All Questions

1
vote
1answer
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
3answers
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
0answers
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
0answers
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
1answer
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
0answers
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
1answer
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
1answer
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
1answer
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
2answers
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
1answer
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
1answer
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
1answer
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
0answers
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
1answer
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
2answers
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
0answers
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
0answers
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
0answers
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
1answer
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
0answers
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
0answers
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
1answer
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
0answers
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
1answer
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
0answers
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
1answer
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
0answers
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
1answer
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
0answers
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
1answer
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
2answers
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
1answer
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
0answers
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
0answers
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
1answer
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
1answer
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
0answers
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
1answer
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
0answers
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
1answer
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
0answers
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
0answers
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
0answers
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
0answers
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
0answers
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
1answer
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
0answers
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
1answer
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
1answer
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 $D$ ...

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