A Formula for Eigenvalues of Jacobi Matrices with a Reflection Symmetry

The spectral properties of two special classes of Jacobi operators are studied. For the first class represented by the 2M-dimensional real Jacobi matrices whose entries are symmetric with respect to the secondary diagonal, a new polynomial identity relating the eigenvalues of such matrices with their matrix entries is obtained. In the limit M→∞ this identity induces some requirements, which should satisfy the scattering data of the resulting infinite-dimensional Jacobi operator in the half-line, of which super- and subdiagonal matrix elements are equal to -1. We obtain such requirements in the simplest case of the discrete Schrödinger operator acting in l2(N), which does not have bound and semibound states and whose potential has a compact support.

The finite basis property of a certain semigroup of upper triangular matrices over a field

Let [Formula: see text] be the semigroup of all upper triangular [Formula: see text] matrices over a field [Formula: see text] whose main diagonal entries are [Formula: see text]s and/or [Formula: see text]s. Volkov proved that [Formula: see text] is nonfinitely based as both a plain semigroup and an involution semigroup under the reflection with respect to the secondary diagonal. In this paper, we shall prove that [Formula: see text] is finitely based for any field [Formula: see text]. When [Formula: see text], this result partially answers an open question posed by Volkov.

Analysis of Image Texture Features Based on Gray Level Co-Occurrence Matrix

Gray level co-occurrence matrix (GLCM) is a second-order statistical measure of image grayscale which reflects the comprehensive information of image grayscale in the direction, local neighborhood and magnitude of changes. Firstly, we analyze and reveal the generation process of gray level co-occurrence matrix from horizontal, vertical and principal and secondary diagonal directions. Secondly, we use Brodatz texture images as samples, and analyze the relationship between non-zero elements of gray level co-occurrence matrix in changes of both direction and distances of each pixels pair by. Finally, we explain its function of the analysis process of texture. This paper can provided certain referential significance in the application of using gray level co-occurrence matrix at quality evaluation of texture image.

Research on Characteristic Properties of Gray Level Co-Occurrence Matrix

Gray level co-occurrence matrix (GLCM) is a second-order statistical measurement. In order to understand the characterization degree of GLCM’s different feature properties, we use images of Brodatz texture images as experimental samples, analyze the change process of feature properties in horizontal, vertical and principal and secondary diagonal directions under the situation of some elements’ dynamic changes such as distance of pixels pair, size of moving window and gray level quantization,. By analyzing the experimental results, this paper can provided certain referential significance in how to select feature properties reasonable in the application of image retrieval and classification and identification which are based on using GLCM as feature.

III.—Note on the Determinant whose Matrix is the Sum of Two Circulant Matrices

Perhaps the most interesting theorem as yet known in regard to circulants concerns the determinant whose matrix is the sum of two circulant matrices, one of the latter being taken symmetric with respect to the primary diagonal and the other with respect to the secondary diagonal: for example, the determinantwhose matrix is the sum of the matricesthat is to say, the matrices of