A new hierarchy of integrable systems associated to Hurwitz spaces

In this paper, we introduce a new class of integrable systems, naturally associated to Hurwitz spaces (spaces of meromorphic functions over Riemann surfaces). The critical values of the meromorphic functions play the role of "times". Our systems give a natural generalization of the Ernst equation; in genus zero, they realize the scheme of deformation of integrable systems proposed by Burtsev, Mikhailov and Zakharov. We show that any solution of these systems in rank 1 defines a flat diagonal metric (Darboux-Egoroff metric) together with a class of corresponding systems of hydrodynamic type and their solutions.     

具有三个公共值的亚纯函数的唯一性

研究具有两个CM公共值和一个IM公共值的亚纯函数的唯一性。     

Finer fractal geometry for analytic families of conformal dynamical systems

We prove several results establishing real analyticity of Hausdorff dimensions of limit sets of analytic families of conformal graph directed Markov systems. With this tool and with iterated function systems resulting from the existence of nice sets in the sense of Rivera-Letelier, we prove that the canonical Hausdorff measure restricted to the radial Julia set of a tame meromorphic function (can be rational) is σ-finite and that the Hausdorff dimension of the radial Julia sets for fairly general families of meromorphic functions (can be rational) is real analytic.     

函数族S的一些性质

本文探讨了亚纯拟星象函数族和传统星象函数族Ｓ之间的对应关系，确定的解析半径，并得到增长、偏差和系数的精确估计。     

某类系数与Fejér缺项级数有关的齐次和非齐次高阶线性微分方程亚纯解的增长性

Nevanlinna理论在复微分方程领域中具有广泛的应用,其中运用该理论研究复线性微分方程亚纯解的增长性和值分布与系数的增长性之间的关系是复微分方程领域中的重要论题．由于缺项级数具有一些特殊性质,当缺项级数作为方程系数时,这些性质即可发挥作用．因此,我们可结合缺项级数的定义和性质研究复线性微分方程亚纯解的性质．在本文中,我们运用Nevanlinna理论并结合Fejér缺项级数的定义和性质对一类齐次和非齐次高阶复线性微分方程进行了研究．当方程的某个系数与Fejér缺项级数有关而其余系数为整函数或亚纯函数时,得到了方程亚纯解的增长级的估计,推广并改进了前人已有结果．     

Locating a straight crack in an infinite elastic medium by using complex path-independent integrals

Summary The Cauchy integral theorem and the relevant formula (or, equivalently, complex path-independent integrals) have been used in a long series of papers for the determination of zeros and poles of analytic and meromorphic functions. Here this approach is generalized to become applicable to the problem of location of a straight crack inside an infinite plane isotropic elastic medium. The complex path-independent integrals used here contain the first complex potential (z) of Kolosov-Muskhelishvili, which can be obtained experimentally. The present method can be modified to apply to a variety of problems where discontinuity intervals of analytic (or, rather, sectionally analytic) functions are sought.     

一类整函数系数线性微分方程解的增长性

本文主要研究某类整函数系数高阶线性微分方程解的增长性,这类方程有一个系数为满足Denjoy猜想极值情况的整函数．运用亚纯函数值分布理论和整函数的渐近值理论,通过比较方程中每一项的模的大小,得到这类方程解的增长级的估计．对只有一个系数起控制作用的方程,当其存在一个系数为二阶微分方程的解时,得到上述方程的非零解都为无穷级．对系数具有相同增长级的方程,当其系数具指数函数形式时,得到上述方程的非零解也为无穷级．文中所得结果是对线性微分方程相关结果的推广和补充．     

Exact solutions of the cubic–quintic Swift–Hohenberg equation and their bifurcations

Many systems of physical interest may be modelled by the bistable Swift–Hohenberg equation with cubic–quintic nonlinearity. We construct a two-parameter family of exact meromorphic solutions of the time-independent equation and use these to construct a one-parameter family of exact periodic solutions on the real line. These are of two types, differing in their symmetry properties, and are connected via an exact heteroclinic solution. We use these exact solutions as initial points for numerical continuation and show that some of these lie on secondary branches while others fall on isolas. The approach substantially enhances our understanding of the solution space of this equation.     

一类具指数函数系数的非线性复差分方程

众所周知,在复微分方程和复差分方程领域中,Malmquist型方程是比Painlev′e方程和Riccati方程形式更一般的非线性方程.在本文中,我们运用Nevanlinna理论的差分模拟结果和微分域理论对一类具指数函数系数的Malmquist型复差分方程进行了研究.当上述Malmquist型复差分方程的有限级超越亚纯解具有较少的零点和极点时,我们得到其增长性和指数函数ez的增长性一致.该结果是对复微分Malmquist定理和复差分Malmquist定理的推广和补充.     

On meromorphic complex differential equations

In this paper, we study first-order, autonomous, complex differential equations of the form i = f (z), where f (z) is the meromorphic function of the complex variable z, defined in a simply connected domain on the Riemann sphere. We concentrate on the phase portraits of such systems, with particular attention being paid to the existence and properties of closed orbits, and orbits which reach the point at infinity (or blow-up) in finite time. Applications of the general theoy are given, as well as discussion of higher dimensional systems     

A Riemann-Hilbert approach to the Akhiezer polynomials

In this paper, we study those polynomials, orthogonal with respect to a particular weight, over the union of disjoint intervals, first introduced by N. I. Akhiezer, via a reformulation as a matrix factorization or Riemann-Hilbert problem. This approach complements the method proposed in a previous paper, which involves the construction of a certain meromorphic function on a hyperelliptic Riemann surface. The method described here is based on the general Riemann-Hilbert scheme of the theory of integrable systems and will enable us to derive, in a very straightforward way, the relevant system of Fuchsian differential equations for the polynomials and the associated system of the Schlesinger deformation equations for certain quantities involving the corresponding recurrence coefficients. Both of these equations were obtained earlier by A. Magnus. In our approach, however, we are able to go beyond Magnus' results by actually solving the equations in terms of the Riemanni Theta-functions. We also show that the related Hankel determinant can be interpreted as the relevant tau-function.     

A C<Superscript><Emphasis Type="Italic">k</Emphasis></Superscript> continuous generalized finite element formulation applied to laminated Kirchhoff plate model

A generalized finite element method based on a partition of unity (POU) with smooth approximation functions is investigated in this paper for modeling laminated plates under Kirchhoff hypothesis. The shape functions are built from the product of a Shepard POU and enrichment functions. The Shepard functions have a smoothness degree directly related to the weight functions adopted for their evaluation. The weight functions at a point are built as products of C ^∞ edge functions of the distance of such a point to each of the cloud boundaries. Different edge functions are investigated to generate C ^k functions. The POU together with polynomial global enrichment functions build the approximation subspace. The formulation implemented in this paper is aimed at the general case of laminated plates composed of anisotropic layers. A detailed convergence analysis is presented and the integrability of these functions is also discussed.     

A new technique to derive the Green’s type integral formula in thermoelasticity

New closed-form influence functions of a unit point heat source on elastic displacements and new Green’s type integral formula for a boundary-value problem (BVP) for a thermoelastic half-space are presented. The main difficulties in obtaining such results are observed in deriving the influence functions of a concentrated unit force onto elastic volume expansion and, also, in Green’s functions in heat conduction. For canonical Cartesian domains, these functions have been derived successfully for hundreds of BVPs and were published in a handbook. So, this paper shows the way to derive not only thermoelastic influence functions and Green’s type integral formulas for the half-space, but also for many new BVPs in thermoelasticity in other Cartesian canonical domains. Moreover, the technique proposed here may be applied in any orthogonal canonical domain provided by the lists of Green’s functions in heat conduction and influence functions for elastic volume expansion that are known.     

Consistent higher degree Petrov–Galerkin methods for the solution of the transient convection–diffusion equation

The solution of the convection–diffusion equation for convection dominated problems is examined using both N + 1 and N + 2 degree Petrov–Galerkin finite element methods in space and a Crank–Nicolson finite difference scheme in time. While traditional N + 1 degree Petrov–Galerkin methods, which use test functions one polynomial degree higher than the trial functions, work well for steady-state problems, they fail to adequately improve the solution for the transient problem. However, using novel N + 2 degree Petrov–Galerkin methods, which use test functions two polynomial degrees higher than the trial functions, yields dramatically improved solutions which in fact get better as the Courani number increases to 1·0. Specifically, cubic test functions with linear trial functions and quartic test functions in conjunction with quadratic trial functions are examined. Analysis and examples indicate that N + 2 degree Petrov–Galerkin methods very effectively eliminate space and especially time truncation errors. This results in substantially improved phase behaviour while not adversely affecting the ratio of numerical to analytical damping.     

The smooth piecewise polynomial particle shape functions corresponding to patch-wise non-uniformly spaced particles for meshfree particle methods

In the previous papers (Kim et al. Submitted for publication, Oh et al. in press), for uniformly or locally non-uniformly distributed particles, we constructed highly regular piecewise polynomial particle shape functions that have the polynomial reproducing property of order k for any given integer k ≥ 0 and satisfy the Kronecker Delta Property. In this paper, in order to make these particle shape functions more useful in dealing with problems on complex geometries, we introduce smooth-piecewise-polynomial Reproducing Polynomial Particle shape functions, corresponding to the particles that are patch-wise non-uniformly distributed in a polygonal domain. In order to make these shape functions with compact supports, smooth flat-top partition of unity shape functions are constructed and multiplied to the shape functions. An error estimate of the interpolation associated with such flexible piecewise polynomial particle shape functions is proven. The one-dimensional and the two-dimensional numerical results that support the theory are resented. June G. Kim is Visiting Professor of the University of North Carolina at Charlotte.     

Study of the monopolar RWG and monopolar n×RWG basis functions for electromagnetic scattering analysis

In this paper, we study the accuracy and the efficiency of the monopolar divergence-conforming Rao–Wilton–Glisson (RWG) and the monopolar curl-conforming n×RWG basis functions for the magnetic field integral equation (MFIE). Similar to cases using RWG and n×RWG basis functions for the MFIE, there are two impedance matrix elements calculation schemes if the monopolar RWG and monopolar n×RWG basis functions are used to the MFIE, respectively. The monopolar basis functions and the implementation schemes used for the MFIE are discussed. The scattering cross section data as well as the CPU time needed to fill the corresponding impedance matrix obtained from numerical solutions of these implementation schemes using monopolar basis functions are investigated. For the monopolar basis functions and the implementation schemes considered, the first scheme of the MFIE using the monopolar curl-conforming n×RWG basis functions gives most accurate results and it is the best choice for the use of the monopolar basis functions to the MFIE.     

Fully tensorial nodal and modal shape functions for triangles and tetrahedra

This paper presents nodal and modal shape functions for triangle and tetrahedron finite elements. The functions are constructed based on the fully tensorial expansions of one‐dimensional polynomials expressed in barycentric co‐ordinates. The nodal functions obtained from the application of the tensorial procedure are the standard h‐Lagrange shape functions presented in the literature. The modal shape functions use Jacobi polynomials and have a natural global C⁰ inter‐element continuity. An efficient Gauss–Jacobi numerical integration procedure is also presented to decrease the number of points for the consistent integration of the element matrices. An example illustrates the approximation properties of the modal functions. Copyright © 2005 John Wiley & Sons, Ltd.     

The determination of the elastic T-term using higher order weight functions

It has been shown in a recent work [1] that the elastic T-term at the tip of a mixed mode crack can be determined by the so-called second order weight functions through a work-conjugate integral that is akin to that of the Bueckner-Rice weight function method for evaluating stress intensity factors. In this paper, the development of the second order weight functions is reviewed. These second order weight functions are determined using a unified finite element method introduced in [2]. The finite element procedure handles both traction and displacement boundaries and it permits the Bueckner-Rice weight functions and the second order weight functions for the elastic T-term to be determined in one single finite element run. The accuracy of the computed weight functions is assessed by comparing the computed results with special closed form solutions. The numerical values of the elastic T-term for single edge notch specimens under tension, pure bending and three-point bend are given. The corresponding second order weight functions are tabulated.