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.     

Some Theorems for Hypersurface of Randers Spaces

In this paper, we consider the hypersurfaces of Randers space with constant flag curvature. (1) Let (Mⁿ⁺¹, F) be a Randers-Minkowski space. If (Mⁿ, F) is a hypersurface of (Mⁿ⁺¹, F) with constant flag curvature K=1, then we can prove that M is Riemannian. (2) Let (Mⁿ⁺¹, F) be a Randers space with constant flag curvature. Assume (M, F) is a compact hypersurface of (Mⁿ⁺¹, F) with constant mean curvature|H|. Then a pinching theorem is established, which generalizes the result of[Proc. Amer. Math. Soc., 120, 1223-1229 (1994)] from the Riemannian case to the Randers space.     

On intersections of independent anisotropic Gaussian random fields

Let XH = {XH(s),s ∈RN1} and X K = {XK(t),t ∈R N2} be two independent anisotropic Gaussian random fields with values in R d with indices H =(H1,...,HN1) ∈(0,1)N1,K =(K1,...,KN2) ∈(0,1) N2,respectively.Existence of intersections of the sample paths of X H and X K is studied.More generally,let E1■RN1,E2■RN2 and FRd be Borel sets.A necessary condition and a sufficient condition for P{(XH(E1)∩XK(E2))∩F≠Ф}>0 in terms of the Bessel-Riesz type capacity and Hausdorff measure of E1×E2×F in the metric space(RN1+N2+d,) are proved,where  is a metric defined in terms of H and K.These results are applicable to solutions of stochastic heat equations driven by space-time Gaussian noise and fractional Brownian sheets.     

Maximal inequalities for a continuous semimartingale

Let X =( X t ) t S 0 be a continuous semimartingale given by d X t = f ( t ) w ( X t )d d M ¢ t + f ( t ) σ ( X t )d M t , X 0 =0, where M =( M t , F t ) t S 0 is a continuous local martingale starting at zero with quadratic variation d M ¢ and f ( t ) is a positive, bounded continuous function on [0, X ), and w , σ both are continuous on R and σ ( x )>0 if x p 0. Denote X 𝜏 * =sup 0 h t h 𝜏 | X t | and J t = Z 0 t f ( s ) } ( X s )d d M ¢ s ( t S 0) for a nonnegative continuous function } . If w ( x ) h 0 ( x S 0) and K 1 | x | n σ 2 ( x ) h | w ( x )| h K 2 | x | n σ 2 ( x ) ( x ] R , n >0) with two fixed constants K 2 S K 1 >0, then under suitable conditions for } we show that the maximal inequalities c p , n log 1 n +1 (1+ J 𝜏 ) p h Á X 𝜏 * Á p h C p , n log 1 n +1 (1+ J 𝜏 ) p (0< p < n +1) hold for all stopping times 𝜏 .     

一类Kirchhoff型系统解的存在性

In this paper,we are interested in the existence of positive solutions for the Kirchhoff type problems{-(a_1 + b_1M_1(∫_?|▽u|~pdx))△_(_pu) = λf(u,v),in ?,-(a_2 + b_2M_2(∫?|▽v|~qdx))△_(_qv) = λg(u,v),in ?,u = v = 0,on ??,where 1 p,q N,M i:R_0~+→ R~+(i = 1,2) are continuous and increasing functions.λ is a parameter,f,g ∈ C~1((0,∞) ×(0,∞)) × C([0,∞) × [0,∞)) are monotone functions such that f_s,f_t,g_s,g_t ≥ 0,and f(0,0) 0,g(0,0) 0(semipositone).Our proof is based on the sub-and super-solutions techniques.     

伪压缩族公共不动点的复合隐格式迭代逼近问题

设H是一实Hillber空间,K是H之一非空间凸子集,设{Ti}Ni=1是N个Lipschitz伪压缩映象使得F=∩Ni=1F(Ti)≠0,其中F(Ti)={x∈K:Tix=x}并且{αn}n∞=1,{βn}∞n=1[0,1]是满足如下条件的实序列(i)∑∞n=1(1-αn)2= ∞;(ii)limn→∞(1-αn)=0;(iii)∑∞n=1(1-βn)< ∞;(iv)(1-αn)L2<1,n1;(v)αn(1-βn)2 αn[βn L(1-βn)]2<1,其中L1是{Ti}iN=1的公共Lipschitz常数,对于x0∈K,设{xn}n∞=1是由下列定义的复合隐格式迭代xn=αnxn-1 (1-αn)Tnyn,yn=βnxn (1-βn)Tnxn,其中Tn=TnmodN,则(i)limn→∞‖xn-p‖存在,对于所有的p∈F;(ii)limn→∞d(xn,f)存在,其中d(xn,F)=infp∈F‖xn-p‖;(iii)liminfn→∞‖xn-Tnxn‖=0.本文的结果推广并且改进H-K.Xu和R.G.Ori在2001年的结果和Osilike在2004年的结果,并且在这篇文章中,主要的证明方法也不同与H-K.Xu和Osilike的方法.     

关于单叶函数系数之—基本引理及其应用

<正> 设函数 f(z)=z+a_2Z~2+…在单位圆|z|<1上是正则的单叶的.这种函数的全体形成一族 S.S 中满足条件|f(z)|1上是单叶的,除开极点ζ=∞是正则的.这种函数的全体形成一族∑.∑中满足条件|F(ζ)|>R的函     

代數拓撲學中(μ,△,γ)-系統的正則同模論Ⅰ

<正> 設K與L為拓撲空間,又設f:K→L為連續映像.由f導出了準同模對應f~n:H~n(L,G)→H~n(K,G),n=1,2,…,其中H~n(L,G),H~n(K,G)表示上同調羣,而G表示係數環或域以γ_p~n(K)或者     

Galois环上本原序列压缩映射的单性

设Ω是 Ｇａｌｏｉｓ环 ＧＲ（２~ｄ,ｒ）的 Ｔｅｉｃｈｍｕｌｌｅｒ代表集,则 ＧＲ（２~ｄ,ｒ）上每条序列ａ有唯一的权位分解,　其中ａ-ｉ是Ω上序列,同时也可自然视为有限域Ｆ-（２~ｒ）,上序列．设ｆ（ｘ）是环 ＧＲ（２~ｄ,ｒ）上强本原多项式,Ｇ（ｆ（ｘ））表示 ＧＲ（２~ｄ,ｒ）上以ｆ（ｘ）为特征多项式的序列的全体,是Ｆ-（２~ｒ）上一类ｄ－１元多项式, 　本文证明了压缩映射是单射,即对 ａ＝ ｂ当且仅当对所有 ａ,ｂ ∈ Ｇ（ｆ（ｘ））．     

直径为5的图的2-导出匹配划分和2-导出匹配覆盖问题的NP-完全性

给定一个简单图G和正整数κ,具有完美匹配的图G的κ-导出匹配划分是对顶点集V(C)的一个κ-划分(V1,V2,...,Vκ),其中对每一个i(1≤i≤κ),由Vi导出的G的子图G[Vi]是1-正则的.κ-导出匹配划分问题是指对给定的图G,判定G是否存在一个κ-导出匹配划分.令M1,M2…,Mκ为图G的κ个导出匹配,如果V(M1)UV(M2)∪...∪V(Mκ)=V(G),则我们称{M1,M2,...,Mκ}是G的κ-导出匹配覆盖.κ-导出匹配覆盖问题是指对给定的图G,判定G是否存在κ-导出匹配覆盖.本文给出了Yang,Yuan和Dong所提出问题的解,证明了直径为5的图的导出匹配2一划分问题和导出匹配2-覆盖问题都是NP-完全的.     

复合形在欧氏空间中的同痕问题(Ⅰ)

<正> 作者曾经指出,一个空间的约化积这一概念对于非同伦性的拓扑问题颇为有用,并曾应用之以研究空间在一欧氏空间中的实现问题,局部实现问题,同痕与同位问题,以及其他一些有关问题(参阅例如[1],与该处文献).近来作者又曾证明对于一个有限复合形在一欧氏空间中的任两线性实现,从复合形的约化积可导出一些不变量来,它们之为0给出了两个实现线性同痕的必要条件,而在临界情形,即欧氏空间的维数等于复合形维数的两倍加1的情形,(复合形维数假定>1)这些必要条件又同时是充分的.本文及以下一