Abstract. Increase or decrease of “the second eigenvalue (the second smallest real part of all eigenvalues)” of the graph Laplacian caused by the addition of an edge to directed graphs is considered. It is proved that the addition of an upstrem edge to a directed path or a directed tree graph decreases the second eigenvalue, while the addition of a downstream edge does not. The extension of those results to the edge-addtion in a directed acyclic graph with a spanning tree is also discussed.

Abstract. We perform numerical verification of subharmonic solutions in the Ikeda delay differential equation with an external force term, a model for optical resonance systems. The Ikeda delay differential equation treated in this paper is characterized by the inclusion of trigonometric functions as nonlinear terms, and conventional numerical verification method, which are effective when the nonlinear terms are of polynomial type, cannot be applied as is. Therefore, we treat the nonlinear terms by numerical integration using the Euler-Maclaurin formula for numerical verification.

Abstract. Logarithms of matrices defined as inverse function of exponentials of matrices arise in control theory and others. One of the algorithm that evaluate the logarithm called inverse scaling and squaring is constructed with matrix square root and Padé approximants, In this algorithm, product form DB iteration that is a variation of Newton’s method is applied to matrix square root. However, the algorithm may cause significant digit loss. So in this article, we propose a new iteration based on product form DB iteration to avoid the significant digit loss, and show the results of numerical experiments to compare their iterations.