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.     

ON GROWTH FACTORS OF THE LU FACTORIZATION

In this article, we derive upper bounds of different growth factors for the LU factorization, which are dominated by A11(k)-1A12(k),A21(k)A11(k)-1, where A11(k), A12(k), A21(k), A22(k) are sub-matrices of A. We also derive upper bounds of growth factors for the Cholesky factorization. Numerical examples are presented to verify our findings.     

Componentwise error analysis for the block LU factorization of totally nonnegative matrices

For the first time, perturbation bounds including componentwise perturbation bounds for the block LU factorization have been provided by Dopico and Molera (2005) [5]. In this paper, componentwise error analysis is presented for computing the block LU factorization of nonsingular totally nonnegative matrices. We present a componentwise bound on the equivalent perturbation for the computed block LU factorization. Consequently, combining with the componentwise perturbation results we derive componentwise forward error bounds for the computed block factors.     

On growth factors of the modified Gram–Schmidt algorithm

Growth factors play a central role in studying the stability properties and roundoff estimates of matrix factorizations; therefore, they have attracted many numerical analysts to study upper bounds of these growth factors. In this article, we derive several upper bounds of row‐wise growth factors of the modified Gram–Schmidt (MGS) algorithm to solve the least squares (LS) problem and the weighted LS problem. We also extend the analysis to the MGS‐like algorithm to solve the constrained LS problem. Copyright © 2008 John Wiley & Sons, Ltd.     

NOTE ON REGULAR D—OPTIMAL MATRICES

Let A be a j x d (0,1) matrix. It is known that if j = 2k - 1 is odd, then det(AAT) ≤ (j+1)((j+1)d/4j)j; if j is even, then det(AAT) ≤ (j+1)((j+2)d/4(j+1))j. A is called a regular D-optimal matrix if it satisfies the equality of the above bounds. In this note, it is proved that if j = 2k - 1 is odd, then A is a regular D-optimal matrix if and only if A is the adjacent matrix of a (2k - 1, k, (j + l)d/4j)-BIBD; if j = 2k is even, then A is a regular D-optimal matrix if and only if A can be obtained from the adjacent matrix B of a (2k + 1,k + 1,(j + 2)d/4(j +1))-BIBD by deleting any one row from B. Three 21 x 42 regular D-optimal matrices, which were unknown in [11], are also provided.     

Error bounds for Newton's iterates derived from the Kantorovich theorem

Summary In this paper, it is shown that the upper and lower bounds of the errors in the Newton iterates recently obtained by Potra-Pták [11] and Miel [7], with the use of nondiscrete induction and majorizing sequence, respectively, follow immediately from the Kantorovich theorem and the Kantorovich recurrence relations. It is also shown that the upper and lower bounds of Miel are finer than those of Potra-Pták.Sponsored by the United States Army under Contract No. DAAG29-80-C-0041     

关于几类图的L(2,1)标号问题

图G的L( 2 ,1 )标号是一个从顶点集V(G)到非负整数集的函数f(x) ,使得若d(x ,y) =1 ,则｜f(x) -f(y) ｜≥ 2 ;若d(x ,y) =2 ,则｜f(x) -f(y) ｜≥ 1 .图G的L( 2 ,1 ) 标号数λ(G)是使得G有max{f(v) ∶v∈V(G) }=k的L( 2 ,1 )标号中的最小数k .Griggs和Yeh猜想对最大度为Δ的一般图G ,有λ(G) ≤Δ2 .本文给出了Kneser图 ,Mycieklski图 ,Descartes图 ,Halin图的λ值的上界 ,并证明了上述猜想对以上几类图成立     

Analysis of transient queues with semidefinite optimization

We derive upper bounds on the tail distribution of the transient waiting time in the GI/GI/1 queue, given a truncated sequence of the moments of the service time and that of the interarrival time. Our upper bound is given as the objective value of the optimal solution to a semidefinite program (SDP) and can be calculated numerically by solving the SDP. We also derive the upper bounds in closed form for the case when only the first two moments of the service time and those of the interarrival time are given. The upper bounds in closed form are constructed by formulating the dual problem associated with the SDP. Specifically, we obtain the objective value of a feasible solution of the dual problem in closed from, which turns out to be the upper bound that we derive. In addition, we study bounds on the maximum waiting time in the first busy period.     

Bounds and Combinatorial Structure of (k,n) Multi-Receiver A-Codes

In the model of(k,n) multi-receiver authentication codes ( A-codes),a transmitter broadcasts a message m to nreceivers in such a way that not only an outside opponent butalso any k-1 receivers cannot cheat any other receiver.In this paper, we derive lower bounds on the cheating probabilitiesand the sizes of keys of (k,n) multi-receiver A-codes.The scheme proposed by Desmedt, Frankel and Yung meets all ourbounds with equalities. This means that our bounds are tightand their scheme is optimum. We further show a combinatorialstructure of optimum (k,n) multi-receiver A-codes.A notion of TWOOAs is introduced. A TWOOA is a pair of orthogonalarrays which satisfy a certain condition. We then prove thatan optimum (k,n) multi-receiver A-codeis equivalent to a TWOOA.     

关于图的L(3,2,1)-标号问题

图G的L( 2 ,1 )标号是一个从顶点集V(G)到非负整数集的函数f(x) .使得若d(x ,y) =1 .则｜f(x) -f(y) ｜≥ 2 ;若d(x ,y) =2 ,则｜f(x) -f(y)｜≥ 1 .图G的L( 2 ,1 )标号数λ(G)是使得G有max{f(v) ∶v∈V(G) }=k的L( 2 ,1 )标号中的最小数k .本文将L( 2 ,1 ) 标号问题推广到更一般的情形即L( 3,2 ,1 ) 标号问题 .我们首先定义了图G的顶点 3 着色及图的 3 色数 χ3 (G)等有关概念 ,并推导出 3 色数 χ3 (G)的上界 ;然后根据 χ3 (G)与λ3 (G)的关系 ,得出了对一般图G ,有λ3 (G) ≤ 3maxH Gδ(H) (Δ2 -Δ 1 )这一一般关系式 ;最后证明了对一般平面图G ,有λ3 (G)≤ 1 5(Δ2 -Δ 1 ) ,并得出了其它几类平面图的λ3 (G)的上界 .     

关于（k,h）-Fibonacci和（k,h）-Lucas数的置换因子循环矩阵的谱范数

给出了置换因子循环矩阵A=Percirc P（F_0^（k,h）,F_1^（k,h）,＊＊＊,F_n-1^（k,h）和B=Percirc P（L_0^（k,h）,L_1^（k,h）,＊＊＊,L_n-1^（k,h）的谱范数的上界与下界,得到了矩阵A与B的Kronecker积与Hadamard积的谱范数的一些界．