Implicit Shape Reconstruction of Unorganized Points Using PDE-Based Deformable 3D Manifolds

<正>In this work we consider the problem of shape reconstruction from an unorganized data set which has many important applications in medical imaging,scientific computing,reverse engineering and geometric modelling.The reconstructed surface is obtained by continuously deforming an initial surface following the Partial Differential Equation(PDE)-based diffusion model derived by a minimal volume-like variational formulation.The evolution is driven both by the distance from the data set and by the curvature analytically computed by it.The distance function is computed by implicit local interpolants defined in terms of radial basis functions.Space discretization of the PDE model is obtained by finite co-volume schemes and semi-implicit approach is used in time/scale.The use of a level set method for the numerical computation of the surface reconstruction allows us to handle complex geometry and even changing topology, without the need of user-interaction.Numerical examples demonstrate the ability of the proposed method to produce high quality reconstructions.Moreover,we show the effectiveness of the new approach to solve hole filling problems and Boolean operations between different data sets.     

Multivariate quasi-tight framelets with high balancing orders derived from any compactly supported refinable vector functions

Generalizing wavelets by adding desired redundancy and flexibility,framelets(i.e.,wavelet frames)are of interest and importance in many applications such as image processing and numerical algorithms.Several key properties of framelets are high vanishing moments for sparse multiscale representation,fast framelet transforms for numerical efficiency,and redundancy for robustness.However,it is a challenging problem to study and construct multivariate nonseparable framelets,mainly due to their intrinsic connections to factorization and syzygy modules of multivariate polynomial matrices.Moreover,all the known multivariate tight framelets derived from spline refinable scalar functions have only one vanishing moment,and framelets derived from refinable vector functions are barely studied yet in the literature.In this paper,we circumvent the above difficulties through the approach of quasi-tight framelets,which behave almost identically to tight framelets.Employing the popular oblique extension principle(OEP),from an arbitrary compactly supported M-refinable vector functionφwith multiplicity greater than one,we prove that we can always derive fromφa compactly supported multivariate quasi-tight framelet such that:(i)all the framelet generators have the highest possible order of vanishing moments;(ii)its associated fast framelet transform has the highest balancing order and is compact.For a refinable scalar functionφ(i.e.,its multiplicity is one),the above item(ii)often cannot be achieved intrinsically but we show that we can always construct a compactly supported OEP-based multivariate quasi-tight framelet derived fromφsatisfying item(i).We point out that constructing OEP-based quasi-tight framelets is closely related to the generalized spectral factorization of Hermitian trigonometric polynomial matrices.Our proof is critically built on a newly developed result on the normal form of a matrix-valued filter,which is of interest and importance in itself for greatly facilitating the study of refinable vector functions and multiwavelets/multiframelets.This paper provides a comprehensive investigation on OEP-based multivariate quasi-tight multiframelets and their associated framelet transforms with high balancing orders.This deepens our theoretical understanding of multivariate quasi-tight multiframelets and their associated fast multiframelet transforms.     

Singular integral operators on product domains along twisted surfaces

We introduce a class of singular integral operators on product domains along twisted surfaces.We prove that the operators are bounded on L^p provided that the kernels satisfy weak conditions.     

Poisson Stable Solutions for Stochastic Differential Equations with Lévy Noise

In this paper,we use a unified framework to study Poisson stable(including stationary,periodic,quasi-periodic,almost periodic,almost automorphic,Birkhoff recurrent,almost recurrent in the sense of Bebutov,Levitan almost periodic,pseudo-periodic,pseudo-recurrent and Poisson stable)solutions for semilinear stochastic differential equations driven by infinite dimensional L′evy noise with large jumps.Under suitable conditions on drift,diffusion and jump coefficients,we prove that there exist solutions which inherit the Poisson stability of coefficients.Further we show that these solutions are globally asymptotically stable in square-mean sense.Finally,we illustrate our theoretical results by several examples.     

A Finite Difference Scheme for Blow-Up Solutions of Nonlinear Wave Equations

<正>We consider a finite difference scheme for a nonlinear wave equation,whose solutions may lose their smoothness in finite time,i.e.,blow up in finite time.In order to numerically reproduce blow-up solutions,we propose a rule for a time-stepping, which is a variant of what was successfully used in the case of nonlinear parabolic equations.A numerical blow-up time is defined and is proved to converge,under a certain hypothesis,to the real blow-up time as the grid size tends to zero.     

手机套餐问题的一个数学模型

随着信息时代的到来,手机在人们日常工作、社交、经营等社会活动中的作用越来越重要.近年来我国通信业务量飞速增长,手机的功劳更是功不可没.手机资费问题也越来越受到人们的关注,并且对原有的各种资费方案越来越质疑.2007年1月以来上海、北京、广东等地相继推出的手机"套餐"琳琅满目,让人眼花缭乱,人们不能理性分辨手机"套餐"究竟优惠在何处.……     

Kobayashi’s and Teichmuller’s Metrics and Bers Complex Manifold Structure on Circle Diffeomorphisms

Given a modulus of continuity ω,we consider the Teichmuller space TC1+ω as the space of all orientation-preserving circle diffeomorphisms whose derivatives are ω-continuous functions modulo the space of Mobius transformations preserving the unit disk.We study several distortion properties for diffeomorphisms and quasisymmetric homeomorphisms.Using these distortion properties,we give the Bers complex manifold structure on the Teichm(u| ")ller space TC^1+H as the union of over all0 <α≤1,which turns out to be the largest space in the Teichmuller space of C1 orientation-preserving circle diffeomorphisms on which we can assign such a structure.Furthermore,we prove that with the Bers complex manifold structure on TC^1+H ,Kobayashi’s metric and Teichmuller’s metric coincide.     

Difference of a Hauptmodul for Γ0(N ) and certain Gross-Zagier type CM value formulas

In this work, we show that the difference of a Hauptmodul for a genus zero group Γ₀(N) as a modular function on Y₀(N) × Y₀(N) is a Borcherds lift of type(2, 2). As applications, we derive the monster denominator formula like product expansions for these modular functions and certain Gross-Zagier type CM value formulas.     

Populations with individual variation in dispersal in heterogeneous environments: Dynamics and competition with simply diffusing populations

We consider a model for a population in a heterogeneous environment, with logistic-type local population dynamics, under the assumption that individuals can switch between two different nonzero rates of diffusion. Such switching behavior has been observed in some natural systems. We study how environmental heterogeneity and the rates of switching and diffusion affect the persistence of the population. The reactiondiffusion systems in the models can be cooperative at some population densities and competitive at others. The results extend our previous work on similar models in homogeneous environments. We also consider competition between two populations that are ecologically identical, but where one population diffuses at a fixed rate and the other switches between two different diffusion rates. The motivation for that is to gain insight into when switching might be advantageous versus diffusing at a fixed rate. This is a variation on the classical results for ecologically identical competitors with differing fixed diffusion rates, where it is well known that "the slower diffuser wins".     

KAM Theory for Partial Differential Equations

In the last years much progress has been achieved in KAM theory concerning bifurcation of quasi-periodic solutions of Hamiltonian or reversible partial differential equations.We provide an overview of the state of the art in this field.     

The generalized hierarchical product of graphs

A generalization of both the hierarchical product and the Cartesian product of graphs is introduced and some of its properties are studied. We call it the generalized hierarchical product. In fact, the obtained graphs turn out to be subgraphs of the Cartesian product of the corresponding factors. Thus, some well-known properties of this product, such as a good connectivity, reduced mean distance, radius and diameter, simple routing algorithms and some optimal communication protocols, are inherited by the generalized hierarchical product. Besides some of these properties, in this paper we study the spectrum, the existence of Hamiltonian cycles, the chromatic number and index, and the connectivity of the generalized hierarchical product.     

The q-deformed harmonic oscillator, coherent states, and the uncertainty relation

For a q-deformed harmonic oscillator, we find explicit coordinate representations of the creation and annihilation operators, eigenfunctions, and coherent states (the last being defined as eigenstates of the annihilation operator). We calculate the product of the “coordinate-momentum” uncertainties in q-oscillator eigenstates and in coherent states. For the oscillator, this product is minimum in the ground state and equals 1/2, as in the standard quantum mechanics. For coherent states, the q-deformation results in a violation of the standard uncertainty relation; the product of the coordinate-and momentum-operator uncertainties is always less than 1/2. States with the minimum uncertainty, which tends to zero, correspond to the values of λ near the convergence radius of the q-exponential. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 147, No. 2, pp. 315–322, May, 2006.     

Gaussian Versus Optimal Integration of Analytic Functions

We consider error estimates for optimal and Gaussian quadrature formulas if the integrand is analytic and bounded in a certain complex region. First, a simple technique for the derivation of lower bounds for the optimal error constants is presented. This method is applied to Szeg?-type weight functions and ellipses as regions of analyticity. In this situation, the error constants for the Gaussian formulas are close to the obtained lower bounds, which proves the quality of the Gaussian formulas and also of the lower bounds. In the sequel, different regions of analyticity are investigated. It turns out that almost exclusively for ellipses, the Gaussian formulas are near-optimal. For classes of simply connected regions of analyticity, which are additionally symmetric to the real axis, the asymptotic of the worst ratio between the error constants of the Gaussian formulas and the optimal error constants is calculated. As a by-product, we prove explicit lower bounds for the Christoffel-function for the constant weight function and arguments outside the interval of integration. September 7, 1995. Date revised: October 25, 1996.     

The permanence and extinction of a nonlinear growth rate single-species non-autonomous dispersal models with time delays

In this paper, we consider the effect of diffusion on the permanence and extinction of a non-autonomous nonlinear growth rate single-species dispersal model with time delays. Firstly, the sufficient conditions of the permanence and extinction of the species are established, which shows if the growth rate and dispersal coefficients is suitable, the species is permanent, on the contrary, it is extinction. Secondly, an interesting result is established, that is, if only the species in some patches even in one patch is permanent, then it is also permanent in other patches. Finally, some examples together with their numerical simulations show the feasibility of our main results.     

Rolling-ball method for estimating the boundary of the support of a point-process intensity

We suggest a generalisation of the convex-hull method, or ‘DEA’ approach, for estimating the boundary or frontier of the support of a point cloud. Figuratively, our method involves rolling a ball around the cloud, and using the equilibrium positions of the ball to define an estimator of the envelope of the point cloud. Constructively, we use these ideas to remove lines from a triangulation of the points, and thereby compute a generalised form of a convex hull. The radius of the ball acts as a smoothing parameter, with the convex-hull estimator being obtained by taking the radius to be infinite. Unlike the convex-hull approach, however, our method applies to quite general frontiers, which may be neither convex nor concave. It brings to these contexts the attractive features of the convex hull: simplicity of concept, rotation-invariance, and ready extension to higher dimensions. It admits bias corrections, which we describe and illustrate through implementation.     

Is prediction possible? Chaotic behavior of Multiple Equilibria Regulation Model in cellular automata topology

In this article, we present the Multiple Equilibria Regulation (MER) Model in cellular automata topology. As argued in previous explorations of the model, for certain parameter values, the behavior of the system exhibits transient chaos (namely, the system is unpredictable but ends in a final steady state). In order to approach empirical reality, we introduce a cellular automata topology. Examining the outcome of the simulations leads us to conclude that for certain parameter values tested, the system yields chaotic behavior. Thus, cellular automata contribution has proven crucial, because the introduced topology converts the behavior of the system from transient chaos to “pure” chaos, i.e., the system is not only unpredictable on the long run but, in addition, it will never rest in a final steady state. According to these findings, authors argue the theoretical hypothesis that the urge for “prediction” in social sciences should be reconsidered in terms of “predictability horizon”. © 2004 Wiley Periodicals, Inc. Complexity 10: 23–36, 2004     

The limiting state of a rigidly fixed nonlinearly elastic multilayer rod

The loss of the load-carrying capacity of a nonlinearly elastic multilayer rod is investigated. The rod, whose layers have various thickness and are made of different materials, is rigidly fixed at both its ends. Rigid contact conditions between the layers are assumed. The problem posed is solved by using the variational method of mixed type in combination with the Rayleigh-Ritz method. The initial analysis is reduced to the solution of the Cauchy problem for a nonlinear ordinary differential equation solved for the first derivative. As the initial condition, the maximum initial eccentricity of the rod is assumed. In the case of zero eccentricity, the Shanley critical force for an axially compressed rod is determined. For a three-layer rod whose outer layers have equal thickness and are made of the same material, numerically, for various degrees of nonlinearity, the effect of physicomechanical and geometric parameters on the critical load of buckling instability is determined. It is found that, by matching the heterogeneity of the rod, it is possible to raise its load-carrying capacity. __________ Translated from Mekhanika Kompozitnykh Materialov, Vol. 42, No. 3, pp. 347–360, May–June, 2006.     

基于线性代数的大规模快速量纲分析算法及其在爆炸与冲击工程研究中的应用

量纲分析是科学研究,特别是工程应用中非常重要的一个理论分析工具.从E.Buckingham提出Π定理开始算起,量纲分析已有一百多年历史,其基本理论和方法已经非常成熟,在各个领域也取得了显著的成果并且仍然有着广泛的应用.然而,随着研究的深入,面对的问题越来越复杂和细致,人们越来越关注在传统量纲分析中忽略掉的一些所谓次要因素的影响,因此涉及的物理量变得越来越多,导致按传统的量纲分析方法处理时常常显得非常繁琐甚至困难.本文从线性代数的观点出发,将量纲分析转换为线性空间问题,通过矩阵运算,完成量纲分析的关键过程.给出了量纲分析对应的线性代数问题的基本定理,并基于这些定理建立了程序化的量纲分析算法,将原本复杂的量纲分析问题转化为借助计算机代数系统能够快速方便解决的矩阵运算问题.最后,结合笔者多年的工作经历,给出了上述方法在爆炸与冲击工程研究领域中的若干应用实例,详细表述了具体操作步骤,验证了算法的优越性.