On the representation formula for well‐ordered elastic composites: a convergence of measure approach

The aim of this paper is to derive an integral representation formula for the effective elasticity tensor for a two‐component well‐ordered composite of elastic materials without using a third reference medium and without assuming the completeness of the eigenspace of the operator ? defined in (2.16) in (J. Mech. Phys. Solids 1984; 32 (1):41–62). As shown in (J. Mech. Phys. Solids 1984; 32 (1):41–62) and (Math. Meth. Appl. Sci. 2006; 29 (6):655–664), this integral representation formula implies a relation which links the effective elastic moduli to the N‐point correlation functions of the microstructure. Such relation not only facilitates a powerful scheme for systematic incorporation of microstructural information into bounds on the effective elastic moduli but also provides a theoretical foundation for de‐homogenization. The analysis presented in this paper can be generalized to an n‐component composite of elastic materials. The relations developed here can be applied to the de‐homogenization for a special class of linear viscoelastic composites. The results will be presented in another paper. Copyright © 2006 John Wiley & Sons, Ltd.     

On the approximate solution of some two‐dimensional singular integral equations

An approximation method for a wide class of two‐dimensional integral equations is proposed. The method is based on using a special function system. Orthonormality and good interaction with fundamental integral operators arising in partial differential equations are remarkable properties of this system. In addition, all the basis elements can easily be calculated by recurrence relations. Taking into account these properties we construct a numerical algorithm which does not require additional effort (such as quadrature) to compute the values of the fundamental operators on the basis elements. Copyright © 2001 John Wiley & Sons, Ltd.     

An integral representation for the weighted geometric mean and its applications

By virtue of Cauchy’s integral formula in the theory of complex functions,the authors establish an integral representation for the weighted geometric mean,apply this newly established integral representation to show that the weighted geometric mean is a complete Bernstein function,and find a new proof of the well-known weighted arithmetic-geometric mean inequality.     

3‐D Contact problems for elastic wedges with Coulomb friction

3‐D quasi‐static contact problems for elastic wedges with Coulomb friction are reduced to integral equations and integral inequalities with unknown contact normal pressures. To obtain these equations and inequalities, Green's functions for the wedges, where one face of the wedges is either stress‐free or fixed, are needed. Using Fourier and Kontorovich–Lebedev integral transformations, all the stresses and displacements in the wedges can be constructed in terms of solutions of Fredholm integral equations of the second kind on the semiaxis. The Green's functions can be calculated as uniformly convergent power series in (1‐2ν), where νis Poisson's ratio. An exponential decay of the kernels and right‐hand sides of the Fredholm integral equations provides the applicability of the collocation method for simple and fast calculation of the Green's functions. For a half‐space, which is a special case of an elastic wedge, the kernels degenerate and the functions reduce to the well‐known Boussinesq and Cerruti solutions. Analysing the contact problems reveals that the Green's functions govern the kernels of the above mentioned integral equations and inequalities. Under the assumption that the punch has a smooth shape, the contact pressure is zero on the boundary of the unknown contact zone. Solving the contact problems with the help of the Galanov–Newton method, the normal contact pressure, the contact zone and the normal displacement around the contact zone can be determined simultaneously. In view of the numerical results, the influence of the friction forces on the punch force and the punch settlement is discussed. Copyright © 2004 John Wiley & Sons, Ltd.     

On the polynomial representation for the number of partitions with fixed length

In this paper, it is shown that the number of partitions of a nonnegative integer with parts can be described by a set of polynomials of degree in , where denotes the least common multiple of the integers and denotes the quotient of when divided by . In addition, the sets of the polynomials are obtained and shown explicitly for and .

     

A new Leray formula for smooth functions on bounded domains in Cn

By means of a new technique of integral representations in Cn given by the authors, we establish a new abstract formula with a vector function W for smooth functions on bounded domains in Cn, which is different from the well-known Leray formula, This new formula eliminates the term that contains the parameter λ from the classical Leray formula, and especially on some domains the uniform estimates for the -equation are very simple. From the new Leray formula, we can obtain correspondingly many new formulas for smooth functions on many domains in Cn, which are different from the classical ones, when we properly select the vector function W.     

On the global weak solutions for a modified two‐component Camassa‐Holm equation

In this paper, we investigate the existence of global weak solutions to the Cauchy problem of a modified two‐component Camassa‐Holm equation with the initial data satisfying lim_{x → ±∞}u₀(x) = u_±. By perturbing the Cauchy problem around a rarefaction wave, we obtain a global weak solution for the system under the assumption u_? ≤ u₊. The global weak solution is obtained as a limit of approximation solutions. The key elements in our analysis are the Helly theorem and the estimation of energy for approximation solutions in $H^1(\mathbb {R})\times H^1(\mathbb {R})In this paper, we investigate the existence of global weak solutions to the Cauchy problem of a modified two‐component Camassa‐Holm equation with the initial data satisfying lim_{x → ±∞}u₀(x) = u_±. By perturbing the Cauchy problem around a rarefaction wave, we obtain a global weak solution for the system under the assumption u_? ≤ u₊. The global weak solution is obtained as a limit of approximation solutions. The key elements in our analysis are the Helly theorem and the estimation of energy for approximation solutions in $H^1(\mathbb {R})\times H^1(\mathbb {R})$ and some a priori estimates on the first‐order derivatives of approximation solutions.     

On the Cauchy problem for the two‐component b‐family system

In this paper, we establish the local well‐posedness for the two‐component b‐family system in a range of the Besov space. We also derive the blow‐up scenario for strong solutions of the system. In addition, we determine the wave‐breaking mechanism to the two‐component Dullin–Gottwald–Holm system. Copyright © 2013 John Wiley & Sons, Ltd.     

On a robust version of the integral representation formula of nonlinear filtering

The paper is concerned with completing “unfinished business” on a robust representation formula for the conditional expectation operator of nonlinear filtering. Such a formula, robust in the sense that its dependence on the process of observations is continuous, was stated in [2] without proof. The main purpose of this paper is to repair this deficiency.The formula is “almost obvious” as it can be derived at a formal level by a process of integration-by-parts applied to the stochastic integrals that appear in the integral representation formula. However, the rigorous justification of the formula is quite subtle, as it hinges on a measurability argument the necessity of which is easy to miss at first glance. The continuity of the representation (but not its validity) was proved by Kushner [9] for a class of diffusions.Here we follow the definition given in [11].     

Stochastic integral representation theorem for quantum semimartingales

The quantum stochastic integral of Itô type formulated by Hudson and Parthasarathy is extended to a wider class of adapted quantum stochastic processes on Boson Fock space. An Itô formula is established and a quantum stochastic integral representation theorem is proved for a class of unbounded semimartingales which includes polynomials and (Wick) exponentials of the basic martingales in quantum stochastic calculus.     

A kinematic formula for integral invariant of degree 4 in real space form

In this paper,kinematic formulae in a real space form are investigated.A kinematic formula for homogeneous polynomial of degree 4 on the second fundamental forms in a real space form is obtained.The formula obtained is a concrete form of the result of Howard and the analogue of the known formula of Chen and Zhou.     

A new Leray formula for smooth functions on bounded domains in Cn

By means of a new technique of integral representations in C ⁿ given by the authors, we establish a new abstract formula with a vector function W for smooth functions on bounded domains in C ⁿ , which is different from the well-known Leray formula. This new formula eliminates the term that contains the parameter A from the classical Leray formula, and especially on some domains the uniform estimates for the are very simple. From the new Leray formula, we can obtain correspondingly many new formulas for smooth functions on many domains in C ⁿ , which are different from the classical ones, when we properly select the vector function W.     

A new computational method for solving two‐dimensional Stratonovich Volterra integral equation

This paper presents a method for computing numerical solutions of two‐dimensional Stratonovich Volterra integral equations using one‐dimensional modification of hat functions and two‐dimensional modification of hat functions. The problem is transformed to a linear system of algebraic equations using the operational matrix associated with one‐dimensional modification of hat functions and two‐dimensional modification of hat functions. The error analysis of the method is given. The method is computationally attractive, and applications are demonstrated by a numerical example. Copyright © 2017 John Wiley & Sons, Ltd.     

On pointwise convergence of the family of Urysohn‐type integral operators

In this paper, we consider a family of nonlinear integral operators of Urysohn‐type and study the pointwise convergence of the family at characteristic points of L₁?function. The kernel K_λ(x,t,u(t)) depends on the positive parameter λ changing on the set of numbers with the accumulation point at infinity and K_λ(x,t,u(t)) is an entire analytic function of variable u, which is a bounded function belonging to L₁(R).     

ψ‐Hyperholomorphic functions and a Kolosov–Muskhelishvili formula

Holomorphic function theory is an effective tool for solving linear elasticity problems in the complex plane. The displacement and stress field are represented in terms of holomorphic functions, well known as Kolosov–Muskhelishvili formulae. In , similar formulae were already developed in recent papers, using quaternionic monogenic functions as a generalization of holomorphic functions. However, the existing representations use functions from to , embedded in . It is not completely appropriate for applications in . In particular, one has to remove many redundancies while constructing basis solutions. To overcome that problem, we propose an alternative Kolosov–Muskhelishvili formula for the displacement field by means of a (paravector‐valued) monogenic, an anti‐monogenic and a ψ‐hyperholomorphic function. Copyright © 2015 John Wiley & Sons, Ltd.     

Quadrature formula for approximating the singular integral of Cauchy type with unbounded weight function on the edges

New quadrature formulas (QFs) for evaluating the singular integral (SI) of Cauchy type with unbounded weight function on the edges is constructed. The construction of the QFs is based on the modification of discrete vortices method (MMDV) and linear spline interpolation over the finite interval [−1,1]. It is proved that the constructed QFs converge for any singular point x not coinciding with the end points of the interval [−1,1]. Numerical results are given to validate the accuracy of the QFs. The error bounds are found to be of order O(h^α|lnh|) and O(h|lnh|) in the classes of functions H^α([−1,1]) and C¹([−1,1]), respectively.     

关于一个完全单调函数

设α>0和β是实数,则函数1+αxx+β在(0,∞)上是完全单调的充要条件是α≤2β.这推广了已有的结果.