# All Questions

**-2**

votes

**0**answers

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

**0**answers

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

**0**answers

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

**0**answers

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

**0**answers

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

**1**answer

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

**1**answer

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

**0**answers

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

**1**answer

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

**0**answers

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

**0**answers

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

**1**answer

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

**2**answers

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

**0**answers

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

**0**answers

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

**1**answer

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

**0**answers

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

**1**answer

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

**2**answers

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

**1**answer

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

**1**answer

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

**0**answers

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

**0**answers

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

**1**answer

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

**0**answers

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

**1**answer

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

**0**answers

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

**0**answers

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

**0**answers

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

**0**answers

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

**0**answers

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

**0**answers

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

**1**answer

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

**1**answer

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

**1**answer

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

**1**answer

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

**1**answer

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

**0**answers

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

**1**answer

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

**0**answers

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

**0**answers

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

**0**answers

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

**1**answer

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

**1**answer

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

**1**answer

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

**1**answer

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

**1**answer

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

**0**answers

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

**0**answers

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

**0**answers

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 $L$ ...