The threshold effects for a family of Friedrichs models under rank one perturbations

A family of Friedrichs models under rank one perturbations h_μ(p), p(−π,π]³, μ>0, associated to a system of two particles on the three-dimensional lattice is considered. We prove the existence of a unique eigenvalue below the bottom of the essential spectrum of h_μ(p) for all non-trivial values of p under the assumption that h_μ(0) has either a threshold energy resonance (virtual level) or a threshold eigenvalue. The threshold energy expansion for the Fredholm determinant associated to a family of Friedrichs models is also obtained.     

Time- and frequency-domain analysis using lumped mass global collocation

Quite recently, a novel global collocation method for the eigenvalue analysis of freely vibrated elastic structures was proposed (Archive of Applied Mechanics: DOI: ). This paper extends the latter methodology on several levels, in both the time and frequency domain. Firstly the formulation is updated so that it can also deal with rods of variable cross section. Then, the fully populated mass matrices of the previous formulation are properly replaced by lumped masses, thus saving still more computer effort. Subsequently, a new general formulation for the transient response analysis is proposed. Finally, a novel procedure for the coupling of two neighboring collinear rods is presented. The theory is supported by six test cases concerning elastic rods of constant and variable cross sections. Among these, transient analysis refers to the response of a single rod due to a Heaviside-type loading as well as to the impact between two collinear rods of different cross sections.     

矩阵方程AV+BW=EVF的一种新的解析通解

该文给出了矩阵方程AV+BW=EVF的一种新的解析通解。该通解是一组自由参向量的显式线性表示，其系数阵是依赖于矩阵F的特征值的数值矩阵。因通解中仅含数值矩阵计算，这为应用计算机求解创造了方便。     

Geometry of Local Adaptive Galerkin Bases

The local adaptive Galerkin bases for large-dimensional dynamical systems, whose long-time behavior is confined to a finite-dimensional manifold, are optimal bases chosen by a local version of a singular decomposition analysis. These bases are picked out by choosing directions of maximum bending of the manifold restricted to a ball of radius ɛ . We show their geometrical meaning by analyzing the eigenvalues of a certain self-adjoint operator. The eigenvalues scale according to the information they carry, the ones that scale as ɛ ² have a common factor that depends only on the dimension of the manifold, the ones that scale as ɛ ⁴ give the different curvatures of the manifold, the ones that scale as ɛ ⁶ give the third invariants, as the torsion for curves, and so on. In this way we obtain a decomposition of phase space into orthogonal spaces E _m , where E _m is spanned by the eigenvectors whose corresponding eigenvalues scale as ɛ ^m . This decomposition is analogous to the Frenet frames for curves. We also discover a practical way to compute the dimension and local structure of the invariant manifold. Accepted 14 October 1998     

On the largest kth eigenvalues of trees with n≡0 (mod k)

We consider the only remaining unsolved case n≡0 (mod k) for the largest kth eigenvalue of trees with n vertices. In 1995, Jia-yu Shao gave complete solutions for the cases k=2,3,4,5,6 and provided some necessary conditions for extremal trees in general cases (cf. [Linear Algebra Appl. 221 (1995) 131]). We further improve Shao's necessary condition in this paper, which can be seen as the continuation of [Linear Algebra Appl. 221 (1995) 131].     

Hamiltonian s-properties and eigenvalues of k-connected graphs

Chvátal and Erdös (1972) [5] proved that, for a k-connected graph G, if the stability number $\alpha (G)\le k?s$, then G is Hamilton-connected ($s=1$) or Hamiltonian ($s=0$) or traceable ($s=?1$). Motivated by the result, we focus on tight sufficient spectral conditions for k-connected graphs to possess Hamiltonian s-properties. We say that a graph possesses Hamiltonian s-properties, which means that the graph is Hamilton-connected if $s=1$, Hamiltonian if $s=0$, and traceable if $s=?1$.For a real number $a\ge 0$, and for a k-connected graph G with order n, degree diagonal matrix $D(G)$ and adjacency matrix $A(G)$, we have identified best possible upper bounds for the spectral radius ${\lambda }_1(aD(\mathrm{\Gamma })+A(\mathrm{\Gamma }))$, where Γ is either G or the complement of G, to warrant that G possesses Hamiltonian s-properties. Sufficient conditions for a graph G to possess Hamiltonian s-properties in terms of upper bounds for the Laplacian spectral radius as well as lower bounds of the algebraic connectivity of G are also obtained. Other best possible spectral conditions for Hamiltonian s-properties are also discussed.     

Dependence of eigenvalues on the diffusion operators with random jumps from the boundary

This paper deals with a non-self-adjoint eigenvalue problem

$\{\begin{array}{c}a(x)y^{″}(x)+b(x)y^{\prime }(x)=\lambda y(x),\hfill \\ y(0)={\int }_0^{1}y(x)\mathrm{d}{\nu }_0(x),y(1)={\int }_0^{1}y(x)\mathrm{d}{\nu }_1(x),\hfill \end{array}$

which is associated with the generator of one dimensional diffusions with random jumps from the boundary. We focus on the dependence of spectral gap, eigenvalues and eigenfunctions on the coefficients a, b and the probability distributions ${\nu }_0$, ${\nu }_1$. To prove this, we show that all the eigenvalues are confined to a parabolic neighborhood of the real axis. Moreover, we also prove that zero is an algebraically simple eigenvalue of the problem.     

A quantummechanical derivation of the eigenvalues of the quadratic Casimir operator of the algebra SU(n) in Young tableau representation

The paper concerns the eigenvalues and eigenfunctions of the quadratic Casimir operator of the algebra $SU(n)$ in the Young tableau representation. Our starting point is a concrete physical problem in second quantized formalism. We use quantummechanical and group theoretical vehicles to determine the aimed quantities. The tableaux are organized according to their eigenvalue. We also investigate the modification rules.