Related papers: Poisson Brackets of Orthogonal Polynomials
The construction of a Laplacian on a class of fractals which includes the Sierpinski gasket ({\bf $SG$}) has given rise to an intensive research on analysis on fractals. For instance, a complete theory of polynomials and power series on…
In this note the notion of Poisson brackets in Kontsevich's "Lie World" is developed. These brackets can be thought of as "universally" defined classical Poisson structures, namely formal expressions only involving the structure maps of a…
A combinatorial theory for type $R_I$ orthogonal polynomials is given. The ingredients include weighted generalized Motzkin paths, moments, continued fractions, determinants, and histories. Several explicit examples in the Askey scheme are…
We continue studying the connection between Jacobi matrices defined on a tree and multiple orthogonal polynomials (MOPs) that was discovered previously by the authors. In this paper, we consider Angelesco systems formed by two analytic…
We explain how to construct a morphism from the group of birational automorphisms of CP^2 preserving the logarithmic Poisson bracket to the Thompson group T. Than we give a linear representation of the former group, provide some information…
We describe the automorphism groups of elliptic Poisson algebras on polynomial algebras in three variables and give an explicit set of generators and defining relations for this group.
The covariant canonical formalism is a covariant extension of the traditional canonical formalism of fields. In contrast to the traditional canonical theory, it has a remarkable feature that canonical equations of gauge theories or gravity…
Orthogonal polynomials on the real line always satisfy a three-term recurrence relation. The recurrence coefficients determine a tridiagonal semi-infinite matrix (Jacobi matrix) which uniquely characterizes the orthogonal polynomials. We…
The Poisson, contact and Nambu brackets define algebraic structures on $C^{\infty}(M)$ satisfying the Jacobi identity or its generalization. The automorphism groups of these brackets are the symplectic, contact and volume preserving…
We develop general expressions for the raising and lowering operators that belong to the orthogonal polynomials of hypergeometric type with discrete and continuous variable. We construct the creation and annihilation operators that…
The general expression for the bicovariant bracket for odd generators of the external algebra on a Poisson-Lie group is given. It is shown that the graded Poisson-Lie structures derived before for $GL(N)$ and $SL(N)$ are the special cases…
In this paper, we present neural networks learning mechanical systems that are both symplectic (for instance particle mechanics) and non-symplectic (for instance rotating rigid body). Mechanical systems have Hamiltonian evolution, which…
A vertical exterior derivative is constructed that is needed for a graded Poisson structure on multisymplectic manifolds over nontrivial vector bundles. In addition, the properties of the Poisson bracket are proved and first examples are…
We consider a hierarchy of Poisson structures defined on rational functions on the Riemann sphere. This hierarchy is originated in the theory of the integrable Camassa-Holm equation associated with the Krein's string spectral problem.…
We consider a set of measures on the real line and the corresponding system of multiple orthogonal polynomials (MOPs) of the first and second type. Under some very mild assumptions, which are satisfied by Angelesco systems, we define…
We compute the Poisson cohomology of a class of Poisson manifolds that are symplectic away from a collection $D$ of hypersurfaces. These Poisson structures induce a generalization of symplectic and cosymplectic structures, which we call a…
Several types of generically-nondegenerate Poisson structures can be effectively studied as symplectic structures on naturally associated Lie algebroids. Relevant examples of this phenomenon include log-, elliptic, $b^k$-, scattering and…
In the context of averaging method, we describe a reconstruction of invariant connection-dependent Poisson structures from canonical actions of compact Lie groups on fibered phase spaces. Some symmetry properties of Wong's type equations…
We classify generic coadjoint orbits for symplectomorphism groups of compact symplectic surfaces with or without boundary. We also classify simple Morse functions on such surfaces up to a symplectomorphism.
We consider an arbitrary Dubrovin-Novikov bracket of degree $k$, namely a homogeneous degree $k$ local Poisson bracket on the loop space of a smooth manifold $M$ of dimension $n$, and show that $k$ connections, defined by explicit linear…