教学研究:介绍量子力学几个基本概念——兼答《关于量子几何相位的评注》中的几个主要问题

介绍了量子绝热定理的物理含义及成立的条件，认为有关主要献（Aharonov-Anandan,Bohm，孙昌璞等）的表述是正确的，而《关于量子几何相位的评注》^[1]（以下简称《评注》）相应的表述不完全正确。在此基础上，认为这些献和教材（R.Shankar)得出的涉及Berry绝热相位的一些论述（不含Berry绝热相因子的瞬时能量本征态不满足含时Schroedinger方程等）也是正确的，而《评注》的论述与此相反。《评注》认为只有γn(C)才是Berry相位。本作则倾向于把γn（t）叫做Berry绝热相位，而把γn（C）=γn（T）-γn（0）叫做几何相位（geometric phase)^[2]。     

The heat kernel and Hardy’s theorem on symmetric spaces of noncompact type

For symmetric spaces of noncompact type we prove an analogue of Hardy’s theorem which characterizes the heat kernel in terms of its order of magnitude and that of its Fourier transform.     

Time-periodic Solutions to the Ginzburg-Landau-BBM Equations

In this paper, the existence and uniqueness of the time-periodic solutions to the Ginzburg-Landau-BBM equations are proved by using a priori estimates and Leray-Schauder fixed point theorem.     

二阶奇异边值问题的正解

利用Leary-Schauder不动点定理讨论了一类二阶奇异边值问题正解的存在性问题,并给出了一个正解存在的必要条件     

Prox-regular functions in variational analysis

The class of prox-regular functions covers all l.s.c., proper, convex functions, lower- functions and strongly amenable functions, hence a large core of functions of interest in variational analysis and optimization. The subgradient mappings associated with prox-regular functions have unusually rich properties, which are brought to light here through the study of the associated Moreau envelope functions and proximal mappings. Connections are made between second-order epi-derivatives of the functions and proto-derivatives of their subdifferentials. Conditions are identified under which the Moreau envelope functions are convex or strongly convex, even if the given functions are not.

     

Nonequilibrium thermodynamics and fluctuation relations for small systems

In this review, we give a retrospect of the recent progress in nonequilibrium statistical mechanics and thermodynamics in small dynamical systems. For systems with only a few number of particles, fluctuations and nonlinearity become significant and contribute to the nonequilibrium behaviors of the systems, hence the statistical properties and thermodynamics should be carefully studied. We review recent developments of this topic by starting from the Gallavotti–Cohen fluctuation theorem, and then to the Evans–Searles transient fluctuation theorem, Jarzynski free-energy equality, and the Crooks fluctuation relation. We also investigate the nonequilibrium free energy theorem for trajectories involving changes of the heat bath temperature and propose a generalized free-energy relation. It should be noticed that the non-Markovian property of the heat bath may lead to the violation of the free-energy relation.     

Compression-expansion fixed point theorem in two norms and applications

In this paper we present a two-norms version of Krasnoselskii's fixed point theorem in cones. The abstract result is then applied to prove the existence of positive Lp solutions of Hammerstein integral equations with better integrability properties on the kernels.     

On Approximation Properties of Generalized q-Bernstein Operators

In this study, we identify a generalization of q-Bernstein type operators and investigate approximation properties of a sequence of these operators . We estimate rate of approximation by modulus of continuity. We prove Voronovskaya type theorem for these operators.     

A generalization of inverse distance weighting and an equivalence relationship to noise-free Gaussian process interpolation via Riesz representation theorem

In this paper, we show the relationship between two seemingly unrelated approximation techniques. On the one hand, a certain class of Gaussian process-based interpolation methods, and on the other hand inverse distance weighting, which has been developed in the context of spatial analysis where there is often a need for interpolating from irregularly spaced data to produce a continuous surface. We develop a generalization of inverse distance weighting and show that it is equivalent to the approximation provided by the class of Gaussian process-based interpolation methods. The equivalence is established via an elegant application of Riesz representation theorem concerning the dual of a Hilbert space. It is thus demonstrated how a classical theorem in linear algebra connects two disparate domains.