4
votes
0answers
52 views

Numerical integration error bounds on the unit sphere

A sequence of points $x_1,x_2,\dots$ on the unit sphere $S^{D-1}$ is said to be uniformly distributed if \begin{align} \lim_{N \rightarrow \infty} \frac{1}{N} \sum_{j=1}^N f(x_j) = \int_{x \in ...
-1
votes
0answers
146 views

Disprove this Piece of Jensen's Inquality “Black Magic”

Jensen's inequality states that if a real valued function $f(x)$ is concave, like $f(x)=\ln |x|,$ then $E(f(X))\le f(E(X)).$ A classic application of this is $E(X) \le \ln |E(e^{X})|.$ Now consider ...
4
votes
0answers
77 views

Categories in which an epimorphism applied to a non-monic epimorphism can be monic

Let $\mathcal{C}$ be a category, and let $A$, $B$, and $C$ be objects. Given $A \xrightarrow{f} B \xrightarrow{g} C$ such that: $f$ is both epic and monic $g$ is epic but not monic $gf$ is epic and ...
-4
votes
0answers
49 views

Pascals triangle [on hold]

I was out sick for a fair bit and I come back and we are doing this! Can someone explain what I'm supposed to do or show me a video?? "Use Pascal's triangle and the regularity of decreasing powers of ...
3
votes
0answers
110 views
+50

Will (general points + small number of arbitrary points) impose independent condtions on plane curves?

It is well known that imposing vanishing at general points of $\mathbb P^2$ gives independent conditions on curves of degree $d$. Also, it is known that a small number ($\le d+1$) points always impose ...
2
votes
1answer
89 views

Largest subgroup of $SU(n)$ for which the adjoint action preserves specific inner product on $\mathfrak{su}(N)$

Given $A,B \in \mathfrak{su}(n)$ such that $K(A, B)=0$, I am looking for the largest subgroup $H$ of $SU(n)$ for which: $K \left(A, Ad_{U}(B) \right) = 0, \ \ \forall U \in H$ where $K$ is the ...
2
votes
1answer
196 views

Real and imaginary part of an holomorphic function

I guess this could be a very elementary question. Anyway I can not find an answer in literature. Let $f:U\rightarrow\mathbb{C}$ be an holomorphic function on an upen subset $U\subseteq\mathbb{C}$. ...
2
votes
0answers
69 views

McDuff's classification of symplectic manifolds

According to a theorem of McDuff, if $(X,\omega)$ is a closed symplectic 4-manifold, which contains a symplectic sphere of self-intersection +1, then $X$ is symplectomorphic to $CP^2$ blown up a few ...
6
votes
3answers
241 views

Nontrivial solutions for $\sum x_i = \sum x_i^3 = 0$

For $x_i \in \mathbb{Z}$, let $\{x_i\}$ be a fundamental solution to the equations: $$ \sum_{i= 1}^N x_i = \sum_{i=1}^N x_i^3 = 0 $$ if $x \in \{x_i\} \Rightarrow -x \notin \{x_i\}$. For instance, a ...
6
votes
2answers
252 views

Charts needed for an atlas

I just read this question link and asked myself, if there is any easy way to decide how many charts you actually need to cover a given compact manifold in $\mathbb{R}^3$, maybe at least in this ...
3
votes
2answers
217 views

Example s.t. the unbased loop-space is not $\Omega X \times X$

For a connected pointed CW-complex $X$, let us write (as usual) $\Omega X$ for the space of based loops at $X$. I am looking for an example where the space $\Omega' X$ of all (unbased) loops in $X$ is ...
6
votes
0answers
102 views

How to show the compatibility between Duflo isomorphism and Harish-Chandra isomorphism for semi-simple Lie algebras?

I was told that the Duflo isomorphism is compatible with the Harish-Chandra isomorphism when the Lie algebra $\mathfrak{g}$ is semi-simple. However I cannot see why this is true. All I can show is ...
5
votes
1answer
62 views

When is a formula preserved under taking factors in a reduced product or the stalk in a Boolean product?

I want to know if there is a nice characterization of when a formula is preserved under taking reduced factors. We say that a formula $\phi$ is closed under taking reduced factors if whenever $I$ is ...
13
votes
1answer
465 views

Must an algebraic variety with trivial tangent bundle be an abelian variety?

Suppose $X$ is a proper algebraic variety with trivial tangent bundle $T_X$ (not only canonical bundle $K_X$), is it true that $X$ is an abelian variety? (For the complex manifold case this is not ...
2
votes
0answers
22 views

LQR solution when there are linear terms in the cost function?

I am trying to solve the following Bellman Equation: $V(s) =\max_u \left[a'u - (u-s)'Q(u-s) + V(u)\right]$ In the equation above, $s,u,a\in \mathbb R^n$, $Q\in \mathbb R^{n\times n}$ is positive ...
1
vote
0answers
25 views

Equivalence of Graphical model selection algorithms

Suppose, a jointly Gaussian random vector is denoted by $X \in \mathbb{R}^{p}$ and $X$ has a distribution given by $\mathcal{N}(\mu,\Sigma)$. It is known that estimating the graphical model that ...
3
votes
0answers
118 views

Why write GRR with the relative tangent sheaf?

The first published version of the Grothendieck-Riemann-Roch theorem, GRR for short, was written in the form $$ \operatorname{ch}(f_!\alpha).\operatorname{Td}(Y) = ...
7
votes
0answers
163 views

Integer solutions of $x^2=4+8y^2+13z^2$

I have been looking for integer solutions of certain Diophantine equations, one of the simplest examples being $x^2=4+8y^2+13z^2$. The ideal answer would be a way to parametrize all the integer ...
2
votes
0answers
132 views

Volume of a complex manifold

Is there a theorem which states the following? Let $\mathcal{M}$ be a $k$ -dimensional complex submanifold of $\mathbb{C}^n, \ 1 \le k \le n$. Let $V \subset M$ be open, relatively compact in $M$, so ...
2
votes
0answers
50 views

Interpolation Operator Bounded in Sobolev Norm

Let $m\in \mathbb{N}$, $p\in [1,\infty]$, $W^{m,p}([0,1])$ the space of all functions $[0,1]\rightarrow \mathbb{R}$ which are $m$ times weakly differentiable and weak derivatives in $L^p$, ...
1
vote
0answers
35 views

Reference request: Topological space of polygonal chains and its properties [migrated]

I'm interested in approximations of $C^1([a,b])$-functions by polygonal chains: image File:NURBstatic.svg by User:WulfTheSaxon licensed under GNU Free Documentation License A polygonal chain can be ...
3
votes
1answer
141 views

A question on the cohomology of elliptic curves over local fields

Let $K$ be a number field,$\nu$ a nonarchimedian prime of $K$, $K_{\nu} $ the completion of $K $ at $\nu $ with maximal unramified extension $K_{\nu}^{unr} $. Let $E $ be an elliptic curve defined ...
5
votes
2answers
89 views

Negatively curved metrics minimizing the length of a homotopy class of simple closed curves

Good afternoon everyone ! I have the following question of Riemannian geometry : Let $M$ be a smooth closed orientable manifold of dimension at least $3$, and let $\mathcal{T} = \{ $ smooth ...
1
vote
2answers
84 views

Sobolev trace map: is the fractional seminorm bounded by just the gradient?

Let $M$ be a compact Riemann manifold. Consider the trace map $T:H^1(M) \to H^{\frac 12}(\partial M)$. Is it always the case that $$|Tu|_{H^{\frac 12}(\partial M)} \leq C\lVert \nabla u ...
-2
votes
0answers
38 views

How do I calculate the Entropy of a vector? [on hold]

I understand the concept of entropy, I even referred to the wikipedia page but I am confused. Can anyone tell me in simple words how I could calculate the entropy of a vector, with an example please? ...
0
votes
0answers
13 views

The mutual information rate spectrum [migrated]

Definition: $\mathbf{X}$ denotes the random vector $({X_1},{X_2},...,{X_n})$. The mutual information between $X$ and $Y$, $I(X;Y)$, is determined by the joint law of $p(X,Y)$, Given two random ...
-4
votes
0answers
29 views

Invariance of absolute determinant under alternating sign changes in columns [on hold]

I (experimentally) notice that for an $MN \times MN$ matrix, where $M$ is even if $N$ is odd and vice-versa, if I multiply each column $c_i$ by the elements of either (i) $T_1 = [t_1^{(1)}, ...
3
votes
1answer
115 views

Existence of state on a C*-algebra satisfying $|\tau(ab)|=\|ab\|$

(This is a repost of a question from math.SE, http://math.stackexchange.com/questions/1240966/existence-of-state-on-a-c-algebra-satisfying-tauab-ab) Let $a,b$ be elements of a unital C*-algebra $A$ ...
-4
votes
0answers
68 views

spheres are not simpletic? [migrated]

Reading some books on diferential geometry, a found that S^2n (with n>1) are not simpletic manifolds. They say it's because the de Rham cohomology of this spheres are R, but I do not understand this ...
15
votes
2answers
345 views

Classification of complex structures on $\mathbb{R}^{2n}$

Is there anything known about classification of complex structures on $\mathbb{R}^{2n}$ up to isomorphism for $n>1$? Say, are there finitely or infinitely many isomorphism classes? If there is a ...
1
vote
0answers
57 views

Is my particular finite dimension Toeplitz matrix always strictly positive?

Let $H(\omega), \; -\pi \leq \omega \leq \pi$ be a real-valued function with a continuous band of zeros, that is (for simplicity) $H(\omega)=0, \; |\omega|\geq \beta \pi$. Define a sequence of banded ...
0
votes
1answer
44 views

What is a fusion system/category without duals or an identity

What do you call a fusion system or fusion category without duals or an identity? For fusion systems I am using Definition 3.1 from On Arithmetic Modular Categories. I want to remove axioms (i) and ...
2
votes
1answer
81 views

Number of connected components of the isometry group of a simply-connected lorentzian manifold

Let $(M,g)$ be a finite-dimensional connected lorentzian manifold. Then the group $G$ of isometries of $M$ (i.e., the group of diffeomorphisms $\varphi : M \to M$ with $\varphi^* g = g$) is a Lie ...
4
votes
2answers
120 views

Orbits of the maximal compact subgroup on the light cone for $p$-adic groups

It is known that if $Q$ is an indefinite non-degenerate quadratic form on $ \mathbb{R}^n$ with $n \ge 3$, then any maximal compact subgroup $K$ of the orthogonal group $SO(Q)$ acts transitively on the ...
3
votes
1answer
85 views

General ballot theorem

I am looking for a version of the Ballot Theorem for general step distributions. Specifically, let $X_1,X_2,\ldots$ be i.i.d. real random variables with some distribution. Let $S_n = S_1 + \cdots + ...
2
votes
0answers
60 views

Pro-G_p*G_q topology, profinite topology

Let $W=G_p*G_q$, where $G_p$ and $G_q$ are pseudovarieties of all finite $p$-groups and all finite $q$-groups respectively, with $p$ and $q$ fixed prime numbers. The pro-$W$ topology on a group $G$ is ...
3
votes
1answer
107 views

Atkin-Lehner theory for nonholomorphic Eisenstein series

I am currently reading something about nonholomorphic Eisenstein series $E_\mathfrak{a}(z,1/2+it)$ for $\Gamma_0(q)$, where $\mathfrak{a}$ is a cusp (cf. Iwaniec, H. Spectral Methods of Automorphic ...
2
votes
1answer
169 views

Uncountable cardinals and Prufer $p$-groups

Let $A$ be an elementary Abelian uncountable $p$-group. Is it known if there is an action of a Prufer $q$-group (here $q$ is a prime not necessarily distinct from $p$) $C_{q^{\infty}}$ onto $A$ such ...
3
votes
1answer
73 views

Polygons with centroid at origin and vertices on compact codimension one submanifold of $\mathbb{R}^{n}-\{0\}$

Let $M$ be a compact codimension one submanifold of $\mathbb{R}^{n}$ which does not contaion $0$ and the origin lies in the bounded component of$\mathbb{R}^{n}-\{0\}$. Is it true to say that: ...
10
votes
1answer
726 views

What is the mathematical structure called if we replace commutative group by commutative monoid in the definition of linear space?

Could anyone tell me what the mathematical structure is called if we replace commutative group by commutative monoid in the definition of linear space? Also, are there any names for "commutative ...
1
vote
2answers
162 views

Surjectivity of trace map

Let $R$ be a closed integral domain with its fraction field $F$. Let $K$ be a finite separable extension field of $F$, and let $A$ be the integral closure of $R$ in $K$. It is well known that the ...
-3
votes
0answers
43 views

Boundary Value System.

The boundary value problem: $$y'' + Q(t)y = f(t)$$ satisfying $$Ay(a) +By(b) = g$$ where A, B and Q are the matrices of order n. After calculation, we can get the form of solution will be $$y(x) = ...
2
votes
1answer
167 views

'Stalk' of vanishing cycles at $k$-point

I have a simple question on notation. Let $S$ be a Henselian trait with closed point $s$ (with finite residue field $k$) and generic point $\eta$. Let $X/S$ be a variety. Then, we have the functor ...
1
vote
1answer
43 views

local bernstein type inequality for multivariate polynomials

Let's say $p(x_1,...,x_n)$ is an n-variate degree d homogenous polynomial. Assume $U \subset S^{n-1}$ and $ vol(U) > 0 $ is there any Bernstein type inequality saying $$ \max_{x \in U , y \in ...
10
votes
1answer
302 views

Is there a unique commutative group structure on $\mathbb{G}_m$?

Let $S$ be a scheme and let $X := \mathrm{Spec}(\mathscr{O}_S[t, t^{-1}])$ be the underlying $S$-scheme of the $S$-group scheme $(\mathbb{G}_m)_S$. Is there only one structure of a commutative ...
1
vote
0answers
76 views

Elliptic regularity Schauder estimates with Dirichlet/Neumann boundary conditions

Consider the linear elliptic equation $Lu = 0$, where $L$ is a second degree elliptic operator with smooth coefficients on a bounded domain $\overline{\Omega} \subset \mathbb{R}^n$, where $\Omega$ is ...
1
vote
0answers
105 views

manifold branched covering space for orbifolds

An orbifold structure on some topological space $X$ is a covering of $X$ with local quotient charts $V/G$, where $V$ is some connected manifold and $G$ effectively acts on $V$ via a finite group of ...
1
vote
0answers
48 views

FPTAS for approximating the permanent of a matrix

My question concerns approximating permanent of an $n$-by-$n$ matrix. Several approximation algorithms have been proposed in the literature for this purpose, whose time complexity depend on $n$ and ...
1
vote
1answer
43 views

Parity-check matrix for code with variable block size and minimum distance

Consider a linear error-correcting code with symbols in $GF(q)$, with codewords of length $k$ generated from messages of length $n$ and minimum distance $d+1 = k-n+1$. In the cases of interest, $q = ...
0
votes
0answers
94 views

Polynomial approximation on affine varieties [migrated]

Let $V,W \subseteq \mathbb{A}^n$ be two affine varieties over an algebraically closed field $k$ of characteristic zero and let $a,b\in k$. Q: Can we find a polynomial $f \in k[X_1,...,X_n]$ such ...

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