An improved method is developed for eigenvalue and eigenvector placement of a closed-loop control system using either state or output feedback. The method basically consists of three steps. First, the singular value or QR decomposition is used to generate an orthonormal basis that spans admissible eigenvector space corresponding to each assigned eigenvalue. Second, given a unitary matrix, the e...

A method for the computation of eigenfrequencies and eigenmodes of fractal drums is presented. The approach involves first mapping the unit disk to a polygon approximating the fractal and then solving a weighted eigenvalue problem on the unit disk by a spectral collocation method. The numerical computation of the complicated conformal mapping was made feasible by the use of the fast multipole m...

Keywords: Upper and lower solutions p-Laplacian operator Fractional differential equation Integral boundary condition Eigenvalue a b s t r a c t In this paper, we are concerned with the eigenvalue problem of a class of singular p-Lapla-cian fractional differential equations involving the Riemann–Stieltjes integral boundary condition. The conditions for the existence of at least one positive sol...

If A is an nxn complex matrix and λ is an eigenvalue of A with geometric multiplicity k, then λ is in at least k of the Geršgorin discs Di of A. Let k, r, t be positive integers with k ≤ r ≤ t . Then there is a t x t complex matrix A and an eigenvalue λ of A such that λ has geometric multiplicity k and algebraic multiplicity t, and λ is in precisely r Geršgorin Discs of A. Some examples and rel...

(1.2) { −∆pu = λa(x)|u|p−2u, u ∈ D 0 (Ω), has the least eigenvalue λ1 > 0 with a positive eigenfunction e1 and λ1 is the only eigenvalue having this property (cf. Proposition 3.1). This gives us a possibility to study the existence of an unbounded branch of positive solutions bifurcating from (λ1, 0). When Ω is bounded, the result is well-known, we refer to the survey article of Amann [2] and t...

The height at which an unloaded column will buckle under its own weight is the fourth root of the least eigenvalue of a certain Sturm–Liouville operator. We show that the operator associated with the column proposed by Keller and Niordson [J. Math. Mech., 16 (1966), pp. 433– 446] does not possess a discrete spectrum. This calls into question their formal use of perturbation theory, so we consid...

We describe extensive computational experiments on spectral properties of random objects random cubic graphs, random planar triangulations, and Voronoi and Delaunay diagrams of random (uniformly distributed) point sets on the sphere). We look at bulk eigenvalue distribution, eigenvalue spacings, and locality properties of eigenvectors. We also look at the statistics of nodal domains of eigenvec...

In this paper, we give necessary and suucient optimality conditions which are easy veriied for the local solution of Celis-Dennis-Tapia sub-problem (CDT subproblem) where the Hessian at this local solution has one negative eigenvalue. If CDT subproblem has no global solution with Hessian of Lagrangian positive semi-deenite, the Hessian of Lagrangian has at least one negative eigenvalue. It is v...

Partially answering a question of Paul Seymour, we obtain a sufficient eigenvalue condition for the existence of k edge-disjoint spanning trees in a regular graph, when k ∈ {2, 3}. More precisely, we show that if the second largest eigenvalue of a d-regular graph G is less than d − 2k−1 d+1 , then G contains at least k edge-disjoint spanning trees, when k ∈ {2, 3}. We construct examples of grap...

An explicit a priori bound for the condition number associated to each of the following problems is given: general linear equation solving, least squares, non-symmetric eigenvalue problems, solving univariate polynomials, and solving systems of multivariate polynomials. It is assumed that the input has integer coefficients and is not on the degeneracy locus of the respective problem (i.e., the ...