Algebraically simple involutive differential systems and the Cauchy problem

We consider systems of polynomial nonlinear partial differential equations (PDEs) possessing certain properties. Such systems were studied by the American mathematician Thomas in the 1930s, and he called them (algebraically) simple. Thomas gave a constructive procedure to split an arbitrary system of PDEs into a finite number of simple subsystems. The class of simple involutive systems of PDEs includes normal, or Kovalevskaya-type, systems and Riquier orthonomic passive systems. Systems of this class admit well-posed Cauchy problems. We discuss the basic features of the splitting algorithm, completion of simple systems to involution, and the well-posedness of the Cauchy problem. Two illustrative examples are given. Bibliography: 17 titles.     

A family of cyclic cubic polynomials whose roots are systems of fundamental units

We give a family of cyclic cubic polynomials whose roots are systems of fundamental units of the splitting fields. These polynomials are constructed by a linear fractional transformation from Shanks’ polynomials with rational coefficients.     

Bivariate orthogonal polynomials in the Lyskova class

Classical orthogonal polynomials in two variables can be characterized as the polynomial solutions of a matrix second-order partial differential equation involving matrix polynomial coefficients. In this work, we study classical orthogonal polynomials in two variables whose partial derivatives satisfy again a second-order partial differential equation of the same type.     

A semiclassical perspective on multivariate orthogonal polynomials

Differential properties for orthogonal polynomials in several variables are studied. We consider multivariate orthogonal polynomials whose gradients satisfy some quasi-orthogonality conditions. We obtain several characterizations for these polynomials including the analogues of the semiclassical Pearson differential equation, the structure relation and a differential-difference equation.     

Orthogonal polynomials on the unit circle and their derivatives

We characterize the sequences of orthogonal polynomials on the unit circle whose derivatives are also orthogonal polynomials on the unit circle. Some relations for the sequences of derivatives of orthogonal polynomials are provided. Finally, we pose some problems about orthogonality-preserving maps and differential equations for orthogonal polynomials on the unit circle.     

Global error estimates for exponential splitting

Global errors of principal exponential splitting formulae, i.e.,first-order splitting, Strang's splitting and parallel splitting,are investigated. It is found that the commutativity of theunderlying matrices in the exponents plays an important rolein the analysis. It is shown that the global error estimatesbehave as polynomials in the splitting steps when the stepsare small and decay exponen tially when the splitting stepsare chosen to be large. They are much more reliable than thosederived from the traditional local truncation error analysiswhen relatively large splitting steps are adopted. This corresponds,for example, to solving partial differential equations usingrelatively large Courant numbers. In this case, the global erroranalysis provides sharp bounds in contrast to the local errorestimates, which give inaccurate or even deceptive results.Numerical examples are given to demonstrate our results.     

Colorings of simplicial complexes and vector bundles over Davis–Januszkiewicz spaces

We show that coloring properties of a simplicial complex K are reflected by splitting properties of a bundle over the associated Davis–Januszkiewicz space whose Chern classes are given by the elementary symmetric polynomials in the generators of the Stanley-Reisner algebra of K.     

Equations aux différences finies et déterminants d'intégrales de fonctions multiformes

We give a formula which computes the determinant of a matrix whose entries are integrals of algebraic differential forms multiplied by a product of complex powers of polynomials. In this formula enters the characteristic polynomial of some monodromies associated with this family of polynomials.     

Discretized Multisplitting AOR Waveform Relaxation Algorithms for Initial Value Problem of Systems of ODEs

1.IntroductionInthedevelopmentofnewelectricalcircuits,thesimulationofthebehaviourofthecircuithasbecomeanessentialtoolforelectricalengineers.Fromthelayoutofthecircuitanonlinearsystemofordinarydifferentialequationsisgeneratedwhichdescribesthedynamicalbehaviourofthecircuit.Inthesimulationofverylargescaleintegrated(VLSI)circuitsthedimensionofthesystemofODEscanbecomeverylarge.Moreoversincethesystemisstiff,solvingthesesystemsisaverycomputionallyintensivetaskandtheuseofsupercomputersbecomesin-evit…     

Z／（p^e）上分裂环的结构

本文研究了环ｚ／（ｐｅ）上多项式的性质和分裂环的结构．主要分析了分裂环中元素的极小多项式，零化理想的结构，和分裂环子环性质．     

On differential equations for orthogonal polynomials on the unit circle

In this paper we characterize sequences of orthogonal polynomials on the unit circle whose corresponding Carathéodory function satisfies a Riccati differential equation with polynomial coefficients, in terms of second order matrix differential equations. In the semi-classical case, a characterization in terms of second order linear differential equations with polynomial coefficients is deduced.     

UNIQUENESS OF MEROMORPHIC OR ENTIRE FUNCTIONS AND THEIR DIFFERENTIAL POLYNOMIALS

This article studies the problem of uniqueness of two entire or meromorphic functions whose differential polynomials share a finite set. The results extendand improve on some theorems given in [3].     

UNIQUENESS OF MEROMORPHIC OR ENTIRE FUNCTIONS AND THEIR DIFFERENTIAL POLYNOMIALS

This article studies the problem of uniqueness of two entire or meromorphic functions whose differential polynomials share a finite set. The results extend and improve on some theorems given in [3].     

─族亚纯函数的正规定则

本文讨论亚纯函数族的正规性，推广并改进了杨乐￣［２，３］，陈怀惠￣［５］和朱秀蓉，龚向宏￣［７］等人的结果。     

Shared Values of Certain Nonlinear Differential Polynomials of Meromorphic Functions

In this paper,by using the idea of truncated counting functions of meromorphic functions,we deal with the problem of uniqueness of the meromorphic functions whose certain nonlinear differential polynomials share one finite nonzero value.     

Block companion Singer cycles, primitive recursive vector sequences, and coprime polynomial pairs over finite fields

We discuss a conjecture concerning the enumeration of nonsingular matrices over a finite field that are block companion and whose order is the maximum possible in the corresponding general linear group. A special case is proved using some recent results on the probability that a pair of polynomials with coefficients in a finite field is coprime. Connection with an older problem of Niederreiter about the number of splitting subspaces of a given dimension are outlined and an asymptotic version of the conjectural formula is established. Some applications to the enumeration of nonsingular Toeplitz matrices of a given size over a finite field are also discussed.     

Appell’s lemma and conservation laws of KdV equation

An alternative method to construct a class of conservation laws of the KdV equation based on the classical Appell’s lemma and the trace formulas of Deift-Trubowitz type is studied. A new type of infinite sequence of conservation laws whose local densities cannot be expressed in terms of differential polynomials is constructed.     

Contributions to diophantine approximation in fields of series

The following topics are taken up: (i) the dimension of certain vector spaces defined in terms of denominations of differential polynomials, (ii) examples of series for which a Thue-Siegel-Roth type theorem holds, (iii) approximation of linear forms whose coefficients satisfy algebraic differential equations, and (iv) rational approximations to solutions of algebraic difference-differential equations.Written with partial support from NSF grant NSF-MCS 78-01770.     

Efficient simulation of discrete stochastic reaction systems with a splitting method

Stochastic reaction systems with discrete particle numbers are usually described by a continuous-time Markov process. Realizations of this process can be generated with the stochastic simulation algorithm, but simulating highly reactive systems is computationally costly because the computational work scales with the number of reaction events. We present a new approach which avoids this drawback and increases the efficiency considerably at the cost of a small approximation error. The approach is based on the fact that the time-dependent probability distribution associated to the Markov process is explicitly known for monomolecular, autocatalytic and certain catalytic reaction channels. More complicated reaction systems can often be decomposed into several parts some of which can be treated analytically. These subsystems are propagated in an alternating fashion similar to a splitting method for ordinary differential equations. We illustrate this approach by numerical examples and prove an error bound for the splitting error.