The problem of reaching consensus in multiagent second-order systems without a spanning outgoing tree in the dependency digraph is considered. A theorem stating that the asymptotic behavior of the system is uniquely determined by the eigenprojector of the Laplacian matrix of the dependency digraph is proved. The earlier results established by the author and independently by Ren and Atkins in their papers are further generalized. For the case in which the dependency digraph contains no spanning outgoing tree, a regularization method is proposed.
The paper is concerned with scheduling the two-way traffic between two stations connected by a single-track railway with a siding. It is shown that if, for each station, the order in which trains leave this station is known or can be found, then for various objective functions an optimal schedule can be constructed in polynomial time using the method of dynamic programming. Based on this result, the paper also presents a polynomial-time algorithm minimising the weighted number of late trains.
Foreword to the thematical issue devoted to the seventieth anniversary of Academician V.S. Tanaev
We consider a sequence of Markov chains that weakly converge to a diffusion process. We assume that the trend contains a linearly growing component. The usual parametrix method does not apply since the trend is unbounded. We show how to modify the parametrix method in order to get local limit theorems in this case.