A Polish group without Lie sums

We construct a Polish group with an invariant metric in which Lie sums and Lie brackets do not exist. The construction of the group and the proof of the main theorem use some facts of combinatorial nature about the free group with two generators equipped with a Graev metric. The second author acknowledges the United States NSF grant DMS-0501039 for the support of his research.     

Graded Multiple Analogs of Lie Algebras

Graded analogs of (n,k,r)-Lie algebras (in particular, of Nambu–Lie algebras), introduced by the authors, are defined and their general property are studied.     

Gerstenhaber brackets on Hochschild cohomology of general twisted tensor products

We present techniques for computing Gerstenhaber brackets on Hochschild cohomology of general twisted tensor product algebras. These techniques involve twisted tensor product resolutions and are based on recent results on Gerstenhaber brackets expressed on arbitrary bimodule resolutions.     

A compressible Hamiltonian electromagnetic gyrofluid model

A Lie-Poisson bracket is presented for a four-field gyrofluid model with magnetic field curvature and compressible ions, thereby showing the model to be Hamiltonian. The corresponding Casimir invariants are presented, and shown to be associated to four Lagrangian invariants advected by distinct velocity fields. This differs from a cold ion limit, in which the Lie-Poisson bracket transforms into the sum of direct and semidirect products, leading to only three Lagrangian invariants.     

The semigroup of values of a one-dimensional local ring with two minimal primes

In this paper, we give a new simple algebraic proof of the involutivity of the characteristic variety after Gabber, which is one of the most important results of D-module theory.     

Chaplygin's Ball Rolling Problem Is Hamiltonian

Mathematical Notes -     

Global analytic and Gevrey hypoellipticity of sublaplacians under Diophantine conditions

In this paper we consider the problem of global Gevrey and analytic regularity for a class of partial differential operators on a torus in the form of a sum of squares of vector fields, which may not satisfy the bracket condition. We show that these operators are globally Gevrey or analytic hypoelliptic on the torus if and only if the coefficients satisfy certain Diophantine approximation properties.

     

Determination of the Electromagnetic Lagrangian from a System of Poisson Brackets

The Lagrangian and Hamiltonian formulations of electromagnetism are reviewed and the Maxwell equations are obtained from the Hamiltonian for a system of many electric charges. It is shown that three of the equations which were obtained from the Hamiltonian, namely the Lorentz force law and two Maxwell equations, can be obtained as well from a set of postulated Poisson brackets. It is shown how the results derived from these brackets can be used to reconstruct the original Lagrangian for the theory aided by some reasoning based on physical concepts.     

泊松括号与Noether定理

从泊松括号出发,利用分析力学知识导出Noether定理,进而讨论了能量、能量和角动量守恒和相应的对称性。