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### Do powers of the shift operator applied to a non-zero vector always yield a total set?

Let $S$ be the (say, left) shift operator on $\ell^2(\mathbb{Z})$. For a non-zero vector $x \in \ell^2(\mathbb{Z})$, consider the set $$X = \{ S^n v \mid n \in \mathbb{Z} \}.$$ Is this always a total ...
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### Categorical Random variable conditional on auxiliarly information variables

Suppose I have a categorical random variable $X\in\{1,2,\ldots n\}$. Suppose also that I have random variables $Y_1, \ldots, Y_n$ that inform the probability of $X$, but only in such a way that the ...
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### An infinite game possibly due to Ernst Specker

I have a vague memory of an infinite game due to Ernst Specker with the following properties: (1) It is a two-person perfect information game, where the players move alternately. (2) The possible ...
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### SPOT as a Conservative Extension of Zermelo-Fraenkel

In a recent article, Hrbacek and Katz have shown that it is possible to formulate an axiomatic theory which provides a formalisation of calculus procedures which make use of infinitesimals (known as ...
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### What is a toric lattice?

What is a toric lattice? and how can I construct one in Macaulay2 and compute its basis? is there any alternative method to make one? Since I went through the whole ...
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### Weak sufficient conditions for non-negative correlation between functions of correlated random variables?

Consider real, nonnegative random variables $A$, $B$, and $X$, and define $Z = \exp(-A X)$ and $W = \exp(-B X)$, and also $U = \exp(-X - A)$ and $V = \exp(-X -B)$. What sorts of minimal sufficient ...
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### Can anything be said about the cohomology class defined by a section of a vector bundle if it is not of the expected dimension?

Let $E$ be a rank $n$ locally free sheaf on a smooth $n$ dimensional variety $X$, and $s\in H^0(X,E)$. If $\dim Z(s)=0$ (which is the expected dimension), then we can understand the cohomology class ...
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### Questions on LND book by Gene Freudenburg

So I was trying to read the first part of Algebraic Theory of Locally Nilpotent Derivations, 2006 by Gene Freudenburg and I have not gone too far. Right in the begging, I struggled with notations and ...
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If $G$ is a graph with edge set $E$, let $W$ be the $\mathbb{Z}/2$-vector space generated by the elements of $E$. If $A = \{a_1, \dots, a_n\} \subset E$, let $\bar{A} = a_1 + \dots + a_n \in V$; then $... 0answers 55 views ### Second derivative of the volume of the$\varepsilon$-neighbourhood of a submanifold Let$M$be a$n$-dimensional compact Riemannian manifold, and$N$a smooth submanifold of$M$of dimension strictly less than$n$. Denote by$N_{\varepsilon}$the$\varepsilon$-neighbourhood of$N$- ... 0answers 70 views ### Monochrome subsemigroup Let$G$be an amenable group. Paint$G$with$k$colors. Is there a monochrome subsemigroup?! Does the analytic structure of being amenable or the algebraic structure of being a group (and ... 1answer 46 views ### Tail bound on the RKHS norm of a zero-mean Gaussian process Let$f \sim \mathcal{GP}(0, K)$be a zero-mean Gaussian process defined on a compact set$\mathcal{D} \subset \mathbb{R}^d$, where$K \colon \mathcal{D} \times \mathcal{D} \rightarrow \mathbb{R} $is ... 0answers 52 views ### The converse of the Cartan theorem on Lie subgroups As stated by Wikipedia https://en.wikipedia.org/wiki/Closed-subgroup_theorem the closed-subgroup theorem (sometimes referred to as Cartan's theorem) is a theorem in the theory of Lie groups. It ... 2answers 155 views ### The moduli space of finite volume hyperbolic 3-manifolds? By finite volume hyperbolic 3-manifold, I do mean$M=\mathbb{H}^{3}/\Gamma$where$\Gamma$is a torsion-free Kleinian group such that the hyperbolic volume$Vol(M)<\infty$. I will call $$\mathcal{M}... 0answers 107 views ### Reference request for an English translation of a book of Tate In this ongoing program, Professor Mahesh Kakde said that the best reference for learning about Stark and Gross-Stark conjecture is this book of John Tate. But this book is in French. Is there any ... 0answers 221 views ### Fibonacci embedded in Catalan? Given a partition \lambda and its Young diagram \pmb{Y}_{\lambda}, we say \lambda is a (t,s)-core partition provided that neither t nor s is a hook length in \pmb{Y}_{\lambda}. We now ... 0answers 15 views ### Reference: Good bounds for Variance of a Random Vector with Known Mean Supported on a Compact Set of Low Metric Entropy Let \emptyset\neq M\subseteq \mathbb{R}^n be a compact set, X:\Omega\rightarrow M be a random vector defined on a complete probability space (\Omega,\mathcal{F},\mathbb{P}) and suppose that \mu:... 0answers 46 views ### Is every nearly rank-1 doubly stochastic matrix a product of pairwise averaging matrices? A doubly stochastic matrix is a square matrix with non-negative real entries where the sum of each row is 1 and the sum of each column is 1. A pairwise averaging matrix is a matrix of the form tA+... 0answers 130 views ### Surfaces in \mathbb{P}^3 swept out by plane curves Fix a line L\subset\mathbb{P}^3 and let \Pi_{t}, for t\in\mathbb{P}^1, be the pencil of planes containing L. Take a general point t\in\mathbb{P}^1, nine general points on \Pi_t and denote ... 2answers 166 views ### Limit of an integral involving Riemann zeta function [closed] Let z\in \mathbb{D} where \mathbb{D} is the unit disc. Considering the non negative real axis (i.e. [0,\infty)) as the branch cut and 0<\arg z<2\pi we define for z=re^{i\theta},$$\... 0answers 108 views ### Eigenbases without the Axiom of Choice I understand that in ZF set theory without the Axiom of Choice (AC), it is consistent to have models in which there exist vector spaces over some (unspecified) field$k$without a basis. So in ... 1answer 166 views ### Injectivity of Keller maps Let$M: \mathbb{C}[x,y] \to \mathbb{C}[x,y]$,$(x,y) \mapsto (p,q)$, with$p,q \in \mathbb{C}[x,y]$satisfying$\operatorname{Jac}(p,q):=p_xq_y-p_yq_x \in \mathbb{C}-\{0\}$. Such a polynomial map is ... 1answer 39 views ### Limiting behavior of$k^{th}$order statistics of n non-i.i.d chi square random variables This is related to one of my previous questions here. Let$(Z_1, Z_2, \ldots, Z_n)\sim N(0, \Omega)$, where$\Omega = (1-\mu) I_{n\times n} + \mu \boldsymbol{1}_n\boldsymbol{1}_n^\top $. Here$\...
A particular case of Dickson's Conjecture states that for $a_1,q_1,a_2,q_2$ with $(a_1,q_1)=(a_2,q_2)=1$, there are infinitely many $n$ for which $q_1 n + a_1$ and $q_2 n+a_2$ are both prime, provided ...