On the Structure of Solutions of the Moisil-Théodoresco System in Euclidean Space

Let be open, let be the Dirac operator in and let be the Clifford algebra constructed over the quadratic space . If for fixed, denotes the space of r-vectors in , then an -valued smooth function W = W ^r  + W ^r+2 in Ω is said to satisfy the Moisil-Théodoresco system if . In terms of differential forms, this means that the corresponding - valued smooth form w = w ^r  + w ^r+2 satisfies in Ω the system d ^* w ^r = 0, dw ^r  + d ^* w ^r+2 = 0; dw ^r+2 = 0. Based on techniques and results concerning conjugate harmonic functions in the framework of Clifford analysis, a structure theorem is proved for the solutions of the Moisil-Théodoresco system.      

Characterization of Polynomial Systems Satisfying Generalized Convolution Differential-Difference Equations

I. Introduction. The present paper has been motivated by the desire to find all polynomial solutions of the convolution type differential -difference equation (1.1) D_xg_n(x)=sum from i=1 to n-1 (g_i(x)g_(n-i)(x),n≥2,) where g_1(x) is assumed to be a constant. This problem arose in work by one of the authors (Kerr) with a differential equation arising in a coal research project     

On Nonexistence of Algebraic Curve Solutions to Second-order Polynomial Autonomous Systems Over the Complex Field

In this paper, by using the method of algebraic analysis, the results in our previous work are generalized. These results are of importance in the qualitative theory of polynomial autonomous systems.     

Isochronous and Strongly Isochronous Foci of Polynomial Liénard Systems

Differential Equations - The real Liénard system $$\dot x=-y$$ , $$\dot y=x+A(x)-B(x)y$$ , where the polynomials $$A(x) $$ and $$B(x) $$ and the derivative $$A^{\prime }(x) $$ satisfy the...     

Painlevé Property and Integrability of Polynomial Dynamical Systems

The main purpose of this paper is to investigate the connection between the Painlev′e property and the integrability of polynomial dynamical systems. We show that if a polynomial dynamical system has Painlev′e property, then it admits certain class of first integrals. We also present some relationships between the Painlev′e property and the structure of the differential Galois group of the corresponding variational equations along some complex integral curve.     

On the Nonexistence of Laurent Polynomial First Integrals for General Nonlinear Systems

The integrability of a system of ODEs 圣=/(x1is defined as the existence of a sufficiently rich set of first integrals such that its solutions canbe expressed by these integrals.Here,a single-valued function西(。)is called a first integralof the system(1)if it is constant along any solution curve of the system(1).If≯(z)isdifferentiable,then this definition can be written as the condition (筹川瑚=0.The singularity analysis has been proved to be a successful tool for finding integrable systems…     

Solutions to Polynomial Generalized Bers–Vekua Equations in Clifford Analysis

In this paper, we mainly study polynomial generalized Vekua-type equation _boxclose)w=0{p(\mathcal{D})w=0} and polynomial generalized Bers–Vekua equation p(D)w=0{p(\mathcal{\underline{D}})w=0} defined in W ì \mathbbRⁿ⁺¹{\Omega\subset\mathbb{R}^{n+1}} where D{\mathcal{D}} and D{\mathcal{\underline{D}}} mean generalized Vekua-type operator and generalized Bers–Vekua operator, respectively. Using Clifford algebra, we obtain the Fischer-type decomposition theorems for the solutions to these equations including (D-l)^kw=0,(D-l)^kw=0(k ? \mathbbN){\left(\mathcal{D}-\lambda\right)^{k}w=0,\left(\mathcal {\underline{D}}-\lambda\right)^{k}w=0\left(k\in\mathbb{N}\right)} with complex parameter λ as special cases, which derive the Almansi-type decomposition theorems for iterated generalized Bers–Vekua equation and polynomial generalized Cauchy–Riemann equation defined in W ì \mathbbRⁿ⁺¹{\Omega\subset\mathbb{R}^{n+1}}. Making use of the decomposition theorems we give the solutions to polynomial generalized Bers–Vekua equation defined in W ì \mathbbRⁿ⁺¹{\Omega\subset\mathbb{R}^{n+1}} under some conditions. Furthermore we discuss inhomogeneous polynomial generalized Bers–Vekua equation p(D)w=v{p(\mathcal{\underline{D}})w=v} defined in W ì \mathbbRⁿ⁺¹{\Omega\subset\mathbb{R}^{n+1}}, and develop the structure of the solutions to inhomogeneous polynomial generalized Bers–Vekua equation p(D)w=v{p(\mathcal{\underline{D}})w=v} defined in W ì \mathbbRⁿ⁺¹{\Omega\subset\mathbb{R}^{n+1}}.     

On the Growth of Components of Meromorphic Solutions of Systems of Complex Differential Equations

This paper investigates the problem of the growth of the components of meromorphic solutions of a class of a system of complex algebraic differential equations, and generalized some of N. Toda＇s results concerning the growth of differential equations to the case of systems of differential equations. The paper considers the existence of admissible solutions of the system of differential equations.     

Three Positive Solutions of the One-Dimensional Generalized Hénon Equation

We study the one-dimensional generalized Hénon equation under the Dirichlet boundary condition. It is known that there exist at least three positive solutions if the coefficient function is even. In this paper, without the assumption of evenness, we prove the existence of at least three positive solutions.     

On the Center of the Generalized Liénard System

In this paper, we discuss the conditions for a center for the generalized Liénard system (E)₁

Application of Generalized Legendre Polynomial in Combinatorial Identities

ＡｐｐｌｉｃａｔｉｏｎｏｆＧｅｎｅｒａｌｉｚｅｄＬｅｇｅｎｄｒｅＰｏｌｙｎｏｍｉａｌｉｎＣｏｍｂｉｎａｔｏｒｉａｌＩｄｅｎｔｉｔｉｅｓＺｈａｎｇＺｈｉｚｈｅｎｇ（张之正）ＬｅｉＺｈｉｊｕｎ；（雷治军）（ＬｕｏｙａｎｇＴｅａｃｈｅｒｓＣａｌｌｅｇｅ，４...     

On Polynomial Representations of Lie Algebras

We study polynomial representations of finite dimensional (R or C) Lie algebras. As a total classification, we show that there are altogether three types of such nontrivial representations and give their subtle structures.     

Painlev′e Prop erty and Integrability of Polynomial Dynamical Systems

The main purpose of this paper is to investigate the connection between the Painlev′e property and the integrability of polynomial dynamical systems. We show that if a polynomial dynamical system has P...     

On the Viscosity Solutions for Nonlinear Systems of Second Order Elliptic Equations

ＯｎｔｈｅＶｉｓｃｏｓｉｔｙＳｏｌｕｔｉｏｎｓｆｏｒＮｏｎｌｉｎｅａｒＳｙｓｔｅｍｓｏｆＳｅｃｏｎｄＯｒｄｅｒＥｌｉｐｔｉｃＥｑｕａｔｉｏｎｓＬｉｕＷｅｉａｎ（刘伟安）（Ｄｅｐｔ．ｏｆＭａｔｈ．，ＷｕｈａｎＵｎｉｖｅｒｓｉｔｙ，Ｗｕｈａｎ，Ｈｕｂｅｉ...     

On a Class of Solutions to the Generalized Derivative Schrödinger Equations

In this work we shall consider the initial value problem associated to the generalized derivative Schrödinger (gDNLS) equations and Following the argument introduced by Cazenave and Naumkin we shall establish the local well-posedness for a class of small data in an appropriate weighted Sobolev space. The other main tools in the proof include the homogeneous and inhomogeneous versions of the Kato smoothing effect for the linear Schrödinger equation established by Kenig-Ponce-Vega.     

On the Nonexistence of Laurent Polynomial First Integrals for General Semi－quasihomogeneous Systems

An autonomous system of ODFs which admitting a quasi-homogeneous group of symmetries is called a quasihomogeneous one.     

Number and Location of Limit Cycles in a Class of Perturbed Polynomial Systems

In this paper,we investigate the number,location and stability of limit cycles in a class of perturbedpolynomial systems with (2n 1) or (2n 2)-degree by constructing detection function and using qualitativeanalysis.We show that there are at most n limit cycles in the perturbed polynomial system,which is similar tothe result of Perko in [8] by using Melnikov method.For n=2,we establish the general conditions dependingon polynomial's coefficients for the bifurcation,location and stability of limit cycles.The bifurcation parametervalue of limit cycles in [5] is also improved by us.When n=3 the sufficient and necessary conditions for theappearance of 3 limit cycles are given.Two numerical examples for the location and stability of limit cycles areused to demonstrate our theoretical results.     

On the Strong Solutions and the Structural Stability of the g-Bénard Problem

In this paper, we study a type of modified Boussinesq equations which is called g-Bénard problem. We show the existence and uniqueness of strong solutions of the problem in two dimensions, and then, we investigate the continuous dependence of the solutions on the viscosity parameter.