Hamiltonian operators and ℓ-coverings

An efficient method to construct Hamiltonian structures for nonlinear evolution equations is described. It is based on the notions of variational Schouten bracket and ℓ^*-covering. The latter serves the role of the cotangent bundle in the category of nonlinear evolution PDEs. We first consider two illustrative examples (the KdV equation and the Boussinesq system) and reconstruct for them the known Hamiltonian structures by our methods. For the coupled KdV–mKdV system, a new Hamiltonian structure is found and its uniqueness (in the class of polynomial (x,t)-independent structures) is proved. We also construct a nonlocal Hamiltonian structure for this system and prove its compatibility with the local one.     

Associative Deformations of Complex Analytic Spaces

A theory of associative deformations is developed for general complex analytic spaces. Deformation–quantization and commutative deformation are particular cases of this concept. Deformation cohomology and obstruction are studied. It is proved that any compact analytic space has a formal versal associative deformation.      

Moore空间P~n(8)及其同伦群

令G是一Abel群,m≥2是一整数．一个型为(G,m)的Moore空间是一单连通的CW-复形X,使得■_i(X)=G(i=m),0(i≠m)．这里J_i为X的第i个整系数的约化同调群．众所周知,Moore空间存在,且任何两个型为(G,n)的Moore空间有相同伦型．取G=Z_k(模k的剩余类加群)．p~n(k)=S~(n-1)∪_(kl_(n-1))e~n为型为(Z_k,n-1)的Moore空间．特别地,考虑k=8,决定了Moore空间p~n(8)的一些同伦群．主要证明工具是Toda引进的复合工具-Toda积,Gray的关于从p~n(8)到n维球面S~n的pinchin映射的同伦纤维的胞腔结构,以及关于亚稳定相对同伦群π_k(X,A)的同伦切割定理,其中A为维数小于n-1的有限CW-复形,X=A∪e~n．     

The Liouville Canonical Form for Compatible Nonlocal Poisson Brackets of Hydrodynamic Type and Integrable Hierarchies

We reduce an arbitrary pair of compatible nonlocal Poisson brackets of hydrodynamic type generated by metrics of constant Riemannian curvature (compatible Mokhov–Ferapontov brackets) to a canonical form, find an integrable system describing all such pairs, and, for an arbitrary solution of this integrable system, i.e., for any pair of compatible Poisson brackets in question, construct (in closed form) integrable bi-Hamiltonian systems of hydrodynamic type possessing this pair of compatible Poisson brackets of hydrodynamic type. The corresponding special canonical forms of metrics of constant Riemannian curvature are considered. A theory of special Liouville coordinates for Poisson brackets is developed. We prove that the classification of these compatible Poisson brackets is equivalent to the classification of special Liouville coordinates for Mokhov–Ferapontov brackets.     

Double quantization of CP type orbits by generalized Verma modules

It is known that symmetric orbits in g^* for any simple Lie algebra g are equipped with a Poisson pencil generated by the Kirillov-Kostant-Souriau bracket and the reduced Sklyanin bracket associated to the “canonical” R-matrix. We realize quantization of the Poisson pencil CPⁿ type orbits (i.e. orbits in sl(n + 1)^* whose real compact form is CPⁿ) by means of q-deformed Verma modules.     

The Schouten-Nijenhuis bracket and interior products

The main properties of the Schouten-Nijenhuis bracket are reviewed and a new formula is proven, which relates that bracket with the right interior product of multivectors by one-forms.     

Gauge-free Hamiltonian structure of the spin Maxwell-Vlasov equations

We derive the gauge-free Hamiltonian structure of an extended kinetic theory, for which the intrinsic spin of the particles is taken into account. Such a semi-classical theory can be of interest for describing, e.g., strongly magnetized plasma systems. We find that it is possible to construct a generalized noncanonical Poisson bracket on the extended phase space, and discuss the implications of our findings, including stability of monotonic equilibria.     

Invitation to composition

In 1963 [Ann. of Math. 78, 267-288], Gerstenhaber invented a comp(osition) calculus in the Hochschild complex of an associative algebra. In this paper, the first steps of the Gerstenhaber theory are exposed in an abstract (comp system) setting. In particular, as in the Hochschild complex, a graded Lie algebra and a pre-coboundary operator can be associated to every comp system. A derivation deviation of the pre-coboundary operator over the total composition is calculated in two ways, (the long) one of which is essentially new and can be seen as an example and elaboration of the auxiliary variables method proposed by Gerstenhaber in the early days of the comp calculus.     

中国古建筑木结构力学研究进展

中国古建筑木结构是我国乃至世界的宝贵文化遗产, 也是中华文明的重要组成部分. 随着时间的延续, 对这些价值连城、失而不可复得的历史文物的保护日益迫切. 本文从古木结构的木材力学性能、关键节点(斗栱、梁柱节点和柱脚节点) 受力机理、木构架整体受力性能研究、结构残损勘查、安全评估及修缮加固等方面, 总结了近30 年来结构学者对古代木结构建筑的研究进展, 以期为今后进行古木建筑的结构研究提供参考.