# All Questions

**0**

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2 views

### Why is it so hard to compute $\pi_n(S^n)$?

Of course it isn't really that hard - nowhere near as hard as $\pi_k(S^n)$ for $n>k$, for instance. The hardness that I'm referring to is based on the observation that apparently nobody knows how ...

**1**

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**0**answers

4 views

### The 4-th generator of $K_1$ group for 3-dimensional NC tori algebra

An $n$-dimensional NC torus algebra $A_\theta^{(n)}$ is defined for any antisymmetric $n\times n$ matrix $\theta$ of real numbers as the universal $C^*$-algebra, generated by unitaries ...

**0**

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10 views

### The behavior of series involving special subsets of the prime numbers

It is well known that the series $\sum_{p\in \mathbb{P}} \frac{1}{p}$ diverges where $\mathbb{P}$ denotes the set of primes. Brun proved that $\sum_{p\in \mathbb{P_2}} \frac{1}{p}$ converges where $ ...

**2**

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**0**answers

36 views

### A curious Gauss-Sum type identity

Let $q=e^{2\pi i/m}$, $a\in\mathbb{R}$ and $1\leq j\leq m-1$. I would like to prove that:
$$(a-1)\sum_{n=0}^{m-1} q^n\frac{\prod_{k=0}^{j-2} (q^{n+k+1}-a)}{\prod_{k=0}^{j} (aq^{n+k}-1)}=0.$$
For ...

**-8**

votes

**0**answers

56 views

### Is there a site where I should post questions about mathematics for which I seek a solution? [on hold]

Is there a site where I should post questions about mathematics for which I seek a solution, without risk that it will be closed for not being "research level"?

**0**

votes

**0**answers

7 views

### Is there effective algorithm for finding “minimal discovery time” for large graphs?

Consider a large, probably sparse graph with Markovian random walkers on it. Define discovery time as time to first
reach a vertex by random walk
from uniform start. Are there effective ways to find ...

**-5**

votes

**0**answers

48 views

### About the Riemann Zeta Function $\zeta(s)$ [on hold]

Could we formulate the Riemann Zeta Function as follows? Sorry for my naive question.
For every odd number on the form $n=pq$, where $p$ and $q$ are primes, there exists $\zeta(s)=0$ such that
...

**0**

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**0**answers

16 views

### Isotropic correlation function for a vector valued random field

I'm having trouble with some of the implications of the following theorem.
Let $\mathbf{T} (\mathbf{x})$ be a mean-square continuous vector valued random field on $\mathbb{R}^3$ satisfying conditions ...

**3**

votes

**0**answers

35 views

### What is known about the chromatic number for minimum-distance graphs in higher dimensions?

For a set of points in $\mathbb{R}^d$ with minimum distance $a$, the minimum-distance graph connect two points iff they are at distance $a$. We can also view it as the tangency graph for a set of ...

**-5**

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65 views

### Which univariate function satisfies $e^{g(x)} + e^{-g(x)} = \alpha x$ for $x>0$ and some constant $\alpha>0$? [on hold]

Which univariate function $x \mapsto g(x)$ satisfies $$e^{g(x)} + e^{-g(x)} = \alpha x$$ for $x>0$ and some constant $\alpha>0$?
How can it be computed? What does it look like? How can it be ...

**-2**

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43 views

### toledo's lecture on cartwirght-steger surface

I am interested in Toledo's lecture given in IAS workshop. I want to find some related reference about his lecture. While actually i am not able to find much. Is someone also interested in this and ...

**0**

votes

**0**answers

46 views

### A probability application question

Suppose there are two possible states $H$ and $L$, with prior probability $p$ and
$1-p$ respectively. There are infinite rounds with a discount factor $ d$. In
round 1, you could choose a value ...

**1**

vote

**0**answers

20 views

### About the $C^{1,1} $regularity of the boundary of a set

I am studying a paper that uses the following property :
Consider $U$ and $V$ open, bounded and convex domains in $R^n$ with $U \supset \overline{V}$ and suppose that $min_{y \in V} |x - y| = \lambda ...

**2**

votes

**1**answer

83 views

### Quantitative stability: Hausdorff distance between subdifferentials

Suppose I have two convex functions $f$ and $g$ mapping $\mathbb{R}^{n}\to\mathbb{R}$ (so they are 'more' than proper). Suppose $\|f-g\|_{\infty}<\epsilon$, the sup-norm. In particular, the ...

**0**

votes

**1**answer

81 views

### Which $\frak{sl}_2$-Representations Arise From Hermitian Metrics

Recal that $\frak{sl}_2$ is the Lie algebra with basis elements $e,f,h$, and bracket
$$
[e,f] = h, ~~~ [h,e] = 2e, ~~~ [h,f] = -2f.
$$
For $M$ a $2n$-complex manifold, the Lefschetz identities tell us ...

**3**

votes

**0**answers

95 views

### Monte Carlo variant of Hilbert's Tenth Problem

Let $k \in \mathbb{N}$. Given an algorithm $\mathcal{A}$ which takes
as argument a polynomial $P \in \mathbb{Z}[x_1,\dots,x_k]$ and either
returns true or false, we say that $\mathcal{A}$ works for ...

**0**

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57 views

### Can (how) one distinguish germs of continuous functions by a countable set of params?

Continuous functions can be distinguished by their values at say rational points of [0 1].
Germs of analytic functions can be distinguished by derivatives at a point.
So in both cases we see ...

**-2**

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**0**answers

37 views

### identity of equation [on hold]

We have the equation ($\partial_{\mu}\partial_{\nu}$-$\eta_{\mu\nu}\Box$)$\phi=0$, where $\phi$ is a scalar field, $\Box=\partial_{\mu}\partial^{\mu}$ is a standart Dalamber operator, $\eta_{\mu\nu}$ ...

**-3**

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110 views

### College - Ecuation to solve in 3 variables (p,q,r) in [1,2] [on hold]

This is the image containing the ecuation

**1**

vote

**1**answer

34 views

### orthonormal basis or Parseval frame for Sobolev spaces

Consider the uni-variate Sobolev space of order $m$:
$$ \mathcal{W}_2^m=\{ f:[0,1] \rightarrow \mathbb{R}|f,f^{(1)},\dots ,f^{(m-1)}\, \text{are absolutely continuous and}\, f^{(m)}\in L^2\}.$$
It is ...

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58 views

### Calculation in From Seiberg Witten to pseudo-holomorphic curve

I am reading the Taubes's paper: From From Seiberg Witten to pseudo-holomorphic curve. I don't know how to get the result (2.17)
\begin{eqnarray*}
...

**6**

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93 views

### scheme of commuting matrices

Let $k$ be any field. Let $r$ and $n$ be two positive integers.
Consider the functor $F$ from the category of $k$-schemes to the category of sets which sends a $k$-scheme $T$ to the set of matrices ...

**1**

vote

**0**answers

18 views

### Properties of solutions of Parabolic type equations

Assume $u\in C([0,1],L^{2})$ satisfies the following Schrodinger equation
$$
\partial_t u=i(\Delta u+Vu), \text{in} ~\mathbb{R}^n\times[0,1],\\
u(0)=u_{0}.
$$
with $V=V_1(x)+V_2(x,t)$, where $V_{1}$ ...

**0**

votes

**2**answers

58 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**

vote

**0**answers

107 views

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

I'm eager to ensure some facts which are elementary for many experts here.
Let $\omega=\frac{-1+i\sqrt{3}}{2}$ be a complex cube root of unity. The Eisenstein integers $\mathbb{Z}[\omega]$ (a unique ...

**4**

votes

**2**answers

399 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

86 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 ...

**6**

votes

**1**answer

103 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

164 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

28 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 $(\sigma, W)$ ...

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vote

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45 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 ...

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16 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 ...

**5**

votes

**1**answer

116 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 ...

**3**

votes

**1**answer

63 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

61 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

150 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 ...

**5**

votes

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80 views

### On the ratio of Gilbreath sequences

Definitions: 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 $0<d_i \le i$. ...

**3**

votes

**2**answers

130 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 ...

**9**

votes

**5**answers

248 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 ...

**2**

votes

**1**answer

106 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

128 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 $\mathbb{A}^n$ ...

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votes

**0**answers

61 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

82 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

176 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

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55 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

118 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

40 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

119 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

102 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 ...