Density of wavelet frames

Density conditions for wavelet systems with arbitrary sampling points to be frames are studied. We show that for a wavelet system generated by admissible functions with irregular affine lattices to be a frame, the sampling points must have a positive lower affine Beurling density. The same is true for wavelet systems with arbitrary sampling points and nice generating functions.     

Beurling Dimension of Gabor Pseudoframes for Affine Subspaces

Pseudoframes for subspaces have been recently introduced by Li and Ogawa as a tool to analyze lower dimensional data with arbitrary flexibility of both the analyzing and the dual sequence. In this paper we study Gabor pseudoframes for affine subspaces by focusing on geometrical properties of their associated sets of parameters. We first introduce a new notion of Beurling dimension for discrete subsets of ℝ ^d by employing a certain generalized Beurling density. We present several properties of Beurling dimension including a comparison with other notions of dimension showing, for instance, that our notion includes the mass dimension as a special case. Then we prove that Gabor pseudoframes for affine subspaces satisfy a certain Homogeneous Approximation Property, which implies invariance under time–frequency shifts of an approximation by elements from the pseudoframe. The main result of this paper is a classification of Gabor pseudoframes for affine subspaces by means of the Beurling dimension of their sets of parameters. This provides us, in particular, with a Nyquist dimension which separates sets of parameters of pseudoframes from those of non-pseudoframes and which links a fixed value to sets of parameters of pseudo-Riesz sequences. These results are even new for the special case of Gabor frames for an affine subspace.      

Σ-Bounded algebraic systems and universal functions. I

We introduce the concept of a Σ-bounded algebraic system and prove that if a system is Σ- bounded with respect to a subset A then in a hereditarily finite admissible set over this system there exists a universal Σ-function for the family of functions definable by Σ-formulas with parameters in A. We obtain a necessary and sufficient condition for the existence of a universal Σ-function in a hereditarily finite admissible set over a Σ-bounded algebraic system. We prove that every linear order is a Σ-bounded system and in a hereditarily finite admissible set over it there exists a universal Σ-function.     

Necessary conditions and sufficient conditions of irregular shearlet frames

In this paper, necessary conditions and sufficient conditions for the irregular shearlet systems to be frames are studied. We show that the irregular shearlet systems to possess upper frame bounds, the space‐scale‐shear parameters must be relatively separated. We prove that if the irregular shearlet systems possess the lower frame bound and the space‐scale‐shear parameters satisfy certain condition, then the lower shearlet density is strictly positive. We apply these results to systems consisting only of dilations, obtaining some new results relating density to the frame properties of these systems. We prove that for a feasible class of shearlet generators introduced by P. Kittipoom et al., each relatively separated sequence with sufficiently hight density will generate a frame. Explicit frame bounds are given. We also study the stability of shearlet frames and show that a frame generated by certain shearlet function remains a frame when the space‐scale‐shear parameters and the generating function undergo small perturbations. Explicit stability bounds are given. Using pseudo‐spline functions of type I and II, we construct a family of irregular shearlet frames consisting of compactly supported shearlets to illustrate our results.     

连续切波变换反演公式的级数表示

我们主要研究连续切波变换反演公式的级数表示.首先引入两类由切波变换反演公式定义的无穷级数和有限级数,并研究了由Kittipoom等人介绍的切波生成空间,得到这个切波生成空间的一些重要性质.其次利用这些结果显示:对于这个切波生成空间,当采样密度趋于无穷时由我们定义的无穷级数按L~2-范数收敛于重构函数;对于可允许函数空间,当采样密度趋于无穷时由我们定义的有限级数按L~2-范数收敛于重构函数.     

Sparse exponential systems: Completeness with estimates

According to A. Beurling and H. Landau, if an exponential system {e ^iλt }_λ∈Λ is a frame in L ² on a set S of positive measure, then Λ must satisfy a strong density condition. We replace the frame concept by a weaker condition and prove that if S is a finite union of segments then the result holds. However, for “generic” S, very sparse sequences Λ are admitted. Supported in part by the Israel Science Foundation.     

Optimal Quarantine Programmes for Controlling an Epidemic Spread

An optimal control problem is formulated with a simple epidemic model in which the control of the epidemic is effected by varying the scale of the quarantine program in a way which minimizes a discounted linear cost over an infinite horizon. An important function of the problem parameters is identified. It is shown that if this function has a value of less than or equal to one, then the optimal policy is not to quarantine at all. While if this functions assume a value in excess of one, then the optimal policy is not to quarantine at all if the initial fraction of infectives is sufficiently high; otherwise, it is optimal to have a full scale quarantine program. Slight modification in these policies are required for the finite horizon version of the problem.     

Embedding Problems of Division Algebras

A finite group G is called admissible over a given field if there exists a central division algebra that contains a G-Galois field extension as a maximal subfield. We give a definition of embedding problems of division algebras that extends both the notion of embedding problems of fields as in classical Galois theory, and the question which finite groups are admissible over a field. In a recent work by Harbater, Hartmann, and Krashen, all admissible groups over function fields of curves over complete discretely valued fields with algebraically closed residue field of characteristic zero have been characterized. We show that also certain embedding problems of division algebras over such a field can be solved for admissible groups.     

Directional maximal operators with smooth densities

We study directional maximal operators on ?ⁿ with smooth densities. We prove that if the classical directional maximal operator in a given set of directions is weak type (1, 1), then the corresponding smooth‐density maximal operator in that set of directions will be bounded on L^q for q suitably large, depending on the order of the stationary points of the density function. In contrast to the classical case, if q is too small, the smooth density operator need not be bounded on L^q. Improving upon previously known results, we also establish that if the density function has only finitely many extreme points, each of finite order, then any maximal operator in a finite sum of diadic directions is bounded on all L^q for q > 1 (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Homogeneous approximation property for wavelet frames with matrix dilations

The homogeneous approximation property (HAP) for wavelet frames was studied recently. The HAP is useful in practice since it means that the number of building blocks involved in a reconstruction of a function up to some error is essentially invariant under time‐scale shifts. In this paper, we prove the HAP for wavelet frames generated by admissible wavelet functions with arbitrary translation parameters and a class of dilation matrices. Moreover, we show that the approximation is uniform to some extent whenever wavelet functions satisfy moderate smooth and decaying conditions (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Some properties of entire functions with nonnegative Taylor coefficients

If f is an entire function of arbitrary finite order and with nonnegative Taylor coefficients, then we prove that its restriction to the positive part of the real axis belongs to de Haan’s class Γ. We also show that f/f′ is a Beurling slowly varying function.     

On lower semicontinuity in BH and 2-quasiconvexification

It is proved that if , with p > 1, if is bounded in , , and if in then provided is 2-quasiconvex and satisfies some appropriate growth and continuity condition. Characterizations of the 2-quasiconvex envelope when admissible test functions belong to BH^p are provided. Received: 10 October 2001 / Accepted: 8 May 2002 / Published online: 17 December 2002     

Burn-in considering yield loss and reliability gain for integrated circuits

This paper presents burn-in effects on yield loss and reliability gain for a lifetime distribution developed from a negative binomial defect density distribution and a given defect size distribution, after assuming that the rate of defect growth is proportional to the power of the present defect size. While burn-in always results in yield loss, it creates reliability gain only if either defects grow fast or the field operation time is long. Otherwise, burn-in for a short time could result in reliability loss. The optimal burn-in time for maximizing reliability is finite if defects grow linearly in time and is infinite if defects grow nonlinearly in time. The optimal burn-in time for minimizing cost expressed in terms of both yield and reliability increases in the field operation time initially but becomes constant as the field operation time is long enough. It is numerically shown that increasing mean defect density or defect clustering increases the optimal burn-in time.     

A nonconforming combined method for solving Laplace's boundary value problems with singularities

Summary For solving Laplace's boundary value problems with singularities, a nonconforming combined approach of the Ritz-Galerkin method and the finite element method is presented. In this approach, singular functions are chosen to be admissible functions in the part of a solution domain where there exist singularities; and piecewise linear functions are chosen to be admissible functions in the rest of the solution domain. In addition, the admissible functions used here are constrained to be continuous only at the element nodes on the common boundary of both methods. This method is nonconforming; however, the nonconforming effect does not result in larger errors of numerical solutions as long as a suitable coupling strategy is used.In this paper, we will develop such an approach by using a new coupling strategy, which is described as follows: IfL+1=O(|lnh|), the average errors of numerical solutions and their generalized derivatives are stillO(h), whereh is the maximal boundary length of quasiuniform triangular elements in the finite element method, andL+1 is the total number of singular admissible functions in the Ritz-Galerkin method. The coupling relation,L+1=O(|lnh|), is significant because only a few singular functions are required for a good approximation of solutions.This material is from Chapter 5 in my Ph.D. thesis: Numerical Methods for Elliptic Boundary Value Problems with Singularities. Part I: Boundary Methods for Solving Elliptic Problems with Singularities. Part II: Nonconforming Combinations for Solving Elliptic Problems with Singularities, the Department of Mathematics and Applied Mathematics, University of Toronto, May 1986     

Ersatz function method for minimizing a finite-valued function on a compact set

A method is proposed for solving optimization problems with continuous variables and a function taking a large finite set of values. Problems of this type arise in the multicriteria construction of a control rule for a discrete-time dynamical system whose performance criteria coincide with the number of violations of requirements imposed on the system. The rule depends on a finite set of parameters whose set of admissible values defines a collection of admissible control rules. An example is the problem of choosing a control rule for a cascade of reservoirs. The optimization method is based on solving a modified problem in which the original function is replaced by a continuous ersatz function. A theorem on the relation between the average-minimal values of the original and ersatz functions is proved. Optimization problems are solved with power-law ersatz functions, and the influence exerted by the exponent on the quality of the solution is determined. It is experimentally shown that the solutions produced by the method are of fairly high quality.     

On the Ershov Upper Semilattice

We find some links between -reducibility and T-reducibility. We prove that (1) if a quasirigid model is strongly -definable in a hereditarily finite admissible set over a locally constructivizable B-system, then it is constructivizable; (2) every abelian p-group and every Ershov algebra is locally constructivizable; (3) if an antisymmetric connected model is -definable in a hereditarily finite admissible set over a countable Ershov algebra then it is constructivizable.     

Guidance problem for a distributed system with incomplete information on the state coordinates and an unknown initial state

We study the problem of guaranteed positional guidance of a linear partially observable control system with distributed parameters to a convex target set at a given time. The problem is considered under incomplete information. More precisely, we assume that the system is subjected to an unknown disturbance; in addition, the initial state is assumed to be unknown as well. Further, the sets of admissible disturbances and the set of admissible initial states, which is assumed to be finite, are known. An algorithm for solving the problem is suggested.