A transform method for linear evolution PDEs on a finite interval

We study initial boundary value problems for linear scalar evolutionpartial differential equations, with spatial derivatives ofarbitrary order, posed on the domain {t > 0, 0 < x <L}. We show that the solution can be expressed as an integralin the complex k-plane. This integral is defined in terms ofan x-transform of the initial condition and a t-transform ofthe boundary conditions. The derivation of this integral representationrelies on the analysis of the global relation, which is an algebraicrelation defined in the complex k-plane coupling all boundaryvalues of the solution. For particular cases, such as the case of periodic boundaryconditions, or the case of boundary value problems for even-orderPDEs, it is possible to obtain directly from the global relationan alternative representation for the solution, in the formof an infinite series. We stress, however, that there existinitial boundary value problems for which the only representationis an integral which cannot be written as an infinite series.An example of such a problem is provided by the linearized versionof the KdV equation. Similarly, in general the solution of odd-orderlinear initial boundary value problems on a finite intervalcannot be expressed in terms of an infinite series.     

Entanglement Evolution of a 2-Qutrit System Interacting with a Fermionic Bath

Based on the algebraic entanglement measure proposed [G. Vidal et al., Phys. Rev. A 65 (2002) 032314],we study the entanglement evolution of both pure quantum states and mixed ones of 2-qutrit system in a symmetrybroken environment consisting of a fermionic bath. Entanglement of states will decrease or remain constant under the influence of environment, and the class of states which remain unchanged has been found out.     

Fixed points in the family of convex representations of a maximal monotone operator

Any maximal monotone operator can be characterized by a convex function. The family of such convex functions is invariant under a transformation connected with the Fenchel-Legendre conjugation. We prove that there exists a convex representation of the operator which is a fixed point of this conjugation.

     

Cyclic Hypomonotonicity,Cyclic Submonotonicity,and Integration

Rockafellar has shown that the subdifferentials of convex functions are always cyclically monotone operators. Moreover, maximal cyclically monotone operators are necessarily operators of this type, since one can construct explicitly a convex function, which turns out to be unique up to a constant, whose subdifferential gives back the operator. This result is a cornerstone in convex analysis and relates tightly convexity and monotonicity. In this paper, we establish analogous robust results that relate weak convexity notions to corresponding notions of weak monotonicity, provided one deals with locally Lipschitz functions and locally bounded operators. In particular, the subdifferentials of locally Lipschitz functions that are directionally hypomonotone [respectively, directionally submonotone] enjoy also an additional cyclic strengthening of this notion and in fact are maximal under this new property. Moreover, every maximal cyclically hypomonotone [respectively, maximal cyclically submonotone] operator is always the Clarke subdifferential of some directionally weakly convex [respectively, directionally approximately convex] locally Lipschitz function, unique up to a constant, which in finite dimentions is a lower C² function [respectively, a lower C¹ function].     

Pseudo-Differential Operators in Algebras of Generalized Functions and Global Hypoellipticity

The aim of this work is to develop a global calculus for pseudo-differential operators acting on suitable algebras of generalized functions. In particular, a condition of global hypoellipticity of the symbols gives a result of regularity for the corresponding pseudo-differential equations. This calculus and this frame are proposed as tools for the study in Colombeau algebras of partial differential equations globally defined on R ⁿ .     

Diffusion in poro‐plastic media

A model is developed for the flow of a slightly compressible fluid through a saturated inelastic porous medium. The initial‐boundary‐value problem is a system that consists of the diffusion equation for the fluid coupled to the momentum equation for the porous solid together with a constitutive law which includes a possibly hysteretic relation of elasto‐visco‐plastic type. The variational form of this problem in Hilbert space is a non‐linear evolution equation for which the existence and uniqueness of a global strong solution is proved by means of monotonicity methods. Various degenerate situations are permitted, such as incompressible fluid, negligible porosity, or a quasi‐static momentum equation. The essential sufficient conditions for the well‐posedness of the system consist of an ellipticity condition on the term for diffusion of fluid and either a viscous or a hardening assumption in the constitutive relation for the porous solid. Copyright © 2004 John Wiley & Sons, Ltd.     

两相同部件温贮备可修的人机系统解的性质分析

本文首先用强连续算子半群理论证明了两相同部件温贮备可修的人机系统动态非负解的存在唯一性 ,然后证明了 0是系统主算子的本征值 ,并得到 0本征值对应的本征向量是正的 ,从而系统存在稳态正解 .     

Additive Maps Preserving Nilpotent Operators or Spectral Radius

Let X be a （real or complex） Banach space with dimension greater than 2 and let B0（X） be the subspace of B（X） spanned by all nilpotent operators on X. We get a complete classification of surjective additive maps Ф on B0（X） which preserve nilpotent operators in both directions. In particular, if X is infinite-dimensional, we prove that Ф has the form either Ф（T） = cATA^-1 or Ф（T） = cAT＇A^-1, where A is an invertible bounded linear or conjugate linear operator, c is a scalar, T＇ denotes the adjoint of T. As an application of these results, we show that every additive surjective map on B（X） preserving spectral radius has a similar form to the above with ｜c｜ = 1.     

与q形变玻色算符逆算符相关的相干态及其量子统计性质

讨论了ｑ形变玻色算符的广义逆算符作用于ｑ－相干态所得到的两类量子态的数学及量子统计性质。结果表明，ｑ－相干态的光子激发态不存在压缩但呈现反聚束效应，而ｑ－相干态的光子湮灭态却存在压缩但不呈现反聚束效应。