# All Questions

**3**

votes

**1**answer

27 views

### Properties to have matrices that commute in $\mathrm{GL}_n(\mathbb C)$

Let $G$ be a finite subgroup of $\mathrm{GL}_n(\mathbb C)$, $A,B \in G$ whose eigenvalues are thus in the unit circle.
Assume that the eigenvalues of $A$ are included in a circle arc of ...

**0**

votes

**0**answers

7 views

### Idempotent fractional ideals of a noetherian domain

Let $R$ be a commutative Noetherian domain, $K$ its fraction field, and $J$ a fractional ideal (i.e. a finitely generated sub-$R$-module of $K$) such that $J^2=J$. Is it true that $J=0$ or $J=R$? If ...

**0**

votes

**0**answers

16 views

### Parabolic bundles on elliptic curves

as a warm up for his thesis I would like a student of mine to read something on parabolic bundles. He is reading the famous Atiyah paper on vector bundles on elliptic curves, so I think it would be ...

**1**

vote

**0**answers

7 views

### Convex interaction energy

Does anybody know examples of absolutely continuous probability measures $\mu_0,\mu_1$ on $\mathbb{R}^n$ with finite 2nd moments such that
$$
\frac{d^2}{dt^2}\left(\int_{\mathbb{R}^n\times ...

**0**

votes

**0**answers

4 views

### TSP: Approximation Ratio of the Double Tree Heuristic after Diagonals have been Removed

In their article "Double-Tree Approximations for the Metric TSP: Is the Best One Good Enough?", Vladimir Deineko and Alexander Tiskin derive a lower bound for the approximation ratio of the ...

**0**

votes

**0**answers

15 views

### Metrizable Coalgebras

A Coalgebra $C$ is called metrizable if there is a base $B$ for $C$(as a vector
space) and a metric $d:B \times B \to \mathbb{R}$ on $B$ such that the linear extension $\tilde{d}: C\otimes C ...

**-2**

votes

**0**answers

38 views

### Prove that (AxB)∩(CxD)=(A∩C)x(B∩D) [on hold]

Prove that $(A\times B)\cap (C\times D) = (A\cap C)\times (B\cap D)$
where $\times$ represents the Cartesian product.

**0**

votes

**1**answer

61 views

### Theorem with an example

i have this theorem
in the paper they gives an example:
but here $H_1$ is not satisfied !
How to correct it please?

**0**

votes

**0**answers

16 views

### What algebras does the hidden subgroup problem for finite abelian groups apply to?

Shor's algorithm is said to solve the hidden subgroup problem for finite Abelian groups, of which factoring and the discrete logarithm problem for integers belong to. Apparently the HSP for finite ...

**0**

votes

**0**answers

17 views

### Quick question about conjugate equivalence bimodules and inner products

let $A$ and $B$ be $W^{*}$-algebras, let $X$ be an $A-B$-equivalence bimodule (according to the definition given in "Morita equivalence for $C^*$-algebras and $W^*$-algebras" by Rieffel, ...

**2**

votes

**0**answers

41 views

### Diameter of $n$-unit-vector closed scribble

Suppose one creates a random, closed, likely self-crossing polygon
from $n$ unit-length vectors arranged head-to-tail,
randomly oriented except for the requirement
that their sum is zero (so the ...

**1**

vote

**1**answer

61 views

### Bound for the Frattini subgroup of a $p$-group

Assume that $G$ is a finite $p$-group, $p$ odd, with a non-trivial elementary abelian Frattini subgroup. Then both $\Phi(G)$ and $G/ \Phi(G)$ are vector spaces over $\mathbb{F}_p$. Is it possible to ...

**0**

votes

**0**answers

31 views

### Reference request: has this semilinear version of Navier Stokes been studied?

I have noticed that the
Navier Stokes equations can be written as a semilinear symmetric first
order system
$$
u_t+A_1u_{x_1}+A_2u_{x_2}+A_3u_{x_3} = f(u)
$$
for a 9 by 1 vector $u$ containing the ...

**1**

vote

**1**answer

67 views

### l-functions of calabi-yau varieties

This question might not be suitable for MO since i know nothing about Calabi-yau varieties aside the fact that they are used in string theory to compactify additional dimensions, but still, it makes ...

**6**

votes

**1**answer

100 views

### Entropy for Haar measure on $O(n)$

Let $G$ be a locally compact group. A measure $\mu$ is the right-Haar measure on $G$ if for every $g\in G$ and $E\subseteq G$ Borel set $\mu(Eg)=\mu(E)$. It is known that every locally compact group ...

**-2**

votes

**0**answers

51 views

### Example of flasque but non-soft sheaves?

Does anyone have an interesting examples of a flasque but not soft $\mathscr{O}_X$-module over a ringed space? Of course with X being non-paracompact.

**1**

vote

**0**answers

52 views

### How to show non-existence of elements in the intersection of two ideals?

Given l, k any two natural numbers, define
$I_1 =\langle y^{l+2k}, (x+y)^{3l+2k} \rangle: x^{l+2k} + \langle y^{2l+3k}, (x+y)^{3l} \rangle: x^{l+2k}; $
$I_2 = \langle x^{l+k} \rangle.$
I want to ...

**0**

votes

**0**answers

44 views

### Characteristic subgroups of the limit group

Let $\{ G_i \}_{i=1}^\infty$ be a direct spectrum of groups with respect to embeddings $\varphi_i:G_i \mapsto G_{i+1}$, $i \in \mathbb{N}$, and let $G$ be the limit group of this spectrum. Suppose ...

**6**

votes

**1**answer

374 views

### Algebraic Closure of a Ring is Not a Ring?

I'm trying to motivate the notion of integrality in a ring extension. It seems that the following would be a good motivation, because it would show that the notion of algebraic elements over a ring is ...

**4**

votes

**2**answers

119 views

### When distance nonincreasing map is an isometry

Let $f: M \to M$ be a distance nonincreasing map between a closed Riemannian manifold $M$
and $f$ is homotopic to the idendity map. Is it then $f$ an isometry?

**8**

votes

**2**answers

333 views

### What was the Question that led Euler to his Investigations on Polyhedra?

The question that led Euler to his investigations on graphs is the well-known question related to the seven bridges of Königsberg, and that story is a must in every introduction to graph theory.
...

**1**

vote

**0**answers

56 views

### Chain homotopy of non-abelian category

How can one define the chain homotopy in non-abelian category? (The category I have in mind is the category of chain complexes of monoids.)

**3**

votes

**1**answer

128 views

### Quotienting $SU(3)$ by $U(1)$?

As is well-known, if we quotient $SU(2)$ by the action of $U_1$, embedded in the diagonal as $(e^{i \theta}, e^{-i \theta})$, we get the $2$-sphere. As is also well-known, if we quotient $SU(3)$ on ...

**-3**

votes

**0**answers

43 views

### Product of Positive Intever Divisors of 6^16 equals 6^k [on hold]

Product of Positive Intever Divisors of 6^16 equals 6^k
How would I find K?
Don't give me the answer, just how to get it
Thanks

**1**

vote

**0**answers

61 views

### Possibilities for dimensions of $\mathfrak{m}^i/\mathfrak{m}^{i+1}$ for a local ring

Let $R$ be a local commutative ring with maximal ideal $\mathfrak{m}$, and denote by $k$ the residue field $R/\mathfrak{m}$. Then we can look at the sequence of $k$-vectorspaces
$$R/\mathfrak{m}, ...

**3**

votes

**0**answers

51 views

### Absolute continuity reflected in Fourier coefficients?

Imagine $\mu$ and $\nu$ are two Borel probabilty measures in the interval $[0,1]$.
We say that $\mu$ is absolutely continuous with respect to $\nu$, if for every measurable set $A$ such that ...

**0**

votes

**0**answers

16 views

### Multiplicity of Minimum Eigenvalue of a Convex Combination of Hermitian matrices?

Let $A_1,\dots,A_L$ be $N\times N$ hermitian matrices. Consider the problem
\begin{align}
\lambda^{\star}=\max_{}&\lambda_{min}\left(\sum_{i=1}^{L}r_iA_i\right) \\
&r_i\geq ...

**-5**

votes

**0**answers

20 views

### identifiability of a linear regression [on hold]

If we have a generative model $X_2=X_1a_1+\varepsilon$ where $\varepsilon \sim \mathcal{N}(0,\sigma_2^2)$
do we have $X_1=X_2a_2+\varepsilon '$ where $\varepsilon \sim \mathcal{N}(0,\sigma_1^2)$
...

**1**

vote

**0**answers

51 views

### Fundamental theorem of calculus for iterated stochastic integrals

I'm trying to find the rate (or a bound for it) with which an iterated integral of the type
$$\int_{-h}^0 \int_{-h}^{t} A_s d B_s A_t d B_t$$
converges to zero (in probability/distribution) for $h ...

**6**

votes

**0**answers

88 views

### Rational Hodge Theory

I am currently learning about the formality result for Kaehler manifolds proved in DGMS. Their result uses some basic Hodge theory, and so only gives formality over $\mathbf{R}$. Later, however, ...

**6**

votes

**1**answer

116 views

### What is the Essential Reason that allows a PTAS for the EUCLIDEAN TSP?

Questions:
Is there some understanding of the reason, why the euclidean TSP allows a PTAS, whereas the metric TSP in general does not and, is the PTAS stable under sufficiently small perturbation ...

**3**

votes

**1**answer

114 views

### Generalization of notion of convexity

I am searching for the correct term for the following, if it exists.
A set $X\subset \mathbb{R}^2$ is called $r$-convex if for any two points $x_1, x_2\in X$ such that there exists an arc of radius ...

**-1**

votes

**1**answer

59 views

### Clique factorization

I'm reading about Clique factorization in wikipedia:
http://en.wikipedia.org/wiki/Gibbs_random_field#Clique_factorization
but I'm unable to understand this:
What is $X_C$ here? Ok I understood ...

**3**

votes

**1**answer

47 views

### introduction books for Dynamic systems of discrete Schrodinger operator for beginner

In this semester, I study in a class of dynamic system. recently the French professor turn to the dynamic system of discrete operator. I find it is difficult to find a book in English. (I have found ...

**3**

votes

**1**answer

72 views

### Motivation for the Preprojective Algebra

Let $Q=(Q_0,Q_1)$ be a quiver and $k$ a field. We construct a new quiver $\bar{Q}$ in the following way. Let the vertices of $\bar{Q}$ be the same as the vertices of $Q$, and let the arrows of ...

**-3**

votes

**0**answers

36 views

### Boolean function resulting in ith bit value? [on hold]

Let's say f:{0,1}^n -> {0,1} is a boolean function. And let's say this function depends only on the ith bit. Namely, it results exactly as the ith bit value of the given input. Is this a valid boolean ...

**6**

votes

**1**answer

111 views

### Properties of the time integral of Wiener process

Let $W_t$ be a Wiener process and consider the time integral
$$ X_T:= \int_0^T W_t dt $$
It is often mentionend in literature that $X_T$ is a Gaussian
with mean 0 and variance $T^3/6$.
I am ...

**-5**

votes

**0**answers

38 views

### Need a Proof -Unbounded function on any open set [on hold]

Lets define f(x)= {if x=m/n then f(x)=n,if x=irrational then f(x)=0}.
Such f(x) is unbounded on any (a,b) .
Can't understand the proof.Can somebody write detailed proof?
Thanks.

**0**

votes

**0**answers

49 views

### The Jordan-Brouwer Separation Theorem for Manifold

I have read the paper The Jordan-Brouwer Seperation Theorem written by Wolfgang Schmaltz. The main result in paper is below
Any compact, connected hypersurface $X$ in $\mathbb R^n$ will divide ...

**14**

votes

**3**answers

634 views

### I would like to have a counter example that Peano's theorem does not apply to spaces with infinite dimension

Does Peano's theorem apply to spaces with infinite dimension? Or is there a counterexample?
Here, Peano's theorem is:
Let $E$ be a space with finite dimension. Consider a point $(t_0,x_0) \in \Re ...

**3**

votes

**0**answers

87 views

### Asymptotics and combinatorics

Wright's expansion of
$$
(1-z)^b\exp[A/(1-z)^c], \text{for } A > 0, 0<c<1\tag{1}
$$
is, in the words of the late, great Mark Kac "well known to those that know it well".
(See, for example, ...

**3**

votes

**1**answer

76 views

### Geometric Intuition of $P^+$ in Modular Tensor Categories

I'm currently reading through Bakalov and Kirillov's "Lectures on Tensor Categories and Modular Functors," and I am having some difficulty understanding the definition of $p^\pm$ given on page 49. ...

**2**

votes

**0**answers

44 views

### $C^1$ stability conjecture on non-compact manifolds

In the study of dynamical systems, the link between structural stability and Smale's Axiom A has been explored by many authors. One of the important outcomes of this endeavor was the proof of $C^1$ ...

**2**

votes

**1**answer

49 views

### Length inequalities in trees and CAT(0) spaces

I have a family of possibly related questions. Let me start with an elementary one:
Question 1. Fix an integer $n$. For which collections of real numbers $a_{ij}$, $i, j = 1, \dots, n$, is it true ...

**4**

votes

**1**answer

110 views

### Dimension of a set detected by a homology class

A colleague asked me a topology question which comes down to this: Suppose that $M$ is a smooth $n$-manifold, and $C\subset M$ is a closed set such that $H_{n-p}(M-C)\to H_{n-p}(M)$ is not surjective. ...

**0**

votes

**0**answers

77 views

### Jacobian change of basis matrix for different dimensions

I am considering a real Lie group $G$ acting transitively on an open set $U$ in a real Euclidean space of lower dimension. Given a smooth, compactly supported function $f: U \rightarrow \mathbb{R}$ ...

**8**

votes

**0**answers

138 views

### Lifting Abelian Varieties to p-adic fields

Assume I have an abelian variety $A$ over a finite field $k$ of characteristic $p$. Work of Norman and Oort (1980) says I can lift $A$ to an abelian variety $\mathscr{A}$ over some characteristic ...

**11**

votes

**2**answers

250 views

### Subgroups of $SL_3(\mathbb{Z})$ that are finitely generated, Zariski-dense, infinite index, and torsion-free

My question stems from Misha's answer of a MathOverflow question. Misha supplied the following question in his answer:
Open question: Does there exist a finitely generated Zariski-dense ...

**10**

votes

**0**answers

207 views

### What is the best lower bound for 3-sunflowers?

A collection of $t$ sets $A_i$ is called a t-sunflower if $A_i \cap A_j = Z $ for all $i \neq j$ for some fixed $Z$. A well-known conjecture of Erdos and Rado says that in any $k$-uniform family of ...

**4**

votes

**0**answers

71 views

### $\Omega$ and $B$ as adjoints between symmetric monoidal $(\infty,n)$- and $(\infty,n-1)$-categories

Given a symmetric monoidal $(\infty,n)$-category $\mathcal{D}$, one obtains a symmetric monoidal $(\infty,n-1)$-category $\Omega \mathcal{D}$ by taking $\Omega \mathcal{D}= ...