On the use of two QMR algorithms for solving singular systems and applications in Markov chain modeling

Recently, Freund and Nachtigal proposed the quasi-minimal residual algorithm (QMR) for solving general nonsingular non-Hermitian linear systems. The method is based on the Lanczos process, and thus it involves matrix—vector products with both the coefficient matrix of the linear system and its transpose. Freund developed a variant of QMR, the transpose-free QMR algorithm (TFQMR), that only requires products with the coefficient matrix. In this paper, the use of QMR and TFQMR for solving singular systems is explored. First, a convergence result for the general class of Krylov-subspace methods applied to singular systems is presented. Then, it is shown that QMR and TFQMR both converge for consistent singular linear systems with coefficient matrices of index 1. Singular systems of this type arise in Markov chain modeling. For this particular application, numerical experiments are reported.     

Lattice vibrational properties of uranium pnictides

Lattice vibrational properties of uranium pnictides have been studied using breathing shell model (BSM) which includes breathing motion of electrons of the U-atoms due tof−d hybridization. The phonon dispersion curves of U-pnicitides calculated from the present model agree reasonably well with the measured data. A comparison has been made between BSM and our results reported earlier obtained from three-body force rigid ion model to reveal the importance of the short-range electron-phonon interactions in these compounds. We also report, for the first time, the two phonon density of states and specific heat for these compounds.     

图和线图的谱性质

Let G be a simple connected graph with n vertices and m edges,Lo be the line graph of G and λ1(LG)≥λ2 (LG)≥...≥λm(LG) be the eigenvalues of the graph LG,.. In this paper, the range of eigenvalues of a line graph is considered. Some sharp upper bounds and sharp lower bounds of the eigenvalues of Lc. are obtained. In oarticular,it is oroved that-2cos(π/n)≤λn-1(LG)≤n-4 and λn(LG)=-2 if and only if G is bipartite.     

Large families of laplacian isospectral graphs

Two graphs are isomorphic only if they are Laplacian isospectral, that is, their Laplacian matrices share the same multiset of eigenvalues. Large families of nonisomorphic Laplacian isospectral graphs are exhibited for which the common multiset of eigenvalues consists entirely of integers.     

单模光纤二阶偏振模色散的Jones矩阵

利用单模光纤PMD矢量与Jones矩阵之间的关系,推导了由该矢量表示的光纤二阶PMD的Jones矩阵解析表达式,利用该矩阵可以确定输出端时域脉冲在二阶PMD近似下的表达式,并由此仿真二阶PMD对信号传输质量的影响.通过比较输入与输出信号脉冲波形和眼图发现,随着传输速率和传输距离的增加输出信号劣化.最后结果表明,当主偏振态的旋转速率为零时,二阶PMD的影响可以忽略;但当其值增大时,输出信号劣化严重.在光纤通信系统设计时,需要考虑偏振模色散影响分析结果,可提供参考依据.     

Stabilization of stochastic singular nonlinear hybrid systems

This paper deals with the control of the class of singular nonlinear stochastic hybrid systems. Under some appropriate assumptions, results on stochastic stability and stochastic stabilization are developed. Two state feedback controllers (linear and nonlinear) that stochastically stabilize the class of systems we are considering are designed. LMI sufficient conditions are developed to compute the gains of these controllers.     

The numerical range of a nonnegative matrix

We offer an almost self-contained development of Perron–Frobenius type results for the numerical range of an (irreducible) nonnegative matrix, rederiving and completing the previous work of Issos, Nylen and Tam, and Tam and Yang on this topic. We solve the open problem of characterizing nonnegative matrices whose numerical ranges are regular convex polygons with center at the origin. Some related results are obtained and some open problems are also posed.     

CRITERION FOR SL（2,Z）－MATRIX TO BE CONJUGATE TO ITS INVERSE

This paper gives a necessary and sufficient condition for a matrix in SL(2, Z) to be conjugate to its inverse. This condition reduces the determination of the conjugation to solving some indeterminate equation of second degree. It yields an algorithm to determine this conjugation in finite steps based on the elementary number theory.     

矩阵系统扰动后的稳定性问题

本文证明：若矩阵系统是广义各态历经的，则系统是稳定的；若矩阵系统是善真的，则系统是条件稳定的。     

A Sharp Upper Bound for the Number of Spanning Trees of a Graph

Let G = (V,E) be a simple graph with n vertices, e edges and d₁ be the highest degree. Further let λ_i, i = 1,2,...,n be the non-increasing eigenvalues of the Laplacian matrix of the graph G. In this paper, we obtain the following result: For connected graph G, λ₂ = λ₃ = ... =  λ_n-1 if and only if G is a complete graph or a star graph or a (d₁,d₁) complete bipartite graph. Also we establish the following upper bound for the number of spanning trees of G on n, e and d₁ only: