Systems Of Linear Diophantine Equations Research Articles

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

In this work, we study the split-complex integer solutions for the split-complex linear Diophantine equation in two variables where are split-complex integers. An algorithm for generating all solutions will be obtained by transforming the split-complex equation to a classical equivalent system of linear Diophantine equations in four variables.

Application of System Linear Diophantine Equations in Balancing Chemical Equations

Abstract: In this manuscript, the step by step procedure for how the system of linear Diophantine equations are applied to balance chemical equations acquired by the reactions of various chemical compounds and their products is scrutinized.

Fixed-point iterative approach for solving linear Diophantine systems with bounds on the variables

ABSTRACT Systems of linear Diophantine equations arise from several applications. Scholars have given attention to such systems and come up with several effective solutions. A new approach, called the fixed-point iterative method, was proposed to solve linear Diophantine equations with lower and upper bounds on the variables. Two steps are involved in solving this problem. First, the problem is transformed into a polytope judgment problem . Then, the approach is used to judge the existence of an integer point in the polytope. Compared with the branch-and-bound method, results show that the approach is feasible and effective for solving linear Diophantine systems.

Parametric Solutions of System of Linear Diophantine Equations by Crushing Method

There are studies on parametric solutions of system of Linear Diophantine equations based on uni-modular reductions of the coefficient matrix. In this paper we generate parametric solutions, with uni-modular row reductions on the coefficient matrix, based on the steps used in obtaining gcd of the coefficients in a row by crushing method. This application of gcd by crushing specifies an order for the row reductions and enables to give algorithm for the computations.

Sparse representation of vectors in lattices and semigroups

We study the sparsity of the solutions to systems of linear Diophantine equations with and without non-negativity constraints. The sparsity of a solution vector is the number of its nonzero entries, which is referred to as the ell _0-norm of the vector. Our main results are new improved bounds on the minimal ell _0-norm of solutions to systems Avarvec{x}=varvec{b}, where Ain mathbb {Z}^{mtimes n}, {varvec{b}}in mathbb {Z}^m and varvec{x} is either a general integer vector (lattice case) or a non-negative integer vector (semigroup case). In certain cases, we give polynomial time algorithms for computing solutions with ell _0-norm satisfying the obtained bounds. We show that our bounds are tight. Our bounds can be seen as functions naturally generalizing the rank of a matrix over mathbb {R}, to other subdomains such as mathbb {Z}. We show that these new rank-like functions are all NP-hard to compute in general, but polynomial-time computable for fixed number of variables.

The Mathematical Safe Problem and Its Solution (Part 2)

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.

Numerical Methods to Solve Problems on a Mathematical Safe

Numerical methods to solve problems on a mathematical safe with an arbitrary finite number of positions of locks are given. The methods are based on TSS-algorithms for constructing sets of basic solutions to systems of linear Diophantine equations in finite fields and rings.

Optimal constraints aggregation method for ILP

In this paper we give a new aggregation method for Integer Linear Program (ILP), that allows to reduce the number of constraints of any ILP. In particular, we study the aggregation of non-negative systems of linear Diophantine equations. We prove that an aggregated system of minimum size can be constructed in polynomial time. We also study the minimum size of the coefficients of the aggregated problem.

Реалізація алгоритму розв’язання системи лінійних діофантових рівнянь в кільці лишків

Реалізація алгоритму розв’язання системи лінійних діофантових рівнянь в кільці лишків

A novel approach to perturbative calculations for a large class of interacting boson theories

We present a method of calculating the interacting S-matrix to an arbitrary perturbative order for a large class of boson interaction Lagrangians. The method takes advantage of a previously unexplored link between the n-point Green's function and a certain system of linear Diophantine equations. By finding all nonnegative solutions of the system, the task of perturbatively expanding an interacting S-matrix becomes elementary for any number of interacting fields, to an arbitrary perturbative order (irrespective of whether it makes physical sense) and for a large class of scalar boson theories. The method does not rely on the position-based Feynman diagrams and promises to be extended to many perturbative models typically studied in quantum field theory. Aside from interaction field calculations we showcase our approach by expanding a pair of Unruh–DeWitt detectors coupled to Minkowski vacuum to an arbitrary perturbative order in the coupling constant. We also link our result to Hafnian as introduced by Caianiello and present a method to list all (2n−1)!! perfect matchings of a complete graph on 2n vertices.

From Feynman rules to conserved quantum numbers, I

In the context of Quantum Field Theory (QFT) there is often the need to find sets of graph-like diagrams (the so-called Feynman diagrams) for a given physical model. If negative, the answer to the related problem ‘Are there any diagrams with this set of external fields?’ may settle certain physical questions at once. Here the latter problem is formulated in terms of a system of linear diophantine equations derived from the Lagrangian density, from which necessary conditions for the existence of the required diagrams may be obtained. Those conditions are equalities that look like either linear diophantine equations or linear modular (i.e. congruence) equations, and may be found by means of fairly simple algorithms that involve integer computations. The diophantine equations so obtained represent (particle) number conservation rules, and are related to the conserved (additive) quantum numbers that may be assigned to the fields of the model.

Constructing graphs with given spectrum and the spectral radius at most 2

Two graphs are cospectral if their spectra coincide. The set of all graphs that are cospectral to a given graph, including the graph by itself, is the cospectral equivalence class of the graph. We say that a graph is determined by its spectrum, or that it is a DS-graph, if it is a unique graph having that spectrum. Given n reals belonging to the interval [−2,2], we want to find all graphs on n vertices having these reals as the eigenvalues of the adjacency matrix. Such graphs are called Smith graphs. Our search is based on solving a system of linear Diophantine equations. We present several results on spectral characterizations of Smith graphs.

Discrete Particle Swarm Optimization in the numerical solution of a system of linear Diophantine equations

This article proposes the use of a discrete version of the well known Particle Swarm Optimization, DPSO, a metaheuristic optimization algorithm for numerically solving a system of linear Diophantine equations. Likewise, the transformation of this type of problem (i.e. solving a system of equations) into an optimization one is also shown. The current algorithm is able to find all the integer roots in a given search domain, at least for the examples shown. Simple problems are used to show its efficacy. Moreover, aspects related to the processing time, as well as to the effect of increasing the population and the search space, are discussed. It was found that the strategy shown herein represents a good approach when dealing with systems that have more unknowns than equations, or when it becomes of considerable size, since a big search domain is required.

Diophantine Approach to Blind Interference Alignment of Homogeneous K-User 2x1 MISO Broadcast Channels

The sufficient condition for a blindly interference-aligned 2-user 2 × 1 broadcast channel (BC) in homogeneous block fading to achieve its maximal 4/3 degree-of-freedom (DoF) is well understood. However, the condition for the general case, i.e., a homogeneous K-user 2 × 1 BC, to achieve the corresponding optimal 2K/2+K-1 DoF remains unsolved and is, thus, the focus of this paper. Here, a K-user 2×1 BC is said to be BIA-feasible if it achieves the optimal DoF via blind interference alignment (BIA). In this paper, we cast this general BIA-feasibility problem in the framework of finding solutions for a system of linear Diophantine equations. Then by studying the solvability of the Diophantine system, we derive the necessary and sufficient conditions on the K users' fading block offsets to ensure the BIA feasibility of the K-user BC. Using the results above, we also examine a 2 × 1 broadcast network with M homogeneous users, in which their offsets are independent of one another and are uniformly distributed over a coherence block. We show by simulations that when the network contains M = 11 homogeneous users, there is more than 95% chance of finding 3 users to form a BIA-feasible 3-user 2 × 1 BC. Furthermore, increasing M to infinity ensures a virtual multiple-input multiple-output (MIMO) channel to be formed from the DoF perspective.

Counting Markov Types, Balanced Matrices, and Eulerian Graphs

The method of types is one of the most popular techniques in information theory and combinatorics. Two sequences of equal length have the same type if they have identical empirical distributions. In this paper, we focus on Markov types, that is, sequences generated by a Markov source (of order one). We note that sequences having the same Markov type share the same so-called balanced frequency matrix that counts the number of distinct pairs of symbols. We enumerate the number of Markov types for sequences of length over an alphabet of size . This turns out to be asymptotically equivalent to estimating the number of the balanced frequency matrices, the number of integer solutions of a system of linear Diophantine equations, and the number of connected Eulerian multigraphs. For fixed , we prove that the number of Markov types is asymptotically equal to d(m) n <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m</sup> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">-m</sup> /(m <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> -m)! where we give an integral representation for d(m). For m →∞, we conclude that asymptotically the number of types is equivalent to √2m <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">3m/2</sup> e <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m</sup> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> /m <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2m2</sup> 2 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m</sup> π <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m/2</sup> n <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m2</sup> -m provided that m = o(n <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1/4</sup> ). These findings are derived by analytical techniques ranging from analytic combinatorics, to multidimensional generating functions, to the saddle point method.

Geometrical regular languages and linear Diophantine equations: The strongly connected case

Given an arbitrarily large alphabet Σ, we consider the family of regular languages over Σ for which the deterministic minimal automaton has a strongly connected state diagram. We present a new method for checking whether such a language is semi-geometrical or not and whether it is geometrical or not. This method makes use of the enumeration of the simple cycles of the state diagram. It is based on the construction of systems of linear Diophantine equations, where the coefficients are deduced from the set of simple cycles.

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

We use the system of linear Diophantine equations introduced by Coolsaet and Jurišić to obtain information about a feasible family of distance-regular graphs with vanishing Krein parameter q221 and intersection arrays {(r+1)(r3−1),r(r−1)(r2+r−1),r2−1;1,r(r+1),(r2−1)(r2+r−1)},r≥2. In this way we are able to calculate certain triple intersection numbers and prove nonexistence for all r≥3. For r=3 nonexistence was not known before, however it is well known that the intersection array for r=2 uniquely determines the halved 7-cube. Then we show how to apply Terwilliger balanced set conditions for Q-polynomial distance-regular graphs to produce additional linear Diophantine equations.

On the existence of solutions in systems of linear Diophantine equations

The question about the existence of solutions in a family of systems of Diophantine linear equations can be always answered by means of a set of functions called ‘Testers’. In this paper, we will propose a procedure to obtain this set of ‘Testers’ (named complete set of testers) which characterizes each family of systems of Diophantine equations.

An efficient algorithm for solving rank one perturbed linear Diophantine systems using Rosser’s approach

Recently, we described a generalization of Rosser’s algorithm for a single linear Diophantine equation to an algorithm for solving systems of linear Diophantine equations. Here, we make use of the new formulation to present a new algorithm for solving rank one perturbed linear Diophantine systems, based on using Rosser’s approach. Finally, we compare the efficiency and effectiveness of our proposed algorithm with the algorithm proposed by Amini and Mahdavi-Amiri (Optim Methods Softw 21:819–831, 2006).

An algorithm for constructing the basis of the solution set for systems of linear Diophantine equations over the ring of integers

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.