A Short Contribution to Split-Complex Linear Diophantine Equations in Two Variables

The main difficulty in solving the systems of linear Diophantine equations is the very rapid growth of the intermediate results. Hence, it is important to design algorithms that restraint the growth of intermediate results. The use of the basis reduction algorithm for solving a linear Diophantine equation is examined in this paper. The motivation behind choosing the basis reduction is twofold. First the basis reduction allows us to work only with integers, which avoids the round-off problems. Second, the basis reduction finds short and nearly orthogonal vectors belonging to the lattice described by the basis. Then, we expect the general solution obtained by this basis will also be short. It is important to note that the basis reduction does not change the lattice, it only derives an alternative way of spanning it. Once we have obtained the vectors given by the reduced basis, we use them to find the general solution of the linear Diophantine equation.

Some methods and algorithms of solution of systems of linear Diophantine equations over the naturals are briefly reviewed. Criteria and an incremental algorithm of efficient solution of the problem of consistency for a system of linear Diophantine equations and inequalities over the naturals are given.

Introduction. The problem of mathematical safe arises in the theory of computer games and cryptographic applications. The article considers numerous variations of the mathematical safe problem and examples of its solution using systems of linear Diophantine equations in finite rings and fields. The purpose of the article. To present methods for solving the problem of a mathematical safe for its various variations, which are related both to the domain over which the problem is considered and to the structure of systems of linear equations over these domains. To consider the problem of a mathematical safe (in matrix and graph forms) in different variations over different finite domains and to demonstrate the work of methods for solving this problem and their efficiency (systems over finite simple fields, finite fields, ghost rings and finite associative-commutative rings). Results. Examples of solving the problem of a mathematical safe, the conditions for the existence of solutions in different areas, over which this problem is considered. The choice of the appropriate area over which the problem of the mathematical safe is considered, and the appropriate algorithm for solving it depends on the number of positions of the latches of the safe. All these algorithms are accompanied by estimates of their time complexity, which were considered in the first part of this paper. Conclusions. The considered methods and algorithms for solving linear equations and systems of linear equations in finite rings and fields allow to solve the problem of a mathematical safe in a large number of variations of its formulation (over finite prime field, finite field, primary associative-commutative ring and finite associative-commutative ring with unit). Keywords: mathematical safe, finite rings, finite fields, method, algorithm.

New algorithms for constructing the Hermite normal form (triangular) and Smith normal form (diagonal) of an integer matrix are presented. A new algorithm for determining the set of solutions to a system of linear diophantine equations is presented. A modification of the Hermite algorithm gives an integer-preserving algorithm for solving linear equations with real-valued variables. Rough bounds for the number of operations are cubic polynomials involving the order of the matrix and the determinant of the matrix. The algorithms are valid if the elements of the matrix are in a principal ideal domain.

Triple intersection numbers of [formula omitted]-polynomial distance-regular graphs

A polynomial algorithm is proposed to construct the minimal generating set of solutions and the basis of the solution set for systems of linear Diophantine equations over the ring of integer. The algorithm is based on a modified TSS method.

A new algorithm for fording the minimal solutions of systems of linear Diophantine equations has recently been published. In its description the emphasis was put on the mathematical aspects of the algorithm. In complement to that, in this paper another presentation of the algorithm is given which may be of use for anyone wanting to implement it.

Another constructive proof is presented for the fact that a system of linear equations with integer coefficients in bounded integer variables is equivalent to a single equation, which is a linear combination of the original ones. The equation is obtained in a number of steps; in each step two equations are replaced by a single one. This replacement is performed subject to the condition that the remaining equations hold and a final equation with relatively small coefficients is obtained. It may be inefficient however to calculate small coefficients, as the original coefficients can be used to represent the final ones in a suitably chosen number system.

Selection probabilities, viz. rank and sample selection probabilities, quantify the impulse suppression, robustness, lowpass filtering ability, and detail preserving ability of stack filters. Closed-form expressions exist for computing selection probabilities from positive Boolean function describing the stack filter. We first present an analysis framework that interprets a Boolean vector to denote membership in a partition set. Using this framework a new algorithm to compute selection probabilities from positive Boolean functions is derived. Next a method to synthesize a stack filter from specified selection probabilities is presented. An underdetermined system of linear Diophantine equations is derived in the unknown elements of the C-matrix and the desired selection probabilities. This system of equations is solved to obtain the candidate solutions for the C-matrix. For each candidate C-matrix, an algorithm based on partition sets is used to derive one or more positive Boolean functions of the stack filter or show that none exists.

Two new methods to solve linear systems of Diophantine equations are proposed - modular (CRT) and p-adic (Hensel). Each of them allows to obtain solutions of a system with the size n x m with the complexity O(nsm). For quasi-square systems, the p-adic method allows to obtain solution with the complexity O(n 3), and the modular method with complexity O(n s+1). Both estimates have the accuracy up to the logarithmic multipliers, s being the power in the estimation of matrix multiplication time.

Linear Diophantine equation are introduced to determine and search for integral solutions according to the associated variables. In this paper, the mathematical techniques are approached to solve Linear Diophantine equation with two, three unknowns and system of linear Diophantine equation by Euclidean algorithmic, congruence modulo n and unimodular row reduction. The modulo arithmetic operation is also applied for linear Diophantine equation in three variables. The Euclidean algorithmic is an efficient method to compute the greatest common divisor (g.c.d) of two integers. The extended Euclidean algorithmic is used also to compute the greatest common divisor (g.c.d) of two positive integers and write this greatest common divisor as an integer linear combination of two positive integers. Moreover, the mathematical methods of Diophantine equations in two and three variables are presented to practical applications in reality life such as buying and selling, traffic flux and the number of atoms of a chemical substance to investigate positive integral solutions.

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New algorithms are described and analysed for solving various problems associated with a large integer matrix: computing the Hermite form, computing a kernel basis, and solving a system of linear diophantine equations. The algorithms are space-efficient and for certain types of input matrices -- for example, those arising during the computation of class groups and regulators--are faster than previous methods. Experiments with a prototype implementation support the running time analyses.

In this chapter we address the problem of finding an x ∈ ℜn which is a solution to the system Ax = b or Ax ≥ b with some or all of the x j required to be integer. This problem has a very long history. Integer equalities were studied by the Greek mathematician Diophantos during the third century A.D. In fact, when all the variables are required to be integer the equations in the system Ax = b are called linear Diophantine equations. We use the terms Diophantine and integer interchangeably. The study of linear and nonlinear Diophantine equations is an important part of number theory. Although linear Diophantine equations were studied in the third century A.D. it was not until the 20th century A.D. (1976 to be precise) that a polynomial algorithm was given for finding an integer solution the system Ax = b.