A NEW DISCRETE INTEGRABLE COUPLING SYSTEM AND ITS HAMILTONIAN STRUCTURE FOR THE MODIFIED TODA LATTICE HIERARCHY

We present a new discrete integrable coupling system by using the matrix Lax pair U, V ∈ sl(4). A novel spectral problem of modified Toda lattice soliton hierarchy is considered. Then, a new discrete integrable coupling equation hierarchy is obtained through the method of the enlarged Lax pair. Finally, we obtain the Hamiltonian structure of the integrable coupling system of the soliton equation hierarchy using the matrix-form trace identity. This discrete integrable coupling system includes a kind of a modified Toda lattice hierarchy.     

变系数Sine-Gordon方程的反散射变换

In this paper, the variable coefficient Sine-Gordon type equation uxt=a(t)sinu β(t)uxx k(t)(xux)x is discussed. It is related to the eigenvalue problem Vx=QV.The structure equation and the evolution laws of scattering data for the second equation are derived and the inverse scattering solution of the first equation is obtained.     

RIEMANN-HILBERT PROBLEMS AND SOLITON SOLUTIONS OF NONLOCAL REVERSE-TIME NLS HIERARCHIES

The paper aims at establishing Riemann-Hilbert problems and presenting soliton solutions for nonlocal reverse-time nonlinear Schrodinger(NLS) hierarchies associated with higher-order matrix spectral problems.The Sokhotski-Plemelj formula is used to transform the Riemann-Hilbert problems into Gelfand-Levitan-Marchenko type integral equations.A new formulation of solutions to special Riemann-Hilbert problems with the identity jump matrix,corresponding to the reflectionless inverse scattering transforms,is proposed and applied to construction of soliton solutions to each system in the considered nonlocal reversetime NLS hierarchies.     

RIEMANN-HILBERT PROBLEMS AND SOLITON SOLUTIONS OF NONLOCAL REVERSE-TIME NLS HIERARCHIES

The paper aims at establishing Riemann-Hilbert problems and presenting soliton solutions for nonlocal reverse-time nonlinear Schrodinger(NLS) hierarchies associated with higher-order matrix spectral problems.The Sokhotski-Plemelj formula is used to transform the Riemann-Hilbert problems into Gelfand-Levitan-Marchenko type integral equations.A new formulation of solutions to special Riemann-Hilbert problems with the identity jump matrix,corresponding to the reflectionless inverse scattering transforms,is proposed and applied to construction of soliton solutions to each system in the considered nonlocal reversetime NLS hierarchies.     

THE SASA-SATSUMA EQUATION ON A NON-ZERO BACKGROUND: THE INVERSE SCATTERING TRANSFORM AND MULTI-SOLITON SOLUTIONS

We concentrate on the inverse scattering transformation for the Sasa-Satsuma equation with 3×3 matrix spectrum problem and a nonzero boundary condition. To circumvent the multi-value of eigenvalues, we introduce a suitable two-sheet Riemann surface to map the original spectral parameter k into a single-valued parameter z. The analyticity of the Jost eigenfunctions and scattering coefficients of the Lax pair for the Sasa-Satsuma equation are analyzed in detail. According to the analyticity of the...     

A New 4 × 4 AKNS Spectral Problem and Its Associated Integrable Decomposition of the AKNS Equation

A new approach to construct a new 4×4 matrix spectral problem from a normal 2×2 matrix spectral problem is presented.AKNS spectral problem is discussed as an example.The isospectral evolution equation of the new 4×4 matrix spectral problem is nothing but the famous AKNS equation hierarchy.With the aid of the binary nonlino earization method,the authors get new integrable decompositions of the AKNS equation. In this process,the r-matrix is used to get the result.     

On initial data of the monopole equation

The space-time monopole equation is the reduction of anti-self-dual Yang-Mills equations in R2,2 to R2,1. This equation is a non-linear wave equation, and can be encoded in a Lax pair. An equivalent Lax pair is used by Dai and Terng to construct monopoles with continuous scattering data, and then the equation can be linearized by the scattering data, allowing one to use the inverse scattering method to solve the Cauchy problem with rapidly decaying small initial data. In this paper, we use the terminology of holomorphic bundle and transversality of certain maps, parametrized by initial data, to give more initial data, with which we can use scattering method to solve the Cauchy problem of the monopole equation up to gauge transformation.     

ANALYSIS OF TWO-DIMENSIONAL ELECTROMAGNETIC SCATTERING BY A PERFECTLY CONDUCTING OBSTACLE IN A HOMOGENEOUS CHIRAL ENVIRONMENT

The time-harmonic electromagnetic plane waves incident on a perfectly conducting obstacle in a homogeneous chiral environment are considered.A two-dimensional direct scat- tering model is established and the existence and uniqueness of solutions to the problem are discussed by an integral equation approach.The inverse scattering problem to find the shape of scatterer with the given far-field data is formulated.Result on the uniqueness of the inverse problem is proved.     

THE PERIODIC HARRY-DYM EQUATION

The development of the inverse scattering transform(I.S.T)has made it possible tosolve certain physically significant nonlinear evolution equations with periodic boundaryconditions.Date and Tanaka have considered kdv equation;Ma and Ablowitz havediscussed the cubic Schrodinger equation.In this paper,following closely the analysis in[2,3]the author considers Harry-Dym eqution(q~2)_t=-2r_(xxx)(Ⅰ)where q(x,t)is periodic in x with period π for all time q(x,t)=q(x π,t),q(x,t)=r~(-1)(x,t)>0     

Regular Solution and Lattice Miura Transformation of Bigraded Toda Hierarchy

The authors give finite dimensional exponential solutions of the bigraded Toda hierarchy （BTH）. As a specific example of exponential solutions of the BTH, the authors consider a regular solution for the （1, 2）-BTH with a 3 × 3-sized Lax matrix, and discuss some geometric structures of the solution from which the difference between the （1, 2）- BTH and the original Toda hierarchy is shown. After this, the authors construct another kind of Lax representation of （N, 1）-BTH which does not use the fractional operator of Lax operator. Then the authors introduce the lattice Miura transformation of （N, 1）-BTH which leads to equations depending on one field, and meanwhile some specific examples which contain the Volterra lattice equation （a useful ecological competition model） are given.     

Boundary-Value Problems of the Theory of Thin and Nonthin Orthotropic Shells with Account of Nonlinearly Elastic Properties and Low Shear Rigidity of Composite Materials

The basic geometric and physical relations and resolving equations of the theory of thin and nonthin orthotropic composite shells with account of nonlinear properties and low shear rigidity of their materials are presented. They are derived based on two theories, namely the theory of anisotropic shells employing the Timoshenko or Kirchhoff-Love hypothesis and the nonlinear theory of elasticity and plasticity of anisotropic media in combination with the Lagrange variational principle. The procedure and algorithm for the numerical solution of nonlinear (linear) problems are based on the method of successive approximations, the difference-variational method, and the Lagrange multiplier method. Calculations of the stress-strain state for a spherical shell with a circular opening loaded with internal pressure are presented. The effect of transverse shear strains and physical nonlinearity of the material on the distribution of maximum deflections and circumferential stresses in the shell, obtained according to two variants of the shell theories, is studied. A comparison of the results of the problem solution in linear and nonlinear statements with and without account of the shell shear strains is given. The numerical data obtained for thin and nonthin (medium thick) composite shells are analyzed.     

On the hyperbolicity of the period-doubling fixed point

We give a new proof of the hyperbolicity of the fixed point for the period-doubling renormalization operator using the local dynamics near a semi-attractive fixed point (in a Banach space) and the theory of holomorphic motions. We also give a new proof of the exponential contraction of the Feigenbaum renormalization operator in the hybrid class of the period-doubling fixed point: our proof uses the non-existence of invariant line fields in the period-doubling tower (C. McMullen), the topological convergence (D. Sullivan), and a new infinitesimal argument.

     

颤振分析中的特征值排序问题

颤振分析中判断颤振临界速度的重要依据是系统V-g和V-f图,即系统特征值随参数的变化曲线.在几乎所有商用软件及自编程序的输出结果中,有时会出现所谓的"窜支"现象,这给颤振临界速度和颤振穿越分支及耦合形式的判断带来很大不便.通过隐函数定理可以证明,除重特征值点以外,系统特征值连续依赖于系统参数变化.依据多元向量值函数连续性,建立对特征值的排列算法,给出系统特征根轨迹的正确曲线,再输出V-g和V-f图数据,从而避免"窜支"现象.编制应用程序,通过几个典型算例对算法进行了验证.该工作能够有效简化颤振分析的后处理工作,提高分析效率.     

Effects of permeability and viscosity in linear polymeric gels

We propose and analyze a mathematical model of the mechanics of gels, consisting of the laws of balance of mass and linear momentum of the polymer and liquid components of the gel. We consider a gel to be an immiscible and incompressible mixture of a nonlinearly elastic polymer and a fluid. The problems that we study are motivated by predictions of the life cycle of body‐implantable medical devices. Scaling arguments suggest neglecting inertia terms, and therefore, we consider the quasi‐static approximation to the dynamics. We focus on the linearized system about stress‐free states, uniform expansions, and compressions and derive sufficient conditions for the solvability of the time‐dependent problems. These turn out to be conditions that guarantee local stability of the equilibrium solutions. We also consider non‐stress free equilibria and states with residual stress and derive an energy law for the corresponding time‐dependent system. The conditions that guarantee stability of solutions provide a selection criteria of the material parameters of devices. The boundary conditions that we consider are of two types, displacement‐traction and permeability of the gel surface to the fluid. We address the cases of viscous and inviscid solvent, assume Newtonian dissipation for the polymer component, and establish existence of weak solutions for the different boundary permeability conditions and viscosity assumptions. We present two‐dimensional, finite element numerical simulations to study stress concentration on edges, this being the precursor to debonding of the gel from its substrate. Copyright © 2015 John Wiley & Sons, Ltd.     

树脂基复合材料粘接的优化设计

本文在试验数据和理论分析的基础上,提出树脂基复合材料单搭接的优化设计的原则,胶粘剂的选择、被粘物的设计,粘接长度的选择、胶层厚度的选择等.认为通过优化设计可以提高粘接强度,减轻产品的重量,提高其质量.     

Clustering Methods in Microarrays

Summary DCT Given a finite set of points in an Euclidean space the \emph{spanning tree} is a tree of minimal length having the given points as vertices. The length of the tree is the sum of the distances of all connected point pairs of the tree. The clustering tree with a given length of a given finite set of points is the spanning tree of an appropriately chosen other set of points approximating the given set of points with minimal sum of square distances among all spanning trees with the given length. DCM A matrix of real numbers is said to be column monotone orderable if there exists an ordering of columns of the matrix such that all rows of the matrix become monotone after ordering. The {\emph{monotone sum of squares of a matrix}} is the minimum of sum of squares of differences of the elements of the matrix and a column monotone orderable matrix where the minimum is taken on the set of all column monotone orderable matrices. Decomposition clusters of monotone orderings of a matrix is a clustering ofthe rows of the matrix into given number of clusters such that thesum of monotone sum of squares of the matrices formed by the rowsof the same cluster is minimal.DCP A matrix of real numbers is said to be column partitionable if there exists a partition of the columns such that the elements belonging to the same subset of the partition are equal in each row. Given a partition of the columns of a matrix the partition sum of squares of the matrix is the minimum of the sum of square of differences of the elements of the matrix and a column partitionable matrix where the minimum is taken on the set of all column partitionable matrices. Decomposition of the rows of a matrix into clusters of partitions is the minimization of the corresponding partition sum of squares given the number of clusters and the sizes of the subsets of the partitions.     

Noncommutative geometry and motives: The thermodynamics of endomotives

We combine aspects of the theory of motives in algebraic geometry with noncommutative geometry and the classification of factors to obtain a cohomological interpretation of the spectral realization of zeros of L-functions. The analogue in characteristic zero of the action of the Frobenius on ?-adic cohomology is the action of the scaling group on the cyclic homology of the cokernel (in a suitable category of motives) of a restriction map of noncommutative spaces. The latter is obtained through the thermodynamics of the quantum statistical system associated to an endomotive (a noncommutative generalization of Artin motives). Semigroups of endomorphisms of algebraic varieties give rise canonically to such endomotives, with an action of the absolute Galois group. The semigroup of endomorphisms of the multiplicative group yields the Bost-Connes system, from which one obtains, through the above procedure, the desired cohomological interpretation of the zeros of the Riemann zeta function. In the last section we also give a Lefschetz formula for the archimedean local L-factors of arithmetic varieties.     

Utilization and maintenance in a model with scrapping

We consider a productive asset, called equipment or capital good, and we examine the properties of, as well as the interactions between, the operating policies, which are determined by its optimal utilization and maintenance, and the capital policy of scrapping, which defines the optimal time when the productive asset is retired from its current use. Starting with an abstract model and using the approach of optimal control, initially we characterize the various types of equipment by assigning to them a single total profit index, which indicates how the above policies affect the flow of operating revenues plus capital gains or losses. This index is a function of market-determined prices. So using it we then investigate how the operating and capital policies are influenced by the rate of discount, the price of new equipment, and the rise or fall of the price of new equipment relative to the value of its output. Among other interesting results, we find that the effects of these prices on the nature and interactions of optimal policies depend crucially on whether the equipment is of the usual profit making type, where output is the main source of revenue, or of the antique type, where the main source of revenue is capital gains.     

EPR制度下的家电以旧换新回收策略优化模型

研究生产者责任延伸制度(EPR)约束下,家电生产销售企业集中决策下实施以旧换新进行废旧产品回收的问题。在对家电消费者市场结构进行分析基础上,建立以企业利润最大化为目标的实施以旧换新回收策略的优化模型,并根据我国家电行业EPR制度的特点,构建政府与企业间竞争的Stackelberg博弈模型,研究在政府环境效益最优时的补贴和基金征收标准下,企业以旧换新回收产品最优定价决策;同时,与企业不直接参与废旧产品回收时单一定价相应结果进行比较。结果表明:在EPR约束下,政府环境效益最优时的补贴和基金征收标准,对企业实施以旧换新回收策略的产品销量、总利润和废旧产品回收量的影响取决于企业成本结构、消费者结构、消费者效用和废旧产品回收处理能力。企业决策者需根据不同情形决定是否参与以旧换新回收实践。     

The Stokes phenomenon in the confluence of the hypergeometric equation using Riccati equation

In this paper we study the confluence of two regular singular points of the hypergeometric equation into an irregular one. We study the consequence of the divergence of solutions at the irregular singular point for the unfolded system. Our study covers a full neighborhood of the origin in the confluence parameter space. In particular, we show how the divergence of solutions at the irregular singular point explains the presence of logarithmic terms in the solutions at a regular singular point of the unfolded system. For this study, we consider values of the confluence parameter taken in two sectors covering the complex plane. In each sector, we study the monodromy of a first integral of a Riccati system related to the hypergeometric equation. Then, on each sector, we include the presence of logarithmic terms into a continuous phenomenon and view a Stokes multiplier related to a 1-summable solution as the limit of an obstruction that prevents a pair of eigenvectors of the monodromy operators, one at each singular point, to coincide.