To any Hamiltonian action of a reductive algebraic group G on a smooth irreducible symplectic variety X we associate certain combinatorial invariants: Cartan space, Weyl group, weight and root lattices. For cotangent bundles these invariants essentially coincide with those arising in the theory of equivarant embeddings. Using our approach we establish some properties of the latter invariants.

We apply a recently introduced reduction procedure based on the embedding of non-crystallographic Coxeter groups into crystallographic ones to Calogero-Moser systems. For rational potentials the familiar generalized Calogero Hamiltonian is recovered. For the Hamiltonians of trigonometric, hyperbolic and elliptic type, we obtain novel integrable dynamical systems with a second potential term whi...

We are motivated by a tantalizing open question in exact algorithms: can we detect whether an n-vertex directed graph G has a Hamiltonian cycle in time significantly less than 2? We present new randomized algorithms that improve upon several previous works: 1. We show that for any constant 0 < λ < 1 and prime p we can count the Hamiltonian cycles modulo pb(1−λ) n 3p c in expected time less than...

We are motivated by a tantalizing open question in exact algorithms: can we detect whether an n-vertex directed graph G has a Hamiltonian cycle in time significantly less than 2? We present new randomized algorithms that improve upon several previous works: a. We show that for any constant 0 < λ < 1 and prime p we can count the Hamiltonian cycles modulo p n 3p ⌋ in expected time less than c for...

We propose a perturbation algorithm for Hamiltonian systems on Lie algebra $\mathbb{V}$, so that it can be applied to non-canonical systems. Given system preserves a subalgebra $\mathbb{B}$ of $\mathbb{V}$, when we add the subalgebra $\mathbb{B}$ will no longer preserved. show how to transform perturbed dynamical to preserve up to terms quadratic in perturbation. apply this method to ...

Abstract. A convenient algebraic structure to describe some forms of dynamics of two hamiltonian systems with nonpotential (magnetic–type) interaction is considered. An algebraic mechanism of generation of such dynamics is explored on simple ”toy” examples and models. Nonpotential chains and their continuum limits are also considered. Examples of hybrid couplings with both potential and nonpote...

In this Communication, we consider a generalised Tavis–Cummings model when the damping process is taken into account. We show that quantum dynamics governed by non-Hermitian Hamiltonian exactly solvable using Quantum Inverse Scattering Method, and Algebraic Bethe Ansatz. The leakage of photons described Lindblad-type master equation. diagonalised state vectors, which are elementary symmetric fu...

Abstract. We define a new class of solutions to the WDVV associativity equations. This class is determined by the property that one of the commuting PDEs associated with such a WDVV solution is linearly degenerate. We reduce the problem of classifying such solutions of the WDVV equations to the particular case of the so-called algebraic Riccati equation and, in this way, arrive at a complete cl...

The Clifford algebraic formulation of the Duffin–Kemmer–Petiau (DKP) algebras is applied to recast De Donder–Weyl Hamiltonian (DWH) theory as an description independent matrix representation DKP algebra. We show that DWH equations for antisymmetric fields arise out action algebra on certain invariant subspaces which carry representations fields. representation-free formula bracket associated wi...