# All Questions

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### Advanced use of commutation matrices

I am aware of matrix operators vec and kronecker product, commutation matrices and various related identities like stated in ...
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### Algebraic independence criterion

Is there any criterion for checking algebraic independence of a set of polynomials in $n$ variables in terms of the leading monomials with respect to some monomial order ? The Jacobian criterion is ...
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### Tight binomial left tail bound

Let $X \sim \text{Bin}(n,p)$. Wikipedia claims $$\mathbf P[X \leq (p-\epsilon)n ] \leq e^{ - 2 \epsilon^2 n}.$$ This follows from Hoeffding's inequality ...
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### Is the positive existential theory undecidable?

Could you tell if the positive existential theory of $\mathbb{C}[e^{\mu x} \mid \mu \in \mathbb{C}]$ is undecidable in the language $\{+, \cdot , \frac{d}{dx} , 0, 1, e^x\}$ ? How can we prove the ...
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### If $k$ is an algebraically closed field of any characteristic, then the fundamental group of $A$ is abelian

This is a followup to my earlier question, see here. I am curious as to whether or not we can, in a similar way, show that if $k$ is an algebraically closed field (of any characteristic) then the ...
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### Simplicial approximation diagram

Let $K$ and $L$ be simplicial complexes as given, and let $\phi:|K|\to|L|$ be the continuous map, where $A=\phi(a)$, $B=\phi(b)$, and so on. Check whether the map $\phi$ has a simplicial ...
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### all subsets borel

Assume Martin's axiom plus $\neg CH$. It is well known, via almost disjoint forcing, that every set of reals of size less than continuum is an example of a metric space whose subsets are all ...
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### F-points of product of closed subgroups vs. product of F-points, F a local field, reference?

Let $F$ be a finite extension of $\mathbb Q_p$, where p is an odd prime. Let $G$ be a connected reductive group defined over $F$. Let $M, H$ be closed $F$-subgroups of $G$ (in particular, I'm ...
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### What characterizations of relative information are known?

Given two probability distributions $p,q$ on a finite set $X$, the quantity variously known as relative information, relative entropy, information gain or Kullback–Leibler divergence is defined ...
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### Equivariant form of Nagata's compactification theorem?

Given a finite group $G$ acting on an algebraic variety $X$ (let's say over $\mathbb C$, if that helps), is there always a proper variety $\bar X$ with a $G$ action such that $X \to \bar X$ is a ...
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### Simple question: different definitions of Bousfield localization

I am not an expert on model categories and I am getting lost with two different definitions I have found on Bousfield localizations. I don't see the link between them. First definition: Let ...
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### Prove or disprove a monotonicity property of a certain convex optimization problem

Let $R = (r_{ij})$ be an $n\times k$ real matrix with only positive entries, and consider the convex optimization problem $\max f(x) = \sum_{i=1}^n \log \sum_{j=1}^k r_{ij} x_j$ such that ...
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### Has every Lusin vector space a stronger Polish vector space topology?

Let $X$ be a topological vector space or even a locally convex space such that its (vector space) topology is Lusin, i.e. there is some stronger Polish topology. Does there also exist a stronger ...
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### Why are these two gamma functions equal? [on hold]

$\gamma_{1}(s)=\pi^{\frac{1}{2}-s}\frac{\Gamma(\frac{s}{2})}{\Gamma(\frac{1}{2}(1-s))}$ $\gamma_{2}(s)=(\frac{b}{2\pi})^{\frac{1}{2}-a} e^{-2i\theta(b)}$ ,where $s=a+bi$ I calculated ...
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### Okounkov-Vershik approach to representation theory of $S_n$

This is a rather soft question. I was wondering if someone could explain on a fundamental and intuitive level, what the Okounkov-Vershik approach to representation theory of $S_n$ is all about. It's ...
For the polynomial kernel, $K(x,y) = (x^Ty+c)^d$, the implicit feature space $\phi$ for which $K(x,y) = \phi(x)^T \phi(y)$ is of finite dimension and well known [1][2]. It is also well known that the ...
### A new result on the Diophantine equation $x^3 + y^3 +z^3 = 3$ [on hold]
The above Diophantine equation is unknown to have any further integer solutions other than $(x, y, z) = (1, 1, 1)$ and $(4, 4, -5)$. I am a prospective undergraduate mathematics student in Zimbabwe ...