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2 views

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 ...
0
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0answers
10 views

Vanishing product of polynomials over finite fields

$(x_1-x_2-x_3+x_4)(x_2-x_4-x_3+x_1)(x_3-x_1)(x_4-x_2)\equiv 0$ over $\mathbb F_3$. Take polynomials $p_1,\dots,p_n$ over variables $x_1,\dots,x_n$ such that $p_i$ does not depend on $x_i$ and $\Pi_{i=...
1
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0answers
10 views

Existence of curves of arbitrary genus on some K3 surface

Voisin uses the fact "If $X$ is a K3 surface with an ample line bundle $\mathcal L$ such that $\mathcal L$ generates $\mathop{\mathrm{Pic}}(X)$ and $(\mathcal L^2) = 4t - 2$, then every smooth ...
0
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0answers
63 views

Can ∞-category be defined in proof assistants?

Can ∞-category be defined in proof assistants? For example, we can directly consider a function such as ...
1
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0answers
74 views

Are there np hard statements which are np complete under the Riemann hypothesis?

Are there np hard problems which are np complete under riemann hypothesis by each of the following cases? Certificate becomes Polynomial sized Reduction to sat becomes derandomized Reduction to sat ...
1
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1answer
116 views

How many ways can $N$ be written as a sum of terms in the form $2^i3^j$?

Given a positive integer $N$, let $f(N)$ be the number of ways $N$ can be decomposed as a sum of terms of the form $2^i3^j$, where each such term appears at most once in the sum. For example, $f(10) = ...
0
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0answers
17 views

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 ...
3
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0answers
67 views

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 ...
0
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0answers
59 views

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 ...
0
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1answer
122 views

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 ...
0
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1answer
77 views

Factor group of direct product by restricted direct product

Let $W:=\prod_{i\in \omega} F_i$ be the (external) unrestricted direct product and $U:=\prod_{i\in \omega}^w F_i$ be the (external) restricted direct product of finite groups $F_i$ such that $|F_{i}|&...
0
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1answer
52 views

Special classes of the arithmetical hierarchy of sentences of finite-order arithmetic

We work in a countable language of finite-order arithmetic, which allows us to quantify over natural numbers, sets of natural numbers, sets of sets of natural numbers, and so on. We measure the ...
0
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0answers
50 views

Lattices containing $A_n$ and $D_n$

How many lattices are there which contain both the $A_n$ and $D_n$ lattices of the same dimension as sublattices? So far, I’ve found examples in 1D, 3D, 8D, and 24D.
5
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0answers
132 views

Verdier duality under more general conditions

In the section 3.2 of Sheaves in Topology by A. Dimca, the author explains that if $f:X\to Y$ is a continuous map (between locally compact, $\sigma$-compact topological spaces with finite homological ...
1
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1answer
55 views

Worst convex compact set for translational packings of $\mathbb R^d$

A (translational) packing of a convex compact subset (with non-empty interior) $\mathcal C$ of $\mathbb R^d$ is a union of translated non-overlapping (but perhaps touching) copies of $\mathcal C$. The ...
1
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0answers
24 views

Subsets of $\mathbb{R}$, every nonempty subset of which generates a disconnected translation-invariant topology

Let $\mathbb{R}$ be the set of real numbers. Given a subset $S$ of $\mathbb{R}$, let $\mathcal{T}_S$ be the translation-invariant topology generated by $S$. That is, $\mathcal{T}_S$ is the topology ...
0
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1answer
103 views

Reference or proof that every finite-rank operator can be written in a specific way

I am asking for a reference or a quick proof of the following classical result: Let $X,Y$ be Banach spaces and let $T:X\to Y$ be a bounded operator such that the image of $T$ has dimension $n\in\...
3
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0answers
22 views

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 ...
3
votes
1answer
141 views

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 ...
-1
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0answers
75 views

The prime factorization of factorial of any integer [closed]

The best known time complexity of the is $O(2^{(N!)^x})$ for all $x>0$ (using the general number field sieve). Is it worthwhile if the factorization of this n factorial can be done in $O((\log N)^{\...
0
votes
1answer
64 views

About the normability of the space of continuous functions

Let $A$ be a subset of $\mathbb{R}^n$, and denote by $C(A)$ the space of complex-valued continuous functions defined on $A$. We know that if $A$ is compact then we can define a norm on $C(A)$ so that ...
3
votes
1answer
98 views

nonabelian $p$-group contains an elementary abelian maximal subgroup

Given an elementary abelian $p$-group $A$ of order $p^n$ for $n\geq 2$ and choose a subgroup $H$ of order $p$ from $Aut(A)\cong GL(n,p)$. We can use semi-direct product $A\rtimes B$ to construct a ...
0
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0answers
76 views

Is solving Diophantine equations $\mathrm{mod}\:p$ NP-complete?

Famously 2-SAT is not NP-complete but 3-SAT is. I was wondering if for a fixed prime $p$ solving systems of Diophantine equations $\mathrm{mod}\:p$ is NP-complete?
1
vote
2answers
79 views

Monotonicity of doubling dimension

Let $(X,d)$ be a metric space with finite Assouad dimension $0<C_X$. It seems intuitive to me that if $\emptyset \subset Y\subseteq X$ then $Y$ is also doubling and its Assouad dimension, denoted ...
2
votes
0answers
53 views

A question about sequences of bounded variation and series convergence

There is a conclusion: For any $x\in \mathbb R^\mathbb N$, we denote by $A_x$ the set $$A_x= \{a\in \mathbb R^\mathbb N:\sum_n x(n)\alpha(n)~\text{converges}\},$$ then for $y,x_1,x_2,\dots,x_k \in \...
0
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0answers
44 views

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 ...
4
votes
1answer
171 views

A question on motivic zeta-function

It's well-known that over $\mathbb F_q$ every smooth projective conic $C$ is isomorphic to a projective line. So the formula for the motivic zeta-function $Z_{mot}(C)$ is evident since $S^n\mathbb P^1 ...
11
votes
2answers
485 views

Minimal set of assumptions for set theory in order to do basic category theory

Consider a normal first course on category theory (say up to and including the statement and proof) of the adjoint functor theorem (AFT). What are the minimal assumptions for the definition of a set ...
9
votes
1answer
189 views

Residue field of point on an algebraic stack

$\DeclareMathOperator{\Spec}{Spec}$ Let $X$ be an algebraic stack. Is there is a well-defined notion of the residue field of a point $x \in |X|$? Attempts: Recall that a point on a stack is an ...
4
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0answers
51 views

Projective objects in the category of simplicial objects in an abelian category

Let $S\mathcal{A}$ be the category of simplicial objects in an abelian category $\mathcal{A}$. In exercise 8.4.5 in Weibel's An Introduction to Homological algebra, it is said that $P \in S\mathcal{A}$...
0
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0answers
43 views

Does there exist an almost surely a.e. differentiable continuous martingale?

Does there exist an a.s. continuous martingale $M_t$, not almost surely constant in t, that is differentiable a.e almost surely? Here the null set of non differentiability is allowed to be random, i.e....
4
votes
0answers
94 views

Largest disk contained in an ellipsoid

It is known that any ellipsoid with principal semi-axes $a$, $b$, $c$ has circular planar sections (https://en.wikipedia.org/wiki/Circular_section). Is the largest circular disk contained within any ...
0
votes
1answer
47 views

Log-concavity of the modified Bessel function of a second kind

I was searching for some results for the log-concavity of the modified Bessel function of a second type, but I failed. Has there been any known work on this? I am not even sure if it is the modified ...
4
votes
0answers
71 views

An example of a projective surface of general type where we know all of the rational points

I am looking for an example of a surface $X \subset \mathbb{P}^3$ defined over $\mathbb{Q}$ with the following properties: 1) $X(\mathbb{Q}) \ne \emptyset$; and 2) we know all of the elements $X(\...
2
votes
1answer
97 views

Dimension of circuit space of a matroid

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 $...
1
vote
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 $N$ - ...
-2
votes
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 ...
1
vote
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 ...
1
vote
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 ...
1
vote
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}...
2
votes
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 ...
7
votes
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 ...
0
votes
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:...
2
votes
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+...
7
votes
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 ...
-4
votes
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 $z=re^{i\theta}$, $$\...
0
votes
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 ...
0
votes
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 ...
2
votes
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 $\...
4
votes
0answers
75 views

Dickson's conjecture for Beatty sequences

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 ...

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