Harmonic morphisms with one-dimensional fibres on Einstein manifolds

We prove that, from an Einstein manifold of dimension greater than or equal to five, there are just two types of harmonic morphism with one-dimensional fibres. This generalizes a result of R.L. Bryant who obtained the same conclusion under the assumption that the domain has constant curvature.

     

On Harmonic Functions on Trees

We study the asymptotic behaviour of harmonic and p-harmonic functions (1<p<) on trees, obtaining estimates about the Hausdorff dimension of radial limits.     

Harmonic analysis for graph refinements and the continuous graph FFT

The discrete Fourier transform and the FFT algorithm are extended from the circle to continuous graphs with equal edge lengths.     

Tangential Boundary Behaviour of Harmonic Functions on Trees

We study the behaviour of harmonic functions on a homogeneous tree from the point of view of the tangential boundary covergence.     

Equality of Lifshitz and van Hove exponents on amenable Cayley graphs

We study the low energy asymptotics of periodic and random Laplace operators on Cayley graphs of amenable, finitely generated groups. For the periodic operator the asymptotics is characterised by the van Hove exponent or zeroth Novikov–Shubin invariant. The random model we consider is given in terms of an adjacency Laplacian on site or edge percolation subgraphs of the Cayley graph. The asymptotic behaviour of the spectral distribution is exponential, characterised by the Lifshitz exponent. We show that for the adjacency Laplacian the two invariants/exponents coincide. The result holds also for more general symmetric transition operators. For combinatorial Laplacians one has a different universal behaviour of the low energy asymptotics of the spectral distribution function, which can be actually established on quasi-transitive graphs without an amenability assumption. The latter result holds also for long range bond percolation models.     

树上马氏链场的若干极限性质

该文引进广义Ｂｅｔｈｅ树和广义Ｃａｙｌｅｙ树的概念,并研究其上马氏链场关于状态和状态序偶出现频率的强极限定理,作为主要结果的推论,得到Ｂｅｔｈｅ树和Ｃａｙｌｅｙ树上马氏链场的ＳｈａｎｎｏｎＭｃＭｉｌｌａｎ定理．证明中采用了研究概率论强极限定理的一种新的方法．     

Harmonic decomposition analysis of contact mechanics of bonded layered elastic solids

The paper presents an iterative method for obtaining footprint, pressure distribution, local deformation and sub-surface stress field for the contact between a rigid cylindrical indenter and an elastic flat substrate. The methodology is applicable for semi-infinite, as well as for thin or thick bonded elastic layered solids with high or low elastic moduli. All findings are in accord with the observed behaviour of hard wear resistant and soft solid lubricating coatings. It is shown that the decomposed contact pressure distribution into a series of harmonic waves induces sub-surface stress fields that decay into the depth of the solid according to their wavelengths. Consequently, conditions vis-à-vis fatigue spalling and adhesion performance may be predicted for given thickness of layered bonded elastic solids.     

Harmonic analysis on discrete Abelian groups

Let be an Abelian group and let denote the linear space of all complex-valued functions defined on equipped with the product topology. We prove that the following are equivalent.

(i) Every nonzero translation invariant closed subspace of contains an exponential; that is, a nonzero multiplicative function.

(ii) The torsion free rank of is less than the continuum.

     

Cayley树上随机场的马尔可夫逼近与一类小偏差定理

通过引进样本相对熵率作为Cayley树上任意随机场与马尔可夫链场之间的偏差的一种度量, 建立了关于状态序偶频率的一类小偏差定理. 证明中应用了研究马尔可夫链强极限定理的一种新的分析方法.     

Fast approximation to Poisson integral based on trigonometric periodic multiresolution analysis

In this article, a novel fast numerical computational algorithm for Poisson integral is developed by means of periodic trigonometric multiresolution analysis (PTMRA). The approximation formula of Poisson integral is derived. Subsequently, we establish some error estimates of approximation Poisson integral. Finally, several numerical results are given. Comparing with the existing wavelet-based method, the proposed method gives superior results.     

Localisation of directional scale-discretised wavelets on the sphere

Scale-discretised wavelets yield a directional wavelet framework on the sphere where a signal can be probed not only in scale and position but also in orientation. Furthermore, a signal can be synthesised from its wavelet coefficients exactly, in theory and practice (to machine precision). Scale-discretised wavelets are closely related to spherical needlets (both were developed independently at about the same time) but relax the axisymmetric property of needlets so that directional signal content can be probed. Needlets have been shown to satisfy important quasi-exponential localisation and asymptotic uncorrelation properties. We show that these properties also hold for directional scale-discretised wavelets on the sphere and derive similar localisation and uncorrelation bounds in both the scalar and spin settings. Scale-discretised wavelets can thus be considered as directional needlets.     

调和分析与信道编解码研究——献给陆善镇教授75华诞

信道编码是通信系统的关键技术之一,其传统的理论工具是代数和有限域GF（p）.本文在有噪信道的信道编码数学模型基础上,介绍基于调和分析的新型信道编码理论,分别应用调和分析的重要工具压缩感知与Walsh-Hadamard变换实现信道编解码和信道编码盲识别.本文综述基于调和分析的信道编码的一些基本结果并介绍最新进展,主要包括：Gilbert-Varshamov界、l1解码、RIP（restricted isometry property）条件和Walsh-Hadamard变换等.     

非线性特征值问题平移幂法的不动点分析

平移对称高阶幂法在求解源自玻色-爱因斯坦凝聚态的非线性特征值问题方面,不仅具有较高的计算效率,而且具有点列收敛性.针对此算法进行不动点分析,区分了使用平移对称高阶幂法可以求得的特征对类型.     

Nonlinear vibration analysis of double-walled carbon nanotubes based on nonlocal elasticity theory

The nonlinear free vibration of double-walled carbon nanotubes based on the nonlocal elasticity theory is studied in this paper. The nonlinear equations of motion of the double-walled carbon nanotubes are derived by using Euler beam theory and Hamilton principle, with considering the von Kármán type geometric nonlinearity and the nonlinear van der Waals forces. The surrounding elastic medium is formulated as the Winkler model. The harmonic balance method and Davidon–Fletcher–Powell method are utilized for the analysis and simulation of the nonlinear vibration. The simulation results show that the nonlocal parameter, aspect ratio and surrounding elastic medium play more important roles in the nonlinear noncoaxial vibration than those in the coaxial vibration of the double-walled carbon nanotubes. The noncoaxial vibration amplitudes of only considering nonlinear van der Waals forces are larger than those of considering both geometric nonlinearity and nonlinear van der Waals forces.     

Nonsmooth analysis on smooth manifolds

We study infinitesimal properties of nonsmooth (nondifferentiable) functions on smooth manifolds. The eigenvalue function of a matrix on the manifold of symmetric matrices gives a natural example of such a nonsmooth function.

A subdifferential calculus for lower semicontinuous functions is developed here for studying constrained optimization problems, nonclassical problems of calculus of variations, and generalized solutions of first-order partial differential equations on manifolds. We also establish criteria for monotonicity and invariance of functions and sets with respect to solutions of differential inclusions.

     

Free vibration and buckling analysis of plates using B-spline wavelet on the interval Mindlin element

A finite element method (FEM) of B-spline wavelet on the interval (BSWI) is used in this paper to solve the free vibration and buckling problems of plates based on Reissner–Mindlin theory. By aid of the high accuracy of B-spline functions approximation for structural analysis, the proposed method could obtain a fast convergence and a satisfying numerical accuracy with fewer degrees of freedoms (DOF). The numerical examples demonstrate that the present BSWI method achieves the high accuracy compared to the exact solution and others existing approaches in the literatures. The BSWI finite element has potential to be used as a numerical method in analysis and design.     

Laplace Operators on Fractal Lattices with Random Blow-Ups

Starting from a finitely ramified self-similar set X we can construct an unbounded set X by blowing-up the initial set X. We consider random blow-ups and prove elementary properties of the spectrum of the natural Laplace operator on X (and on the associated lattice). We prove that the spectral type of the operator is almost surely deterministic with the blow-up and that the spectrum coincides with the support of the density of states almost surely (actually, our result is more precise). We also prove that if the density of states is completely created by the so-called Neuman–Dirichlet eigenvalues, then almost surely the spectrum is pure point with compactly supported eigenfunctions.