Varying methods exist for computing a presentation of a finitely generated commutative cancellative monoid.We use an algorithm of Contejean andDevie [An efficient incremental algorithm for solving systems of linear diophantine equations, Inform. andComput. 113 (1994) 143–172] to show how these presentations can be obtained from the nonnegative integer solutions to a linear system of equations. ...

Given a linear equation L, a set A of integers is L-free if A does not contain any ‘nontrivial’ solutions to L. This notion incorporates many central topics in combinatorial number theory such as sum-free and progression-free sets. In this paper we initiate the study of (parameterised) complexity questions involving L-free sets of integers. The main questions we consider involve deciding whethe...

The linearizability of differential equations was first considered by Lie for scalar second order semi-linear ordinary differential equations. Since then there has been considerable work done on the algebraic classification of linearizable equations and even on systems of equations. However, little has been done in the way of providing explicit criteria to determine their linearizability. Using...

We are going to solve this problem in three steps. Step 1 In the first step, we find all solutions of (1) that are of the special form u(x, t) = X(x)T (t) for some function X(x) that depends on x but not t and some function T (t) that depends on t but not x. This is where the name “separation of variables” comes from. It is of course too much to expect that all solutions of (1) are of this form...

This paper studies the possibility of achieving indistinguishabilitybased security in privately outsourcing linear equation systems over real numbers. The particular task is to solve a full-rank n× n system Ax = b. Since the most complex part of this task is inverting A, the problem can be reduced to outsourcing of a square matrix inverse computation. Although outsourcing matrix inverse is triv...

We consider the half-linear second-order difference equation Δ rkΦ Δxk ckΦ xk 1 0, Φ x : |x|p−2x, p > 1, where r, c are real-valued sequences. We associate with the above-mentioned equation a linear second-order difference equation and we show that oscillatory properties of the abovementioned one can be investigated using properties of this associated linear equation. Themain tool we use is a l...

we will apply the successive approximation method forproving the hyers--ulam stability of a linear integral equation ofthe second kind.

In this research, a theoretical study has been undertaken on the density and also on the relationship between refractive index and density for aqueous solutions. A simple linear equation is suggested to show this relationship. Also, a semi-empirical equation has been developed for estimating the constant of this linear equation. By using our suggested equations, the refractive indices of aq...

In this paper, we use the continuous Legendre wavelets on the interval [0,1] constructed by Razzaghi M. and Yousefi S. [6] to solve the linear second kind integral equations. We use quadrature formula for the calculation of the products of any functions, which are required in the approximation for the integral equations. Then we reduced the integral equation to the solution of linear algebraic ...

This article clarifies what it means to solve a system of fuzzy linear equations, relying on the fact that they are a direct extension of interval linear systems of equations, already studied in a specific interval mathematics literature. We highlight four distinct definitions of a systems of linear equations where coefficients are replaced by intervals, each of which based on a generalization ...