Related papers: On positional representation of integer vectors

We discuss the following extremal problem and its relevance to the sum of the so-called superoptimal singular values of a matrix function: Given an $m\times n$ matrix function $\Phi$ on the unit circle $\mathbb{T}$, when is there a matrix…

Motivated by the problem of integer sparse recovery we study the following question. Let $A$ be an $m \times d$ integer matrix whose entries are in absolute value at most $k$. How large can be $d=d(m,k)$ if all $m \times m$ submatrices of…

For every $2n\times 2n$ real positive definite matrix $A,$ there exists a real symplectic matrix $M$ such that $M^TAM=\diag(D,D),$ where $D$ is the $n\times n$ positive diagonal matrix with diagonal entries $d_1(A)\le \cdots\le d_n(A).$ The…

Let $G$ be a connected semisimple simply connected Lie group with a compact Cartan subgroup and let $\Gamma$ be a uniform lattice in $G$. Let $\widehat{G}_d$ denote the set of equivalence classes of unitary discrete series representations…

A set A of natural numbers is finitely embeddable in another such set B if every finite subset of A has a rightward translate that is a subset of B. This notion of finite embeddability arose in combinatorial number theory, but in this paper…

We study the maximal number of pairwise distinct columns in a $\Delta$-modular integer matrix with $m$ rows. Recent results by Lee et al. provide an asymptotically tight upper bound of $O(m^2)$ for fixed $\Delta$. We complement this and…

We introduce convex integrals of molecules in Lipschitz-free spaces $\mathcal{F}(M)$ as a continuous counterpart of convex series considered elsewhere, based on the de Leeuw representation. Using optimal transport theory, we show that these…

Let $X$ be a $2$-dimensional subshift of finite type generated by a finite set of forbidden blocks (of finite size). We give an algorithm for generating the elements of the shift space using sequence of finite matrices (of increasing size).…

We prove that the Smith forms of the powers of an integer square matrix behave in an eventually periodic manner. More precisely, if $\mathrm{SF}(M)$ denotes the Smith form of $M \in \Z^{m \times m}$, then for every $A \in \Z^{m \times m}$…

Let $S$ be a semigroup, let $n\in\mathbb{N}$ be a positive natural number, let $A,B\subseteq S$, let $\mathcal{U},\mathcal{V}\in\beta S$ and let let $\mathcal{F}\subseteq\{f:S^{n}\rightarrow S\}$. We say that $A$ is $\mathcal{F}$-finitely…

We prove that any multi-variate Hasse-Schmidt derivation can be decomposed in terms of substitution maps and uni-variate Hasse-Schmidt derivations. As a consequence we prove that the bracket of two $m$-integrable derivations is also…

Let $f\colon X\to Y$ be a perfect surjective map of metrizable spaces. It is shown that if $Y$ is a $C$-space (resp., $\dim Y\leq n$ and $\dim f\leq m$), then the function space $C(X,\uin^{\infty})$ (resp., $C(X,\uin^{2n+1+m})$) equipped…

Let $X$ be an arbitrary separable symmetric space on $[0,1]$. By using a combination of the frame approach and the notion of the multiplicator space $\mathscr{M}(X)$ of $X$ with respect to the tensor product, we investigate the problem when…

Given an integer nonsingular $n\times n$ matrix $M$ and a point $y \in \mathbb{R}^n/\mathbb{Z}^n$, consider the set $\tilde E(M,y)$ of vectors $x\in \mathbb{R}^n$ such that $y$ is not a limit point of the sequence $\{M^k x \mod…

Let X be a smooth hypersurface in projective space. We discuss in this paper when X can be defined by an equation det M = 0 (resp. pf M = 0), where M is a matrix (resp. a skew-symmetric matrix) with homogeneous entries. Standard homological…

We show that for any finitely generated group of matrices that is not virtually solvable, there is an integer m such that, given an arbitrary finite generating set for the group, one may find two elements a and b that are both products of…

Let $W_{k,n}^{i}(m)$ denote a matrix with rows and columns indexed by the $k$-subsets and $n$-subsets, respectively, of an $m$-element set. The row $S$, column $T$ entry of $W_{k,n}^{i}(m)$ is $1$ if $|S \cap T| = i$, and is $0$ otherwise.…

The main aim of this paper is to characterize ideals I in the power series ring R=K[[x1,...,xs]] that are finitely determined up to contact equivalence by proving that this is the case if and only if I is an isolated complete intersection…

We prove that for any connected open set $\Omega\subset \R^n$ and for any set of matrices $K=\{A_1,A_2,A_3\}\subset M^{m\times n}$, with $m\ge n$ and rank$(A_i-A_j)=n$ for $i\neq j$, there is no non-constant solution $B\in…

Let $A$ be an $(m \times n)$ integral matrix, and let $P=\{ x : A x \leq b\}$ be an $n$-dimensional polytope. The width of $P$ is defined as $ w(P)=min\{ x\in \mathbb{Z}^n\setminus\{0\} :\: max_{x \in P} x^\top u - min_{x \in P} x^\top v…