Proper actions on cohomology manifolds

Essential results about actions of compact Lie groups on connected manifolds are generalized to proper actions of arbitrary groups on connected cohomology manifolds. Slices are replaced by certain fiber bundle structures on orbit neighborhoods. The group dimension is shown to be effectively finite. The orbits of maximal dimension form a dense open connected subset. If some orbit has codimension at most , then the group is effectively a Lie group.

     

Algebraic -actions of entropy rank one

We investigate algebraic -actions of entropy rank one, namely those for which each element has finite entropy. Such actions can be completely described in terms of diagonal actions on products of local fields using standard adelic machinery. This leads to numerous alternative characterizations of entropy rank one, both geometric and algebraic. We then compute the measure entropy of a class of skew products, where the fiber maps are elements from an algebraic -action of entropy rank one. This leads, via the relative variational principle, to a formula for the topological entropy of continuous skew products as the maximum of a finite number of topological pressures. We use this to settle a conjecture concerning the relational entropy of commuting toral automorphisms.

     

Associativity of crossed products by partial actions, enveloping actions and partial representations

Given a partial action of a group on an associative algebra , we consider the crossed product . Using the algebras of multipliers, we generalize a result of Exel (1997) on the associativity of obtained in the context of -algebras. In particular, we prove that is associative, provided that is semiprime. We also give a criterion for the existence of a global extension of a given partial action on an algebra, and use crossed products to study relations between partial actions of groups on algebras and partial representations. As an application we endow partial group algebras with a crossed product structure.

     

柱状装药预制破片缩比战斗部爆炸冲击波和破片的作用时序

针对柱状装药的周向预制破片战斗部,结合无量纲分析方法和爆炸驱动理论,确定了影响破片和冲击波相遇位置的关键参数,给出了由缩比战斗部推广预测原型战斗部爆炸产生的破片冲击波作用时序的方法。采用ANSYS/LS-DYNA有限元软件进行数值模拟,对比验证了理论分析和数值试验结果,分析了战斗部缩比比例对冲击波和破片作用时序的影响。结果表明:缩比模型与原型战斗部爆炸产生的破片和冲击波的相遇位置之比和相遇时间之比主要取决于两模型的质量比,在不考虑破片速度衰减时,两模型中载荷相遇位置之比和相遇时间之比等于其质量比的0.33次方。受破片速度衰减影响,该方法仅适用于质量缩比不小于0.2的模型。     

On Schreier graphs of gyrogroup actions

The notion of a group action can be extended to the case of gyrogroups. In this article, we examine a digraph and graph associated with a gyrogroup action on a finite nonempty set, called a Schreier digraph and graph. We show that algebraic properties of gyrogroups and gyrogroup actions such as being gyrocommutative, being transitive, and being fixed-point-free are reflected in their Schreier digraphs and graphs. We also prove graph-theoretic versions of the three fundamental theorems involving actions: the Cauchy–Frobenius lemma (also known as the Burnside lemma), the orbit-stabilizer theorem, and the orbit decomposition theorem. Finally, we make a connection between gyrogroup actions and actions of symmetric groups by evaluation via Schreier digraphs and graphs.     

Optical shuttering action in nematic phase of SMHBLC: observation of a ribbon-like texture

In this study, complexes belonging to supramolecular hydrogen-bonded liquid crystal homologous series are synthesized and characterized. Hydrogen bond is formed between p-n-alkyloxy benzoic acids (nBAO, where n?=?5–11) and chlorobenzoic acid (ClBAO), respectively. The isolated homologues are characterized by various techniques such as polarizing optical microscopic (POM) studies, differential scanning calorimetry (DSC), and Fourier transform–infrared spectroscopy. Based on the POM and DSC studies, the phase diagram has been constructed and discussed. A new smectic ordering, labeled as smectic R, has been characterized, which exhibits a ribbon-like texture. This phase is observed in the complexes pertaining to the higher homologous series. Tilt angle in this phase has been experimentally deduced and the results are fitted to the power law which concurs with the mean-field theory predicted value. Optical shuttering action in the homologue has been detected in the nematic phase and the results are also discussed.     

Multiplicative Invariants and Semigroup Algebras

Let G be a finite group acting by automorphism on a lattice A, and hence on the group algebra S=k[A]. The algebra of G-invariants in S is called an algebra of multiplicative invariants. We present an explicit version of a result of Farkas stating that multiplicative invariants of finite reflection groups are semigroup algebras.     

Spherical Tropical Geometry: a Survey of Recent Developments

This is a survey of some recent results on spherical tropical geometry.     

Hopf cyclic cohomology and Hodge theory for proper actions on complex manifolds

We introduce two Hopf algebroids associated to a proper and holomorphic Lie group action on a complex manifold. We prove that the cyclic cohomology of each Hopf algebroid is equal to the Dolbeault cohomology of invariant differential forms. When the action is cocompact, we develop a generalized complex Hodge theory for the Dolbeault cohomology of invariant differential forms. We prove that every cyclic cohomology class of these two Hopf algebroids can be represented by a generalized harmonic form. This implies that the space of cyclic cohomology of each Hopf algebroid is finite dimensional. As an application of the techniques developed in this paper, we generalize the Serre duality and prove a Kodaira type vanishing theorem.     

Some number-theoretic identities from group actions

We apply group actions to some natural situations like the natural ‘linear’ action of GL _r (Z _n ) and some of its subgroups to derive number-theoretic identities like