Shapiro polynomials of the second kind

In this work by replacing the initial data in recursive relations the Shapiro polynomials of the second kind were introduced and their relation to the Shapiro polynomials of the first kind was established. In addition to the results similar to the known for the Shapiro polynomials of the first kind the following was defined: the relation was established between the coefficients of the basic and complementary polynomials of the first and second kind, namely, their pairwise orthogonality was proved and the number of the positive coefficients for all polynomials was counted.     

Integrability of Hamiltonian systems and differential Galois groups of higher variational equations

Given a complex analytical Hamiltonian system, we prove that a necessary condition for its meromorphic complete integrability is the commutativity of the identity component of the Galois group of each variational equation of arbitrary order along any integral curve. This was conjectured by the first author based on a suggestion by the third author. The first-order non-integrability criterion, obtained by the first and second authors using only first variational equations, is extended to higher orders by the present criterion. Using this result (at order two, three or higher) it is possible to solve important open problems of integrability which escaped the first order criterion.     

A Banach Lattice Having the Approximation Property,but not Having the Bounded Approximation Property

Mathematical Notes - The first example of a Banach space with the approximation property but without the bounded approximation property was given by Figiel and Johnson in 1973. We give the first...     

Most homeomorphisms of the circle are semiperiodic

This paper was written while the first three authors were visiting the University of Dortmund in 1992. The first two authors thankfully acknowledge the support offered by TEMPUS and the University of Dortmund.     

Vibration and stability of ring-stiffened Euler–Bernoulli tie-bars

Tie-bars are frequently used in structural and mechanical engineering applications. To satisfy requirements like weight reduction, stability improvement, etc., the tie-bars are stiffened with rings. In this paper a method is developed to calculate the natural frequencies, buckling axial force, etc., of the ring-stiffened tie-bars. The dynamics of the ringed and the unringed portions of the beam are treated separately. The mode shape of the first portion was expressed as the superposition of two functions multiplied by constants. Consideration of continuity of deflection and of slope and compatibility of bending moment and shearing force at the first step enabled the mode shape of the second portion to be expressed as the superposition of two functions but multiplied by the same constants as in the first portion. This procedure was recursively carried out up to the last portion. The frequency equation was then derived from the boundary conditions at the end. Buckling of the tie-bar was considered as the case when one of the natural frequencies is zero. The first three frequency parameters and the first two buckling dimensionless axial forces are tabulated for tie-bars stiffened with various number of rings and for various combinations of boundary conditions. The calculation procedure can handle any number and any type of ring-stiffeners.     

Mean convergence of hermite interpolation revisited

The first author is supported by the Hungarian National Foundation for Scientific Research Grant No. 1910 and No. T7570, and the second author is supported by the National Science Foundation Grant No. 9302721. The work was done during the first author's visit in Eugene, Oregon in 1993, and was completed during the second author's visit to the Mathematisches Institut, University of Erlange-Nürnberg, supported by the Alexander von Humboldt Foundation.     

On using estimates of Lipschitz constants in global optimization

Several authors have proposed estimating Lipschitz constants in global optimization by a multiple of the largest slope (in absolute value) between successive evaluation points. A class of univariate functions is exhibited for which the global optimum will be missed when using such a procedure, even if the multiple is arbitrarily large.Research of the first and third authors was supported by AFOSR Grants 0271 and 0066 to Rutgers University. Research of the second author was supported by NSERC Grant GP0036426 and FCAR Grant 90NC0305.This research was done while the first author was Professor and the third author was Graduate Student at RUTCOR, Rutgers University.     

An improved boundary element Galerkin method for three-dimensional crack problems

In this paper we analyze the solution of crack problems in three-dimensional linear elasticity by equivalent integral equations of the first kind on the crack surface. Besides existence and uniqueness we give sharp regularity results for the solution of these pseudodifferential equations. Two versions of Eskin's Wiener-Hopf technique are presented: the first one requires the factorization of matrix-valued symbols which is avoided in the second case. Based on these regularity results we show how to improve the boundary element Galerkin method for our integral equations by using special singular trial functions. We apply the approximation property and inverse assumption of these elements together with duality arguments and derive quasi-optimal asymptotic error estimates in a scale of Sobolev spaces.Dedicated to Prof. Dr.-Ing. W. L. Wendland on the occasion of his 50th birthday.A part of this work was done while the first author was a guest at the Georgia Institute of Technology and while the second author was partially supported by the NSF grant DMS-8501797.     

A note on the first nonzero eigenvalue of the laplacian acting on P-forms

The first author was supported by the Swiss National Sciences Foundation and the second author was partially supported by contrat CEE GADGET.SC1-0105-C.     

Laplace's method for the laws of heat processes on loop spaces

In this paper, we will consider Laplace's method for a class of heat processes on loop spaces. We will obtain the first term of the asymptotics under assumptions that the function under consideration attains its minimum at a unique point and that the Hessian at the point is non-degenerate. This kind of process was first introduced by P. Malliavin in 1990 for the loop group case and then gradually generalized by various authors. Our tool is the rough path theory of T. Lyons. This technique was pioneered by S. Aida for finite-dimensional processes in his unpublished paper.     

Circle and divisor problems,and double series of Bessel functions

In approximately 1915, Ramanujan recorded two identities involving doubly infinite series of Bessel functions. The identities were brought to the mathematical public for the first time when his lost notebook was published in 1988, and are connected with the classical, long-standing circle and divisor problems, respectively. We provide a proof of the first identity for the first time by analytically continuing a new kind of Dirichlet series. Delicate estimates of exponential sums are needed, and the new methods we introduce may be of independent interest.     

Optimal Control in Heterogeneous Domain Decomposition Methods for Advection-Diffusion Equations

New domain decomposition methods (DDM) based on optimal control approach are introduced for the coupling of first and second order equations on overlapping subdomains. Several cost functionals and control functions are proposed. Uniqueness and existence results are proved for the coupled problem, and the convergence of iterative processes is analyzed. The work was supported by the Russian Foundation for Basic Research (04-01-00615) and it was partly carried out while the first author was visiting the IACS at EPFL.     

On a result of Birkhoff

The research of the first two authors was supported by the NSERC of Canada.     

Well posed cauchy problems for complex nonlinear equations must be semilinear

The first author was supported in part by CNPq     

Group actions,gauge transformations,and the calculus of variations

The first author was supported by various research projects of the DFG     

On Gage-Hamilton's entropy formula and Harnack inequality for the curve shortening flow

By the first two derivatives of the Boltzmann entropy of the curvature, which was first studied by Gage and Hamilton for the curve shortening flow in the plane, we define a monotonicity formula which is strictly increasing unless on a shrinking circle. By calculating pointwisely we give an alternate proof of Gage-Hamilton's Harnack inequality.     

On groups of hyperbolic length

We investigate the recently introduced notion of rotation numbers for periodic orbits of interval maps. We identify twist orbits, that is those orbits that are the simplest ones with given rotation number. We estimate from below the topological entropy of a map having an orbit with given rotation number. Our estimates are sharp: there are unimodal maps where the equality holds. We also discuss what happens for maps with larger modality. In the Appendix we present a new approach to the problem of monotonicity of entropy in one-parameter families of unimodal maps. This work was partially done during the first author’s visit to IUPUI (funded by a Faculty Research Grant from UAB Graduate School) and his visit to MSRI (the research at MSRI funded in part by NSF grant DMS-9022140) whose support the first author acknowledges with gratitude. The second author was partially supported by NSF grant DMS-9305899, and his gratitude is as great as that of the first author.     

Squares of characters with few irreducible constituents in finite groups

The research of the first author was partially supported by a grant from NSF.     

Ergodicity of reversible reaction diffusion processes

Summary Reaction-diffusion processes were introduced by Nicolis and Prigogine, and Haken. Existence theorems have been established for most models, but not much is known about ergodic properties. In this paper we study a class of models which have a reversible measure. We show that the stationary distribution is unique and is the limit starting from any initial distribution.The work was begun while the first author was visiting Cornell and supported by the Chinese government. The initial results (for Schlögl's first model) was generalized while the three authors were visiting the Nankai Institute for Mathematics, Tianjin, People's Republic of ChinaPartially supported by the National Science Foundation and the Army Research Office through the Mathematical Sciences Institute at Cornell UniversityPartially supported by NSF grant DMS 86-01800     

输流粘弹性曲管的稳定性分析

根据变质量弹性系统Hamilton原理,用变分法建立了输流粘弹性曲管的运动微分方程,并用归一化幂级数法导出了输流粘弹性曲管的复特征方程组．以两端固支Kelvin-Voigt模型粘弹性输流圆管为例,分析了无量纲延滞时间和质量比对输流管道无量纲复频率和无量纲流速之间的变化关系的影响．在无量纲延滞时间较大时,粘弹性输流圆管的特点是它的第1、2、3阶模态不再耦合,而是在第1、第2阶上先发散失稳,然后在1阶模态上再发生单一模态颤振．