Cluster formation in mesoscopic systems

Graph-theoretical approach is used to study cluster formation in mesocsopic systems. Appearance of these clusters are due to discrete resonances which are presented in the form of a multigraph with labeled edges. This presentation allows to construct all non-isomorphic clusters in a finite spectral domain and generate corresponding dynamical systems automatically. Results of MATHEMATICA implementation are given and two possible mechanisms of cluster destroying are discussed

Discretization strategies for computing Conley indices and Morse decompositions of flows

Conley indices and Morse decompositions of flows can be found by using algorithms which rigorously analyze discrete dynamical systems. This usually involves integrating a time discretization of the flow using interval arithmetic. We compare the old idea of fixing a time step as a parameters to a time step continuously varying in phase space. We present an example where this second strategy necessarily yields better numerical outputs and prove that our outputs yield a valid Morse decomposition of the given flow

Generalized Integer Partitions, Tilings of Zonotopes and Lattices

In this paper, we study two kinds of combinatorial objects, generalized integer partitions and tilings of two dimensional zonotopes, using dynamical systems and order theory. We show that the sets of partitions ordered with a simple dynamics, have the distributive lattice structure. Likewise, we show that the set of tilings of zonotopes, ordered with a simple and classical dynamics, is the disjoint union of distributive lattices which we describe. We also discuss the special case of linear integer partitions, for which other dynamical systems exist. These results give a better understanding of the behaviour of tilings of zonotopes with flips and dynamical systems involving partitions.Comment: See http://www.liafa.jussieu.fr/~latapy

Graph Kernels via Functional Embedding

We propose a representation of graph as a functional object derived from the power iteration of the underlying adjacency matrix. The proposed functional representation is a graph invariant, i.e., the functional remains unchanged under any reordering of the vertices. This property eliminates the difficulty of handling exponentially many isomorphic forms. Bhattacharyya kernel constructed between these functionals significantly outperforms the state-of-the-art graph kernels on 3 out of the 4 standard benchmark graph classification datasets, demonstrating the superiority of our approach. The proposed methodology is simple and runs in time linear in the number of edges, which makes our kernel more efficient and scalable compared to many widely adopted graph kernels with running time cubic in the number of vertices