Stability of localized operators

Let ?p, 1?p?∞, be the space of all p-summable sequences and Ca be the convolution operator associated with a summable sequence a. It is known that the ?p-stability of the convolution operator Ca for different 1?p?∞ are equivalent to each other, i.e., if Ca has ?p-stability for some 1?p?∞ then Ca has ?q-stability for all 1?q?∞. In the study of spline approximation, wavelet analysis, time-frequency analysis, and sampling, there are many localized operators of non-convolution type whose stability is one of the basic assumptions. In this paper, we consider the stability of those localized operators including infinite matrices in the Sjöstrand class, synthesis operators with generating functions enveloped by shifts of a function in the Wiener amalgam space, and integral operators with kernels having certain regularity and decay at infinity. We show that the ?p-stability (or Lp-stability) of those three classes of localized operators are equivalent to each other, and we also prove that the left inverse of those localized operators are well localized.     

Pushing the envelope of the test functions in the Szegö and Avram–Parter theorems

The Szegö and Avram–Parter theorems give the limit of the arithmetic mean of the values of certain test functions at the eigenvalues of Hermitian Toeplitz matrices and the singular values of arbitrary Toeplitz matrices, respectively, as the matrix dimension goes to infinity. The question on whether these theorems are true whenever they make sense is essentially the question on whether they are valid for all continuous, nonnegative, and monotonously increasing test functions. We show that, surprisingly, the answer to this question is negative. On the other hand, we prove the two theorems in a general form which includes all versions known so far.     

Matrix and vector models in the strong coupling limit

We consider matrix and vector models in the large-N limit: we study N × N matrices and vectors with N² components. In the case of a zero-dimensional model (D = 0), we prove that in the strong coupling limit (g → ∞), the partition functions of the two models coincide up to a coefficient. This also holds for D = 1. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 155, No. 2, pp. 236–243, May, 2008.     

奇异协方差阵下有效前沿及有效组合的解析解

利用广义逆矩阵研究了协方差阵奇异时的投资组合问题,突破了传统方法中要求协方差阵可逆的限制,得到了证券市场存在有效组合的充要条件,并给出了有效前沿和有效组合的解析解,成功地推广了经典Markowitz模型,同时还将有助于证券组合有效子集的深入研究.     

光纤—自聚焦棒—反射器耦合系统的后向耦合系数

用光线矩阵方法推导了光纤-自聚焦棒-反射器在共轴耦合情况下的后向耦合系数;分析了它与耦合系统结构参数之间的关系.用计算机模拟画出了反射器为平面镜的系统的后向耦合系数和后向耦合损耗的曲线分布,计算曲线与实验结果大致符合.     

Estimating the covariance matrix and the generalized variance under a symmetric loss

For estimating the power of a generalized variance under a multivariate normal distribution with unknown means, the inadmissibility of the best affine equivariant estimator relative to the symmetric loss is shown, and a class of improved estimators is given. The problem of estimating the covariance matrix is also discussed.     

任意除环上一矩阵方程组的一般解与中心和反中心对称解

Necessary and sufficient conditions are given for the existence of the general solution, the centrosymmetric solution, and the centroskewsymmetric solution to a system of linear matrix equations over an arbitrary skew field. The representations of such the solutions of the system are also derived.     

Photocatalytic membrane of a novel high surface area TiO2 synthesized from titanium triisopropanolamine precursor

Photocatalytic membrane was successfully prepared using an efficiently high surface area TiO₂ catalyst, dispersed into polyacrylonitrile matrix. The catalyst was directly synthesized using titanium triisopropanolamine as a precursor. The membranes were characterized using FT‐IR, TGA, SEM and their photocatalytic performance tested, viz. stability, permeate flux and photocatalytic degradation of 4‐NP. We find that polyacrylonitrile is an effective matrix, showing high stability and low permeate flux. The amount of TiO₂ loaded in the membrane was varied between 1, 3 and 5 wt% to explore the activity and stability of membranes in the photocatalytic reaction of 4‐NP. As expected, the higher the loading of TiO₂ loaded, the higher the resulting catalytic activity. Copyright © 2006 John Wiley & Sons, Ltd.     

Decentralized Guaranteed Cost Control for Uncertain Large-Scale Systems Using Delayed Feedback: LMI Optimization Approach

In this paper, we propose a design method of guaranteed cost controllers for uncertain large-scale systems with time delays in subsystem interconnections using delayed feedback. Using the Lyapunov method, a linear matrix inequality (LMI) optimization problem is formulated to design a delayed feedback controller which minimizes the upper bound of a given quadratic cost function. A numerical example is included to illustrate the design procedures.Communicated by Q. C. ZhaoThe authors thank the Associate Editor and three anonymous referees for careful reading and useful suggestions.     

Determining Causal Skeletons with Information Theory

Modeling a causal association as arising from a communication process between cause and effect, simplifies the discovery of causal skeletons. The communication channels enabling these communication processes, are fully characterized by stochastic tensors, and therefore allow us to use linear algebra. This tensor-based approach reduces the dimensionality of the data needed to test for conditional independence, e.g., for systems comprising three variables, pair-wise determined tensors suffice to infer the causal skeleton. The only thing needed is a minor extension to information theory, namely the concept of path information.