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The purpose of this paper is to study multi-derivations of Poisson algebras. First, we show that the space of multi-derivations of a Lie algebra 픤 with the Nijenhuis–Richardson bracket is a differential graded Lie algebra. Then we introduce the notion of a multi-derivation of a Poisson algebra, and show that the space of multi-derivations of a Poisson algebra with the Nijenhuis–Richardson bracket is also a graded Lie algebra. Finally, we study multi-derivations of three concrete Poisson algebras coming from linear Poisson manifolds, symplectic manifolds and Poisson manifolds with vanishing first cohomology groups. We establish the relationship between multi-derivations of a Lie algebra 픤 and multi-derivations of the linear Poisson manifold 픤∗, and show that a two-derivation of a connected symplectic manifold (M,ω) is λπ, where π is the Poisson bivector field induced by the symplectic structure ω and λ is a real number. We also prove that a two-derivation of a connected Poisson manifold (M,π) with trivial Poisson cohomology group H1(M,π) is fπ, where f is a Casimir function. [ABSTRACT FROM AUTHOR]

The algebraic method of singular reduction is applied for non regular group action on manifolds which provides singular symplectic spaces. The problem of deformation quantization of the singular surfaces is the focus. For some examples of singular Poisson spaces Gr¨onewold–Moyal series is explicitly constructed and convergence is checked. Some examples of deformation quantization of singular Poisson spaces are considered in detail. [ABSTRACT FROM AUTHOR]

Limited by resources, single centralised ZigBee cannot be deployed in very large-scale application. To solve this problem, the paper presents a strategy design of ZigBee network that can be centralised and decentralised deploying in a wide area. There are two kinds of networks structure: the mobile independent network and the multi-sink large-scale network. The paper analyzes their performance, and then focuses on researching the main mathematics problems of the mobile independent network. In the paper, its mathematical model is multi-domain stochastic process.In the end, its main metric are simulated with MATLAB, the results of simulation and experiments show the feasibility of this network. [ABSTRACT FROM AUTHOR]

We show that a Minkowski phase space endowed with a bracket relatively to a conformable differential realizes a Poisson algebra, confering a bi-Hamiltonian structure to the resulting manifold. We infer that the related Hamiltonian vector field is an infinitesimal Noether symmetry, and compute the corresponding deformed recursion operator. Besides, using the Hamiltonian–Jacobi separability, we construct recursion operators for Hamiltonian vector fields in conformable Poisson–Schwarzschild and Friedmann–Lemaître–Robertson–Walker (FLRW) manifolds, and derive the related constants of motion, Christoffel symbols, components of Riemann and Ricci tensors, Ricci constant and components of Einstein tensor. We highlight the existence of a hierarchy of bi-Hamiltonian structures in both the manifolds, and compute a family of recursion operators and master symmetries generating the constants of motion. [ABSTRACT FROM AUTHOR]

We study various properties of polarized Poisson structure subordinate to the polarized symplectic structure.We study also the notion of a polarized Poisson manifold, i.e., a Poisson manifold foliated by coisotropic submanifolds. We show that the characteristic distribution of a polarized Poisson structure is completely integrable and its leaves are symplectic. In the particular case when the foliation is Lagrangian, we show that, the polarized Poisson manifold is also foliated by polarized symplectic leaves, and we prove Darboux's theorem corresponding to a Lagrangian foliation with respect to a polarized Poisson manifold. [ABSTRACT FROM AUTHOR]

We study equivariant bi-vector fields on a toric variety. We prove that, on a smooth toric variety of dimension n, the locus where the rank of an equivariant bi-vector field is ≤ 2k is not empty and has at least a component of dimension ≥ 2k + 1, for all integers k > 0 such that 2k < n. The same is true also for k = 0, if the toric variety is smooth and compact. While for the non compact case, the locus in question has to be assumed to be non empty.

We show that a Minkowski phase space endowed with a bracket relatively to a conformable differential realizes a Poisson algebra, confering a bi-Hamiltonian structure to the resulting manifold. We infer that the related Hamiltonian vector field is an infinitesimal Noether symmetry, and compute the corresponding deformed recursion operator. Besides, using the Hamiltonian–Jacobi separability, we construct recursion operators for Hamiltonian vector fields in conformable Poisson–Schwarzschild and Friedmann–Lemaître–Robertson–Walker (FLRW) manifolds, and derive the related constants of motion, Christoffel symbols, components of Riemann and Ricci tensors, Ricci constant and components of Einstein tensor. We highlight the existence of a hierarchy of bi-Hamiltonian structures in both the manifolds, and compute a family of recursion operators and master symmetries generating the constants of motion.

We present a new look at the classification of real low-dimensional Lie algebras based on the notion of a linear bundle of Lie algebras. Belonging to a suitable family of Lie bundles entails the compatibility of the Lie–Poisson structures with the dual spaces of those algebras. This gives compatibility of bi-Hamiltonian structure on the space of upper triangular matrices and with a bundle at the algebra level. We will show that all three-dimensional Lie algebras belong to two of these families and four-dimensional Lie algebras can be divided in three of these families.

The paper surveys open problems and questions related to different aspects of integrable systems with finitely many degrees of freedom. Many of the open problems were suggested by the participants of the conference 'Finite-dimensional Integrable Systems, FDIS 2017' held at CRM, Barcelona in July 2017. [ABSTRACT FROM AUTHOR]

We prove a convexity theorem for the image of the moment map of a Hamiltonian torus action on a bm-symplectic manifold. [ABSTRACT FROM AUTHOR]

We associate to any integrable Poisson manifold a stack, i.e., a category fibered in groupoids over a site. The site in question has objects Dirac manifolds and morphisms pairs consisting of a smooth map and a closed 2-form. We show that two Poisson manifolds are symplectically Morita equivalent if and only if their associated stacks are isomorphic. We also discuss the non-integrable case. [ABSTRACT FROM AUTHOR]