AN ADAPTIVE TRUST-REGION METHOD FOR GENERALIZED EIGENVALUES OF SYMMETRIC TENSORS

For symmetric tensors,computing generalized eigenvalues is equivalent to a homogenous polynomial optimization over the unit sphere.In this paper,we present an adaptive trustregion method for generalized eigenvalues of symmetric tensors.One of the features is that the trust-region radius is automatically updated by the adaptive technique to improve the algorithm performance.The other one is that a projection scheme is used to ensure the feasibility of all iteratives.Global convergence and local quadratic convergence of our algorithm are established,respectively.The preliminary numerical results show the efficiency of the proposed algorithm.

Generalized Inverse Eigenvalue Problem for Centrohermitian Matrices

In this paper we first consider the existence and the general form of solution to the following generalized inverse eigenvalue problem(GIEP): given a set of n-dimension complex vectors {x j}m j=1 and a set of complex numbers {λ j}m j=1, find two n×n centrohermitian matrices A,B such that {x j}m j=1 and {λ j}m j=1 are the generalized eigenvectors and generalized eigenvalues of Ax=λBx, respectively. We then discuss the optimal approximation problem for the GIEP. More concretely, given two arbitrary matrices, , ∈C n×n, we find two matrices A and B such that the matrix (A*,B*) is closest to (,) in the Frobenius norm, where the matrix (A*,B*) is the solution to the GIEP. We show that the expression of the solution of the optimal approximation is unique and derive the expression for it.

Improved Classification Approach via GEPSVM

A modified multisurface ＂proximal support vector machine classifier via generalized eigenvalues （GEPSVM for short）＂ was proposed. By defining a new principle, we designed a new classification approach via GEPSVM, namely, maximum or minimum plane distance GEPSVM （MPDGEPSVM）. Unlike GEPSVM, our approach obtains two planes by solving two simple eigenvalue problems, such that it can avoid occurrence of singular problems. Our approach, compared with GEPSVM, has better classification performalce. Moreover, MPDGEPSVM is over one order of magnitude faster than GEPSVM, and almost two orders of magnitude faster than SVM. Computational results on public datasets from UCI database illustrated the efficiency of MPDGEPSVM.

Efficient Anomalous Forcings for Linear Problems

For linear forcing problems, a method is developed to provide a set of forcing modes which form a complete orthonormal basis for the finite-time response to steady forcing in the energy inner product space. The forcing modes are found by calculating eigenvectors of a positive definite and symmetric matrix determined from given equations of motion. The amplitude of responses to forcing modes is given in terms of the associated eigenvalues. This method is used in a nondivergent barotropic model linearized about the 300 hPa zonally-varying climatological flow both for northern summertime and wintertime. The results show that the amplitude of response varies considerably with different forcing modes. Only a few of forcing modes associated with the leading eigenvalues, called efficient forcing mode, can excite significant response. The efficient forcing modes possess highly localized spatial structure with wavetrain appearance. Most of the efficient forcings are located to the south of regions of the jet cores. The forcings located over polar regions are also efficient. In addition, the response is larger in wintertime than in summertime for a given forcing.

Tripartite continuous-variable entanglement of NOPA system

We investigate the fundamental limits to the achievable tripartite continuous-variable （CV） entanglement criterion of a generalized Vl criterion. Our numerical simulation results show that the non-degenerate eigenvalues do effect the performances of the estimated minimum variances. From below the threshold to above the threshold, with the increase of the pump parameter, the tripartite CV entanglement gradually disappears. The different off-diagonal elements seriously distort the weights for entanglement. We can obtain a good tripartite CV entanglement by appropriately controlling the values of off-diagonal elements eij.

The Magnetic Field as Trigger for Disturbances of a Flow of a Newtonian Fluid through a Cylindrical Pipe with a Horizontal Axis

A Fourrier Petrov Galerkin spectral method is described for high accuracy computation of linearized dynamics for flow in a circular pipe. The code used here is based on solenoidal velocity variables. It is written in FORTRAN. Systematic studies are presented on the dependence of eigenvalues and other quantities on the axial and azimuthal wave number;the reynolds’ number Re and a new none-dimensional number Ne. The flow will be considered stable if all the real parts of the eigenvalues are negative and unstable if only one of them is positive.

Generalized Inverse Eigenvalue Problem for (P,Q)-Conjugate Matrices and the Associated Approximation Problem

In this paper,the generalized inverse eigenvalue problem for the（P,Q）-conjugate matrices and the associated approximation problem are discussed by using generalized singular value decomposition（GSVD）.Moreover,the least residual problem of the above generalized inverse eigenvalue problem is studied by using the canonical correlation decomposition（CCD）.The solutions to these problems are derived.Some numerical examples are given to illustrate the main results.

A symmetric substructuring method for analyzing the natural frequencies of conical origami structures

Conical origami structures are characterized by their substantial out-of-plane stiffness and energy-absorptioncapacity.Previous investigations have commonly focused on the static characteristics of these lightweight struc-tures.However,the efficient analysis of the natural vibrations of these structures is pivotal for designing conicalorigami structures with programmable stiffness and mass.In this paper,we propose a novel method to analyzethe natural vibrations of such structures by combining a symmetric substructuring method(SSM)and a gener-alized eigenvalue analysis.SSM exploits the inherent symmetry of the structure to decompose it into a finiteset of repetitive substructures.In doing so,we reduce the dimensions of matrices and improve computationalefficiency by adopting the stiffness and mass matrices of the substructures in the generalized eigenvalue analysis.Finite element simulations of pin-jointed models are used to validate the computational results of the proposedapproach.Moreover,the parametric analysis of the structures demonstrates the influences of the number of seg-ments along the circumference and the radius of the cone on the structural mass and natural frequencies of thestructures.Furthermore,we present a comparison between six-fold and four-fold conical origami structures anddiscuss the influence of various geometric parameters on their natural frequencies.This study provides a strategyfor efficiently analyzing the natural vibration of symmetric origami structures and has the potential to contributeto the efficient design and customization of origami metastructures with programmable stiffness.

A Note on the Hoffman-Wielandt Theorem for Generalized Eigenvalue Problems

Let {A, B} and {C, D} be diagonalizable pairs of order n, i.e., there exist invertible matrices P, Q and X, Ysuchthat A = P∧Q, B = PΩQ, C =XГY, D= X△Y, where∧ = diag（α1, α2, …, αn）, Ω= diag（βl, β2, …βn）,Г=diag（γ1,γ2,…,γn）, △=diag（δl,δ2,…,δn）.Let ρ（（α,β）, （γ,δ））=｜αδ-βγ｜/√｜α｜^2＋｜β｜^2√｜γ｜^2＋｜δ｜^2.In this paper, it will be proved that there is a permutation τ of {1,2,... ,n} such thatn∑i=1[ρ（（αi,βi）,（γτ（i）,δτ（i）））]^2≤n[1-1/κ^2（Y）κ^2（Q）（1-d2F（Z,W）/n）],where κ（Y） = ｜｜Y｜｜2｜｜Y^-1｜｜2,Z= （A,B）,W= （C, D） and dF（Z,W） = 1/√2｜｜Pz＊ -Pw＊｜｜F.

Novel passive localization algorithm based on double side matrix-restricted total least squares

In order to solve the bearings-only passive localization problem in the presence of erroneous observer position, a novel algorithm based on double side matrix-restricted total least squares （DSMRTLS） is proposed. First, the aforementioned passive localization problem is transferred to the DSMRTLS problem by deriving a multiplicative structure for both the observation matrix and the observation vector. Second, the corresponding optimization problem of the DSMRTLS problem without constraint is derived, which can be approximated as the generalized Rayleigh quotient minimization problem. Then, the localization solution which is globally optimal and asymptotically unbiased can be got by generalized eigenvalue decomposition. Simulation results verify the rationality of the approximation and the good performance of the proposed algorithm compared with several typical algorithms.

Spreading Speeds of Nonlocal KPP Equations in Heterogeneous Media

This paper is devoted to studying the asymptotic behavior of the solution to nonlocal Fisher-KPP type reaction diffusion equations in heterogeneous media.The kernel K is assumed to depend on the media.First,we give an estimate of the upper and lower spreading speeds by generalized principal eigenvalues.Second,we prove the existence of spreading speeds in the case where the media is periodic or almost periodic by showing that the upper and lower generalized principal eigenvalues are equal.