Related papers: Linear Equations for Solving Partial Latin Squares…
A quantum Latin square of order $n$ (denoted as QLS$(n)$) is an $n\times n$ array whose entries are unit column vectors from the $n$-dimensional Hilbert space $\mathcal{H}_n$, such that each row and column forms an orthonormal basis. Two…
In polynomial optimization problems, nonnegativity constraints are typically handled using the sum of squares condition. This can be efficiently enforced using semidefinite programming formulations, or as more recently proposed by Papp and…
Given two integers $m$ and $n$ with $m\leq n$, a Latin rectangle of size $m\times n$ is a bi-dimensional array with $m$ rows and $n$ columns filled with symbols from an alphabet with $n$ symbols, such that each row contains a permutation of…
We provide a necessary and sufficient condition for existence of Gaussian cubature formulas. It consists of checking whether some overdetermined linear system has a solution and so complements Mysovskikh's theorem which requires computing…
Starting from the results of Charles Fefferman and Janos Koll\`ar in Continuous Solutions of Linear Equations [1], we adopt a new approach based on Fefferman's techniques of Glaeser refinement to show a more general result than the one…
A linear equation with coefficients in $\mathbb{F}_q$ is common if the number of monochromatic solutions in any two-coloring of $\mathbb{F}_q^n$ is asymptotically (as $n \to \infty$) at least the number expected in a random two-coloring.…
We study the problem of solvability of linear differential systems with small coefficients in the Liouvillian sense (or, by generalized quadratures). For a general system, this problem is equivalent to that of solvability of the Lie algebra…
Most recent results in matrix completion assume that the matrix under consideration is low-rank or that the columns are in a union of low-rank subspaces. In real-world settings, however, the linear structure underlying these models is…
We present an algorithm to solve a system of diagonal polynomial equations over finite fields when the number of variables is greater than some fixed polynomial of the number of equations whose degree depends only on the degree of the…
In this paper, we introduce a numerical solution of a stochastic partial differential equation (SPDE) of elliptic type using polynomial chaos along side with polynomial approximation at Sinc points. These Sinc points are defined by a…
A three-step method due to Nijhoff and Bobenko & Suris to derive a Lax pair for scalar partial difference equations (P\Delta Es) is reviewed. The method assumes that the P\Delta Es are defined on a quadrilateral, and consistent around the…
Linear systems are the bedrock of virtually all numerical computation. Machine learning poses specific challenges for the solution of such systems due to their scale, characteristic structure, stochasticity and the central role of…
Given an integer partition $(h_1,h_2,\dots,h_k)$ of $n$, is it possible to find an order $n$ latin square with $k$ disjoint subsquares of orders $h_1,\dots,h_k$? This question was posed by L.Fuchs and is only partially solved. Existence has…
We propose a method by which to examine all possible partial difference Lax pairs that consist of 'two by two' discrete linear problems, where the matrices contain one separable term in each entry. We thereby derive new, higher-order…
This is a companion note to the paper "Almost all Steiner triple systems have perfect matchings (arXiv:1611.02246). That paper contains several general lemmas about random Steiner triple systems; in this note we record analogues of these…
In our work, we consider the linear least squares problem for $m\times n$-systems of linear equations $Ax = b$, $m\geq n$, such that the matrix $A$ and right-hand side vector $b$ can vary within an interval $m\times n$-matrix and an…
Suppose that $k$ is a function of $n$ and $n\to\infty$. We show that with probability $1-O(1/n)$, a uniformly random $k\times n$ Latin rectangle contains no proper Latin subsquare of order $4$ or more, proving a conjecture of Divoux, Kelly,…
We study a class of combinatorial scheduling problems characterized by a particular type of constraint often associated with electrical power or gas energy. This constraint appears in several practical applications and is expressed as a sum…
In this work we present a new simple but efficient scheme - Subsquares approach - for development of algorithms for enclosing the solution set of overdetermined interval linear systems. We are going to show two algorithms based on this…
A distributed discrete-time algorithm is proposed for multi-agent networks to achieve a common least squares solution of a group of linear equations, in which each agent only knows some of the equations and is only able to receive…