Related papers: Sparse solutions of linear Diophantine equations

We investigate the number of integer solutions to a multiplicative Diophantine approximation problem and show that the associated counting function converges in distribution to a normal law. Our approach relies on the analysis of…

Mixed dictionaries generated by cosine and B-spline functions are considered. It is shown that, by highly nonlinear approaches such as Orthogonal Matching Pursuit, the discrete version of the proposed dictionaries yields a significant gain…

We present an ordinary differential equations approach to the analysis of algorithms for constructing $l_1$ minimizing solutions to underdetermined linear systems of full rank. It involves a relaxed minimization problem whose minimum is…

In this short note we study the existence and number of solutions in the set of integers ($Z$) and in the set of natural numbers ($N$) of Diopahntine Equations of second degree with two variables of the general form $ax^2-by^2=c$.

In this paper, we first prove the existence of solutions to Dirichlet problems involving the fractional $g$-Laplacian operator and lower order terms by appealing to sub- and supersolution methods. Moreover, we also state the existence of…

This paper deals with the following question: Suppose that there exist an integer or a non-negative integer solution $x$ to a system $Ax = b$, where the number of non-zero components of $x$ is $n$. The target is, for a given natural number…

We present a linear-system solver that, given an $n$-by-$n$ symmetric positive semi-definite, diagonally dominant matrix $A$ with $m$ non-zero entries and an $n$-vector $\bb $, produces a vector $\xxt$ within relative distance $\epsilon$ of…

We consider fractional differential equations of order $\alpha \in (0,1)$ for functions of one independent variable $t\in (0,\infty)$ with the Riemann-Liouville and Caputo-Dzhrbashyan fractional derivatives. A precise estimate for the order…

In this paper we discuss three symbolic approaches for the generation of a finite difference scheme of a partial differential equation (PDE). We prove, that for a linear PDE with constant coefficients these three approaches are equivalent…

The linear nonhomogeneous fractional difference system with constant coefficients is introduced. An explicit solution to the system is acquired by proposing a newly discrete retarded perturbation of the nabla Mittag-Leffer-type function…

We deal with the problem to find the number $P(b)$ of integer non-negative solutions of an equation $\sum_{i=1}^{n} a_i x_i=b$, where $a_1,a_2,...,a_n$ are natural numbers and $b$ is a non-negative integer. As different from the traditional…

This article discusses the question - how to estimate the number of solutions of algebraic Diophantine equations with natural coefficients using Circular method developed by Hardy and Littlewood. This paper considers the estimate of the…

We provide new results on the existence of nonzero positive weak solutions for a class of second order elliptic systems. Our approach relies on a combined use of iterative techniques and classical fixed point index. Some examples are…

Diophantine subsets of $\mathbb{Z}$ play a key role in the negative answer to Hilbert's tenth problem. The definition of diophantine set generalizes in several ways to other commutative rings. We compare these definitions. Along the way, we…

We consider the number of solutions in positive integers $(x,y,z)$ for the purely exponential Diophantine equation $a^x+b^y =c^z$ (with $\gcd(a,b)=1$). Apart from a list of known exceptions, a conjecture published in 2016 claims that this…

In this paper, we solve the simultaneous Diophantine equations m.(x_1^k+....+x_{t_1}^k)=n.(y_1^k+....+y_{t_2}^k); k=1,3, where t_1, t_2>3, and m, n are fixed arbitrary and relatively prime positive integers. This is done by choosing two…

We present a novel approach to the formulation and the resolution of sparse Linear Discriminant Analysis (LDA). Our proposal, is based on penalized Optimal Scoring. It has an exact equivalence with penalized LDA, contrary to the multi-class…

This paper is concerned with the study of diagonal Diophantine inequalities of fractional degree $ \theta ,$ where $ \theta >2$ is real and non-integral. For fixed non-zero real numbers $ \lambda_i $ not all of the same sign we write…

In this paper we use the values of the Riemann $Z(t)$-function in order to construct certain quasi-orthonormal system of vectors. On this basis we prove a formula for microscopic interpolation of the function $Z(t)$. Simultaneously we have…

Most adaptive finite element strategies employ the D\"orfler marking strategy to single out certain elements $\mathcal{M} \subseteq \mathcal{T}$ of a triangulation $\mathcal{T}$ for refinement. In the literature, different algorithms have…