1
vote
0answers
21 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
1answer
31 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
0answers
18 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
1answer
45 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 ...
5
votes
1answer
70 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 ...
-1
votes
0answers
39 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
0answers
36 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
0answers
41 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
1answer
309 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 ...
3
votes
2answers
107 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?
7
votes
2answers
267 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
0answers
51 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.)
2
votes
1answer
120 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
0answers
41 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
0answers
52 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}, ...
2
votes
0answers
43 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
0answers
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
0answers
19 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
0answers
41 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
0answers
83 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
1answer
93 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
0answers
91 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 ...
0
votes
1answer
47 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
1answer
43 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
1answer
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
0answers
34 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
1answer
108 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
0answers
37 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
0answers
47 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 ...
11
votes
3answers
581 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
0answers
86 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
1answer
67 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
0answers
43 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
1answer
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
1answer
107 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
0answers
76 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
0answers
134 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
2answers
242 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
0answers
196 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
0answers
63 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}= ...
2
votes
1answer
116 views

Intersection theory on M_{g,n}

Is there a paper\book that lists the top intersections of Hodge classes and tautological classes on $\overline{\mathcal{M}}_{g,n}$ for small $g$ and $k$, e.g. $g=2,3$ and $k=0,1,2$ ?
1
vote
0answers
77 views

Could one recover the relative K-theory from the quotient derived category?

Let $A\to B$ be a full embedding of exact categories that induces an embedding $D^b(A)\to D^b(B)$. My question is: what can one say about the relation of the homotopy cofibre $K(A)\to K(B)$ (the ...
-3
votes
0answers
77 views

Veronese surface [on hold]

I have a question(Hartshorne ,page 13,exercise 13): If Y be the image of the 2-uple embedding(described in exercise 12) of P^2 in P^5. and if Z(a subset of Y) is a closed curve(variety of dim 1) show ...
4
votes
0answers
102 views

About the convergence rate for an approximation to the heat kernel

Let $G(t,x)$ be the heat kernel $$ G(t,x)=\frac{1}{\sqrt{2\pi t}}e^{-\frac{x^2}{2t}}, \quad t>0, \:x\in\mathbb{R}. $$ Here is one approximation to $G(t,x)$: $$ G_\epsilon(t,x)=e^{-t/\epsilon} ...
2
votes
1answer
62 views

Characterizing and counting boolean functions with all influences 1/2

Is there a characterization of boolean functions $f:\{-1,1\}^n \longrightarrow \{-1,1\}$, so that $\mathbf{Inf_i}[f]=\frac{1} {2}$, for all $1\leq i\leq n$? Is it known how many such functions there ...
-2
votes
0answers
7 views

Trigonometric substitution [migrated]

Been out of touch with trigonometry for some time now. Need help proving this expression. Sin2x/2 = 1/2(1-Cosx) Any help will be appreciated. Thanks.
1
vote
1answer
103 views

Bott formula for projective bundles

For a projective space one has Bott formula to compute $h^q(ℙ^n,Ω^p(k))$, where $Ω^p(k)$ is the k-twisted sheaf of sections in the p-th power of the cotangent bundle of $ℙ^n$. I am wondering if there ...
4
votes
1answer
210 views

Existence of a certain subset of natural numbers equidistributed modulo $m$ for every $m$

I was talking to a friend and the following set $S$ came up. Let $f$ be some real valued function tending to infinity. Let $S$ be a subset of natural numbers such that $|S \cap [1,N]| = N^{\delta}+ ...
3
votes
0answers
44 views

Matching in the Boolean Algebra

We consider the Boolean Algebra of the set $[n]=\{1,\dots,n\}$ and the matching $\psi$ from the $m+1$ subsets of $[n]$ into the $m$ subsets of $[n]$ for $(n+1)/2\leq m+1\leq n$, which is used to show ...
1
vote
0answers
63 views

Where are there defined objects between gerbes and bundle gerbes?

Consider a special kind of/something like a gerbe where there is first given local trivialization data with equivalence over 2-fold overlaps but not isomorphism. Does this exist in the literature?

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