η-INVARIANT AND CHERN-SIMONS CURRENT

The author presents an alternate proof of the Bismut-Zhang localization formula of ηinvariants, when the target manifold is a sphere, by using ideas of mod k index theory instead of the difficult analytic localization techniques of Bismut-Lebeau. As a consequence, it is shown that the R/Z part of the aualytically defined η invariant of Atiyah-Patodi-Singer for a Dirac operator on an odd dimensional closed spin manifold can be expressed purely geometrically through a stable Chern-Simons current on a higher dimensional sphere. As a preliminary application, the author discusses the relation with the Atiyah-Patodi-Singer R/Z index theorem for unitary flat vector bundles,and proves an R refinement in the case where the Dirac operator is replaced by the Signature operator.     

Reflectance distribution in optimal transmittance cavities: The remains of a higher dimensional space

One of the few examples in which the physical properties of an incommensurable system reflect an underlying higher dimensionality is presented. Specifically, we show that the reflectivity distribution of an incommensurable one-dimensional cavity is given by the density of states of a tight-binding Hamiltonian in a two-dimensional triangular lattice. Such effect is due to an independent phase decoupling of the scattered waves, produced by the incommensurable nature of the system, which mimics a random noise generator. This principle can be applied to design a cavity that avoids resonant reflections for almost any incident wave. An optical analogy, by using three mirrors with incommensurable distances between them, is also presented. Such array produces a countable infinite fractal set of reflections, a phenomena which is opposite to the effect of optical invisibility.     

关于3阶Carmichael数

朱文余和孙琦（见《数学进展》，2004，33（4）：505-507）提出了关于3阶Carmichael数的三个问题，我们（见《四川大学学报（自然科学版）》，2006，43（6）：1197-1201）肯定地回答了问题1．本文模仿Howe的寻找严格2阶Carmichael数（见Mathematics of Computation，2000，69（232）：1711—1719）的方法，提出寻找满足某种条件的3阶Carmichael数的方法，并用这种方法确实找到了几百个这样的数，因而完全肯定地回答了问题2．     

Thermofield dynamics and Casimir effect for fermions

A generalization of the Bogoliubov transformation is developed to describe a space compactified fermionic field. The method is the fermionic counterpart of the formalism introduced earlier for bosons [Phys. Rev. A 66 (2002) 052101], and is based on the thermofield dynamics approach. We analyze the energy-momentum tensor for the Casimir effect of a free massless fermion field in a d-dimensional box at finite temperature. As a particular case the Casimir energy and pressure for the field confined in a three-dimensional parallelepiped box are calculated. It is found that the attractive or repulsive nature of the Casimir pressure on opposite faces changes depending on the relative magnitude of the edges. We also determine the temperature at which the Casimir pressure in a cubic box changes sign and estimate its value when the edge of the cube is of the order of the confining lengths for baryons.     

Orthogonal similarity transformation into block‐semiseparable matrices of semiseparability rank k

Very recently, an algorithm, which reduces any symmetric matrix into a semiseparable one of semi‐ separability rank 1 by similar orthogonality transformations, has been proposed by Vandebril, Van Barel and Mastronardi. Partial execution of this algorithm computes a semiseparable matrix whose eigenvalues are the Ritz‐values obtained by the Lanczos' process applied to the original matrix. Also a kind of nested subspace iteration is performed at each step. In this paper, we generalize the above results and propose an algorithm to reduce any symmetric matrix into a similar block‐semiseparable one of semiseparability rank k, with k ∈ ?, by orthogonal similarity transformations. Also in this case partial execution of the algorithm computes a block‐semiseparable matrix whose eigenvalues are the Ritz‐values obtained by the block‐Lanczos' process with k starting vectors, applied to the original matrix. Subspace iteration is performed at each step as well. Copyright © 2005 John Wiley & Sons, Ltd.     

Zero-point momentum in complex media

In this work we apply field regularization techniques to formulate a number of new phenomena related to momentum induced by electromagnetic zero-point fluctuations. We discuss the zero-point momentum associated with magneto-electric media, with moving media, and with magneto-chiral media.     

Blow-up solutions and global solutions for nonlinear parabolic problems

This paper deals with the blow-up of positive solutions for a nonlinear parabolic equation subject to nonlinear boundary conditions. We obtain the conditions under which the solutions may exist globally or blow up in a finite time, by a new approach. Moreover, upper estimates of the “blow-up time”, blow-up rate and global solutions are obtained also.     

关于n维单形的两个轨迹定理

在n维欧氏空间En中,应用向量方法,给出了关于n维单形的两个优美的轨迹定理.     

L（1, 2）-edge-labelings for lattices

For a graph G and two positive integers j and k, an m-L(j, k)-edge-labeling of G is an assignment on the edges to the set {0, 1, 2,..., m}, such that adjacent edges which receive labels differ at least by j, and edges which are distance two apart receive labels differ at least by kThe λ j,k-number of G is the minimum m such that an m-L(j, k)-edge-labeling is admitted by GIn this article, the L(1, 2)-edge-labeling for the hexagonal lattice, the square lattice and the triangular lattice are studied, and the bounds for λ j,k-numbers of these graphs are obtained.