On the Number of Elementary Submodels of an Unsuperstable Homogeneous Structure

We show that if M is a stable unsuperstable homogeneous structure, then for most κ ? |M|, the number of elementary submodels of M of power κ is 2^κ.     

On Nonstructure of Elementary Submodels of an Unsuperstable Homogeneous Structure

In the first part of this paper we let M be a stable homogeneous model and we prove a nonstructure theorem for the class of all elementary submodels of M, assuming that M is ‘unsuperstable’ and has Skolem functions. In the second part we assume that M is an unstable homogeneous model of large cardinality and we prove a nonstructure theorem for the class of all elementary submodels of M.     

Jacobi structures of evolutionary partial differential equations

In this paper we introduce the notion of infinite dimensional Jacobi structure to describe the geometrical structure of a class of nonlocal Hamiltonian systems which appear naturally when applying reciprocal transformations to Hamiltonian evolutionary PDEs. We prove that our class of infinite dimensional Jacobi structures is invariant under the action of reciprocal transformations that only change the spatial variable. The main technical tool is in a suitable generalization of the classical Schouten–Nijenhuis bracket to the space of the so called quasi-local multi-vectors, and a simple realization of this structure in the framework of supermanifolds. These constructions are used to compute the Lichnerowicz–Jacobi cohomologies and to prove a Darboux theorem for Jacobi structures with hydrodynamic leading terms. We also introduce the notion of bi-Jacobi structures, and consider the integrability of a system of evolutionary PDEs that possesses a bi-Jacobi structure.     

Countably categorical and autostable Boolean algebras with distinguished ideals

We study countable Boolean algebras with finitely many distinguished ideals (countable I-algebras) whose elementary theory is countably categorical, and autostable I-algebras which form their subclass. We propose a new characterization for the former class that allows to answer a series of questions about the structure of countably categorical and autostable I-algebras.     

Finiteness of U‐rank implies simplicity in homogeneous structures

A (large) superstable homogeneous structure is said to be simple if every complete type over any set A has a free extension over any B ? A. In this paper we give a characterization for this property in terms of U‐rank. As a corollary we get that if the structure has finite U‐rank, then it is simple. (© 2003 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

On the complexity of categoricity in computable structures

We investigate the computational complexity the class of Γ‐categorical computable structures. We show that hyperarithmetic categoricity is Π¹₁‐complete, while computable categoricity is Π⁰₄‐hard. (© 2003 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Complex Product Structures and Affine Foliations

A complex product structure on a manifold is an appropriate combination of a complex structure and a product structure. The existence of such a structure determines many interesting properties of the underlying manifold, notably that the manifold admits a pair of complementary foliations whose leaves carry affine structures. This is due to the existence of a unique torsion-free connection which preserves both the complex and the product structure; this connection is not necessarily flat. We study the existence of complex product structures on tangent bundles of smooth manifolds, and we investigate the structure of manifolds admitting a complex product structure and a compatible hypersymplectic metric, showing that the foliations mentioned earlier are either symplectic or Lagrangian, depending on the symplectic form under consideration.     

Exotic smooth structures and symplectic forms on closed manifolds

In this paper we discuss relations between symplectic forms and smooth structures on closed manifolds. Our main motivation is the problem if there exist symplectic structures on exotic tori. This is a symplectic generalization of a problem posed by Benson and Gordon. We give a short proof of the (known) positive answer to the original question of Benson and Gordon that there are no Kähler structures on exotic tori. We survey also other related results which give an evidence for the conjecture that there are no symplectic structures on exotic tori.     

双重结构失衡、消费增长与产业结构升级

在中国经济发展进入新常态阶段,原来主要依靠投资和出口拉动的经济增长模式难以为继,消费不仅要成为中国经济增长的重要拉动力,而且也要成为促进产业结构调整的重要力量.当前中国存在的问题是,需求结构失衡导致产业结构失衡,拉动经济发展的消费比重偏低从而导致第三产业发展不足.解决问题的应对之策是,通过扩大消费比重、促进消费结构升级来提高第三产业比重并发展该产业,通过升级消费结构从而优化调整产业结构.     

Several Cohomology Algebras Connected with the Poisson Structure

The structure of a Lie superalgebra is defined on the space of multiderivations of a commutative algebra. This structure is used to define some cohomology algebra of Poisson structure. It is shown that when a commutative algebra is an algebra of C -functions on the C -manifold, the cohomology algebra of Poisson structure is isomorphic to an algebra of vertical cohomologies of the foliation corresponding to the Poisson structure.     

Index sets of decidable models

We study the index sets of the class of d-decidable structures and of the class of d-decidable countably categorical structures, where d is an arbitrary arithmetical Turing degree. It is proved that the first of them is m-complete ∑ ₃ ^{0, d} , and the second is m-complete ∑ ₃ ^{0, d} \∑ ₃ ^{0, d} in the universal computable numbering of computable structures for the language with one binary predicate.     

Share functions for cooperative games with levels structure of cooperation

In a standard TU-game it is assumed that every subset of the player set N can form a coalition and earn its worth. One of the first models where restrictions in cooperation are considered is the one of games with coalition structure of Aumann and Drèze (1974). They assumed that the player set is partitioned into unions and that players can only cooperate within their own union. Owen (1977) introduced a value for games with coalition structure under the assumption that also the unions can cooperate among them. Winter (1989) extended this value to games with levels structure of cooperation, which consists of a game and a finite sequence of partitions defined on the player set, each of them being coarser than the previous one.     

Infinitesimal variations of Hodge structure at infinity

By analyzing the local and infinitesimal behavior of degenerating polarized variations of Hodge structure the notion of infinitesimal variation of Hodge structure at infinity is introduced. It is shown that all such structures can be integrated to polarized variations of Hodge structure and that, conversely, all are limits of infinitesimal variations of Hodge structure at finite points. As an illustration of the rich information encoded in this new structure, some instances of the maximal dimension problem for this type of infinitesimal variation are presented and contrasted with the “classical” case of IVHS at finite points.      

Almost Complex Structures Which Are Compatible with Kähler or Symplectic Structures

In this note we prove that half of all homotopy classes of almost complex structures on M is not compatible with any symplectic structure for a certain class of oriented compact 4-manifolds M. In particular, half of all homotopy classes of almost complex structures on an oriented 4-manifold is not compatible to any Kähler structure.     

Binding Number,Minimum Degree,and Cycle Structure in Graphs

A well‐known theorem of Woodall states that if a graph G has binding number at least 3/2, then G is hamiltonian. We generalize Woodall's theorem as follows.