# All Questions

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### Prove that (AxB)∩(CxD)=(A∩C)x(B∩D)

Prove that $(A\times B)\cap (C\times D) = (A\cap C)\times (B\cap D)$ where $\times$ represents the Cartesian product.
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### 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?
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### Who are the modern day “Grassmans”, and what are their theories?

Grassmann undeniably has a bit of a sad history, given the lame credit his work received during his lifetime, and much, much later. Gian-Carlo Rota in his book Indiscrete Thoughts theorizes: To ...
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### 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 ...
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### 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, ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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.
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### 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 ...
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### 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 ...
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### 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 ...
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### 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?
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### 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. ...
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### 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.)
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### 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 ...
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### 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
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}, ... 0answers 44 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 ... 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 ... 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) ... 0answers 44 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 ... 0answers 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, ... 1answer 105 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 ... 0answers 96 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 ... 1answer 57 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 ... 1answer 45 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 ... 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 ... 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 ... 1answer 109 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 $$It is often mentionend in literature that X_T is a Gaussian with mean 0 and variance T^3/6. I am ... 0answers 38 views ### Need a Proof -Unbounded function on any open set [on hold] Lets define f(x)= {if x=m/n then f(x)=n,if x=irrational then f(x)=0}. Such f(x) is unbounded on any (a,b) . Can't understand the proof.Can somebody write detailed proof? Thanks. 0answers 49 views ### The Jordan-Brouwer Separation Theorem for Manifold I have read the paper The Jordan-Brouwer Seperation Theorem written by Wolfgang Schmaltz. The main result in paper is below Any compact, connected hypersurface X in \mathbb R^n will divide ... 3answers 618 views ### I would like to have a counter example that Peano's theorem does not apply to spaces with infinite dimension Does Peano's theorem apply to spaces with infinite dimension? Or is there a counterexample? Here, Peano's theorem is: Let E be a space with finite dimension. Consider a point (t_0,x_0) \in \Re ... 0answers 87 views ### Asymptotics and combinatorics Wright's expansion of$$ (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, ... 1answer 74 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. ... 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 ... 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 ... 1answer 109 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. ... 0answers 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} ... 0answers 137 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 ... 2answers 244 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 ... 0answers 203 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 ... 0answers 65 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}= ... 1answer 118 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 ? 0answers 78 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 ... 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 ... 0answers 111 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} ...
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 ...