On local spectral properties of complex symmetric operators

In this paper we study properties of complex symmetric operators. In particular, we prove that every complex symmetric operator having property (β) or (δ) is decomposable. Moreover, we show that complex symmetric operator T has Dunford?s property (C) and it satisfies Weyl?s theorem if and only if its adjoint does.     

Common properties of the operator products in local spectral theory

Let X, Y be Banach spaces, A, D : X → Y and B, C : Y → X be the bounded linear operators satisfying operator equation set AC D = DBD,DBA = AC A.In this paper, we show that AC and BD share some basic operator properties such as the injectivity and the invertibility. Moreover, we show that AC and BD share many common local spectral properties including SVEP, Bishop property(β) and Dunford property(C).     

Newton’s method as a tool for finding the eigenvalues of certain two-parameter (multiparameter) spectral problems

An iterative algorithm is examined for finding the eigenvalues of the two-parameter (multiparameter) algebraic eigenvalue problem. This algorithm uses Newton’s method and an efficient numerical procedure for differentiating determinants. Some numerical examples are given.     

A Note on the Browder＇s and Weyl＇s Theorem

Let T be a Banach space operator, E（T） be the set of all isolated eigenvalues of T and π（T） be the set of all poles of T. In this work, we show that Browder＇s theorem for T is equivalent to the localized single-valued extension property at all complex numbers λ in the complement of the Weyl spectrum of T, and we give some characterization of Weyl＇s theorem for operator satisfying E（T） = π（T）. An application is also given.     

On generalized criss‐cross and near‐commutativity for n‐tuples

Here, we study multivariable versions of a generalization of Jacobson's lemma and give common spectral properties for n‐tuples that satisfy generalized criss‐cross or near commutativity.     

Epr-bohm experiment and Bell’s inequality: Quantum physics meets probability theory

Our main aim in this paper is to inform the physics community (and especially experts in quantum information) about investigations of the problem of the probabilistic compatibility of a family of random variables: the possibility of realizing such a family based on a single probability measure (of constructing a single Kolmogorov probability space). These investigations were started a hundred years ago by Boole. The complete solution of the problem was obtained by the Soviet mathematician Vorobiev in the 1960s. It turns out that probabilists and statisticians obtained inequalities for probabilities and correlations that include the famous Bell’s inequality and its generalizations. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 157, No. 1, pp. 99–115, October, 2008.     

(ω)性质及Weyl型定理

(ω)性质是Rakocevic给出的Weyl定理的一种变化.本文通过定义新的谱集,给出了有界线性算子同时满足(ω)性质和a-Weyl定理的充要条件.同时,利用所得的主要结论,研究了H(p)算子的(ω)性质.     

Gevrey-hypoellipticity for a Class of Parabolic Type Operators

This paper studies the micro-local version of the Gevrey-hypoellipticity for a class of parabolic type operator, ∂_t-a(t, x; D_s,) (a(t, x;D_s) (a(t, x; ξ) ∈ C^∞([0, T], S^m_{G^m}(R^n)) (m ≥ 1/s) is a Gevrey-pseudoanaiytic symbol of class s(s > 1)), and we obtain the following main result: Under the condition (I), the operator stated above is micro-local Gevrey-hypoelliptic. In order to prove our main result, the auther have used (α, β) method in this paper.     

On the Equivalence and Generalized of Weyl Theorem Weyl Theorem

We know that an operator T acting on a Banach space satisfying generalized Weyl＇s theorem also satisfies Weyl＇s theorem. Conversely we show that if all isolated eigenvalues of T are poles of its resolvent and if T satisfies Weyl＇s theorem, then it also satisfies generalized Weyl＇s theorem. We give also a sinlilar result for the equivalence of a-Weyl＇s theorem and generalized a-Weyl＇s theorem. Using these results, we study the case of polaroid operators, and in particular paranormal operators.     

The theoretic design of NMR pulses program of arbitrary N-qubit Grover’s algorithm and the NMR experiment proof

Grover’s quantum searching algorithm is most widely studied in the current quantum computation research, and has been implemented experimentally by NMR (Nuclear Magnetic Resonance) technique. In this article, we design arbitrary N-qubit NMR pulses program of Grover’s algorithm based on the multiple-quantum operator algebra theory and demonstrate 2-qubit pulses program experimentally. The result also proves the validity of the multiple-quantum operator algebra theory     

Modified Camassa–Holm and Degasperis–Procesi Equations Solved by Adomian’s Decomposition Method and Comparison with HPM and Exact Solutions

Many physical and scientific phenomena are modeled by nonlinear partial differential equations (NPDEs); it is difficult to handle nonlinear part of these equations. Recently some analytical methods are applied to solve such equations. In this work, modified Camassa–Holm and Degasperis–Procesi equation is studied. Adomian’s decomposition method (ADM) is applied to obtain solution of this equation. The results are compared to those of homotopy perturbation method (HPM) and exact solution. The study highlights the significant features of the employed method and its ability to handle nonlinear partial differential equations.     

On the structure of the class A(ϕ)

In this paper, we study whether the set A(?) is closed under multiplication f · g, where f and g belong to the class A(?). We also study the problem of the existence of a solution of the equation Bx = C (where B,C ∈ A(?) and B ≠ 0) on the set A(?).     

广义(ω)性质的一个注记

通过定义新的谱集,研究了Weyl定理的一个变形—广义(w)性质,给出了Banach空间上有界线性算子满足广义(w)性质的充要条件.同时,利用所得的主要结论,我们研究了广义(w)性质的摄动.     

加权Hardy空间与加权L^p空间之间的算子内插

利用Calderon-Zygmund分解,给出了当w∈A1时加权Hardy空间与加权L^p空间之间的算子内插。     

The prime-counting function and its analytic approximations

The paper describes a systematic computational study of the prime counting function π(x) and three of its analytic approximations: the logarithmic integral \({\text{li}}{\left( x \right)}: = {\int_0^x {\frac{{dt}}{{\log \,t}}} }\), \({\text{li}}{\left( x \right)} - \frac{1}{2}{\text{li}}{\left( {{\sqrt x }} \right)}\), and \(R{\left( x \right)}: = {\sum\nolimits_{k = 1}^\infty {{\mu {\left( k \right)}{\text{li}}{\left( {x^{{1 \mathord{\left/ {\vphantom {1 k}} \right. \kern-\nulldelimiterspace} k}} } \right)}} \mathord{\left/ {\vphantom {{\mu {\left( k \right)}{\text{li}}{\left( {x^{{1 \mathord{\left/ {\vphantom {1 k}} \right. \kern-\nulldelimiterspace} k}} } \right)}} k}} \right. \kern-\nulldelimiterspace} k} }\), where μ is the Möbius function. The results show that π(x)x) for 2≤x≤10¹⁴, and also seem to support several conjectures on the maximal and average errors of the three approximations, most importantly \({\left| {\pi {\left( x \right)} - {\text{li}}{\left( x \right)}} \right|} < x^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}}\) and \( - \frac{2}{5}x^{{3 \mathord{\left/ {\vphantom {3 2}} \right. \kern-\nulldelimiterspace} 2}} < {\int_2^x {{\left( {\pi {\left( u \right)} - {\text{li}}{\left( u \right)}} \right)}du < 0} }\) for all x>2. The paper concludes with a short discussion of prospects for further computational progress.     

Gaussian approximation of local empirical processes indexed by functions

Summary. An extended notion of a local empirical process indexed by functions is introduced, which includes kernel density and regression function estimators and the conditional empirical process as special cases. Under suitable regularity conditions a central limit theorem and a strong approximation by a sequence of Gaussian processes are established for such processes. A compact law of the iterated logarithm (LIL) is then inferred from the corresponding LIL for the approximating sequence of Gaussian processes. A number of statistical applications of our results are indicated. Received: 11 January 1995/In revised form: 12 July 1996     

Cyclically stationary Brownian local time processes

Summary. Local time processes parameterized by a circle, defined by the occupation density up to time T of Brownian motion with constant drift on the circle, are studied for various random times T. While such processes are typically non-Markovian, their Laplace functionals are expressed by series formulae related to similar formulae for the Markovian local time processes subject to the Ray–Knight theorems for BM on the line, and for squares of Bessel processes and their bridges. For T the time that BM on the circle first returns to its starting point after a complete loop around the circle, the local time process is cyclically stationary, with same two-dimensional distributions, but not the same three-dimensional distributions, as the sum of squares of two i.i.d. cyclically stationary Gaussian processes. This local time process is the infinitely divisible sum of a Poisson point process of local time processes derived from Brownian excursions. The corresponding intensity measure on path space, and similar Lévy measures derived from squares of Bessel processes, are described in terms of a 4-dimensional Bessel bridge by Williams’ decomposition of It?’s law of Brownian excursions. Received: 28 June 1995     

一致Fredholm指标算子及广义(ω′)性质

本文给出了一致Fredholm指标算子的定义及判定,同时定义了Weyl型定理的一种新变化:广义(ω')性质.根据一致Fredholm指标性质定义出一种新的谱集,通过该谱集给出了Hilbert空间上有界线性算子满足广义(ω')性质的充要条件,并且研究了广义(ω')性质的摄动,还研究了算子的亚循环性和广义(ω')性质之间的关系.     

The mean value property for harmonic functions on graphs and trees

Following the Euclidean example, we introduce the strong and weak mean value property for finite variation measures on graphs. We completely characterize finite variation measures with bounded support on radial trees which have the strong mean value property. We show that for counting measures on bounded subsets of a tree with root o, the strong mean value property is equivalent to the invariance of the subset under the action of the stabilizer of o in the automorphism group. We finally characterize, using the discrete Laplacian, the finite variation measures on a generic graph which have the weak mean value property and we give a non-trivial example. Received: July 21, 2000; in final form: March 13, 2001?Published online: March 19, 2002     

On two questions of the theory of retracts

We establish that condition (Γ) on brick decomposition is indecomposable. This answers K. Borsuk’s question [1]. We prove that there exist metric spaces X and Y and a point (a, b) ∈ X × Y such that (a, b) is an r-point of the product X × Y; moreover, a is not an r-point of X. This answers A. Kosinski’s question [2].