# All Questions

**-1**

votes

**0**answers

21 views

### subsequence problem [on hold]

give you two sequence A,B which length is N.A_i>=1,A_i<=N,B_i>=1,B_i<=N.Prove that there exists a subsequence of A and B that the sum of two subsequence is equal? How to solve it?

**0**

votes

**2**answers

35 views

### Motivation for weak solution of a PDE (initial condition)

The following question came to me when reading the famous paper of ALT and LUCKHAUS: "Quasilinear elliptic-parabolic differential equations"
When looking at a (nonlinear degenerate) PDE like
$$ ...

**-1**

votes

**0**answers

57 views

### Mellin transform on $\mathbb{Z}[\omega]$

I'm eager to ensure some facts which are elementary for many experts here.(I also post the question on Stack exchange to discuss more effectively. ...

**3**

votes

**1**answer

169 views

### Who first defined quantum integers?

Who first gave the defintion of quantum integers
$$ [m]_q = \frac{1 - q^m}{1 - q} $$ and addition as
$$ [m]_q \oplus_q [n]_q = [m]_q + q^m [n]_q $$ and multiplication as
$$ [m]_q \otimes_q [n]_q = ...

**0**

votes

**1**answer

66 views

### In a finite field with characteristic 2, how can I check if a given polynomial is divisible by (x^2+1)? [on hold]

Given the 0 or 1 coefficients of a very high degree polynomial $P(x)$ over GF(2), where x is an element of $GF(1024)$, is there a simple algorithmic way to find out if this polynomial is divisible ...

**4**

votes

**1**answer

64 views

### cohomology of classifying space of permutation groups

Let $\Sigma_k$ be the permutation group of order $k$.
Let $r: \Sigma_k\to GL(k)$ be the regular representation by permuting the order of the standard basis of $\mathbb{R}^n$.
Let $\rho: ...

**1**

vote

**1**answer

109 views

### A proof from Lang's undergraduate analysis

This is from P.580 of Serge Lang's undergraduate analysis (2nd edition).
$\textbf {Proposition 2.3.}$ Let $A$ be an admissible set in $\mathbb R^n$ and assume that its closure $\bar{A}$ is contained ...

**0**

votes

**0**answers

21 views

### Projection formula for smooth representations of locally profinite groups

Let $G$ be a locally profinite (i.e., locally compact, Hausdorff, and totally disconnected) topological group, $H \le G$ a closed subgroup, $(\pi, V)$ a smooth representation of $G$, and $(\sigma, W)$ ...

**1**

vote

**0**answers

28 views

### Fundamental invariants for root subsystems

Let $\Phi$ be an irreducible root system of rank $\ell$. The fundamental invariants of $\Phi$ is a set of $\ell$ integers $d_1, \cdots, d_\ell$ canonically attached to $\Phi$.
Now suppose $\Psi$ is ...

**0**

votes

**0**answers

11 views

### How can the Cayley-table for the elements of basis of a Cayley-Dickson algebra be summarized in an algebraic expression?

One would be able to construct a Cayley table that has all $e_i$ elements of the basis of algebra $A$ where $0<i<\dim A$ such that $e_0=1$ and $e_1=i$ and $e_2=j$ and so on. I'm looking for an ...

**4**

votes

**1**answer

56 views

### What version of the wreath product embedding theorem is actually stated in the famous paper of Kaloujnine and Krasner?

This question is inspired by Terry Tao's blog post and the comments there. I have always cited M. Krasner and L. Kaloujnine, "Produit complet des groupes de permutations et le problème d'extension de ...

**1**

vote

**0**answers

37 views

### Lower bound of first moment of $L$-function on $\mathrm{GL}(3)$

Let $\pi$ be an automorphic form on $\mathrm{GL}(3,\mathbb{A}_{\mathbb{Q}})$.
Do we know any case that
\[\int_0^{T} \left|L(\frac{1}{2} + it, \pi)\right| dt \gg T\]
holds unconditionally?
I know the ...

**3**

votes

**0**answers

48 views

### Weinstein's local classification of Lagrangian foliations

In the paper "Symplectic manifolds and their Lagrangian submanifolds", Weinstein showed that locally all the Lagrangian foliations are symplectomorhic to the fiber foliation of a cotangent bundle.
I ...

**0**

votes

**1**answer

110 views

### Probability of the maximum of a throw of an infinite number of $n$-sided dice being $k$

Let $X$ be the random variable obtained as the maximum of a throw of $m$ dice (each of which is $n$-sided). In other words, $X = \max\{l_1,\cdots, l_m\}$ where $l_i$ can take any value between $1$ and ...

**4**

votes

**0**answers

40 views

### On the ratio of Gilbreath sequences

Let $n \in \mathbb{N}_{>0} \cup \{ \infty \}$ and let $E_n$ be the set of sequences $(d_i)_{i=1}^n$ such that $d_1=1$, $d_i$ is an even integer (for $i > 1$) and $0<d_i \le i$. Let ...

**2**

votes

**2**answers

109 views

### Map between stacks and automorphism groups

I know that the Torelli morphism $t_g:\mathcal{M}_g\rightarrow \mathcal{A}_g$ between the stacks of smooth curves of genus $g$ and principally polarized abelian varieties of dimension $g$ is of order ...

**7**

votes

**5**answers

134 views

### The limit of edge-midpoint convex polyhedra

Starting with a convex polyhedron $P_1 \subset \mathbb{R}^3$,
replace that with $P_2$, the convex hull of the midpoints of the edges of $P_1$.
Continuing this process, we obtain a ...

**3**

votes

**1**answer

72 views

### A question on Gandy-Jensen system and the rudimentary functions

Let $\mathrm{R}_0,\cdots,\mathrm{R}_8$ be the following functions:
$\mathrm{R}_0(x,y)=\{x,y\}$
$\mathrm{R}_1(x,y)=x-y$
$\mathrm{R}_2(x)=\bigcup x$
$\mathrm{R}_3(x,y)=x\times y$
...

**0**

votes

**0**answers

13 views

### Dirichlet distribution: Normalization of alpha values [migrated]

I'm a programmer and currently trying to apply the Latent Dirichlet Allocation algorithm by Blei et al. on a text mining problem. I am using a library called gensim for this, which takes, among ...

**1**

vote

**1**answer

107 views

### Is every polynomial ring over any field regular?

Is every polynomial ring over any field regular?
For a field that is algebraically closed, it is true as any maximal ideal of $k[x_1,...,x_n]$corresponds to a point $(t_1,...,t_n)$ in $\mathbb{A}^n$ ...

**2**

votes

**0**answers

52 views

### quotient a scheme by a stratified vector bundle

Let $k$ be a field.
Let $X$ be a $k$-scheme of finite type, normal and integral. We consider $f,g:R\rightarrow X$ an equivalence relation, surjective and such that it is a stratified vector bundle, ...

**1**

vote

**1**answer

75 views

### hypergeometric at nearest singularity

Reference request. A prototype case:
In
$$
{}_2F_1\left(\frac{1}{12},\frac{5}{12};\frac{1}{2};x\right) =
A\log\left(\frac{1}{1-x}\right) + B + o(1),
\qquad x \to 1^-
$$
what can we say about the ...

**-1**

votes

**0**answers

116 views

### A metric on $S^{2}$ [on hold]

Let $p:S^{3}\to S^{2}$ be the Hopf fibration $p(z,w)= (\parallel z\parallel^{2}-\parallel w\parallel^{2},\;2z\bar{w})$.
Define a metric on $S^{2}$ as follows:
$$d(x,y)=Hd(p^{-1}(x), ...

**2**

votes

**0**answers

51 views

### A 1-D random variable from a random distribution

I have a random variable $X$ that is drawn from the pdf
$$
f(x; \mu, \sigma, \sigma_{\mu}, \sigma_{\sigma}) = \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} \frac{1}{|\hat{\sigma}|\sqrt{2\pi }} ...

**5**

votes

**1**answer

115 views

### O-minimal Theories with Non-Dense Order Type

I asked this question on MSE, but I haven't received any comments or responses (also, it has a very low view count), so I thought I would also ask it here.
In this paper, Knight, Pillay, and ...

**0**

votes

**0**answers

38 views

### Existence and Uniqueness of Volterra integral equations of the first kind

$$
\int_0^t k(s,t)f(s)ds=g(t)
$$
To prove the existence and uniqueness of solutions of Volterra integral equation(VIE) of the first kind, we usually differentiate it and convert to the VIE of the ...

**2**

votes

**0**answers

98 views

### A possible generalization of the Borsuk Ulam theorem via action of symmetric groups

The symmetric group $S_{m}$ is equiped with the counting Har measure $\mu$ and the obvious sgn character. Assume that $S_{m}$ acts on $S^{n}$, $n\geq m-1$ and $f:S^{n}\to \mathbb{R}^{n}$ ...

**3**

votes

**1**answer

80 views

### The complex heat kernel on a Riemann manifold

There is a vast literature available for the heat kernel. Nevertheless, I haven't been able to find almost anything useful about the kernel of the equation $\frac{1}{\mathbb{i}} \frac{\partial ...

**1**

vote

**0**answers

104 views

### is there an analogy between fractals and automorphic forms? [on hold]

Disclaimer: this question is rather vague and thus might not be suitable for this site. Still, as I already asked a similar question on MSE and got no feedback, I finally decided to take the plunge ...

**1**

vote

**0**answers

54 views

### Homology and Exterior Square

Let $G$ be a finite group and $G \wedge G$ denote the exterior square of $G$. It is well known that the second integral homology $H_2(G,\mathbb{Z})$ is the kernel of homomorphism $x \wedge y \mapsto ...

**-4**

votes

**1**answer

87 views

### A criterion of norm null sequences in Banach space [on hold]

I would like to know if for a weak* null sequence $\left( f_{n}\right) $ in a Banach space $X$,
the following characterisation is true and what about its proof:
$\left( f_{n}\right) $ is norm null ...

**-4**

votes

**0**answers

73 views

### an question about number theory [on hold]

Let $s_i=\frac{(q^n-1)...(q^n-q^{i-1})}{(q^{i-1})...(q^i-q^{i-1})}$, where $q$ is prime and $n$ is a positive integer.
Now can anyone tell me this, $\lim_{n\mapsto \infty}\frac{\sum_{1\leq i\leq ...

**6**

votes

**2**answers

183 views

### Guises of the Stasheff polytopes, associahedra for the Coxeter $A_n$ root system?

Richard Stanley keeps a famous running compilation of different guises of the celebrated Catalan numbers. The number of vertices of the associahedron is one instantiation among the multitude, and the ...

**7**

votes

**1**answer

159 views

### On $V$-decisive and weakly homogeneous forcings

Suppose that $\Bbb P$ is a forcing in $V$, we say that $\Bbb P$ is $V$-decisive if whenever $\varphi(x_1,\ldots,x_n)$ is a statement in the language of forcing, and $u_1,\ldots,u_n\in V$ then $1_{\Bbb ...

**1**

vote

**1**answer

61 views

### Connections between loops (algebraic structure) and graphs

I would like to know whether there are known constructions which provide a bijection between loops (isomorphism classes) and (possibly directed) graphs. Any reference to a useful paper in this ...

**1**

vote

**0**answers

74 views

### Clarifications on twisted forms

Suppose $F = F(\bar{k})$ is a finite algebraic group over a number field $k$. The absolute Galois group $\Gamma_k$ of $k$ acts on $F$ by group automorphisms via a homomorphism $\rho: \Gamma_k \to ...

**-1**

votes

**0**answers

105 views

### Where can I find the classification of groups of order 8p? [on hold]

I need to classify the groups of order $8p$ up to isomorphism. We know that one of these groups is $G=\langle a,b| a^p=b^8=1, b^{-1}ab=a^{-1}\rangle$. Can I find other groups of this classification ...

**3**

votes

**0**answers

120 views

### Canonical functions in set theory and their applications

Given regular cardinal $\kappa>\omega,$ we can define the canonical functions $f_\alpha: \kappa\to \kappa,$ for $\alpha<\kappa^+.$
Some of their properties are presented in Chapter 22 of the ...

**2**

votes

**0**answers

78 views

### Is the kissing number in $n$ dimensions always divisible by $n$? And what is the base of exponential growth of the kissing number?

And why are the kissing numbers for 1, 2, 3, 4 and 8 dimensions all highly composite numbers?

**1**

vote

**1**answer

116 views

### How to extend index theorem to infinite dimensional manifolds?

I was thinking about the following "obvious" construction the other day. Let
$$
\mathbb{S}^{\infty}_{2}=\bigcup_{i=1}^{\infty}\mathbb{S}^{2i}
$$
Then because we know that $\chi(\mathbb{S}^{2i})=2$ for ...

**4**

votes

**0**answers

139 views

### Separating points in the plane II

Let A be a set of $2m$ points on the plane so that no open set of diameter $2$ has more than m of them. Define $A+A+...+A$ ($k$ times) to be the multiset of $k$-sums from $A$. That is, we consider all ...

**9**

votes

**2**answers

374 views

### Parity of $\lfloor 1/(x y) \rfloor$ not equally distributed

A curious puzzle for which I would appreciate an explanation.
For $x$ and $y$ both uniformly and independently distributed in $[0,1]$,
the value of $\lfloor 1/(x y) \rfloor$ has a bias toward odd ...

**2**

votes

**0**answers

82 views

### Do the algebras for a $\infty$-monad form a stable $\infty$-category?

I'm wondering if a monad $T$ on a stable $\infty$-category $\cal C$ has a stable $\infty$-categories of algebras, provided $T$ preserves finite limits/colimits.
Is this true?

**1**

vote

**1**answer

127 views

### Is the notation ${}^t g$ for the transpose of a linear transformation intended to be suggestive?

The notation ${}^t g$ for the transpose of a linear transformation is, in my view, quite unusual: otherwise (at least in many areas of math), one almost never sees subscripts or superscripts appearing ...

**2**

votes

**1**answer

90 views

### Which nonnegative matrices have exact nonnegative matrix factors of smaller dimensionality?

The nonnegative matrix
$V = \left( \begin{array}{cc}
1 & 1 \\
1 & 1 \end{array} \right)$
has nonnegative matrix factors $W = \left( \begin{array}{c} 1 \\ 1 \end{array} \right)$ and $H = ...

**2**

votes

**0**answers

58 views

### Actions of amenable groups on graphs with uncountably many ends

Let $G$ be a finitely generated amenable group acting transitively on an amenable Schreier graph $S$. Is it possible for $S$ to have uncountably many ends? An amenable graph with uncountably many ends ...

**8**

votes

**0**answers

94 views

### Extended TFT with coefficients in spans in any $\infty$-topos

In the TFT classification article by Jacob Lurie (arXiv:0905.0465) the (∞,n)-category of correspondences (there: $Fam_n$) plays a key role, whose $k$-morphisms are $k$-fold spans of ...

**1**

vote

**0**answers

99 views

### Simply connected pencils

Is it true/false that for every complex projective variety $V$, there is a smooth simply-connected one parameter family $\pi:X\to \mathbb{P}^1$, one of whose smooth fibers is $V$?
(i.e. we want ...

**2**

votes

**3**answers

189 views

### Representations of the two dimensional non-abelian Lie algebra

I would like to know a complete description of the indecomposable representations of the two dimensional non-abelian Lie algebra over the complex numbers. The finite dimensional representations would ...

**2**

votes

**1**answer

173 views

### “Nice” functions on infinite-dimensional space of germs of continuous functions at a point

Consider set of all germs of continuous functions at some point.
Question: What are some functions ("any/nice/constructive/whatever") from this set to R (reals) ? (Except evalution at point and made ...