Means of unitaries, conjugations, and the Friedrichs operator

If C is a conjugation (an isometric, conjugate-linear involution) on a separable complex Hilbert space H, then T∈B(H) is called C-symmetric if T=CT^∗C. In this note we prove that each C-symmetric contraction T is the mean of two C-symmetric unitary operators. We discuss several corollaries and an application to the Friedrichs operator of a planar domain.     

Failed disk recovery in double erasure RAID arrays

Reliability is a major concern in the design of large disk arrays. In this paper, we examine the effect of encountering more failures than that for which the RAID array was initially designed. Erasure codes are incorporated to enable system recovery from a specified number of disk erasures, and strive beyond that threshold to recover the system as frequently, and as thoroughly, as is possible. Erasure codes for tolerating two disk failures are examined. For these double erasure codes, we establish a correspondence between system operation and acyclicity of its graph model. For the most compact double erasure code, the full 2-code, this underlies an efficient algorithm for the computation of system operation probability (all disks operating or recoverable).When the system has failed, some disks are nonetheless recoverable. We extend the graph model to determine the probability that d disks have failed, a of which are recoverable by solving one linear equation, b of which are further recoverable by solving systems of linear equations, and d−a−b of which cannot be recovered. These statistics are efficiently calculated for the full 2-code by developing a three variable ordinary generating function whose coefficients give the specified values. Finally, examples are given to illustrate the probability that an individual disk can be recovered, even when the system is in a failed state.     

Combinatorial data analysis: Confirmatory comparisons between sets of matrices

The task of assessing the similarity of pattern between the entries of two square matrices has been discussed extensively over the last decade, as a unifying strategy for approaching a variety of seemingly disparate statistical problems. As typically defined, the comparison depends on a measure of matrix correspondence, usually a normalized cross-product measure of some form, that is evaluated for relative size by the use of a reference distribution constructed through an equally likely permutation hypothesis defined at the level of the objects corresponding to the rows and columns of the two matrices. The extreme generality provided by this very simple framework subsumes a variety of different statistical problems, ranging from the study of spatial autocorrelation for variables observed over a set of geographic locations, to the topics of analysis of variance, the measurement of rank correlation, and confirmation techniques concerned with various conjectures of combinatorial structure that might be posited for an empirically determined measure of relationship between pairs of a given set of objects. The comparison strategies extant always assume that both matrices are fixed, and in those cases where one of the matrices codifies a given theoretical structure to be evaluated according to a second, this assumption can lead to substantial arbitrariness in how matrix similarity might be indexed, and thus, in how the comparison is implemented. As developed in this paper, exactly the same principles appropriate for use in the fixed comparison context can be extended to include matrices constructed through optimally weighted linear combinations of other sets of matrices. This generalization provides one mechanism for developing comparison strategies that allow assessment against very broad classes of matrices, which in turn serve to represent very general conjectures of possible combinatorial structure. This paper reviews some of these extensions in detail, with a particular emphasis on categorical and ordered categorical variables and whether they may reflect an empirically generated measure of object relationship.     

Butyltin speciation in sediments and biological samples by hydride generation gas chromatography quartz furnace atomic absorption spectrometry; a study of acid leaching procedures and hydride generation

The efficiency of the extraction of butyltin compounds from sediments and biological matter by acid leaching procedures has been evaluated on real samples using the hydride generation gas chromatography quartz furnace atomic absorption (GC AA) speciation method. The most efficient method uses cold pure acetic acid over a period of four hours. Hydride generation using sodium tetrahydroborate (NaBH₄) is dependent on matrix composition and care must be taken to adapt the amount of reactant to the organic content of the sample.     

Matrix models, complex geometry, and integrable systems: I

We consider the simplest gauge theories given by one-and two-matrix integrals and concentrate on their stringy and geometric properties. We recall the general integrable structure behind the matrix integrals and turn to the geometric properties of planar matrix models, demonstrating that they are universally described in terms of integrable systems directly related to the theory of complex curves. We study the main ingredients of this geometric picture, suggesting that it can be generalized beyond one complex dimension, and formulate them in terms of semiclassical integrable systems solved by constructing tau functions or prepotentials. We discuss the complex curves and tau functions of one-and two-matrix models in detail. [This article was written at the request of the Editorial Board. It is based on several lectures presented at schools of mathematical physics and talks at the conference “Complex Geometry and String Theory” and the Polivanov memorial seminar.] __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 147, No. 2, pp. 163–228, May, 2006.     

混合互补判断矩阵一致性研究

给出混合互补判断矩阵一致性的定义和判别加性一致性的方法.定义了核算子、核矩阵,对带有精确数、三角模糊数和梯形模糊数的混合互补判断矩阵给出基于核矩阵的一致性调整方法,调整量可以是精确数也可以是模糊数.最后给出一个应用实例.     

EXISTENCE OF SOLUTIONS OF A FAMILY OF NONLINEAR BOUNDARY VALUE PROBLEMS IN L~2-SPACES

§ 1　Introduction and preliminariesThe equations related to the p- L aplacian operatorΔp have been studied in differentaspects[1— 4 ] sinceΔp arises from a variety of physical phenomena such as non Newtonianfluids,reaction- diffusion problems,etc.In this paper,the following equation(1) which canbe regarded as an extension of those in[1— 4 ]will be discussed:- div(α(gradu) ) +|u(x) |p- 2 u(x) +g(x,u(x) ) =f (x)　 a.e. onΩ-〈n,α(gradu)〉∈βx(u(x) ) ,a.e.onΓ . (1)　　More details can b…     

实对称矩阵的迹的几点性质

研究实对称矩阵的迹的几点性质，给出６个引理，推得３个定理     

Minimum Cycle of Row Vector of a Generalized Circulant Fuzzy Matrix

51.IntroductionLetF=[O,1],foranya,beF,definea b=max{a,b},a*b=min{a,b}.Mn(F)standsforthesetofa1lthenthordermatrixesinF,matrixAinMn(F)iscalledFuzzymatrixA.Letrbeanynon-negativeinteger.lnannthordermatrix,exclusiveofthefirstrow,theelementsofeveryotherrowarethee1ementsoftheprecedingrowthatcirculaterightwardr-co1umns.Thisnthordermatrixisannthorderr-circulantFuzzymatrix(alsocalledgenearLizedcirculantFuzzymatrix)-Wecallrthecircu1antindexofr-circulantFuzzymatrix.Inthebooklistedin[1j,thisisgiven:…