We consider a model of neural and gene networks where the nonlinearities in the system of differential equations are discontinuous and piecewise constant. We develop a framework for study of such systems. As a first step we associate to the system a graph G on a hypercube and show how the collection of strongly connected components of G relates to the dynamics of the flow on the set of rays thr...

Kengo Matsumoto has introduced λ-graph systems and strong shift equivalence of λ-graph systems [Doc.Math.4 (1999), 285-340]. We associate to a subshift of a subshift a λ-graph system. The strong shift equivalence class of the associated λ-graph system is an invariant of subsystem equivalence. Wolfgang Krieger and Kengo Matsumoto have introduced the λ-entropy of a λ-graph system and have shown i...

This paper deals with the modeling of bond graph buck converter systems. The bond graph formalism, which represents a heterogeneous formalism for physical modeling, is used to design a sub-model of a power MOSFET and PiN diode switchers. These bond graph models are based on the device’s electrical elements. The application of these models to a bond graph buck converter permit us to obtain an in...

My research involves finding bounds of graph theoretic invariants, in particular the independence number of a graph; constructing algorithms for calculating values of these invariants; investigating the structure of independent sets of vertices in graphs; and identifying and programming successful heuristics for automating conjecture-making. of these conjectures have involved the independence n...

We generalize the order-theoretic variant of the Myhill-Nerode theorem to graph languages, and characterize the recognizable graph languages as the class of languages for which the Myhill-Nerode quasi order is a well quasi order. In the second part of the paper we restrict our attention to graphs of bounded interface size, and use Myhill-Nerode quasi orders to verify that, for such bounded grap...

Symbolic dynamics is partly the study of walks in a directed graph. By a walk, here we mean a morphism to the graph from the Cayley graph of the monoid of non-negative integers. Sets of these walks are also important in other areas, such as stochastic processes, automata, combinatorial group theory, C∗-algebras, etc. We put a Quillen model structure on the category of directed graphs, for which...

We study diffusions, variational principles and associated boundary value problems on directed graphs with natural weightings. We associate to certain subgraphs (domains) a pair of sequences, each of which is invariant under the action of the automorphism group of the underlying graph. We prove that these invariants differ by an explicit combinatorial factor given by Stirling numbers of the fir...

The embedded template is a geometric tool in dynamics being used to model knots and links as periodic orbits of $3$-dimensional flows. We prove that for an $S^3$ with fixed homeomorphism type, its boundary trivalent spatial graph complete isotopic invariant. Moreover, we construct invariant templates by Kauffman's graphs, which set links. As application, the classification simple Smale flows on...

Invariant manifolds provide the geometric structures for describing and understanding dynamics of nonlinear systems. The theory of invariant manifolds for both finite and infinite dimensional autonomous deterministic systems, and for stochastic ordinary differential equations is relatively mature. In this paper, we present a unified theory of invariant manifolds for infinite dimensional random ...

Colin de Verdi ere introduced an interesting linear algebraic invariant (G) of graphs. He proved that (G) 2 if and only if G is outerplanar, and (G) 3 if and only if G is planar. We prove that if the complement of a graph G on n nodes is outerplanar, then (G) n ? 4, and if it is planar, then (G) n ? 5. We give a full characterization of maximal planar graphs with (G) = n ? 5. In the opposite di...