On the Structure of the B—cocleft Module Coalgebra

§１．　ＴｈｅＥｑｕｉｖａｌｅｎｔＴｈｅｏｒｅｍｏｆｔｈｅＣｒｏｓｓｅｄＣｏｐｒｏｄｕｃｔＬｅｔＣｂｅａｌｅｆｔＨ－ｗｅａｋｌｙｃｏｍｏｄｕｌｅｃｏａｌｇｅｂｒａ［４］ｗｉｔｈｔｈｅｓｔｒｕｃｔｕｒｅρ－Ｃ（ｃ）＝∑ｃ（１）ｃ（２）．ＤｂｅｌｅｆｔＨ－ｍｏｄｕｌｅｃｏａｌｇｅｂｒａ［２］ｗｉｔｈｔｈｅｓｔｒｕｃｔｕｒｅ“”．Ｆｏｒα∈Ｈｏｍκ（Ｃ，ＨＨ）ｄｅｎｏｔｅα（ｃ）＝∑α１（ｃ）α２（ｃ）．Ｄｅｆｉｎｅ△－：ＣＤ→（ＣＤ）（ＣＤ）ａｎｄε－：ＣＤ→κａｓｆｏｌｌｏｗ－ｉ…     

The Convergence Theorems for Conormed－Seminormed Fuzzy Integralto⊥－decomposable Fuzzy Measure

ＴｈｅＣｏｎｖｅｒｇｅｎｃｅＴｈｅｏｒｅｍｓｆｏｒＣｏｎｏｒｍｅｄ－ＳｅｍｉｎｏｒｍｅｄＦｕｚｚｙＩｎｔｅｇｒａｌｔｏ⊥－ｄｅｃｏｍｐｏｓａｂｌｅＦｕｚｚｙＭｅａｓｕｒｅＬｉｕＸｕｅｃｈｅｎｇ（ＤｅｐａｒｔｍｅｎｔｏｆＭａｔｈｅｍａｔｉｃｓ，Ｈｅｂｅ...     

RadicalsofGronpG－gradedΓ－rings

ＲａｄｉｃａｌｓｏｆＧｒｏｎｐＧ－ｇｒａｄｅｄΓ－ｒｉｎｇｓＲａｄｉｃａｌｓｏｆＧｒｏｎｐＧ－ｇｒａｄｅｄΓ－ｒｉｎｇｓＧｕｏＬｉｎｇｚｈｏｎｇ，ＦａｎＦｎｓｈｅｎｇ（ＤｅｐａｒｔｍｅｎｔｏｆＡｕｔｏｍａｔｉｃＣｏｎｔｒｏｌＤｅｐａｒｔｍｅｎｔｏｆＭ...     

NolinearEvolutionofFerroelectricDomains

ＮｏｌｉｎｅａｒＥｖｏｌｕｔｉｏｎｏｆＦｅｒｒｏｅｌｅｃｔｒｉｃＤｏｍａｉｎｓ１ＷｅｉＬＵ，Ｄａｉ－ＮｉｎｇＦＡＮＧＫｅｈ－ＣｈｉｈＨＷＡＮＧ（ＤｅｐａｒｔｍｅｎｔｏｆＥｎｇｉｎｅｅｒｉｎｇＭｅｃｈａｎｉｃｓ，ＴｓｉｎｇｈｕａＵｎｉｖｅｒｓｉｔｙ，Ｂ...     

A Formula Involving Convolrtions and Partition-Sums

ＡＦｏｒｍｕｌａＩｎｖｏｌｖｉｎｇＣｏｎｖｏｌｎｔｉｏｎｓａｎｄＰａｒｔｉｔｉｏｎ－ＳｕｍｓＬｅｅｔｓｃｈＣ．Ｈｓｕ（Ｉｎｓｔ．ｏｆＭａｔｈ．Ｓｃｉ．，ＤａｌｉａｎＵｎｉｖ－ｏｆＴｅｃｈ．，Ｄａｌｉａｎ１１６０２４）ＡＦｏｒｍｕｌａＩｎｖｏｌｖｉｎｇ...     

MicromechanismofFerroelectrics

ＭｉｃｒｏｍｅｃｈａｎｉｓｍｏｆＦｅｒｒｏｅｌｅｃｔｒｉｃｓＸｉＣＨＥＮ，Ｄａｉ－ＮｉｎｇＦＡＮＧＫｅｈ－ＣｈｉｈＨＷＡＮＧ（ＤｅｐａｒｔｍｅｎｔｏｆＥｎｇｉｎｅｅｒｉｎｇＭｅｃｈａｎｉｃｓ，ＴｓｉｎｇｈｕａＵｎｉｖｅｒｓｉｔｙ，Ｂｅｉｊｉｎｇ１００...     

Perturbation Theory for Dual Cosine Function（Ⅱ）—Time－dependent Perturbation in the Sun－reflexive Case

ＰｅｒｔｕｒｂａｔｉｏｎＴｈｅｏｒｙｆｏｒＤｕａｌＣｏｓｉｎｅＦｕｎｃｔｉｏｎ（Ⅱ）—Ｔｉｍｅ－ｄｅｐｅｎｄｅｎｔＰｅｒｔｕｒｂａｔｉｏｎｉｎｔｈｅＳｕｎ－ｒｅｆｌｅｘｉｖｅＣａｓｅＤｉｎｇＴｉａｎｂｉａｏ（丁天彪）（ＮｏｒｔｈＣｈｉｎａＩｎｓｔｉｔ...     

Travelling Waves for a Cross-Diffusion Model of Forest Boundary Dynamics

ＴｒａｖｅｌｌｉｎｇＷａｖｅｓｆｏｒａＣｒｏｓｓ－ＤｉｆｆｕｓｉｏｎＭｏｄｅｌｏｆＦｏｒｅｓｔＢｏｕｎｄａｒｙＤｙｎａｍｉｃｓ￥（吴雅萍）ＹａｐｉｎｇＷｕ（ＩｎｓｉｔｕｔｅｏｆａｐｐｌｉｅｄＰｈｙｓｉｃｓａｎｄＣｏｍｐｕｔａｔｉｏｎａｌＭａｔｈｅｍａ...     

Fixed Point Theorems for Increasing Operators of None－continuity and None－compactness Conditions

ＦｉｘｅｄＰｏｉｎｔＴｈｅｏｒｅｍｓｆｏｒＩｎｃｒｅａｓｉｎｇＯｐｅｒａｔｏｒｓｏｆＮｏｎｅ－ｃｏｎｔｉｎｕｉｔｙａｎｄＮｏｎｅ－ｃｏｍｐａｃｔｎｅｓｓＣｏｎｄｉｔｉｏｎｓＺｈａｎｇＱｉｎｇｚｈｅｎｇ（ＤｅｐａｒｔｍｅｎｔｏｆＭａｔｈｅｍａｔｉｃｓ，...     

Concerning a Kind of Integrals of Cinmplex-Valued Functions of Large Numbers

ＣｏｎｃｅｒｎｉｎｇａＫｉｎｄｏｆｉｎｔｅｇｒａｌｓｏｆＣｏｍｐｌｅｘ－ＶａｌｎｅｄＦｕｎｃｔｉｏｎｓｏｆＬａｒｇｅＮｎｍｂｅｒｓＬ．Ｃ．Ｈｓｕ（Ｄｅｐｔ．ｏｆＭａｔｈ．Ｓｃｉ．，ＤａｌｉａｎＵｎｉｖ．ｏｆＴｅｃｈ，Ｄａｌｉａｎ１１００２３）Ｃｏｎｃ...     

Constructing infra-nilmanifolds admitting an Anosov diffeomorphism

In this paper we establish an algebraic characterization of those infra-nilmanifolds modeled on a free c-step nilpotent Lie group and with an abelian holonomy group admitting an Anosov diffeomorphism. We also develop a new method for constructing examples of infra-nilmanifolds having an Anosov diffeomorphism.     

On properly essential classical conformal diffeomorphism groups

In this article, we prove that various classical conformal diffeomorphism groups, which are known to be essential (Banyaga, J Geom 68(1–2):10–15, 2000), are in fact properly essential. This is a consequence of a local criterion on a conformal diffeomorphism in the form of a cohomological equation. Furthermore, we study the orbit of a tensor field under the action of the conformal diffeomorphism group for these classical conformal structures. On every closed contact manifold, we find conformal contact forms that are not diffeomorphic.     

Anosov diffeomorphisms on infra-nilmanifolds associated to graphs

Anosov diffeomorphisms on closed Riemannian manifolds are a type of dynamical systems exhibiting uniform hyperbolic behavior. Therefore, their properties are intensively studied, including which spaces allow such a diffeomorphism. It is conjectured that any closed manifold admitting an Anosov diffeomorphism is homeomorphic to an infra-nilmanifold, that is, a compact quotient of a 1-connected nilpotent Lie group by a discrete group of isometries. This conjecture motivates the problem of describing which infra-nilmanifolds admit an Anosov diffeomorphism. So far, most research was focused on the restricted class of nilmanifolds, which are quotients of 1-connected nilpotent Lie groups by uniform lattices. For example, Dani and Mainkar studied this question for the nilmanifolds associated to graphs, which form the natural generalization of nilmanifolds modeled on free nilpotent Lie groups. This paper further generalizes their work to the full class of infra-nilmanifolds associated to graphs, leading to a necessary and sufficient condition depending only on the induced action of the holonomy group on the defining graph. As an application, we construct families of infra-nilmanifolds with cyclic holonomy groups admitting an Anosov diffeomorphism, starting from faithful actions of the holonomy group on simple graphs.     

On closed manifolds admitting an Anosov diffeomorphism but no expanding map

A few years ago, the first example of a closed manifold admitting an Anosov diffeomorphism but no expanding map was given. Unfortunately, this example is not explicit and is high-dimensional, although its exact dimension is unknown due to the type of construction. In this paper, we present a family of concrete 12-dimensional nilmanifolds with an Anosov diffeomorphism but no expanding map, where a nilmanifold is defined as the quotient of a 1-connected nilpotent Lie group by a cocompact lattice. We show that this family has the smallest possible dimension in the class of infra-nilmanifolds, which is conjectured to be the only type of manifolds admitting Anosov diffeomorphisms up to homeomorphism. The proof shows how to construct positive gradings from the eigenvalues of the Anosov diffeomorphism under some additional assumptions related to the rank, using the action of the Galois group on these algebraic units.     

On a Lagrangian Formulation of the Incompressible Euler Equation

In this paper we show that the incompressible Euler equation on the Sobolev space $H^s(\mathbb{R}^n), s › n ⁄ 2+1$, can be expressed in Lagrangian coordinates as a geodesic equation on an infinite dimensional manifold. Moreover the Christoffel map describing the geodesic equation is real analytic. The dynamics in Lagrangian coordinates is described on the group of volume preserving diffeomorphisms, which is an analytic submanifold of the whole diffeomorphism group. Furthermore it is shown that a Sobolev class vector field integrates to a curve on the diffeomorphism group.     

Right-angled Artin groups in the C∞ diffeomorphism group of the real line

We prove that every right-angled Artin group embeds into the C^∞ diffeomorphism group of the real line. As a corollary, we show every limit group, and more generally every countable residually RAAG group, embeds into the C^∞ diffeomorphism group of the real line.     

Pseudo-distances on symplectomorphism groups and applications to flux theory

Starting from a given norm on the vector space of exact 1-forms of a compact symplectic manifold, we produce pseudo-distances on its symplectomorphism group by generalizing an idea due to Banyaga. We prove that in some cases (which include Banyaga’s construction), their restriction to the Hamiltonian diffeomorphism group is equivalent to the distance induced by the initial norm on exact 1-forms. We also define genuine “distances to the Hamiltonian diffeomorphism group” which we use to derive several consequences, mainly in terms of flux groups.     

The abundance of strange attractor in the families of nearby singular diffeomorphism

In this paper, we consider the families of nearby singular diffeomorphism and the measure of a set in the parameter space, such that for each point of the set the corresponding diffeomorphism possesses strange attractor. For some families of one-dimensional mapping satisfying certain transversality condition, we prove that there is a positive measure set in the parameter space, such that the system in the corresponding families of nearly singular diffeomorphism has strange attractor. Furthermore, we study the dynamics of this type of strange attractor. Project Supported by Fund of National Science of China     

Exotic torus manifolds and equivariant smooth structures on quasitoric manifolds

In 2006 Masuda and Suh asked if two compact non-singular toric varieties having isomorphic cohomology rings are homeomorphic. In the first part of this paper we discuss this question for topological generalizations of toric varieties, so-called torus manifolds. For example we show that there are homotopy equivalent torus manifolds which are not homeomorphic. Moreover, we characterize those groups which appear as the fundamental groups of locally standard torus manifolds. In the second part we give a classification of quasitoric manifolds and certain six-dimensional torus manifolds up to equivariant diffeomorphism. In the third part we enumerate the number of conjugacy classes of tori in the diffeomorphism group of torus manifolds. For torus manifolds of dimension greater than six there are always infinitely many conjugacy classes. We give examples which show that this does not hold for six-dimensional torus manifolds.     

A special class of nilmanifolds admitting an Anosov diffeomorphism

A nilmanifold admits an Anosov diffeomorphism if and only if its fundamental group (which is finitely generated, torsion-free and nilpotent) supports an automorphism having no eigenvalues of absolute value one. Here we concentrate on nilpotency class 2 and fundamental groups whose commutator subgroup is of maximal torsion-free rank. We prove that the corresponding nilmanifold admits an Anosov diffeomorphism if and only if the torsion-free rank of the abelianization of its fundamental group is greater than or equal to 3.