Spectra of 2 ×2 Upper-Triangular Operator Matrices

In [Perturbation of Spectrums of 2 × 2 Operator Matrices, Proceedings of the American Mathematical Society, Vol. 121, 1994], the authors asked whether there was an operator ?such that ?for a given pair?(A,B)?of operators, where the operator ?was defined by . In this note, a partial answer for the question is given.

Completeness of system of root vectors of upper triangular infinitedimensional Hamiltonian operators appearing in elasticity theory

This paper deals with a class of upper triangular infinite-dimensional Hamilto- nian operators appearing in the elasticity theory. The geometric multiplicity and algebraic index of the eigenvalue are investigated. Furthermore, the algebraic multiplicity of the eigenvalue is obtained. Based on these properties, the concrete completeness formulation of the system of eigenvectors or root vectors of the Hamiltonian operator is proposed. It is shown that the completeness is determined by the system of eigenvectors of the operator entries. Finally, the applications of the results to some problems in the elasticity theory are presented.

Eigenfunction expansion method of upper triangular operator matrixand application to two-dimensional elasticity problems based onstress formulation

This paper studies the eigenfunction expansion method to solve the two dimensional （2D） elasticity problems based on the stress formulation. The fundamental system of partial differential equations of the 2D problems is rewritten as an upper tri angular differential system based on the known results, and then the associated upper triangular operator matrix matrix is obtained. By further research, the two simpler com plete orthogonal systems of eigenfunctions in some space are obtained, which belong to the two block operators arising in the operator matrix. Then, a more simple and conve nient general solution to the 2D problem is given by the eigenfunction expansion method. Furthermore, the boundary conditions for the 2D problem, which can be solved by this method, are indicated. Finally, the validity of the obtained results is verified by a specific example.

Upper Triangular Matrix of Lie Algebra and a New Discrete Integrable Coupling System

The upper triangular matrix of Lie algebra is used to construct integrable couplings of discrete solition equations. Correspondingly, a feasible way to construct integrable couplings is presented. A nonlinear lattice soliton equation spectral problem is obtained and leads to a novel hierarchy of the nonlinear lattice equation hierarchy. It indicates that the study of integrable couplings using upper triangular matrix of Lie algebra is an important step towards constructing integrable systems.

Decomposition of Jordan automorphism of two-order upper triangular matrix algebra over certain semilocal rings

Let T(R) be a two-order upper matrix algebra over the semilocal ring R which is determined by R=F×F where F is a field such that CharF=0. In this paper, we prove that any Jordan automorphism of T(R) can be decomposed into a product of involutive, inner and diagonal automorphisms.

考虑基质吸力影响的非饱和路堤三维稳定性上限分析

采用极限上限分析定理构建了路堤失稳的三维旋转破坏机构,继而根据Bishop非饱和土抗剪强度理论,考虑非饱路堤内部基质吸力的空间分布及其随地下水位的变化,建立了路堤破坏土体外力功率与内部能量耗散功率的能量守恒方程,并利用遗传算法编写了非饱和路堤最小上限解的高效搜索算法。通过将非饱和土路堤基底破坏模式退化为边坡破坏模式,并与现有非饱和土边坡稳定性计算结果对比,验证了所提上限解的正确性和遗传搜寻算法的准确性。进一步地,对路堤三维稳定性的关键影响因素展开了系统分析,研究了路堤填土孔径分布、基质吸力、进气值倒数、路堤倾角以及有效内摩擦角等因素对路堤三维稳定性的影响规律。研究表明,非饱和土质路堤的稳定性不仅取决于路堤填土性质,而且依赖于影响土体吸力大小与分布的填土孔径参数和进气值等因素。地下水位升降引起的基质吸力变化对路堤稳定性存在显著影响。研究结果为路堤稳定性精细化分析提供了重要的理论依据。

一种L~U波段RF MEMS单刀十掷开关的设计

论文提出了一种适用于1GHz~60GHz频段的RF MEMS单刀十掷开关。该开关采用三角形上电极结构,以减小悬臂梁等效弹性系数和闭合接触面积,达到低驱动电压、高隔离度的目的。通过渐进传输线结构,改善阶跃补偿引起的信号损耗问题。仿真结果显示:该单刀十掷开关的驱动电压为13V,在1GHz~60GHz频段范围内,输入端两侧输出端口的插入损耗<2dB,其余八个端口插入损耗≤0.59dB,所有端口隔离度均>30dB。此开关可适用于无线通信系统、雷达系统和仪器测量系统等对射频性能要求高的领域内。

常数项为零的多项式在严格上三角矩阵代数上的像

定义了多项式的最小次数。使用多项式的最小次数和Zariski拓扑,给出了代数闭域上常数项为零的多项式在严格上三角矩阵代数上像的完整刻画。

Generalized Drazin spectrum of upper triangular matrices in Banach algebras

Let A be a Banach algebra with unit e and a,b,c∈A,Mc=(a c 0 b)∈M_(2)(A).The concepts of left and right generalized Drazin invertible of elements in a Banach algebra are proposed.A generalized Drazin spectrum of is defined byσ_(gD)(α)={λ∈C:α-λe is not generalized Drazin invertible}.It is shown thatσ_(gD)(a)∪σ_(gD)(b)=σ_(gD)(M_(C))∪W_(2),where W_(g) is a union of certain holes σ_(gD) and W_(g)■σ_(gD)(a)∩σ_(gD)(b),or more finely.In addition,some properties of generalized Drazin spectrum of elements in a Banach algebra are studied.

具有未知增长率的一类上三角非线性多智能体系统的协同控制

基于有向领导者-追随者型的网络拓扑结构,研究了一类具有未知增长率的上三角非线性多智能体系统的协同控制问题。首先,通过引入坐标变换,将原系统转换为带有时间增益的等效系统;其次,针对标称系统进行控制器设计,获得原系统的分布式时变状态反馈控制器。所设计的控制器能够保证追随者与领导者的输出跟踪误差有界,且闭环系统所有信号有界。最后,利用仿真实例验证了控制方案的有效性。

Property(R)for Upper Triangular Operator Matrices

Property(R)holds for an operator when the complement in the approximate point spectrum of the Browder essential approximate point spectrum coincides with the isolated points of the spectrum which are eigenvalues of finite multiplicity.Let A∈B(H)and B∈B(K),where H and K are complex infinite dimensional separable Hilbert spaces.We denote by M_(C)the operator acting on H⊕K of the form M_(C)=(AC0B).In this paper,we give a sufficient and necessary condition for M_(C)∈(R)for all C∈B(K,H).

单位上三角矩阵群的多项式自同构

幂零群是群论中的一类重要研究对象,其中单位上三角矩阵群由于其幂零类为n-1,在幂零群中有极为重要的地位.而多项式自同构由于其多项式特性,更是受到群论学者的特别关注,本文通过换位子计算得出有限域上的3阶单位上三角矩阵群的每个多项式自同构一定是内自同构.

Obtaining Crisp Priorities for Triangular and Trapezoidal Fuzzy Judgments

This paper proposes anoptimal fuzzy-based model for obtaining crisp priorities for Fuzzy-AHP comparison matrices.Crisp judgments cannot be given for real-life situations,as most of these include some level of fuzziness and com-plexity.In these situations,judgments are represented by the set of fuzzy numbers.Most of the fuzzy optimization models derive crisp priorities for judgments repre-sented with Triangular Fuzzy Numbers(TFNs)only.They do not work for other types of Triangular Shaped Fuzzy Numbers(TSFNs)and Trapezoidal Fuzzy Numbers(TrFNs).To overcome this problem,a sum of squared error(SSE)based optimization model is proposed.Unlike some other methods,the proposed model derives crisp weights from all of the above-mentioned fuzzy judgments.A fuzzy number is simulated using the Monte Carlo method.A threshold-based constraint is also applied to minimize the deviation from the initial judgments.Genetic Algorithm(GA)is used to solve the optimization model.We have also conducted casestudiesto show the proposed approach’s advantages over the existingmethods.Results show that the proposed model outperforms other models to minimize SSE and deviation from initial judgments.Thus,the proposed model can be applied in various real time scenarios as it can reduce the SSE value upto 29%compared to the existing studies.

Lie Triple Derivations on Upper Triangular Matrices over a Commutative Ring

Let T（n, R） be the Lie algebra consisting of all n× n upper triangular matrices over a commutative ring R with identity 1 and M be a 2-torsion free unital T（n, R）-bimodule. In this paper, we prove that every Lie triple derivation d ： T（n, R） →M is the sum of a Jordan derivation and a central Lie triple derivation.

Weyl定理与(R)性质的判定

利用半Fredholm算子的扰动不变性,研究有界线性算子与上三角算子矩阵的Weyl型定理。首先,给出有界线性算子同时满足Browder定理和(R_(1))性质,或者同时满足Weyl定理和(R)性质的充要条件;然后,讨论上三角算子矩阵同时满足Weyl定理和(R)性质的条件。

矩阵之和Drazin逆的表示及其应用

通过将矩阵之和转化为矩阵之积的思想,利用矩阵Drazin逆的定义、性质,将和矩阵Drazin逆问题转化为三角分块矩阵的Drazin逆问题,给出了在一定条件下和矩阵Drazin逆新的表示,进而给出分块矩阵在更弱条件下Drazin逆的表示,最后通过算例来验证结果的科学性。

Possible Spectrums of 3×3 Upper Triangular Operator Matrices

Let H1, H2 and H3 be infinite dimensional separable complex Hilbert spaces. We denote by M（D,V,F） a 3×3 upper triangular operator matrix acting on Hi ＋H2＋ H3 of theform M（D,E,F）=（A D F 0 B F 0 0 C）.For given A ∈ B（H1）, B ∈ B（H2） and C ∈ B（H3）, the sets ∪D,E,F^σp（M（D,E,F））,∪D,E,F ^σr（M（D,E,F））,∪D,E,F ^σc（M（D,E,F）） and ∪D,E,F σ（M（D,E,F）） are characterized, where D ∈ B（H2,H1）, E ∈B（H3, H1）, F ∈ B（H3,H2） and σ（·）, σp（·）, σr（·）, σc（·） denote the spectrum, the point spectrum, the residual spectrum and the continuous spectrum, respectively.

无界上三角算子矩阵的谱等式

研究了无穷维复可分Hilbert空间中的2×2无界上三角算子矩阵■是满射、下方有界及可逆的充要条件,进而得到了等式σ*(T)=σ*(A)∪σ*(D)成立的充要条件,其中σ*∈{σδ,σap,σ}。这些结论推广了Du,Han及Barraa等学者在有界算子矩阵的情形下给出的充分条件。作为应用,给出了对角占优的上三角无穷维Hamilton算子可逆及谱等式成立的充要条件,并辅以实例佐证。

Minimal Rank Preserving Additive Mappings on Upper Triangular Matrices

The additive mappings that preserve the minimal rank on the algebra of all n × n upper triangular matrices over a field of characteristic 0 are characterized.

Biderivations of the Algebra of Strictly Upper Triangular Matrices over a Commutative Ring

Let Nn（R）be the algebra consisting of all strictly upper triangular n × n matrices over a commutative ring R with the identity.An R-bilinear map φ ：Nn（R）×Nn（R）→ Nn（R）is called a biderivation if it is a derivation with respect to both arguments.In this paper,we define the notions of central biderivation and extremal biderivation of Nn（R）,and prove that any biderivation of Nn（R）can be decomposed as a sum of an inner biderivation,central biderivation and extremal biderivation for n ≥ 5.