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Rank of Linear and Quadratic Combinations of Matrices
No full textIn this paper, the rank of some combinations of matrices is analysed. In particular, the rank of all matrices on the line joining two rank matrices is characterized, and the rank of convex combinations of two matrices and quadratic combinations of three matrices is studied. Presented results concern the problem of robustness of rank under certain kinds of perturbations of a matrix
On the Existence of Hurwitz Polynomials with no Hadamard Factorization
No full textA Hurwitz stable polynomial of degree has a Hadamard factorization if it is a Hadamard product (i.e., element-wise multiplication) of two Hurwitz stable polynomials of degree . It is known that Hurwitz stable polynomials of degrees less than four have a Hadamard factorization. It is shown that, for arbitrary , there exists a Hurwitz stable polynomial of degree which does not have a Hadamard factorization
On the vertex-face graphs of triangulations
No full textLet be a triangulation with vertex set and edge set embedded on an orientable surface with genus . Define to be the graph obtained from by inserting a new vertex to each face of and adding three new edges and , where and are the three vertices on the boundary of . Let be the graph obtained from by deleting all edges in of . In this paper, first some spectral properties of and are considered, then it is proved that and , where is the number of spanning trees of . As applications, the number of spanning trees and Kirchhoff indices of some lattices in the context of statistical physics are obtained
A general method to obtain the spectrum and local spectra of a graph from its regular partitions
No full textIt is well known that, in general, part of the spectrum of a graph can be obtained from the adjacency matrix of its quotient graph given by a regular partition. In this paper, a method that gives all the spectrum, and also the local spectra, of a graph from the quotient matrices of some of its regular partitions, is proposed. Moreover, from such partitions, the -local multiplicities of any class of vertices is also determined, and some applications of these parameters in the characterization of completely regular codes and their inner distributions are described. As examples, it is shown how to find the eigenvalues and (local) multiplicities of walk-regular, distance-regular, and distance-biregular graphs.
 
On Properties of Semipositive Cones and Simplicial Cones
No full textFor a given nonsingular matrix , the cone , and its subcone lying on the positive orthant, called as semipositive cone, are considered. If the interior of the semipositive cone is not empty, then is named as semipositive matrix. It is known that is a proper polyhedral cone. In this paper, it is proved that is a simplicial cone and properties of its extremals are analyzed. An one-one relation between simplicial cones and invertible matrices is established. For a proper cone in , denotes the collection of matrices that leave invariant. For a given minimally semipositive matrix (no column-deleted submatrix is semipositive) , it is shown that the invariant cone is a simplicial cone
Rewilding the night sky: Mitigating the costs of light pollution for bats and insects
No full textAltering the LED street lighting regime in Colter Bay, Grand Teton National Park from warm white to red, in short-term blocks (3-7 days per color) substantially reduces attraction of nocturnal arthropods but has little influence on bat space use. We recommend research on long-term application of this mitigation approach and investigation of lower intensity levels.
Featured photo taken from Figure 2 of the report
Graphs that are cospectral for the distance Laplacian
No full textThe distance matrix of a graph is the matrix containing the pairwise distances between vertices, and the distance Laplacian matrix is , where is the diagonal matrix of row sums of . Several general methods are established for producing -cospectral graphs that can be used to construct infinite families. Examples are provided to show that various properties are not preserved by -cospectrality, including examples of -cospectral strongly regular and circulant graphs. It is established that the absolute values of coefficients of the distance Laplacian characteristic polynomial are decreasing, i.e., , where is the coefficient of .
 
The interesting spectral interlacing property for a certain tridiagonal matrix
No full textIn this paper, a new tridiagonal matrix, whose eigenvalues are the same as the Sylvester-Kac matrix of the same order, is provided. The interest of this matrix relies also in that the spectrum of a principal submatrix is also of a Sylvester-Kac matrix given rise to an interesting spectral interlacing property. It is proved alternatively that the initial matrix is similar to the Sylvester-Kac matrix
Moore-Penrose inverse of some linear maps on infinite-dimensional vector spaces
No full textThe aim of this work is to characterize linear maps of infinite-dimensional inner product spaces where the Moore-Penrose inverse exists. This MP inverse generalizes the well-known Moore-Penrose inverse of a matrix . Moreover, a method for the computation of the MP inverse of some endomorphisms on infinite-dimensional vector spaces is given. As an application, the least norm solution of an infinite linear system from the Moore-Penrose inverse offered is studied
On inequalities for A-numerical radius of operators
No full textLet be a positive operator on a complex Hilbert space Inequalities are presented concerning upper and lower bounds for -numerical radius of operators, which improve on and generalize the existing ones, studied recently in [A. Zamani. A-Numerical radius inequalities for semi-Hilbertian space operators. Linear Algebra Appl., 578:159--183, 2019]. Also, some inequalities are obtained for -numerical radius of operator matrices, where is the diagonal operator matrix whose diagonal entries are . Further, upper bounds are obtained for -numerical radius for product of operators, which improve on the existing bounds