Nonlinear Subdivision Schemes on Irregular Meshes

The present article deals with convergence and smoothness analysis of geometric, nonlinear subdivision schemes in the presence of extraordinary points. We discuss when the existence of a proximity condition between a linear scheme and its nonlinear analogue implies convergence of the nonlinear scheme (for dense enough input data). Furthermore, we obtain C ¹ smoothness of the nonlinear limit function in the vicinity of an extraordinary point over Reif’s characteristic parametrization. The results apply to the geometric analogues of well-known subdivision schemes such as Doo–Sabin or Catmull–Clark schemes.     

三进制Loop细分曲面的C~1连续性分析

本文利用计算机代数系统,通过对三进制Loop细分算法的细分矩阵和特征映射的构造与分析,证明Loop给出的掩模设计能够保证细分曲面在奇异点是C1连续的,还给出细分矩阵的次优势特征值的一个取值范围,在此范围内利用三进制Loop细分算法生成的细分曲面都是C1连续的.最后给出一种三进制Loop细分算法的新的边点掩模设计方法,在保证细分曲面是C1连续的前提下,比Loop给出的计算公式更简单,细分算法在奇异点附近收敛更快.     

Smoothness of Stationary Subdivision on Irregular Meshes

We derive necessary and sufficient conditions for tangent plane and C ^k -continuity of stationary subdivision schemes near extraordinary vertices. Our criteria generalize most previously known conditions. We introduce a new approach to analysis of subdivision surfaces based on the idea of the universal surface . Any subdivision surface can be locally represented as a projection of the universal surface, which is uniquely defined by the subdivision scheme. This approach provides us with a more intuitive geometric understanding of subdivision near extraordinary vertices. February 16, 1998. Date revised: January 27, 1999. Date accepted: April 2, 1999.     

Decompositions of trigonometric polynomials with applications to multivariate subdivision schemes

We study multivariate trigonometric polynomials satisfying the “sum-rule” conditions of a certain order. Based on the polyphase representation of these polynomials relative to a general dilation matrix, we develop a simple constructive method for a special type of decomposition of such polynomials. These decompositions are of interest in the analysis of convergence and smoothness of multivariate subdivision schemes associated with general dilation matrices. The approach presented in this paper leads directly to constructive algorithms, and is an alternative to the analysis of multivariate subdivision schemes in terms of the joint spectral radius of certain operators. Our convergence results apply to arbitrary dilation matrices, while the smoothness results are limited to two classes of dilation matrices.     

Estimates for Green's matrices of elliptic systems byL p theory

Summary We prove existence and optimal decay properties of a Green's matrix for elliptic systems of second order. The results follow from regularity theorems in weak Lebesgue spaces which can be obtained from the classicalL ^p theory using interpolation theorems. This article was processed by the author using the LATEX style filecljour1 from Springer-Verlag.     

Subdivision schemes with general dilation in the geometric and nonlinear setting

We establish results on convergence and smoothness of subdivision rules operating on manifold-valued data which are based on a general dilation matrix. In particular we cover irregular combinatorics. For the regular grid case results are not restricted to isotropic dilation matrices. The nature of the results is that intrinsic subdivision rules which operate on geometric data inherit smoothness properties of their linear counterparts.     

Exponentials Reproducing Subdivision Schemes

This paper is concerned with a family of nonstationary, interpolatory subdivision schemes that have the capability of reproducing functions in a finite-dimensional subspace of exponential polynomials. We give conditions for the existence and uniqueness of such schemes, and analyze their convergence and smoothness. It is shown that the refinement rules of an even-order exponentials reproducing scheme converge to the Dubuc—Deslauriers interpolatory scheme of the same order, and that both schemes have the same smoothness. Unlike the stationary case, the application of a nonstationary scheme requires the computation of a different rule for each refinement level. We show that the rules of an exponentials reproducing scheme can be efficiently derived by means of an auxiliary orthogonal scheme , using only linear operations. The orthogonal schemes are also very useful tools in fitting an appropriate space of exponential polynomials to a given data sequence.     

Transient and Stationary Distributions for the GI/G/k Queue with Lebesgue-Dominated Inter-Arrival Time Distribution

In this paper, the multi-server queue with general service time distribution and Lebesgue-dominated iid inter-arival times is analyzed. This is done by introducing auxiliary variables for the remaining service times and then examining the embedded Markov chain at arrival instants. The concept of piecewise-deterministic Markov processes is applied to model the inter-arrival behaviour. It turns out that the transition probability kernel of the embedded Markov chain at arrival instants has the form of a lower Hessenberg matrix and hence admits an operator–geometric stationary distribution. Thus it is shown that matrix–analytical methods can be extended to provide a modeling tool even for the general multi-server queue.     

Interpolatory wavelets for manifold-valued data

Geometric wavelet-like transforms for univariate and multivariate manifold-valued data can be constructed by means of nonlinear stationary subdivision rules which are intrinsic to the geometry under consideration. We show that in an appropriate vector bundle setting for a general class of interpolatory wavelet transforms, which applies to Riemannian geometry, Lie groups and other geometries, Hölder smoothness of functions is characterized by decay rates of their wavelet coefficients.     

A result analogous to G. Szego’s theorem

The paper proves a new result similar to the well known theorems of G. Szego and A. Kolmogorov on the best approximation by analytic polynomials in weighted L ^p spaces. Such results are essential in prediction theory for stationary processes. It is well known, that for one step prediction, the size of the best approximation is the same for all p. The paper proves that for two step prediction the best approximation sizes are different for p = 2 and p = ∞.     

Subdivision Schemes and Irregular Grids

The present paper deals with subdivision schemes associated with irregular grids. We first give a sufficient condition concerning the difference scheme to obtain convergence. This condition generalizes a necessary and sufficient condition for convergence known in the case of uniform and stationary schemes associated with a regular grid. Through this sufficient condition, convergence of a given subdivision scheme can be proved by comparison with another scheme. Indeed, when two schemes are equivalent in some sense, and when one satisfies the sufficient condition for convergence, the other also satisfies it and it therefore converges too. We also study the smoothness of the limit functions produced by a scheme which satisfies the sufficient condition. Finally, the results are applied to the study of Lagrange interpolating subdivision schemes of any degree, with respect to particular irregular grids.     

On the complexity of the approximate table representation of discrete analogs of functions of finite smoothness in the metric ofL P

For discrete analogs of classes of functions of finite smoothness, we study the quantity log Approx characterizing the minimal necessary length of tables that allow us to reconstruct functions from these classes with error not exceeding 1 in the metric of the spaceL ^p . Translated fromMatematicheskie Zametki, Vol. 64, No. 5, pp. 643–647, November, 1998. The author wishes to express his gratitude to his scientific adviser O. B. Lupanov for setting the problem and attention to the author's work. This research was supported by the Russian Foundation for Basic Research under grant No. 96-01-01068.     

Concerning Order of Convergence for Subdivision

We discuss order of convergence for subdivision algorithms, in the scalar-valued and the vector-valued case. In order to find the generic order, the usual definition of convergence order is extended, refering to a proper quasi interpolant operator whose representation on polynomial spaces can be constructively determined with recourse to properties of the subdivision mask. Assuming stability and smoothness of the limit functions, the approximation order of the quasi interpolant operator determines the order of convergence of subdivision.     

Lie groupoids as generalized atlases

Starting with some motivating examples (classical atlases for a manifold, space of leaves of a foliation, group orbits), we propose to view a Lie groupoid as a generalized atlas for the “virtual structure” of its orbit space, the equivalence between atlases being here the smooth Morita equivalence. This “structure” keeps memory of the isotropy groups and of the smoothness as well. To take the smoothness into account, we claim that we can go very far by retaining just a few formal properties of embeddings and surmersions, yielding a very polymorphous unifying theory. We suggest further developments.     

Finite Energy Solutions of Maxwell''s Equations and Constructive Hodge Decompositions on Nonsmooth Riemannian Manifolds

We consider two basic potential theoretic problems in Riemannian manifolds: Hodge decompositions and Maxwell's equations. Here we are concerned with smoothness and integrability assumptions. In the context of L^p forms in Lipschitz domains, we show that both are well posed provided that 2−<p<2+, for some >0, depending on the domain. Our approach is constructive (in the sense that we produce integral representation formulas for the solutions) and emphasizes the intimate connections between the two problems at hand. Applications to other related PDEs, such as boundary problems for the Hodge Dirac operator, are also presented.     

F*-Rings Are O*

O ^*-rings were introduced by Fuchs and recently characterized by Steinberg. A ring R is called O ^* if every partial order on R extends to a total order. We weaken the condition on the ordering of the ring by requiring that every partial order on R extends to an f-order. We call those rings F ^*-rings. We show that the two classes of rings coincide.     

Constructing a sequence of discrete Hessian matrices of an SC 1 function uniformly convergent to the generalized Hessian matrix

We construct a uniform approximation for generalized Hessian matrix of an SC ¹ function. Using the discrete gradient and the extended second order derivative, we define the discrete Hessian matrix. We construct a sequence of sets, where each set is composed of discrete Hessian matrices. We first show some new properties of SC ¹ functions. Then, we prove that for SC ¹ functions the sequence of the set of discrete Hessian matrices is uniformly convergent to the generalized Hessian matrix.      

Smoothness equivalence properties of univariate subdivision schemes and their projection analogues

We study the following modification of a linear subdivision scheme S: let M be a surface embedded in Euclidean space, and P a smooth projection mapping onto M. Then the P-projection analogue of S is defined as T := P ◦ S. As it turns out, the smoothness of the scheme T is always at least as high as the smoothness of the underlying scheme S or the smoothness of P minus 1, whichever is lower. To prove this we use the method of proximity as introduced by Wallner et al. (Constr Approx 24(3):289–318, 2006; Comput Aided Geom Design 22(7):593–622, 2005). While smoothness equivalence results are already available for interpolatory schemes S, this is the first result that confirms smoothness equivalence properties of arbitrary order for general non-interpolatory schemes.     

The smallness problem for -algebras

Akemann showed that any von Neumann algebra with a weak^* separable dual space has a faithful normal representation on a separable Hilbert space. He posed the question: If a C^*-algebra has a weak^* separable state space, must it have a faithful representation on a separable Hilbert space? Wright solved this question negatively and showed that a unital C^*-algebra has the weak^* separable state space if and only if it has a unital completely positive map, into a type I factor on a separable Hilbert space, whose restriction to the self-adjoint part induces an order isomorphism. He called such a C^*-algebra almost separably representable. We say that a unital C^*-algebra is small if it has a unital complete isometry into a type I factor on a separable Hilbert space. In this paper we show that a unital C^*-algebra is small if and only if the state spaces of all n by n matrix algebras over the C^*-algebra are weak^*-separable. It is natural to ask whether almost separably representable algebras are small or not. We settle this question positively for simple C^*-algebras but the general question remains open.     

Estimation of elastic parameters in beams and certain plates:H 1 regularization

The estimation of elastic parameters in beams and certain types of plates is discussed using anH ¹-regularization technique that easily accommodates pointwise constraints. The optimal coefficient is shown to enjoy more regularity than that assumed in the formulation of the problem. This additional smoothness is useful for analyzing the limit behavior of finite-dimensional problems. Numerical results are presented.This work was supported in part by the Air Force Office of Scientific Research, Grant AFOSR-84-0271.