Abstract:
This paper presents some proofs of Kharitonov's theorem and its generalized version,andmakes some comparisons among them.By resorting to the concept of positive pair of polyno-mials,we give a quite simple proof of Kharitonov's theorem.We naturally verify the general-ized Kharitonov's theorem by introducing the so-called Kharitonov's combination.A Nyquist-typeproof of the generalized version is also given,which is the simplest proof appeared in litera-ture so far.The proofs of the generalized version can easily be reduced to the proofs of theoriginal theorem.These proofs provide some insights into the intrinsic characteristics ofKharitonov's theorem.

Abstract:
According to Kharitonov theorem, an expected characteristic polynomial of the close loop systems is given and the simultaneous stabilization problem of linear systems with same order is transformed into solving problem of a set of inequalities. And then a linear programing method for the inequalities is studied and third kinds of solutions of the problem and their properties are discussed. Examples demonstrate the effectiveness of the stabilization method.

Abstract:
We consider $N\times N$ Hermitian or symmetric random matrices with independent entries. The distribution of the $(i,j)$-th matrix element is given by a probability measure $\nu_{ij}$ whose first two moments coincide with those of the corresponding Gaussian ensemble. We prove that the joint probability distribution of the components of eigenvectors associated with eigenvalues close to the spectral edge agrees with that of the corresponding Gaussian ensemble. For eigenvectors associated with bulk eigenvalues, the same conclusion holds provided the first four moments of the distribution $\nu_{ij}$ coincide with those of the corresponding Gaussian ensemble. More generally, we prove that the joint eigenvector-eigenvalue distributions near the spectral edge of two generalized Wigner ensembles agree, provided that the first two moments of the entries match and that one of the ensembles satisfies a level repulsion estimate. If in addition the first four moments match then this result holds also in the bulk.

Abstract:
In this paper, we study the frequency response of uncertain systems using Kharitonov stability theory on first order complex polynomial set. For an interval transfer function, we show that the minimal real part of the frequency response at any fixed frequency is attained at some prescribed vertex transfer functions. By further geometric and algebraic analysis, we identify an index for strict positive realness of interval transfer functions. Some extensions and applications in positivity verification and robust absolute stability of feedback control systems are also presented.

Abstract:
in this short note a sensitivity result for quadratic semidefinite programming is presented under a weak form of second order sufficient condition. based on this result, also the local convergence of a sequential quadratic semidefinite programming algorithm extends to this weak second order sufficient condition. mathematical subject classification: 90c22, 90c30, 90c31, 90c55.

Abstract:
We propose a model-independent analysis of the neutrino mass matrix through an expansion in terms of the eigenvectors defining the lepton mixing matrix, which we show can be parametrized as small perturbations of the tribimaximal mixing eigenvectors. This approach proves to be powerful and convenient for some aspects of lepton mixing, in particular when studying the sensitivity of the mass matrix elements to departures from their tribimaximal form. In terms of the eigenvector decomposition, the neutrino mass matrix can be understood as originating from a tribimaximal dominant structure with small departures determined by data. By implementing this approach to cases when the neutrino masses originate from different mechanisms, we show that the experimentally observed structure arises very naturally. We thus claim that the observed deviations from the tribimaximal mixing pattern might be interpreted as a possible hint of a ``hybrid'' nature of the neutrino mass matrix.

Abstract:
How close are Galerkin eigenvectors to the best approximation available out of the trial subspace ? Under a variety of conditions the Galerkin method gives an approximate eigenvector that approaches asymptotically the projection of the exact eigenvector onto the trial subspace -- and this occurs more rapidly than the underlying rate of convergence of the approximate eigenvectors. Both orthogonal-Galerkin and Petrov-Galerkin methods are considered here with a special emphasis on nonselfadjoint problems. Consequences for the numerical treatment of elliptic PDEs discretized either with finite element methods or with spectral methods are discussed and an application to Krylov subspace methods for large scale matrix eigenvalue problems is presented. New lower bounds to the $sep$ of a pair of operators are developed as well.

Abstract:
We investigate the evolution of a given eigenvector of a symmetric (deterministic or random) matrix under the addition of a matrix in the Gaussian orthogonal ensemble. We quantify the overlap between this single vector with the eigenvectors of the initial matrix and identify precisely a "Cauchy-flight" regime. In particular, we compute the local density of this vector in the eigenvalues space of the initial matrix. Our results are obtained in a non perturbative setting and are derived using the ideas of [O. Ledoit and S. P\'ech\'e, Prob. Th. Rel. Fields, {\bf 151} 233 (2011)]. Finally, we give a robust derivation of a result obtained in [R. Allez and J.-P. Bouchaud, Phys. Rev. E {\bf 86}, 046202 (2012)] to study eigenspace dynamics in a semi-perturbative regime.

Abstract:
We characterize the contractions that are similar to the backward shift in the Hardy space $H^2$. This characterization is given in terms of the geometry of the eigenvector bundles of the operators.

Abstract:
We describe two algorithms for the eigenvalue, eigenvector problem which, on input a Gaussian matrix with complex entries, finish with probability 1 and in average polynomial time.