Related papers: On linear equations arising in Combinatorics (Part…
In the first two papers, the author embarked on a study of classes of linear equations over integers satisfying a "Farkas-type" property. As the third paper in this study, the present paper deals with another class of linear equations over…
In one of his papers, the author introduces the class of Farkas-related vectors for which a version of Farkas' lemma over integers is derived. In this paper, two similar classes are introduced and studied.
An equivalence relation in the symmetric group, where is a positive integer has been considered. An algorithm for calculation of the number of the equivalence classes by this relation for arbitrary integer has been described.
Based on the Gale-Ryser theorem for the existence of suitable $(0,1)$-matrices for different partitions of a natural number, we revisit the classical result of G. G. Lorentz regarding the characterization of a plane measurable set, in terms…
The equivalence test is a main part in any classification problem. It helps to prove bounds for the main parameters of the considered combinatorial structures and to study their properties. In this paper, we present algorithms for…
The purpose of this note is to survey a methodology to solve systems of polynomial equations and inequalities. The techniques we discuss use the algebra of multivariate polynomials with coefficients over a field to create large-scale linear…
This paper considers the problem of testing whether there exists a non-negative solution to a possibly under-determined system of linear equations with known coefficients. This hypothesis testing problem arises naturally in a number of…
A new proof for adjoint systems of linear equations is presented. The argument is built on the principles of Algorithmic Differentiation. Application to scalar multiplication sets the base line. Generalization yields adjoint inner vector,…
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…
Solving polynomial equations is a subtask of polynomial optimization. This article introduces systems of such equations and the main approaches for solving them. We discuss critical point equations, algebraic varieties, and solution counts.…
In the paper, some special linear combinations of the terms of rational cycles of generalized Collatz sequences are studied. It is proved that if the coefficients of the linear combinations satisfy some conditions then these linear…
We examine the convergence properties of sequences of nonnegative real numbers that satisfy a particular class of recursive inequalities, from the perspective of proof theory and computability theory. We first establish a number of results…
This paper studies the problem of testing whether a system of linear equality and inequality constraints admits a solution when the coefficients of that system may have to be estimated. We show that a wide range of inferential questions in…
Comparative prime number theory is the study of the {\em{discrepancies}} of distributions when we compare the number of primes in different residue classes. This work presents a list of the problems being investigated in comparative prime…
The equivalence group is determined for systems of linear ordinary differential equations in both the standard form and the normal form. It is then shown that the normal form of linear systems reducible by an invertible point transformation…
We explore various combinatorial problems mostly borrowed from physics, that share the property of being continuously or discretely integrable, a feature that guarantees the existence of conservation laws that often make the problems…
Bilinear systems of equations are defined, motivated and analyzed for solvability. Elementary structure is mentioned and it is shown that all solutions may be obtained as rank one completions of a linear matrix polynomial derived from…
In the present article, real number representations, that are generalizations of classical positive and alternating representations of numbers, are introduced and investigated. The main metric relation, properties of cylinder sets are…
The goal of this Section is to formulate some of the basic results on the theory of integral equations and mention some of its applications. The literature of this subject is very large. Proofs are not given due to the space restriction.…
New iterative methods for solving linear equations are presented that are easy to use, generalize good existing methods, and appear to be faster. The new algorithms mix two kinds of linear recurrence formulas. Older methods have either high…