In this paper, Jordan–Kronecker invariants are calculated for all nilpotent 6- and 7-dimensional Lie algebras. We consider the Poisson bracket family, depending on lambda parameter a coalgebra, i.e., linear space dual to algebra. For some

Time-changed Lévy processes include the fractional Poisson process, and the scaling limit of a continuous time random walk. They are obtained by replacing the deterministic time variable by a positive non-decreasing random process. The use of time-changed processes in modeling often requires the knowledge of their second order properties such as the correlation function. This paper provides the...

Consider a Poisson structure on C∞(R3,R) with bracket {, } and suppose that C is a Casimir function. Then {f, g} =< ∇C, (∇g × ∇f) > is a possible Poisson structure. This confirms earlier observations concerning the Poisson structure for Hamiltonian systems that are reduced to a one degree of freedom system and generalizes the Lie-Poisson structure on the dual of a Lie algebra and the KKS-symple...

We define the (second) Adler-Gelfand-Dickey Poisson structure on differential operators over an elliptic curve and classify symplectic leaves of this structure. This problem turns out to be equivalent to classification of coadjoint orbits for double loop algebras, conjugacy classes in loop groups, and holomorphic vector bundles over the elliptic curve. We show that symplectic leaves have a fini...

Following the revival of interest in the behavior of classical, nonlinear systems, a variety of new theoretical tools was developed to allow for a description of the essential properties of those systems. One of the most powerful mathematical methods developed recently is the Lie-Poisson bracket technique [1], the use of which has led to considerable progress in our understanding of conservativ...

We study a renewal risk model in which the surplus process of the insurance company is modeled by a compound fractional Poisson process. We establish the long-range dependence property of this non-stationary process. Some results for the ruin probabilities are presented in various assumptions on the distribution of the claim sizes. AMS 2000 subject classifications: Primary 60G22, 60G55, 91B30; ...

As a generalization of Postnikov’s construction [P], we define a map from the space of edge weights of a directed network in an annulus into a space of loops in the Grassmannian. We then show that universal Poisson brackets introduced for the space of edge weights in [GSV3] induce a family of Poisson structures on rational-valued matrix functions and on the space of loops in the Grassmannian. I...

This paper provides a precise sense in which the time t map for the Euler equations of an ideal fluid in a region in Rn (or a smooth compact n-manifold with boundary) is a Poisson map relative to the Lie-Poisson bracket associated with the group of volume preserving diffeomorphism group. This is interesting and nontrivial because in Eulerian representation, the time t maps need not be C from th...

New physical and mathematical applications of recently invented fractional Poisson probability distribution have been presented. As a physical application, a new family of quantum coherent states have been introduced and studied. Mathematical applications are related to the number theory. We have developed fractional generalization of the Bell polynomials, the Bell numbers, and the Stirling num...