# All Questions

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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 $U\subseteq\mathbb{C}$. ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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? ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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: ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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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, $q = ... 0answers 94 views ### Polynomial approximation on affine varieties [migrated] Let$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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