$ is quantizable?. I want to define ...
-1
votes
0answers
64 views

Automorphism group of a class of abelian varieties

Given two abelian varieties $V$ and $V'$ sharing the same Hasse-Weil L-function, is there a well known, 'canonical' notion of automorphism groups thereof such that $Aut(V)\cong Aut(V')$? If so, does ...
3
votes
1answer
152 views

“theta characteristics” on general motives?

Yesterday I read in some texts on theta characteristics of algebraic curves, which are structures on their Jacobians... Do you know if someone has thought if such, but more general, things can exist ...
1
vote
1answer
43 views

Trace of the inverse sample covariance as the number of samples and dimension scale to infinity

Let $x_1,\dots,x_n$ be i.i.d. $N(0,I_{p\times p})$, with $n>p$. Let $\hat S=\frac1n\sum_{i=1}^n x_i x_i^T$ be the sample covariance. Assume the asymptotic setting where $\frac pn\to \alpha<1$. ...
1
vote
1answer
155 views

Counterexamples to Kunneth formula in algebraic K-theory

Let $X$ be a smooth projective variety with an action of linear algebraic group $G$. Theorem 5.6.1 in Criss/Ginzburg (Representation Theory and Complex Geometry) lists a bunch of equivalent ...
0
votes
1answer
51 views

Algo for covering maximum surface of a polygon with rectangles

I'm looking to an algorithm to covering maximum surface of a polygon with rectangles. Rectangles have to have a specific width, a rectangle can't overlap an other one and each one has to fit in the ...
4
votes
1answer
344 views

Inverse galois problem and étale homotopy

Is there any relation between étale homotopy theory (Grothendieck-Galois theory) and the inverse Galois problem?...I mean...in classical homotopy theory, every finite group $G$ realizes as a "Galois ...
1
vote
0answers
75 views

Induced structure of topological group [on hold]

If we consider a closed Jordan curve $\mathcal{C}$, I know that it's homeomorphic to the circle $S^1$. Now I take an homeomorphism $\phi:S^1\longrightarrow\mathcal{C}$ and this homeomorphism induces a ...
1
vote
1answer
59 views

Non-trivial homomorphisms to finite groups on fixed generating set

A group $G$ is residually finite if for each element $g\in G$ there exists a (surjective) homomorphism $f_g: G \rightarrow H_g$ such that $H_g$ is finite and $f_g(g)\ne 1$. Consider the weaker ...
0
votes
0answers
41 views

Questions about the regularity of the solution of the heat equation in a bounded domain [on hold]

I have questions about the proof of the following theorem: Theorem 8 (Smoothness). Suppose $u\in C^2_1(U_T)$ solves the heat equation in $U_T$. Then $u\in C^\infty(U_T)$ Here is the statement and ...
0
votes
0answers
32 views

How to find $K,W,S$ in the Mostow decomposition theorem?

The Mostow decomposition theorem states: Let $Z$ a nonsingular complex matrix, then $Z$ can be factored as: $$Z=We^{iK}e^S$$ where: $W$ is unitary, $K$ is real and skew symmetric and $S$ is real and ...
4
votes
0answers
58 views

A right adjoint to the truncated Witt functor?

For any ring $A$, let $\mathrm{wEt}_A$ be the category of weakly etale $A$-algebras ; it is a cocomplete category. By a theorem of Van der Kallen, the truncated Witt vector functor $$ W_r : \mathrm{...
6
votes
1answer
98 views

Stochastic Analogue of Stokes Theorem

Dynkin's formula can be thought of as the stochastic version of the Fundamental Theorem of calculus, $$E^x[f(X_{\tau})]=f(x)+E^x\left[\int_0^{\tau}Af(X_u)du\right],$$ where $\tau$ is a first exit time ...
0
votes
1answer
35 views

Length of longest directed circuit in random tournament

Build a random tournament $T=(V,E)$ on $V=\{1,\ldots, n\}$ in the following fashion: for $i < j\in \{1,\ldots, n\}$ let the probability be $0.5$ whether $(i,j)\in E$ or $(j,i)\in E$ (in a ...
5
votes
1answer
144 views

Random Cantor sets on the unit interval

Denote $A=\{0\}, B=\{0,1\}$. Then any subset of $\Omega:=\{A,B\}^{\mathbb N}$ is a continuum provided the number of $B$'s is infinite. We treat these as binary expansions of numbers in $[0,1]$. For ...
1
vote
1answer
108 views

Yoneda extension of a faithful functor is faithful

Let $F: \mathcal C \to \mathcal D$ be a functor with $\mathcal D$ cocomplete, and let $\mathscr P \mathcal C$ be the free cocompletion of $\mathcal C$ (i.e., the category of small presheaves on $\...
0
votes
1answer
105 views

Necessary and sufficient conditions for Kolmogorov's Extension Theorem

Let $(X_n,\mathcal{X}_n)$, $n=1,2,\ldots$ be measurable spaces. Define $Y_n = \prod_{k=1}^n X_k$ and let $\mathcal{Y}_n$ be the corresponding product $\sigma$-algebra. Similarly let $Y=\prod_{k=1}^\...
-1
votes
0answers
60 views

Estimating an exponential sum of a particular type

I was trying to estimate the following exponential sum: For given irrationals $\alpha$ and $\beta$, and given integer $x$ , let $$S(x,\alpha,\beta)=\sum_{n\leq N}\sum_{m\le M}A(m)B(m,n)e(x(m\alpha-n\...
0
votes
0answers
61 views

2D sequence of integers [on hold]

I found the following sequence of integers $a_n^k$, where $n$ and $k$ to extend to infinity. Each row are coefficients of a polynomial, similar to the coefficients of the Legendre polynomials. The ...
4
votes
0answers
100 views

Automorphisms of an infinite graph built from a finite motif

Suppose we have a lattice $L$ in $\mathbb{R}^n$ for which we choose some fundamental domain $D\subset \mathbb{R}^n$ homeomorphic to a closed ball. Translates of $D$ by distinct elements of $L$ ...
4
votes
1answer
192 views

Mapping a group to a finite group s.t. the image of each generator is nontrivial

Recall that a group $G$ is called residually finite if for any nontrivial element $g\in G$ there exists a finite group $H$ and a homomorphism $f$ from $G$ to $H$ such that $f(g)\neq1$. My question is ...
0
votes
1answer
33 views

Convergence of an inhomogeneous markov chain

A markov chain is defined as $X_t=F(X_{t-1})X_{t-1}$, where $X_t$ and $X_{t-1}$ are both vector. So the transition matrix depends on the current states. I want to show that for any given initial ...
3
votes
1answer
120 views

When does every $\infty$-localization correspond to a Bousfield localization?

Let $\mathcal{M}$ be a model category presenting an $\infty$-category $\mathcal{C}$. I believe that every left Bousfield localization $\widetilde{\mathcal{M}}$ of $\mathcal{M}$ corresponds to a ...
3
votes
0answers
63 views

A color interpolation lemma

I need the following "color interpolation lemma". Actually I know a way to prove it, but I'm not very satisfied with that proof. Lemma. Let $G=(V,E)$ be a (properly) colored graph with colors $1, \...
1
vote
0answers
17 views

Examples of partially permutative left-distributive algebras

An algebra $(X,*)$ that satisfies the identity $x*(y*z)=(x*y)*(x*z)$ is said to be a left-distributive algebra. Let $L:X^{2}\rightarrow X^{2}$ be the mapping defined by $L(x,y)=(x*y,x)$ and let $T:X^{...
5
votes
0answers
92 views

For which Ramsey type results density versions are wrong?

I look for examples of Ramsey-type statements, for which the density counterparts do not hold. Example: usual Ramsey theorem. If all edges of a complete graph $K_n$ are colored in $c$ colors, there ...
-3
votes
0answers
62 views

Determinant of a tensor product [closed]

Let V and W be two vector spaces over a field of characteristic zero. Give a formula for the top exterior power of V tensor W.
5
votes
2answers
103 views

Stable unions without stable images

A regular category is one with finite limits and pullback-stable images (i.e. (regular epi, mono) factorizations). A coherent category is a regular category that also has pullback-stable finite ...
0
votes
1answer
95 views

Doubling metrics, doubling measures, Lebesgue density

As stated in this question, Lebesgue differentiation theorem holds on locally doubling space? and proved here, http://www.math.uiuc.edu/~tyson/595f15lecture2.pdf the Lebesgue differentiation theorem (...
4
votes
1answer
130 views

Asymptotics on the number of ways to pair off $\{1, 2, \dots, 2n\}$ into primes

Given $S = \{1, 2, \dots, 2n\}$, we can always pair off elements into $n$ pairs such that each sum to a prime. The proof of this fact is easy and follows from Bertrand's postulate. Now, let $\gamma(n)...
0
votes
0answers
47 views

When are these sums consecutive integers? [closed]

It is possible to construct $\frac{n(n-1)}{2}$ sums which each contain two distinct summands chosen from a set $n$ numbers. For which $n\geq 3$ do there exist a set of $n$ (distinct) integers such ...
2
votes
0answers
140 views

Fields generated by torsion points of CM elliptic curves

I'm using the same setup as Corollary 1.7 on p. 44 of de Shalit manuscript (Iwasawa theory of elliptic curves with complex multiplication). I think there is a mistake in his Corollary 1.7 and I'm ...
2
votes
0answers
83 views

Threshold for prophet inequality

The prophet inequality is related to the following scenario: Suppose there are $n$ independent positive random variables $X_1,\dots,X_n$. They might not be identically distributed. We reveal them ...
1
vote
2answers
154 views

Maps between symmetric powers of the natural module for $SL_2 (k)$ in prime characteristic

Let $G=SL_2(k)$ considered as a linear algebraic group over an algebraically closed field of prime characteristic. Let $E$ be the natural module for $G$ and denote by $S^r (E)$ its $r-$th symmetric ...
2
votes
0answers
67 views

An question about Cauchy Problem in General Relativity [closed]

Yesterday, in Brazilian School on Differential Geometry, a friend asked me the question: Given an (non-trivial) initial data set $(M,g,k)$ for the Cauchy problem in General Relativity. Is there ...
-5
votes
0answers
47 views

Is a continuous two variables function also continuous with respect to each variable? [closed]

I have a simple question, let $f:X\times Y\rightarrow Z$ be a map with two variables, and $X,Y,Z$ are topological spaces, I want to know if $f$ is continuous, then how about $f_{x_{0}}:Y\rightarrow Z$ ...
3
votes
1answer
270 views

Geometric intuition for the condition of Galois descent

Continuing in my attempts to understand bits and pieces of Borceux and Janelidze's Galois Theories, I've just realized that I don't have any geometric intuition for the most convenient ...
3
votes
0answers
45 views

Which blow ups in the base of a conic bundle preserve the “standard” condition?

Assume we are given a nontrivial standard conic bundle $\pi: X\rightarrow S$, that is $X$ and $S$ are smooth projective varieties (say over $\mathbb{C}$), $\pi$ is flat and furthermore we have $Pic(X)=...
-4
votes
0answers
43 views

How can i integral of this function? [closed]

I want to know how can i solve this function. $\int (1-y^d)^n \, dy$ Is it possible to solve it? If you know the method, please teach me.
1
vote
0answers
53 views

Is the category of prederivators cartesian closed?

The question is in the title. ${\bf PDer} = Fun({\bf Cat}^\text{op}, {\bf CAT})$ is obviously cartesian since $\bf CAT$ is. The usual argument for presheaf categories does not apply directly since 1-...
1
vote
0answers
29 views

Strict/strong functors are co/reflective inside lax functors, the coendy way

Bozapalides' remarks on lax presheaves show that the category $[{\cal A}^\text{op}, {\bf Cat}]$ is reflective and coreflective inside the category of lax functors, lax natural transformations and ...
2
votes
1answer
53 views

How to compute bounding coefficients for McDiarmid's inequality?

I am trying to understand the proof in Sec. A2 of Gretton et al.. To make the question self-contained, I summarize below the key ingredients. At the end of the post, I state my question. Given a ...

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