# All Questions

98,519 questions

**1**

vote

**1**answer

40 views

### Exact solution of two coupled transport equations

I want to solve the following system
$$\eqalign{
& y_t = -y_x + z\text{ in }(0,\text{T}) \times (0,1) \cr
& z_t = z_x + y\text{ in }(0,\text{T}) \times (0,1) \cr
& y(0,x) = y_0,\,\,z(...

**10**

votes

**3**answers

200 views

### Mean maximum distance for N random points on a unit square

Following up on Mean minimum distance for N random points on a one-dimensional line and Mean minimum distance for N random points on a unit square (plane), I have the following questions.
Given N ...

**4**

votes

**0**answers

140 views

### Reference request: ramified and local geometric class field theory

There are lots of references on global unramified geometric class field theory (following Deligne's $\ell$-adic sheaves approach). There are also some notes talking about how to extend Deligne's ...

**2**

votes

**0**answers

85 views

### Calculating the expectation of a sum of dependent random variables

Let $(X_i)_{i=1}^m$ be a sequence of i.i.d. Bernoulli random variables such that $\Pr(X_i=1)=p<0.5$ and $\Pr(X_i=0)=1-p$. Let $(Y_i)_{i=1}^m$ be defined as follows: $Y_1=X_1$, and for $2\leq i\leq ...

**3**

votes

**1**answer

168 views

### Homotopy type of $SO(4)/SO(2)$

A classical result states that the quotient $SO(4)/SO(3)$ is homotopy equivalent to $S^3$. In fact, this can be stated in more general terms since $SO(n+1)/SO(n)$ has the homotopy type of $S^n$. What ...

**-2**

votes

**0**answers

86 views

### Existence of a function $\xi: \Omega \to \mathbb R$

Does there exists a function $\xi: \Omega \to \mathbb R$, which is not a random variable such that for all $x \in \mathbb R$ $\xi^{-1}(x) = \{ \omega \mid \xi(\omega) = x \}$ is a Borel set?
where $\...

**5**

votes

**1**answer

146 views

### Comonad for normalized pseudofunctors for strict higher categories

Garner constructed (in [1] ) for the category of strict $n$-categories a comonad $Q$ as the left part of a cofibrantly generated algebraic weak factorization system such that $$n\text{-}\operatorname{...

**3**

votes

**1**answer

123 views

### Is $\dim_k M/xM$ a multiple of $\dim_k R/xR$ for $M$ finitely generated, torsion-free $R$-module?

Let $R$ be a one-dimensional, reduced and noetherian $k$-algebra (we may also assume that $R$ is a finite $k[x]$-algebra). Let $M$ be a finitely generated, torsion-free module over $R$, i.e. no ...

**0**

votes

**1**answer

62 views

### How good is the LP relaxation?

Consider the optimization problem
\begin{align}
\max_{x\in\mathbb{R}^n}~c^Tx~, \text{ s.t. } Ax=b,~x_i\in\{0,1\}~\forall i
\end{align}where $c,b\in\mathbb{R}_{+}^n$ and $A\in\mathbb{R}_{+}^{n\times n}$...

**-2**

votes

**2**answers

85 views

### Is every graph an incompatibility graph?

Let $G=(V,E)$ be a simple, undirected graph. Is there a partial ordering $\leq\subseteq (V\times V)$ with the following property? $$\{v,w\} \in E \text{ if and only if } v||y$$
(We write $v||w$ in ...

**10**

votes

**1**answer

157 views

### Does every section of the map Gal$(\overline{k(\!(t)\!)}/k(\!(t)\!))\rightarrow$ Gal$(\overline{k}/k)$ stabilize a compatible system of roots of $t$?

There may be some technical issues with the question, but hopefully what I mean is clear...
Let $k$ be a number field (or maybe any finitely generated field over $\mathbb{Q}$ of characteristic 0)
...

**4**

votes

**1**answer

106 views

### Can smoothness of curves into a convenient locally convex vector space be tested with just a dense subspace of the dual?

Let $E$ be a (Hausdorff) locally convex vector space (from now on just "lcs" for short). We say that $E$ is convenient (also called locally complete, Mackey-complete or $c^\infty$-complete) if, given ...

**5**

votes

**1**answer

244 views

### On limits of manifolds

This question should be fairly elementary. I’d just like to check I’m not missing anything.
Let $\{M_n\}_{n\ge 0}$ be an inverse system of smooth manifolds with transition maps $f_{t,s} : M_t\to M_s$,...

**-2**

votes

**0**answers

57 views

### one-dimension differentiability [on hold]

Is the following 2 situations possible?
$f(x)$ is differentiable at $x_{0}$
$
\forall \epsilon > 0,\exists \sigma >0,\exists\breve{x}\in \breve{U}(x_{0},\sigma)
$
$f(\breve{x})$ is not ...

**4**

votes

**1**answer

117 views

### The existence of the extension of a non-trivial line bundle

In Three Dimensional Gravity Revisted, Witten studied the Abelian Chern-Simons theory in three dimensions.
Let $W$ be a three dimensional manifold. Let $\mathcal{L}$ be a non-trivial line-bundle over ...

**14**

votes

**2**answers

585 views

### Units in group rings.

Let $G$ be a finite solvable group of order $n$, and let $g_1 ... g_n$ be an enumeration of its elements. Let $a_1 ... a_n$ be a sequence of integers, such that $\sum a_i$ is relatively prime to $n$.
...

**-5**

votes

**0**answers

81 views

### Problem with a lot of computation [on hold]

This is my first post here. I have trouble with expanding my function to expanded form. I was wondering is there any easy way to do it using WolframAplha or anything else. Here is problem.
We have ...

**12**

votes

**0**answers

223 views

### Hensel lemma and rational points in complete noetherian local ring

Let $A$ be a complete noetherian local ring and $\mathfrak{m}$ be its maximal ideal.
If we have several polynomials $f_i \in A[X_1, \dots, X_m]$ which have a common zero $x_n$ in $A/\mathfrak{m}^n$ ...

**1**

vote

**0**answers

37 views

### Volume of caps of a polytope

Let $K$ be a polytope in $\mathbb R^d$, blow it up by a factor $\lambda>0$. For a unit vector $u \in \mathbb S^{d-1}$, $\lambda K$ has 2 support hyperplanes $H_1$ and $H_2$ with corresponding ...

**1**

vote

**1**answer

92 views

### Mostow rigidity for complex hyperbolic manifolds

A Riemannian manifold $(X,g)$ is hyperbolic if the sectional curvatures are constant and negative. A theorem of Mostow says that these manifolds are determined by their fundamental group.
Theorem (...

**2**

votes

**0**answers

38 views

### Maximal tori of a symmetric subgroup

Suppose $G$ is a complex connected reductive algebraic group, $K$ is a symmetric subgroup of $G$ (i.e. the fixed points of an involution $\theta$ of $G$), and $T$ is a $\theta$-stable maximal torus in ...

**1**

vote

**0**answers

67 views

### Do we have $K \cap P = (K \cap M)(K \cap N)$?

Let $G$ be a connected, reductive group over a $p$-adic field $k$, let $P$ be a parabolic subgroup with Levi $M$ and radical $N$. Let $K$ be a maximal open compact subgroup of $G$ in good position ...

**6**

votes

**1**answer

63 views

### How to compute the Kahler potential of a Sasaki metric

The Question
Given Hessian manifold $M$, there is a natural Kahler structures on $TM$. Is it possible to write the Kahler potential of these in terms of the Hessian potential?
Background
To ...

**3**

votes

**0**answers

68 views

### Comparison of Kodaira dimensions in semistable degenerations

Let $\pi \colon Z\to \Delta$ be a semistable degeneration with a simple normal crossings central fiber $\pi^{-1}(0)$. Here $Z$ is a smooth complex projective variety (or Kähler manifold), $\Delta$ is ...

**2**

votes

**1**answer

75 views

### The blow-up rate of a nonlinear oscillator

(Related to this Math.SE question.)
For $p>1$, let $u$ be a solution to $$\tag{1}\frac{d^2 u}{dt^2} + u = |u|^{p-1}u$$ that blows up at $T>0$, that is $$\lim_{t\nearrow T}u(t)=+\infty.$$
...

**-1**

votes

**0**answers

63 views

### Weak closure of subsets of the unitary sphere of a Banach space

Assume that $(X,\|\cdot\|)$ is a Banach space with $\|\cdot\|$ strictly convex. Define $S=\{x\in X:\|x\|=1\}$. Suppose that $\varepsilon>0$ and $x_0\in S$ and define
$$
B_\varepsilon=\{x\in X:\|x-...

**1**

vote

**1**answer

118 views

### Descending almost-contained subsets of $\omega$

Let $A$ be an infinite subset of $\omega$ such that $\omega\setminus A$ is also infinite.
Under the Continuum Hypothesis is there a sequence $(A_\alpha)_{\alpha<\omega_1}$ of subsets of $\omega$ ...

**3**

votes

**0**answers

32 views

### Optimizing a multivariate symmetric (permutation-invariant) function

Let $\ell$ and $d$ be two integers such that $\ell \le d$.
I would like to find the global maxima of the following symmetric function $f\colon (0,1]^n \to \mathbb{R}$,
$$f(x_1, \ldots, x_n) := \sum_{\...

**5**

votes

**1**answer

75 views

### Examples of non-isomorphic $C^\ast$ algebras with isomorphic quasi-state spaces

Let $A$ (resp. $B$) be a unital $C^\ast$-algebra, $\mathcal{Q}(A)$ (resp. $\mathcal{Q}(B)$) the compact convex subset of $A^\ast$ equipped with the $\sigma(A^\ast, A)$ (resp. $\sigma(B^\ast, B)$) ...

**4**

votes

**0**answers

46 views

### Explicit parametrization of closed space curves of constant curvature

Joining arcs of helices it is pretty easy to obtain closed curves with constant curvature and $C^2$ regularity (see http://www.heldermann-verlag.de/jgg/jgg01_05/jgg0203.pdf). Joining arcs of Salkowski ...

**4**

votes

**1**answer

107 views

### Global dimension of the tensor algebra

Let $R$ be a semisimple ring with a non-zero $R$-bimodule V. Let $T_R(V):= \bigoplus\limits_{k=0}^{\infty}{V^{\otimes_k}}$ be the tensor algebra of $V$.
Question 1: Is there a simple proof that $...

**3**

votes

**2**answers

222 views

### Is there any Lie algebra structure on the sheaf of sections of adjoint bundle

Let $X$ be an irreducible smooth projective variety over $\mathbb{C}$.
Let $G$ be an affine algebraic group over $\mathbb{C}$.
Let $p : E_G \longrightarrow X$ be a holomorphic principal $G$-bundle ...

**1**

vote

**1**answer

125 views

### Hodge numbers of compact Ricci-flat Kaehler manifold

Assume that $M$ is a closed connected Ricci-flat Kaehler manifold $M$ of complex dimension $n\geq 3$ with $h^{2,0}(M)=0$. Is is possible that
$h^{n, 0}(M)\neq 1$
$h^{p, 0}(M)\neq 0$ for some $0< p&...

**1**

vote

**0**answers

31 views

### Spherical code for interesection of $k$-sparse vectors and unit sphere

Let us assume $X\in\mathbb{R}^{n\times d}, rank(X)=d$, integer $k\in\mathbb{N},k\ll d$, positive constant $0<\epsilon<1$, and $\mathcal{S}\subset \mathbb{R}^d$ denotes the unit sphere. We also ...

**-1**

votes

**0**answers

136 views

### a question about the notation in the book “Opera de Cribro”

When I study the book "Opera De Cribro" by John Friedlander, Henryk Iwaniec-(2010), in Sections 1.2 and 1.3, I confused with notation used there. In page 3 it is defined:
$$\cal{A} = (a_n) , n\le x$$,...

**4**

votes

**1**answer

122 views

### $Td_p$ notation of Kotschick

In this paper, notation $Td_p$ is used without explicit definition (it is stated that it is a certain combination of Chern numbers). It is claimed that HRR theorem implies
$$
Td_p(M)=\sum_{q}(-1)^q h^{...

**-2**

votes

**1**answer

79 views

### Probability doubt [on hold]

Four cards are randomly selected from a pack of 52 cards. If the first two cards are kings, what is the probability that the third card is a king?
The given answer is 2/50.
I was thinking 4 cards ...

**1**

vote

**0**answers

59 views

### Homology of SL(2,R) with finite coefficients

Consider the third homology group of a real special linear group
$H_3 (SL(2,\mathbb{R}),\mathbb{F}_p)$. It is known$[1]$ that for $p=2$ the third homology group of $SL(2,\mathbb{R})$ vanishes.
...

**3**

votes

**1**answer

101 views

### Is such singularity Gorenstein?

Let $X$ be an integral normal scheme over $\mathbb{C}$ with isolated singularity at a closed point $p$. Suppose that $X$ admits a rational resolution $f:Y\to X$ with the exceptional set equals to the ...

**-1**

votes

**0**answers

87 views

### Hadamard matrix research [on hold]

I need to find out whether something new has happened in the study of Hadamard matrices for the last 10 years.
If someone was engaged in matrices, can you recommend literature?
K. Horadam is the ...

**3**

votes

**1**answer

107 views

### Global dimension of a graded algebra

Let $A= \bigoplus\limits_{n=0}^{\infty}{A_n}$ be an $\mathbb{N}$-graded algebra with semisimple $A_0$.
Question: Do we have that the global dimension of $A$ is equal to $\sup \{i \geq 0 | Ext_A^i(...

**3**

votes

**0**answers

117 views

### Computing Bohr Radii

The Bohr radius $R$ for $\mathcal{H}(\mathbb{D})$ is defined as $$R = \sup\limits_{0<r<1} \left\{ r\ |\ \sum\limits_{k=0}^{\infty}|a_k|r^k \leq |f|_\mathbb{D} \text{ for all }f(z)=\sum\limits_{k=...

**1**

vote

**0**answers

19 views

### Second order necessary and sufficient conditions for convex nonsmooth optimization

For convex smooth optimization, first and second order necessary and sufficient conditions are well known. Does such standard second order necessary and sufficient conditions exist for convex ...

**0**

votes

**0**answers

42 views

### Show the spectral radius of a matrix is smaller than 1

Let $\hat{\bf H}$ be a $p\hat{N}\times p \hat{N}$ sparse matrix consisting of $p\times p$ blocks, where each block is of size $\hat{N}\times\hat{N}$. The values in $\hat{\bf H}$ is illustrated below (...

**0**

votes

**0**answers

30 views

### k-ary necklaces with conserved/fixed indexes

I asked a question in a previous post about enumerating all possible k-ary bracelets with certain positions fixed to a specific bead/character. I asked a few mathematicians, and it seems its quite ...

**5**

votes

**0**answers

144 views

### Non reduced projective schemes

Let $X$ be separated noetherian scheme. Suppose that $X_{red}$ (the associated reduced scheme) is projective. Does it follow that $X$ is also projective ? Otherwise what are the minimal conditions on $...

**1**

vote

**1**answer

84 views

### Is every finite graph isomorphic to the proximity graph of some $S\subseteq \mathbb{R}^n$?

This is the question that I should have asked before asking this older question.
If $(X,d)$ is a metric space, we associate with it a simple, undirected graph, called its proximity graph $G(X,d)$ ...

**0**

votes

**0**answers

55 views

+50

### Bifurcations due to a nonlinearity parameter

Suppose we want to analyze the behavior of the system
$$\dot{\mathbf{x}}=\mathbf{f}(\mathbf{x},t;\varepsilon),\quad \mathbf{x}\in\mathbb{R}^n,\quad t\in\mathbb{R}^+,\quad\varepsilon\in\mathbb{R}^+,
$$
...

**7**

votes

**1**answer

125 views

### Finite subgroups of $PSU(3)$

I'm looking for a reference to a classification or description of finite subgroups of $SU(3)$ that contain the center, or equivalently $PSU(3)$. Can anyone point me in the right direction?

**4**

votes

**1**answer

81 views

### Semi-continuous fields of C*-algebras having dimension one on a dense set

Given a Hausdorff, locally compact space $X$, let us consider a semi-continuous field $\{A_x\}_{x\in X}$ of C*-algebras over $X$, such that $A_x$ is one-dimensional for every $x$ in a dense subset $D$ ...