CONSTRUCTION OF COMPACTLY SUPPORTED BIVARIATE ORTHOGONAL WAVELETS BY UNIVARIATE ORTHOGONAL WAVELETS

After some permutation of conjugate quadrature filter, new conjugate quadrature filters can be derived. In terms of this permutation, an approach is developed for constructing compactly supported bivariate orthogonal wavelets from univariate orthogonal wavelets. Non-separable orthogonal wavelets can be achieved. To demonstrate this method, an example is given.     

AVAILABILITY ANALYSIS OF A TWO-UNIT SERIES SYSTEM WITH SHUT-OFF RULE AND “FIRST-FAIL,FIRST-REPAIRED”

Ⅰ.Intreduction It is Well known that series system is one of the most essential and important medels in reliability theory and application.A series system of twe units operates if and only if both units operate.However,in practice,unit 1 upon failure may shut off unit 2 but not vice versa.For example,failure of the power supply may shut down an electronic device but not vice versa.Barlow and Hudes first discussed the asymptotic measures of such a system performance.Next Khalil reviewed the results on availability of series systems with various shut-off rules,introduced combination concepts of shut-off rules,and extended the results     

FAST DENSE MATRIX METHOD FOR THE SOLUTION OF INTEGRAL EQUATIONS OF THE SECOND KIND

We present a fast algorithm based on polynomial interpolation to approximate matrices arising from the discretization of second-kind integral equations where the kernel function is either smooth, non-oscillatory and possessing only a finite number of singularities or a product of such function with a highly oscillatory coefficient function. Contrast to wavelet-like approximations, ourapproximation matrix is not sparse. However, the approximation can be construced in O(n) operations and requires O(n) storage, where n is the number of quadrature points used in the discretization. Moreover, the matrix-vector multiplication cost is of order O(nlogn). Thus our scheme is well suitable for conjugate gradient type methods. Our numerical results indicate that the algorithm is very accurate and stable for high degree polynomial interpolation.     

Adic Attractor and Topological Entropy

1 IntroductionLet M be a manifold (possibly with boundary), and F : M → M be continuous. Call a closed invariant set A (?) M an adic attractor of f if it attracts almost all points (in the sense of Lebesgue measure) and the restriction f|A is topologically conjugate to an adic system. Such an attractor A is called n-adic if the restriction f|A can be topologically conjugate the n-adic system.     

进位吸引子与拓扑熵

Let M be a manifold (possibly with boundary), and f:M→M be continuous. Call a closed invariant set A包含M an adic attractor of f if it attracts almost all points (in the sense of Lebesgue measure) and the restriction f|A is topologically conjugate to an adic system. Such an attractor A is called n-adic if the restriction flA can be topologically conjugate the n-adic system.     

A DIRECT SEARCH FRAME-BASED CONJUGATE GRADIENTS METHOD

A derivative-free frame-based conjugate gradients algorithm is presented.Convergenceis shown for C~1 functions,and this is verified in numerical trials.The algorithm is tested ona variety of low dimensional problems,some of which are ill-conditioned,and is also testedon problems of high dimension.Numerical results show that the algorithm is effectiveon both classes of problems.The results are compared with those from a discrete quasi-Newton method,showing that the conjugate gradients algorithm is competitive.Thealgorithm exhibits the conjugate gradients speed-up on problems for which the Hessian atthe solution has repeated or clustered eigenvalues.The algorithm is easily parallelizable.     

The best quadrature based on given Hermite information for the Sobolev class KW~r[a,b]

As usual, denote by KWr[a,b] the Sobolev class consisting of every function whose (r-1)th derivative is absolutely continuous on the interval [a,b] and rth derivative is bounded by K a.e. in [a, b]. For a function f∈KWr[a, b], its values and derivatives up to r -1 order at a set of nodes x are known. These values are said to be the given Hermite information. This work reports the results on the best quadrature based on the given Hermite information for the class KWr[a. b]. Existence and concrete construction issue of the best quadrature are settled down by a perfect spline interpolation. It turns out that the best quadrature depends on a system of algebraic equations satisfied by a set of free nodes of the interpolation perfect spline. From our another new result, it is shown that the system can be converted in a closed form to two single-variable polynomial equations, each being of degree approximately r/2. As a by-product, the best interpolation formula for the class KWr[a, b] is also obtained.     

Chaotic invariant sets of a delayed discrete neural network of two non-identical neurons

In this paper, we show that a delayed discrete Hopfield neural network of two nonidentical neurons with no self-connections can demonstrate chaotic behavior in a region away from the origin. To this end, we first transform the model, by a novel way, into an equivalent system which enjoys some nice properties. Then, we identify a chaotic invariant set for this system and show that the system within this set is topologically conjugate to the full shift map on two symbols. This confirms chaos in the sense of Devaney. Our main result is complementary to the results in Kaslik and Balint (2008) and Huang and Zou (2005), where it was shown that chaos may occur in neighborhoods of the origin for the same system. We also present some numeric simulations to demonstrate our theoretical results.     

Strictly nonnegative tensors and nonnegative tensor partition

We introduce a new class of nonnegative tensors—strictly nonnegative tensors.A weakly irreducible nonnegative tensor is a strictly nonnegative tensor but not vice versa.We show that the spectral radius of a strictly nonnegative tensor is always positive.We give some necessary and su?cient conditions for the six wellconditional classes of nonnegative tensors,introduced in the literature,and a full relationship picture about strictly nonnegative tensors with these six classes of nonnegative tensors.We then establish global R-linear convergence of a power method for finding the spectral radius of a nonnegative tensor under the condition of weak irreducibility.We show that for a nonnegative tensor T,there always exists a partition of the index set such that every tensor induced by the partition is weakly irreducible;and the spectral radius of T can be obtained from those spectral radii of the induced tensors.In this way,we develop a convergent algorithm for finding the spectral radius of a general nonnegative tensor without any additional assumption.Some preliminary numerical results show the feasibility and effectiveness of the algorithm.     

Constructing super Gabor frames: the rational time-frequency lattice case

For a time-frequency lattice Λ = A Z d B Z d , it is known that an orthonormal super Gabor frame of length L exists with respect to this lattice if and only if |det( AB) | = 1 L . The proof of this result involves various techniques from multi-lattice tiling and operator algebra theory, and it is far from constructive. In this paper we present a very general scheme for constructing super Gabor frames for the rational lattice case. Our method is based on partitioning an arbitrary fundamental domain of the lattice in the frequency domain such that each subset in the partition tiles R d by the lattice in the time domain. This approach not only provides us a simple algorithm of constructing various kinds of orthonormal super Gabor frames with flexible length and multiplicity, but also allows us to construct super Gabor (non-Riesz) frames with high order smoothness and regularity. Several examples are also presented.     

α尺度r重正交共轭滤波器的构造

对于α尺度r重紧支撑正交多小波系统,给出了由长为L的α尺度r重正交共轭滤波器构造长为L+1的α尺度r重正交共轭滤波器的一般方法,也给出了由低阶矩阵滤波器构造高阶矩阵滤波器的方法.若给定的正交共轭滤波器满足完全重构条件,则利用算法构造新的滤波器也满足完全重构条件,算法还保持正交共轭滤波器对称性,这一点在信号处理方面具有很好的应用价值.     

Symmetric–Antisymmetric Orthonormal Multiwavelets and Related Scalar Wavelets

For compactly supported symmetric–antisymmetric orthonormal multiwavelet systems with multiplicity 2, we first show that any length-2Nmultiwavelet system can be constructed from a length-(2N+1) multiwavelet system and vice versa. Then we present two explicit formulations for the construction of multiwavelet functions directly from their associated multiscaling functions. This is followed by the relationship between these multiscaling functions and the scaling functions of related orthonormal scalar wavelets. Finally, we present two methods for constructing families of symmetric–antisymmetric orthonormal multiwavelet systems via the construction of the related scalar wavelets.     

Hermite Interpolation in Loop Groups and Conjugate Quadrature Filter Approximation

A classical result of Weierstrass ensures that any continuous finite length trajectory in a vector space can be uniformly approximated by one whose coordinates are trigonometric functions. We derive an analogous result for trajectories in spheres and apply it to show that a continuous frequency response of a conjugate quadrature filter can be uniformly approximated by the frequency response of a finitely supported conjugate quadrature filter. We also extend this result, so as to preserve specified roots of the frequency response, and derive an approximation result for refinable functions whose integer translates are orthonormal. Our methods utilize properties of loop groups, jets, and the Brouwer topological degree. Mathematics Subject Classifications (2000) 22E67(Primary), 41A29(Secondary), 42A10, 42A11, 42C40, 47H10, 47H11.     

一类具有相同对称/反对称中心的正交平衡向量小波系统的不存在性

本文得到两个结果:首先证明尺度因子m与重数r的乘积为奇数时,具有相同对称/反对称中心1/2(1+μ+μ/m-1)(μ∈N)的正交向量小波系统的不存在性;其次证明尺度因子m=3,重数r为偶数时,具有相同对称/反对称中心1/2(1+μ+μ/m-1)的正交平衡向量小波系统的不存在性,这里N是正整数集合.     

Extension Principles for Dual Multiwavelet Frames of $$L_2(\mathbb {R}^s)$$ constructed from Multirefinable Generators

In this work we prove that any pair of homogeneous dual multiwavelet frames of \(L_2(\mathbb {R}^s)\) constructed from a pair of refinable function vectors gives rise to a pair of nonhomogeneous dual multiwavelet frames and vice versa. We also prove that the Mixed Oblique Extension Principle characterizes dual multiwavelet frames. Our results extend recent characterizations of affine dual frames derived from scalar refinable functions obtained in [3].     

多小波框架的构造理论

本文给出了多小波框架的sub-QMF条件,提出了多小波框架低通滤波器的参数化设计,由正交分解和矩阵的酉扩张得到其相应的高通滤波器表示的整套多小波框架设计的参数化方法,同时针对多描述编码的需求,构造了两个长折叠对称带参数的多小波紧框架.     

Multiwavelet packets and frame packets ofL 2(ℝ d )

The orthonormal basis generated by a wavelet ofL ²(ℝ) has poor frequency localization. To overcome this disadvantage Coifman, Meyer, and Wickerhauser constructed wavelet packets. We extend this concept to the higher dimensions where we consider arbitrary dilation matrices. The resulting basis ofL ²(ℝ ^d ) is called the multiwavelet packet basis. The concept of wavelet frame packet is also generalized to this setting. Further, we show how to construct various orthonormal bases ofL ²(ℝ ^d ) from the multiwavelet packets.     

Design of compactly supported trivariate orthogonal wavelets

In this paper, a method is developed for constructing compactly supported trivariate orthogonal wavelets from univariate orthogonal wavelets, essential idea of the approach is permutation of conjugate quadrature filter. Nonseparable and separable wavelets can be achieved from univariate orthogonal wavelets. Two examples are given to demonstrate this method.     

一个简单的维尔斯特拉斯函数

从数学分析知函数在某区间上可微则必连续,但反之未必;本文构造一个函数,并证明了它在[0,1]连续且处处不可微.