Solutions Of Linear Diophantine Equations Research Articles

Integral geometry on discrete matrices

Abstract In this note, we study the Radon transform and its dual on the discrete matrices by defining hyperplanes as being infinite sets of solutions of linear Diophantine equations. We then give an inversion formula and a support theorem.

On Some Methods for Solution of Linear Diophantine Equations

The paper considers a linear Diophantine equation. A method (algorithm) for finding a general class of solutions of equation is proposed. The proposed algorithm is explained by examples of equations with two and three variables, trying to direct the reader to a general idea that describes the essence of the method used.

Partitions, diophantine equations, and control systems

Ordered partitions of elements of a reduced abelian monoid are defined and studied by means of the solutions of linear diophantine equations. Links to feedback classification of linear dynamical systems over certain commutative rings are given in the same way as partitions of integers are related to feedback classification of linear dynamical systems over fields in the classical literature.

Kuttaka method–a technical view point

Kuttaka is an algorithm for finding integer solution of Linear Diophantine Equations. Bhaskara simplified Aryabhatta’s “Kuttaka Method” to solve equations of the type ax+c=by. “Kuttaka Method” for solving quadratic indeterminate equations was given by him in 12th century well before the renaissance. European mathematicians got it in the 17th century. In the present article, the technical view point of “Kuttaka Method” has been highlighted.

Sparse Solutions of Linear Diophantine Equations

We present structural results on solutions to the Diophantine system $A{\boldsymbol y} = {\boldsymbol b}$, ${\boldsymbol y} \in \mathbb Z^t_{\ge 0}$ with the smallest number of non-zero entries. Our tools are algebraic and number theoretic in nature and include Siegel's Lemma, generating functions, and commutative algebra. These results have some interesting consequences in discrete optimization.

Solution sets for equations over free groups are EDT0L languages

We show that, given an equation over a finitely generated free group, the set of all solutions in reduced words forms an effectively constructible EDT0L language. In particular, the set of all solutions in reduced words is an indexed language in the sense of Aho. The language characterization we give, as well as further questions about the existence or finiteness of solutions, follow from our explicit construction of a finite directed graph which encodes all the solutions. Our result incorporates the recently invented recompression technique of Jeż, and a new way to integrate solutions of linear Diophantine equations into the process. As a byproduct of our techniques, we improve the complexity from quadratic nondeterministic space in previous works to [Formula: see text] here.

A Class of PID Controllers Tuned in Fractional Representation

The contribution is focused on design and tuning of simple controllers for continuous-time SISO systems without and with time delays by algebraic methods. The control synthesis is based on general solutions of linear Diophantine equations in the ring of proper and Hurwitz stable rational functions (RPS). Both, feedback (1DOF) a feedback and feed forward (2DOF) structures of the control system are considered. A scalar positive parameter is introduced for tuning and influencing control responses. In the paper, this parameter is used for aperiodic tuning of PI controllers. The methodology is utilized for autotuning principles and for robust applications. Interval perturbations in controlled systems and robustness of proposed algorithms are outlined through the value set concept, zero exclusion condition and Kharitonov theorem. For both applications user-friendly program packages were developed in the Matlab+Simulink environment with the support of Polynomial Toolbox.

Hardness of approximating the Minimum Solutions of Linear Diophantine Equations

Let 1 ≤ p < ∞ be any fixed real. We show that assuming P ≠ N P , it is hard to approximate the Minimum Solutions of Linear Diophantine Equations in ℓ p norm within any constant factor and it is also hard to approximate the Minimum Solutions of Linear Diophantine Equations in ℓ p norm within the factor n c / log log n for some constant c > 0 where n is the number of variables in the equations.

Integer Solutions of Linear Diophantine Equations Form a Group

It is shown that the set of integer solutions of a single Diophantine equation, or of several simultaneous linear Diophantine equations, in an arbitrary number of variables, say $n$, is a group with respect to a suitably defined binary operation. Further, the aforesaid group is the direct sum of $n$ cyclic groups.

Discrete truncated powers and lattice points in rational polytope

Discrete truncate power is very useful for studying the number of nonnegative integer solutions of linear Diophantine equations. In this paper, some detail information about discrete truncated power is presented. To study the number of integer solutions of linear Diophantine inequations, the generalized truncated power and generalized discrete truncated power are defined and discussed, respectively. We use generalized discrete truncated powers and multivariate splines to investigate the lattice points in rational polytopes. In particular, we present the degree and period of multidimensional Ehrhart polynomial.

A Sharp Bound on Positive Solutions of Linear Diophantine Equations

Let $Ax = b$ be an $m \times n$ system of linear equations with rank $m < n$ and integer coefficients. Denote by Y the maximum of the absolute values of the $m \times m$ minors of the augmented matrix $(A,b)$. It is proved that if the system has an integral solution $x = (x_i )$ with each $x_i \geqq 0$, and either $Ax = 0$ has no such solution which is nontrivial or there is an $m \times (m + 1)$ submatrix $A'$ of A with rank m such that $A'y = 0$ has a solution with positive integer components, then $Ax = b$ has an integral solution with each $0\leqq x_i \leqq Y$. The bound is sharp.

Efficient solution of linear diophantine equations

This paper presents a new method for finding complete information about the set of all nonnegative integer solutions of homogeneous and inhomogeneous linear diophantine equations. Such solutions are fundamental for associative-commutative unification. Our algorithm finds all minimal solutions as “monotone” paths in a graph which encodes the linear diophantine equation. This encoding makes repeated arithmetic operations obsolete and allows inexpensive tests for minimality of solutions. This graph algorithm compares favourably with the known methods, namely lexicogragraphic algorithm and completion procedure. A PASCAL implementation can be found in the Appendix.

A Sharp Bound for Solutions of Linear Diophantine Equations

A Sharp Bound for Solutions of Linear Diophantine Equations

A sharp bound for solutions of linear Diophantine equations

Let $Ax = b$ be an $m \times n$ system of linear equations with rank $m$ and integer coefficients. Denote by $Y$ the maximum of the absolute values of the $m \times m$ minors of the augmented matrix $\left ( {A,b} \right )$. It is proved that if the system has an integral solution, then it has an integral solution $x = \left ( {{x_i}} \right )$ with $\max \left | {{x_i}} \right | \leq Y$. The bound is sharp.

Algebraic properties of discrete box splines

The central objective of this paper is to discuss linear independence of translates of discrete box splines which we introduced earlier as a device for the fast computation of multivariate splines. The results obtained here allow us to draw conclusions about the structure of such discrete splines which have, in particular, applications to counting the number of nonnegative integer solutions of linear diophantine equations.

A new enumeration scheme for the knapsack problem

This paper presents a new enumeration scheme to solve the one-dimensional knapsack problem motivated by some observations on number theory, more specifically on the determination of the number of solutions of linear diophantine equations. This new algorithm is pseudopolynomial and its special features provide a reduction in running time and in the computational memory requirements as compared with other exact (dynamic programming) methods.

Small solutions of linear Diophantine equations

Let Ax = B be a system of m × n linear equations with integer coefficients. Assume the rows of A are linearly independent and denote by X (respectively Y) the maximum of the absolute values of the m × m minors of the matrix A (the augmented matrix ( A, B)). If the system has a solution in nonnegative integers, it is proved that the system has a solution X = ( x i ) in nonnegative integers wity x i ⩽ X for n - m variables and x i ⩽ ( m - m + 1) Y for m variables. This improves previous results of the authors and others.

Pairwise relatively prime solutions of linear diophantine Equations

AbstractIt is shown that if a1,…,am are relatively prime integers then for every integer n the equation has infinitely many solutions in pairwise relatively prime integers x1,…,xm.

Numbers of solutions of linear diophantine equations and their applications in the theory of invariant cubature formulas

Numbers of solutions of linear diophantine equations and their applications in the theory of invariant cubature formulas

Bounds on Positive Integral Solutions of Linear Diophantine Equations

Assuming the existence of a solution, we find bounds for small solutions $x$ of the finite matrix equation $Ax = B$, where each entry of $A,B$ is an integer, and $x$ is a nontrivial column vector with nonnegative integer entries.