Introduction. In this note we develop a multistep formulation of the optimized Lax-Wendroff method for hyperbolic systems. This scheme was derived by Strang [6], [7]. The present formulation extends in a natural way, the two-step formulation of Richtmyer [5] for systems in one-space variable. We summarise this case in the next section. One Space Dimension. We consider the first-order conservati...

Progresses on the dynamical system theory of 2D Euler equation are reported. An open problem is presented. The isospectral theory on Lax pairs of Euler equations is initiated.

We discuss algebraic and analytic structure of rational Lax operators. With algebraic reductions of Lax equations we associate a reduction group-a group of twisted automor-phisms of the corresponding infinite dimensional Lie algebra. We present a complete study of dihedral reductions for sl(2, C) Lax operators with simple poles and corresponding integrable equations. In the last section we give...

Probabilistic bisimulation is a widely studied equivalence relation for stochastic systems. However, it requires the behavior of the states to match on actions with matching labels. This does not allow bisimulation to capture symmetries in the system. In this thesis we define lax probabilistic bisimulation, in which actions are only required to match within given action equivalence classes. We ...

BACKGROUND The importance of membrane compartmentalization into specific membrane microdomains has been shown in many biological processes such as immunoreceptor signaling, membrane trafficking, pathogen infection, and tumor progression. Microdomains like lipid rafts, caveolae and tetraspanin enriched microdomains are relatively resistant to solubilization by some detergents. Large detergent-re...

We define a Lax operator as monic pseudodifferential L(∂) of order N ≥ 1, such that the equations $$\frac{\partial L(\partial)}{\partial t_k}=[(L^\frac{k}{N}(\partial))_+,L(\partial)]$$ are consistent and non-zero for infinitely many positive integers k. Consistency an equation means its flow is defined by evolutionary vector field. In present paper we demonstrate traditional theory KP N-th KdV...

Having as starting point Barr’s description of topological spaces as lax algebras for the ultrafilter monad [2], in this paper we present further topological examples of lax algebras – such as quasi-metric spaces, approach spaces and quasi-uniform spaces – and show that, in a suitable setting, the categories of lax algebras have indeed a topological nature. Furthermore, we generalize to this se...

We propose a Lax equation for the non-linear sigma model which leads directly to the conserved local charges of the system. We show that the system has two infinite sets of such conserved charges following from the Lax equation, much like dispersionless systems. We show that the system has two Hamiltonian structures which are compatible so that it is truly a bi-Hamiltonian system. However, the ...

A three-step method due to Nijhoff and Bobenko & Suris to derive a Lax pair for scalar partial difference equations (P Es) is reviewed. The method assumes that the P Es are defined on a quadrilateral, and consistent around the cube. Next, the method is extended to systems of P Es where one has to carefully account for equations defined on edges of the quadrilateral. Lax pairs are presented for ...

A supersymmetric version is proposed for the well known Harry Dym system. A general class super Lax operator which leads to consistent equations is considered.