-2
votes
0answers
35 views

In algebraic system, what consequences brings having two elements whose powers greater than 1 coincide? [on hold]

In algebraic system, what consequences brings having two elements (say, $w_1$ and $w_2$) whose powers greater than 1 coincide? Will the exponentiation or taking root become ambiguiuos given that $w_1$...
1
vote
0answers
58 views

Existence of a $\lambda$-generated Model Category Structure

Apologies if this is a stupid question: Let $C$ be a cofibrantly generated model category ($\mathbf{Edit}$: Combinatorial) and let $[X,C]$ be functor category equipped with the projective model ...
0
votes
0answers
24 views

A sum involving binomial coefficients and its evaluation using the Gamma function [migrated]

Does anyone know how to prove (or a reference for) the following identity for positive integers $r$: $$\sum_{i=0}^r (-1)^i{r\choose i}\frac{1}{ir+1}= \frac{\Gamma(1+1/r)\Gamma(r+1)}{\Gamma(r+1+1/r)}$$
3
votes
0answers
46 views

Action of longest element of Weyl group on zero weight space

Let: $G$ be a real semisimple Lie group; $\rho$ be an irreducible representation of $G$ on a finite-dimensional real vector space; $A$ be a "Cartan subspace" of $G$ (a Lie subalgebra which is ...
0
votes
0answers
58 views

Example of bundle-mapping over $S^4$ with singularity $S^2$

Could anyone give a non-trivial example of a bundle-mapping over $S^4$, i.e. find two complex rank 2 vector bundles $E_0,E_1$ over $S^4$ and a bundle mapping $$0\to E_0\overset{v}{\to}E_1\to0$$ such ...
0
votes
1answer
58 views

Domain of the Stokes operator

Let $\Omega\subseteq\mathbb R^d$ be open ($d\in\mathbb N$) $\mathcal D:=C_c^\infty(\Omega)^d$ and $$\mathfrak D:=\left\{\phi\in\mathcal D:\nabla\cdot\phi=0\right\}$$ $\mathcal H:=\overline{\mathfrak ...
4
votes
1answer
99 views

Betweenness in permutations

Let us consider permutations $\pi$ on $\{1,\dots,n\}$ as sequences $\pi(1),\pi(2),\dots,\pi(n)$. For a permutation $\pi$ let $R(\pi)$ be the ternary relation with $(x,y,z)\in R(\pi)$ whenever element $...
-2
votes
0answers
37 views

metrizable topology and furstenberg's topology [on hold]

Let $X$ be a metrizable topological space. Are there methods for constructing a metric which induces the topology of $X$? And,pls, Is anyone aware of any problem related to opened Furstenberg's ...
3
votes
1answer
155 views

Rank 2 vector bundles over $\mathbb CP^2$

Is there any classification of the rank 2 complex vector bundles over $\mathbb CP^2$ up to diffeomorphism? Thank you.
1
vote
0answers
65 views

Question on Hochschild cohomology

Let given ring $A$ without zero divisors and subring $\mathbf{k}\subset Z(A)$. Let also given $M~-$ $A~-$ bimodule, such that $xm = mx, \forall x\in \mathbf{k}, m\in M$. Is it true that if for $\...
0
votes
0answers
63 views

If $X_0$ is very dense in $X$ and $A \cap X$ is closed, then what is $A$?

Let $X$ be a scheme and let $X_0$ be a very dense subset (e.g., $X$ a finite type scheme over a field and $X_0$ the closed points). If $A$ is an arbitrary subset of $X$ such that $X_0 \cap A$ is ...
6
votes
1answer
125 views

On a result of Frankl and Wilson

In the paper 'Intersection theorems with geometric consequences' (Combinatorica 1981) P. Frankl and R. M. Wilson consider families $\mathcal{F}$ of $k$-subsets of $\{1,\dots,n\}$ with the restriction ...
3
votes
2answers
163 views

$\sum_{m\in \mathbb{Z}} e^{-im^2 t} e^{i m z} =? $

Can anyone sum up this series? $f(z, t) = \sum_{m\in \mathbb{Z}} e^{-im^2 t} e^{i m z} . $ In the mathematical sense, each term of this series is of modulus 1, and the series is not convergent. ...
4
votes
0answers
106 views

Hierarchical (Recursive) Random Walk Model

Consider the following hierarchical (recursive) random walk model. For the first level $\ell=1$, let $\{X_t^{(\ell)}\}_{t=1}^{T}$ be a (discrete) random walk. For the next level $\ell=2$, we ...
0
votes
0answers
23 views

Max number of points in a grid with distance to the origin between $d$ and $d+\sqrt{2}/2$, for some distance $d$? [on hold]

Consider the square grid centered at the origin $\{-n,\ldots,n\}\times\{-n,...,n\}$. What is the value or an upper bound, as a function of $n$, of $\max f(d)$, where $d$ is the distance from some ...
1
vote
1answer
121 views

Rank 2 complex vector bundles over $S^2\times S^2$

How could people classify all rank $2$ complex vector bundles over $S^2\times S^2$ up to isomorphism? Could you give a rank 2 complex vector bundle which cannot be split as a sum of two line bundles?
8
votes
0answers
103 views

Penrose transform and general wave equations

In the late 1960's Penrose developed twistor theory, which (amongst other things) lead to an exceptional description for solutions to the wave equation on Minkowski space via the so-called Penrose ...
21
votes
1answer
990 views

What did Yu Jianchun discover about Carmichael numbers?

There's a news story going around (see for example [1]; other accounts are even more breathless) about an amateur mathematician, Yu Jianchun, finding an "alternative method to verify Carmichael ...
4
votes
2answers
188 views

Rank 2 complex vector bundles over $S^4$

On $S^4$, we know that rank 2 complex vector bundles are classified by $\pi_3(U(2))=\mathbb Z$. Any element $g\in\pi_3(U(2))=\mathbb Z$ determines a complex vector bundle $E$ over $S^4$. Can we say ...
1
vote
1answer
151 views

Partial Universes and the Axioms of $ZF$ Set Theory Without Choice

In his Senior Thesis, Samuel Coskey answered the question of which axioms of $ZFC$ hold at each stage of the cumulative hierarchy. Here is the list of his results: Axioms that always hold: ...
1
vote
1answer
34 views

Polynomial with subset of critical points and values prescribed

Motivated by this question I am motivated to pose the following question: Given $k$ points $(x_1, y_1), \ldots (x_k, y_k)$ with (WLOG) $x_i < x_{i+1}$, can we find a polynomial $p(x)\in\Bbb R[x]$ ...
-4
votes
0answers
27 views

Reccurence relation for [on hold]

Hi guys I am a little confused on a question I am working on. "Write the recurrence relation for: 5, 0, -8, -17, -25, -30,…" I am getting Cn= {5 if n=1} {Cn-1+(2n-1) if n > 1} ...
0
votes
0answers
17 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
126 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
58 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
112 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
70 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
116 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
114 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
67 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
149 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
154 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
49 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
327 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
74 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
40 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
31 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
96 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
33 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
141 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
107 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
58 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
98 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 $L$ ...

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