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.     

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".     

A SPECTRAL METHOD FOR A WEAKLY SINGULAR VOLTERRA INTEGRO-DIFFERENTIAL EQUATION WITH PANTOGRAPH DELAY

In this paper,a Jacobi-collocation spectral method is developed for a Volterraintegro-differential equation with delay,which contains a weakly singular kernel.We use a function transformation and a variable transformation to change the equation into a new Volterra integral equation defined on the standard interval [-1,1],so that the Jacobi orthogonal polynomial theory can be applied conveniently.In order to obtain high order accuracy for the approximation,the integral term in the resulting equat...     

Solitary Wave Solutions of Some Nonlinear Physical Models Using Riccati Equation Approach

Riccati equation approach is used to look for exact travelling wave solutions of some nonlinear physical models.Solitary wave solutions are established for the modified KdV equation,the Boussinesq equation and the Zakharov-Kuznetsov equation.New generalized solitary wave solutions with some free parameters are derived.The obtained solutions,which includes some previously known solitary wave solutions and some new ones,are expressed by a composition of Riccati differential equation solutions followed by a polynomial.The employed approach,which is straightforward and concise,is expected to be further employed in obtaining new solitary wave solutions for nonlinear physical problems.     

带振荡核奇异积分算子交换子在加权Morrey空间中的有界性质

本文研究Morrey空间中的交换子有界性的问题.利用John-Nirenberg不等式等工具建立带振荡核的奇异积分算子与BMO函数生成交换子在加权Morrey空间中的加权估计.     

Weighted Estimates for Commutators of Strongly Singular Calderón–Zygmund Operators

In this paper,the authors establish the boundedness of commutators generated by strongly singular Calderón–Zygmund operators and weighted BMO functions on weighted Herz-type Hardy spaces.Moreover,the corresponding results for commutators generated by strongly singular Calderón–Zygmund operators and weighted Lipschitz functions can also be obtained.     

Multiple weighted estimates for commutators of multilinear fractional integral operators

Multilinear commutators and iterated commutators of multilinear fractional integral operators with BMO functions are studied. Both strong type and weak type endpoint weighted estimates involving the multiple weights for such operators are established and the weak type endpoint results are sharp in some senses. In particular, we extend the results given by Cruz-Uribe and Fiorenza in 2003 and 2007 to the multilinear setting. Moreover, we modify the weak type of endpoint weighted estimates and improve the strong type of weighted norm inequalities on the multilinear commutators given by Chen and Xue in 2010 and 2011.     

单边奇异积分交换子的加权变差不等式

本文主要研究了单边Calder\''{o}n-Zygmund奇异积分交换子的加权变差不等式,即建立了单边Calder\''{o}n-Zygmund奇异积分与Lipschitz函数生成的交换子的加权变差不等式, 同时,利用权的外推法得到了该交换子在Triebel-Lizorkin空间上的加权变差不等式.     

带非光滑核的多线性奇异积分算子的极大交换子

本文研究了具有非光滑核的m-线性Calderon-Zygmund算子的极大交换子的Cotlar不等式,建立了上述m-线性Calderon-Zygmund算子的交换子和极大交换子的加权不等式.     

单边Triebel-Lizorkin空间及其应用

引进了一类单边Tiebel-Lizorkin空间.在单边的意义下,利用权的外推法,得到了由CalderónZygmund奇异积分算子与Lipschjtz函数生成的交换子从加权Lebesgue空间到加权Triebel-Lizorkin 空间的有界性质.同时研究了单边分数次积分算子交换子的有关性质.     

Commutators of maximal functions on spaces of homogeneous type and their weighted,local versions

We obtain the characterizations of commutators of several versions of maximal functions on spaces of homogeneous type. In addition, with the aid of interpolation theory, we provide weighted version of the commutator theorems by establishing new characterizations of the weighted BMO space. Finally, a concrete example shows that the local version of commutators also has an independent interest.     

Compactness for the commutators of multilinear singular integral operators with non-smooth kernels

In this paper, the behavior for commutators of a class of bilinear singular integral operators associated with non-smooth kernels on the product of weighted Lebesgue spaces is considered. By some new maximal functions to control the commutators of bilinear singular integral operators and CMO(Rn) functions, compactness for the commutators is proved.     

Oscillation and variation inequalities for singular integrals and commutators on weighted Morrey spaces

This paper is devoted to investigating the bounded behaviors of the oscillation and variation operators for Calderón-Zygmund singular integrals and the corresponding commutators on the weighted Morrey spaces. We establish several criterions of boundedness, which are applied to obtain the corresponding bounds for the oscillation and variation operators of Hilbert transform, Hermitian Riesz transform and their commutators with BMO functions, or Lipschitz functions on weighted Morrey spaces.     

强奇异积分算子的多线性交换子的Sharp极大函数估计和连续性

本文对由强奇异积分算子和加权Lipschitz函数生成的多线性交换子证明了其sharp极大函数估计,作为应用,得到了该多线性交换子的连续性.