# All Questions

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### Let $A\in\operatorname{M}_n(F)$ be a matrix, how to prove $\bigcap_{X\in C(A)}C(X)=F[A]=\frac{F[x]}{(m_A(x))}$

I asked the following question on Math Stack Exchange, but no people reply. I know MO is more professional and it is for mathematicians to discuss research problems. Maybe this question is unsuitable ...
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### Cancellation and splitting theorems for vector bundles etc over schemes

It is not too hard, in the theory of vector bundles over manifolds (or nice topological spaces, say locally contractible with finite covering dimension), to arrive at a splitting theorem. This ...
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### Stricter Notion of Crossing in a Partition

Let $k$ be an integer. Traditionally a partition $\pi=V_1\cup \dots \cup V_n$ of the set $[k]:=\{1,\dots, k\}$ is called crossing when there exist $a,c\in V_i$ and $b,d\in V_j\not= V_i$ such that ...
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### contraction of tubular neighborhoods in Berkovich spaces

a question about Berkovich spaces: There is "Tubular Theorem, 1" on page 27 of the slide http://math.univ-lyon1.fr/~thuillier/articles/Toric_Leuven.pdf which says: If D be a simple normal crossing ...
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### Symmetry on a sphere

Let $u$ be a smooth function on the sphere $S^2$. Suppose there exists $C>0$ such that for all $R \in SO(3)$, the area of every nonempty connected component of $\{x\in S^2: u(x)> u(Rx)\}$ is at ...
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### Tannakian formalism for topological Hopf algebras

Tannaka-Krein duality allows, under the appropriate assumptions, to reconstruct a Hopf algebra from its category of modules. This method was found to be powerful for instance in the work of ...
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### Switching Algebra Simplification - Beginner - how to continue with this problem? [on hold]

I'm currently trying to simplify the following expression: f = w'xz + wy + wy'z'+ w'x'z' + xyz There are five terms, nine literals. Assume this notation: xyz = x*y*z, and z' = the complement of z. ...
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### Can any reductive $k$-group be written as a semidirect product of $k$-linear groups?

This might be an easy question for the experts, so I apologise in advance. By a reductive group over a field $k$, I mean a linear algebraic group (not necessarily connected) such that the unipotent ...
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### Density of function spaces

Let $\Omega$ be a subset of either $\mathbb{R}^n$ ($n\geq 3$, if it matters) or of a compact manifold. In either case, we'll call the manifold $M$. Let $V_i\subset V_{i+1} \subset \Omega$ be an ...
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### Number of edges in linklessly embeddable graphs

Consider graphs over $n$ nodes. What is the maximum number of edges of a linklessly embeddable graph? A more general question is the following. Given $\mu$ what is the maximum number of edges of ...
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### Reference for maps whose pushouts are also homotopy pushouts

Consider a category C with weak equivalences, e.g., a model category. For the purposes of this question, let's say that a morphism f in C is an i-cofibration if any pushout (alias cobase change) ...
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### Derivative of softmax loss function [on hold]

I am trying to wrap my head around backpropagation in a neural network with a softmax classifier, which uses the softmax function: $$p_j = \frac{e^o_j}{\sum_k e^{o_k}}$$ ...
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### Linear least squares with unordered response variable

In the classical linear regression model one considers the equation $$y = X \beta + \epsilon.$$ I was wondering whether there are also results when the ordering of the response variable $y$ is not ...
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A pencil $l$ of quadrics in $\mathbb{P}^4_{<x_0,\cdots,x_4>}$ consists of singular quadrics only if: (1) quadrics in $l$ have a common singular point; or (2) quadrics in $l$ contain a common ...
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### metrizable neighborhoods of compact subsets

This is a question about general topology: Assume we are given a first countable Hausdorff space and a compact subset K. Is it possible to find a countable basis of open neighborhoods of K ? ...
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### a generalization of the annihilator of cokernel ideal

Let $R$ be a (commutative, associative, unital) ring, consider a homomorphism of some (finitely generated) free $R$-modules $F\stackrel{A}{\rightarrow}G$. Its basic invariants are the Fitting ideals, ...
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### Bases of surface groups with length restrictions

This question asks for a generalization of Bases of surface groups following the notation and definitions given therein. Let $\Gamma_g$ be a surface group of genus $g \geq 2$, $B$ a surface basis of ...
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### Conditional Form of Rosenthal's Inequality

Rosenthal's Inequality as stated in the book "Martingale Limit Theory and Its Application" by Hall and Heyde states the following: If $\{S_i, \mathcal{F}_i, 1\leq i \leq n\}$ is a martingale and ...
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### Handelman's positivstellensatz for symmetric matrix-valued polynomials

For certain classes of sets $S \subseteq \mathbb{R}^n$, there exist algebraic characterizations of real valued polynomials $p: \mathbb{R}^n \rightarrow \mathbb{R}$ that are positive on $S$. Several ...