For a general vector space V , and a linear operator T , we have already asked the question “when is there a basis of V consisting only of eigenvectors of T?” – this is exactly when T is diagonalizable. Now, for an inner product space V , we know how to check whether vectors are orthogonal, and we know how to define the norms of vectors, so we can ask “when is there an orthonormal basis of V co...

This note deals with the following problem: Let A be an n X n Hermitian matrix, and Q and 0 be two n X rn (n > m > 1) matrices both with orthonormal column vectors. How do the eigenvalues of the m X m Hermitian matrix Q”AQ differ from those of the m X m Hermitian matrix QHAQ? We give a positive answer to one of the unsolved problems raised recently by Sun. In what follows, we will consider the ...

An application of the new formulation of the eigenchannel method [R. Szmytkowski, Ann. Phys. (N.Y.) 311, 503 (2004)] to quantum scattering of Dirac particles from non-local separable potentials is presented. Eigenchannel vectors, related directly to eigenchannels, are defined as eigenvectors of a certain weighted eigenvalue problem. Moreover, negative cotangents of eigenphase-shifts are introdu...

Abstract: In this paper we study the spectrum and solution expansion of the differential equation with multiple delays. Firstly, we present explicitly the asymptotic expressions of the eigenvalues under certain conditions. Then we prove that the root vectors of the system fail to form a basis for the state Hilbert space. However, by a trick, we expand the solution of the system according to the...

we investigate a class of fourth-order differential operators with eigenparameter dependent boundary conditions and transmission conditions. a self-adjoint linear operator a is defined in a suitable hilbert space h such that the eigenvalues of such a problem coincide with those of a . we discuss asymptotic behavior of its eigenvalues and completeness of its eigenfunctions. finally, we obtain th...

In this paper, we demonstrate how PCA and ICA can be used for embedding graphs in pattern-spaces. Graph spectral feature vectors are calculated from the leading eigenvalues and eigenvectors of the unweighted graph adjacency matrix. The vectors are then embedded in a lower dimensional pattern space using both the PCA and ICA decomposition methods. Synthetic and real sequences are tested using th...

In recent years, many types of research have continued to improve the environment human speech and emotion recognition. As facial recognition has gradually matured through recognition, result this study provided more accurate complex emotional performance, identification will be derived from subjective interpretation into use computers automatically interpret speaker’s expression. Focused on in...

This paper describes a technique for characterization and segmentation of anisotropic patterns that exhibit a single local orientation. Using Gaussian derivatives we construct a gradient-square tensor at a selected scale. Smoothing of this tensor allows us to combine information in a local neighborhood without canceling vectors pointing in opposite directions. Whereas opposite vectors would can...

the purpose of this paper is to study the higher order asymptotic distributions of the eigenvalues associated with a class of sturm-liouville problem with equation of the form w??=(?2f(x)?r(x)) (1), on [a,b, where ? is a real parameter and f(x) is a real valued function in c2(a,b which has a single zero (so called turning point) at point 0x=x and r(x) is a continuously differentiable function. ...

This chapter serves as a brief introduction to the supervised learning setting and kernel methods. Moreover, several results from linear algebra, probability theory, and functional analysis are reviewed which will be used throughout the thesis. 2.1 Some notational conventions We begin by introducing some basic notational conventions. The sets N, Z, R, C denote the natural, integer, real, and co...