Constructing two-weight codes with prescribed groups of automorphisms

We construct new linear two-weight codes over the finite field with q elements. To do so we solve the equivalent problem of finding point sets in the projective geometry with certain intersection properties. These point sets are in bijection to solutions of a Diophantine linear system of equations. To reduce the size of the system of equations we restrict the search for solutions to solutions with special symmetries.Two-weight codes can be used to define strongly regular graphs. We give tables of the two-weight codes and the corresponding strongly regular graphs. In some cases we find new distance-optimal two-weight codes and also new strongly regular graphs.     

LP narrowing: A new strategy for finding all solutions of nonlinear equations

An efficient algorithm is proposed for finding all solutions of systems of n nonlinear equations. This algorithm is based on interval analysis and a new strategy called LP narrowing. In the LP narrowing strategy, boxes (n-dimensional rectangles in the solution domain) containing no solution are excluded, and boxes containing solutions are narrowed so that no solution is lost by using linear programming techniques. Since the LP narrowing is very powerful, all solutions can be found very efficiently. By numerical examples, it is shown that the proposed algorithm could find all solutions of systems of 5000-50,000 nonlinear equations in practical computation time.     

Stability by the linear approximation for discrete equations

This paper is concerned with exponential stability of solutions of perturbed discrete equations. For a given m>1 we will provide necessary and sufficient conditions for exponential stability of all perturbed systems with perturbation of order m under the assumption that the unperturbed linear system is exponentially stable. Basing on this result we obtained necessary and sufficient conditions for exponential stability of the perturbed system for all perturbations of order m>1 for regular systems. Our results are expressed in terms of regular coefficients of the unperturbed system.     

Nonnegative elements of subgroups of

We give a method to compute all nonnegative integer solutions of a homogeneous system of linear equations. We use this method to provide an algorithm to compute all nonnegative integer solutions of homogeneous systems of equations having some of the equations in congruences.     

On zeros of solutions of linear differential equations

We consider linear differential equations with regular coefficients in $\text{¦ z ¦ < 1}$. We obtain sufficient conditions for all the solutions of these equations to vanish a given number of times at the most. First the results are obtained for differential equations of second order, then for differential equations of nth order, n > 2.     

Tropical differential equations

Tropical differential equations are introduced and an algorithm is designed which tests solvability of a system of tropical linear differential equations within the complexity polynomial in the size of the system and in the absolute values of its coefficients. Moreover, we show that there exists a minimal solution, and the algorithm constructs it (in case of solvability). This extends a similar complexity bound established for tropical linear systems. In case of tropical linear differential systems in one variable a polynomial complexity algorithm for testing its solvability is designed.We prove also that the problem of solvability of a system of tropical non-linear differential equations in one variable is NP-hard, and this problem for arbitrary number of variables belongs to NP. Similar to tropical algebraic equations, a tropical differential equation expresses the (necessary) condition on the dominant term in the issue of solvability of a differential equation in power series.     

The Mirror Systems of Integrable Equations

We provide an algorithm to convert integrable equations to regular systems near noncharacteristic, movable singularity manifolds of solutions. We illustrate how the algorithm is equivalent to the Painlevé test. We also use thealgorithm to prove the convergence of the Laurent series obtained from the Painlevé test.     

Computer Algebra and Bifurcations

In this paper we will present the family of Newton algorithms. From the computer algebra point of view, the most basic of them is well known for the local analysis of plane algebraic curves f(x,y)=0 and consists in expanding y as Puiseux series in the variable x. A similar algorithm has been developped for multi-variate algebraic equations and for linear differential equations, using the same basic tools: a “regular” case, associated with a “simple” class of solutions, and a “simple” method of calculus of these solutions; a Newton polygon; changes of variable of type ramification; changes of unknown function of two types y=ct ^μ+? or y=exp?(c/t ^μ)?. Our purpose is first to define a “regular” case for nonlinear implicit differential equations f(t,y,y′)=0. We will then apply the result to an explicit differential equation with a parameter y′=f(y,α) in order to make a link between the expansions of the solutions obtained by our local analysis and the classical theory of bifurcations.     

Asymptotic behavior for a nematic liquid crystal model with different kinematic transport properties

We study the asymptotic behavior of global classical solutions to hydrodynamical systems modeling the nematic liquid crystal flows under kinematic transports for molecules of different shapes. The coupling system consists of Navier–Stokes equations and kinematic transport equations for the molecular orientations. We prove the convergence of global solutions to single steady states as time tends to infinity as well as estimates on the convergence rate both in 2D for arbitrary regular initial data and in 3D for certain particular cases.     

On the shape of solution sets of systems of (functional) equations

Solution sets of systems of linear equations over fields are characterized as being affine subspaces. But what can we say about the “shape” of the set of all solutions of other systems of equations? We study solution sets over arbitrary algebraic structures, and we give a necessary condition for a set of n-tuples to be the set of solutions of a system of equations in n unknowns over a given algebra. In the case of Boolean equations we obtain a complete characterization, and we also characterize solution sets of systems of Boolean functional equations.     

Systems of differential equations with periodic coefficients

We consider linear systems of differential equations with periodic coefficients. We prove the solvability of nonhomogeneous systems in the Sobolev space W ₂ ¹ (R) and establish estimates for the solutions. This result implies a perturbation theorem for the exponential dichotomy of systems of differential equations with periodic coefficients.     

Exact slow-fast decomposition of the singularly perturbed matrix differential Riccati equation

In this paper the Hamiltonian matrix formulation of the Riccati equation is used to derive the reduced-order pure-slow and pure-fast matrix differential Riccati equations of singularly perturbed systems. These pure-slow and pure-fast matrix differential Riccati equations are obtained by decoupling the singularly perturbed matrix differential Riccati equation of dimension n₁+n₂ into the pure-slow regular matrix differential Riccati equation of dimension n₁ and the pure-fast stiff matrix differential Riccati equation of dimension n₂. A formula is derived that produces the solution of the original singularly perturbed matrix differential Riccati equation in terms of solutions of the pure-slow and pure-fast reduced-order matrix differential Riccati equations and solutions of two reduced-order initial value problems. In addition to its theoretical importance, the main result of this paper can also be used to implement optimal filtering and control schemes for singularly perturbed linear time-invariant systems independently in pure-slow and pure-fast time scales.     

On Periodic Solutions of One Class of Systems of Differential Equations

We study the problem of the existence of periodic solutions of two-dimensional linear inhomogeneous periodic systems of differential equations for which the corresponding homogeneous system is Hamiltonian. We propose a new numerical-analytic algorithm for the investigation of the problem of the existence of periodic solutions of two-dimensional nonlinear differential systems with Hamiltonian linear part and their construction. The results obtained are generalized to systems of higher orders. __________ Translated from Ukrains'kyi Matematychnyi Zhurnal, Vol. 57, No. 4, pp. 483–495, April, 2005.     

On the dichotomy of linear autonomous systems of functional-differential equations

We study the exponential dichotomy of solutions to the Cauchy problem for a subclass of linear autonomous systems of functional-differential equations of neutral type by reducing this problem to the same problem for a difference equation of the form x _n = Γx _n?1 with a compact operator Γ, in Banach space. The spectrum of Γ is described. The dichotomy of the solutions of the Cauchy problem is a consequence of the dichotomy of the spectrum. In a particular case, the result obtained provides a criterion for the dichotomy for a linear autonomous system of functional-differential equations of retarded type of general form.     

Presentations of finitely generated cancellative commutative monoids and nonnegative solutions of systems of linear equations

Varying methods exist for computing a presentation of a finitely generated commutative cancellative monoid. We use an algorithm of Contejean and Devie [An efficient incremental algorithm for solving systems of linear diophantine equations, Inform. and Comput. 113 (1994) 143-172] to show how these presentations can be obtained from the nonnegative integer solutions to a linear system of equations. We later introduce an alternate algorithm to show how such a presentation can be efficiently computed from an integer basis.     

EG-eliminations

We propose an algorithm to put linear recurrent systems in a form which is convenient for using the systems to search for polynomial, power series, Laurent series, and other types of solutions of various linear functional systems (differential,difference and q-difference). Some algorithms to search for solutions of functional systems are described. None of the proposed algorithms requires preliminary uncoupling of linear systems     

A communication-less parallel algorithm for tridiagonal Toeplitz systems

Diagonally dominant tridiagonal Toeplitz systems of linear equations arise in many application areas and have been well studied in the past. Modern interest in numerical linear algebra is often focusing on solving classic problems in parallel. In McNally [Fast parallel algorithms for tri-diagonal symmetric Toeplitz systems, MCS Thesis, University of New Brunswick, Saint John, 1999], an m processor Split & Correct algorithm was presented for approximating the solution to a symmetric tridiagonal Toeplitz linear system of equations. Nemani [Perturbation methods for circulant-banded systems and their parallel implementation, Ph.D. Thesis, University of New Brunswick, Saint John, 2001] and McNally (2003) adapted the works of Rojo [A new method for solving symmetric circulant tri-diagonal system of linear equations, Comput. Math. Appl. 20 (1990) 61–67], Yan and Chung [A fast algorithm for solving special tri-diagonal systems, Computing 52 (1994) 203–211] and McNally et al. [A split-correct parallel algorithm for solving tri-diagonal symmetric Toeplitz systems, Internat. J. Comput. Math. 75 (2000) 303–313] to the non-symmetric case. In this paper we present relevant background from these methods and then introduce an m processor scalable communication-less approximation algorithm for solving a diagonally dominant tridiagonal Toeplitz system of linear equations.     

The formation of trapped surfaces in spherically-symmetric Einstein–Euler spacetimes with bounded variation

We study the evolution of a self-gravitating compressible fluid in spherical symmetry and we prove the existence of weak solutions with bounded variation for the Einstein–Euler equations of general relativity. We formulate the initial value problem in Eddington–Finkelstein coordinates and prescribe spherically symmetric data on a characteristic initial hypersurface. We introduce here a broad class of initial data which contain no trapped surfaces, and we then prove that their Cauchy development contains trapped surfaces. We therefore establish the formation of trapped surfaces in weak solutions to the Einstein equations. This result generalizes a theorem by Christodoulou for regular vacuum spacetimes (but without symmetry restriction). Our method of proof relies on a generalization of the “random choice” method for nonlinear hyperbolic systems and on a detailed analysis of the nonlinear coupling between the Einstein equations and the relativistic Euler equations in spherical symmetry.     

The <Emphasis Type="Italic">N</Emphasis>-wave equations with <Emphasis Type="Italic">PT</Emphasis> symmetry

We study extensions of N-wave systems with PT symmetry and describe the types of (nonlocal) reductions leading to integrable equations invariant under the P (spatial reflection) and T (time reversal) symmetries. We derive the corresponding constraints on the fundamental analytic solutions and the scattering data. Based on examples of three-wave and four-wave systems (related to the respective algebras sl(3,C) and so(5,C)), we discuss the properties of different types of one- and two-soliton solutions. We show that the PT-symmetric three-wave equations can have regular multisoliton solutions for some specific choices of their parameters.