Higher rank numerical ranges and low rank perturbations of quantum channels

For a positive integer k, the rank-k numerical range Λk(A) of an operator A acting on a Hilbert space H of dimension at least k is the set of scalars λ such that PAP=λP for some rank k orthogonal projection P. In this paper, a close connection between low rank perturbation of an operator A and Λk(A) is established. In particular, for 1?r<k it is shown that Λk(A)⊆Λk−r(A+F) for any operator F with rank(F)?r. In quantum computing, this result implies that a quantum channel with a k-dimensional error correcting code under a perturbation of rank at most r will still have a (k−r)-dimensional error correcting code. Moreover, it is shown that if A is normal or if the dimension of A is finite, then Λk(A) can be obtained as the intersection of Λk−r(A+F) for a collection of rank r operators F. Examples are given to show that the result fails if A is a general operator. The closure and the interior of the convex set Λk(A) are completely determined. Analogous results are obtained for Λ_∞(A) defined as the set of scalars λ such that PAP=λP for an infinite rank orthogonal projection P. It is shown that Λ_∞(A) is the intersection of all Λk(A) for k=1,2,…. If A−μI is not compact for all μ∈C, then the closure and the interior of Λ_∞(A) coincide with those of the essential numerical range of A. The situation for the special case when A−μI is compact for some μ∈C is also studied.     

Linear interval equations with symmetric solution sets

Criteria for symmetry and boundedness are found for the combined solution set of a system of linear algebraic equations with interval coefficients. It is shown that the problem of the best inner interval estimation of a symmetric solution set can be exactly solved by linear programming methods.     

Smoothness of solutions of almost hypoelliptic equations

A two-dimensional linear differential operator P(D) = P(D ₁, D ₂) is called almost hypoelliptic if all derivatives D ^α P of the characteristic polynomial P(ζ) = P(ζ ₁, ζ ₂) are estimated by P(ζ). Assuming that {Ω _κ = (x ₁, x ₂) ∈ E ² : |x ₁| < κ, x ₂ ∈ R ¹}, the paper proves that if the width κ of the strip Ω _κ exceeds some C = C(P) > 0, then all solutions {u} of the almost hypoelliptic equation P(D)u = 0 in a Sobolev space are infinitely smooth functions with respect to x ₁.     

On the Space bvp（F） of Sequences of p-bounded Variation of Fuzzy Numbers

Recently, the space bvp of real or complex numbers consisting of all sequences whose differences are in the space lp has been studied by Basar, Altay [Ukrainian Math. J. 55（1）（2003）, 136-147], where 1 ≤ p ≤ ∞. The main purpose of the present paper is to introduce the space bvp（F） of sequences of p-bounded variation of fuzzy numbers. Moreover, it is proved that the space bvp（F） includes the space lp（F） and also shown that the spaces bvp（F） and lp（F） axe isomorphic for 1 ≤ p ≤∞. Furthermore, some inclusion relations have been given.     

THREE-POINT BOUNDARY VALUE PROBLEM FOR p-LAPLACIAN DIFFERENTIAL EQUATION AT RESONANCE

By topological degree theory,the three-point boundary value problem for p-Laplacian differential equation at resonance is studied. Some new results on the existence of solutions are obtained,which improve and extend some known ones in the previous literatures.     

On the asymptotic behaviour for an electromagnetic system with a dissipative boundary condition

In this work we study some properties of solutions for the systemdescribing a three-dimensional non-homogeneous non-conductingdielectric with a general boundary condition with memory. Wefirst show the existence of the inverse of this boundary condition,which allows us to introduce a boundary free energy, similarto the one considered by Fabrizio & Morro (1996, Arch. Rat.Mech. Anal., 136, 359–381). Then, we prove existence anduniqueness theorems for weak and strong solutions of the evolutiveproblem in a finite time interval. Moreover, following Rivera& Olivera (1997, Boll. U.M.I., 11-A, 115–127), weexamine some dissipative properties of the boundary conditionand of its inverse and we give a useful energy estimate. Finally,when there is no memory in the boundary condition the exponentialdecay of the solution is proved.     

一种计算工作时间参数新方法

为了编制和优化施工进度计划,计算构成施工项目的各项工作最早开始时间、最迟开始时间、最早完成时间、最迟完成时间、总时差和自由时差等时间参数十分重要.提出了一种计算工作时间参数新方法.该方法以工作完成时间为决策变量,通过建立和求解线性规划模型来得到各种工作时间参数.其建模思路清晰,不需绘制网络图,能用通用办公软件EXCEL求解.模拟计算表明,用该方法与用标准网络计划技术计算出的工作时间参数完全一致.     

Totally skew embeddings of manifolds

We study a version of Whitney’s embedding problem in projective geometry: What is the smallest dimension of an affine space that can contain an n-dimensional submanifold without any pairs of parallel or intersecting tangent lines at distinct points? This problem is related to the generalized vector field problem, existence of non-singular bilinear maps, and the immersion problem for real projective spaces. We use these connections and other methods to obtain several specific and general bounds for the desired dimension.     

模糊线性系统的扰动分析

使用谱范数分析了模糊线性系统在三种情形下的扰动: (1)右端模糊向量有扰动, 系数矩阵不变; (2)系数矩阵有扰动,右端模糊向量不变; (3)系数矩阵和右端模糊向量都有扰动,并通过数值实例验证给出的扰动界的估计.