关于李群胚的几点讨论

文章讨论了李群胚作为丛的一些性质,得出李群胚的内子群胚是主丛的结论;研究了李群胚在其内子群胚上的作用,并证明了李群胚上的Maurer-cartan形式在其任意左不变向量场上作用的结果为常数.文末推广了关于李代数胚态射的一个结论.     

关于李群胚和泊松作用的讨论

定义了纤维丛的相配群胚的概念,从作用的角度研究了李群胚与主丛的关系;给出了一个泊松群胚在泊松流形上的作用是泊松作用的充要条件;文末得到了一些关于泊松流形上Casimir函数的结果.     

Locally Vacant Double Lie Groupoids and the Integration of Matched Pairs of Lie Algebroids

We prove a general integrability result for matched pairs of Lie algebroids. (Matched pairs of Lie algebras are also known as double Lie algebras or twilled extensions of Lie algebras.) The method used is an extension of a method introduced by Lu and Weinstein in the case of Poisson Lie groups, and yields double groupoids which satisfy an étale form of the vacancy condition.     

Geometric structures as deformed infinitesimal symmetries

A general model for geometric structures on differentiable manifolds is obtained by deforming infinitesimal symmetries. Specifically, this model consists of a Lie algebroid, equipped with an affine connection compatible with the Lie algebroid structure. The curvature of this connection vanishes precisely when the structure is locally symmetric.

This model generalizes Cartan geometries, a substantial class, to the intransitive case. Simple examples are surveyed and corresponding local obstructions to symmetry are identified. These examples include foliations, Riemannian structures, infinitesimal -structures, symplectic and Poisson structures.

     

Obstruction classes of crossed modules of Lie algebroids and Lie groupoids linked to existence of principal bundles

Let K be a Lie group and P be a K-principal bundle on a manifold M. Suppose given furthermore a central extension of K. It is a classical question whether there exists a -principal bundle on M such that . Neeb (Commun. Algebra 34:991–1041, 2006) defines in this context a crossed module of topological Lie algebras whose cohomology class is an obstruction to the existence of . In the present article, we show that is up to torsion a full obstruction for this problem, and we clarify its relation to crossed modules of Lie algebroids and Lie groupoids, and finally to gerbes.      

The localized longitudinal index theorem for Lie groupoids and the van Est map

We define the “localized index” of longitudinal elliptic operators on Lie groupoids associated with Lie algebroid cohomology classes. We derive a topological expression for these numbers using the algebraic index theorem for Poisson manifolds on the dual of the Lie algebroid. Underlying the definition and computation of the localized index, is an action of the Hopf algebroid of jets around the unit space, and the characteristic map it induces on Lie algebroid cohomology. This map can be globalized to differentiable groupoid cohomology, giving a definition of the “global index”, that can be computed by localization. This correspondence between the “global” and “localized” index is given by the van Est map for Lie groupoids.     

Cartan geometries and multiplicative forms

In this paper we show that Cartan geometries can be studied via transitive Lie groupoids endowed with a special kind of vector-valued multiplicative 1-forms. This viewpoint leads us to a more general notion, that of Cartan bundle, which encompasses both Cartan geometries and G-structures.     

Basic forms and Morse–Novikov cohomology of Lie groupoids

Given a basic closed 1‐form on a Lie groupoid , the Morse–Novikov cohomology groups are defined in this paper. They coincide with the usual de Rham cohomology groups when θ is exact and with the usual Morse–Novikov cohomology groups when is the unit groupoid generated by a smooth manifold M. We prove that the Morse–Novikov cohomology groups are invariant under Morita equivalences of Lie groupoids. On orbifold groupoids, we show that these groups are isomorphic to sheaf cohomology groups. Finally, when θ is not exact, we extend a vanishing theorem from smooth manifolds to orbifold groupoids.     

Geometric lattice actions, entropy and fundamental groups

Let be a lattice in a noncompact simple Lie Group G, where . Suppose acts analytically and ergodically on a compact manifold M preserving a unimodular rigid geometric structure (e.g. a connection and a volume). We show that either the action is isometric or there exists a "large image" linear representation of . Under an additional assumption on the dynamics of the action, we associate to a virtual arithmetic quotient of full entropy. Received: December 14, 2000     

On the geometry of Kawaguchi spaces

In the paper, with the use of the E. Cartan exterior forms method, the theory of linear and affine connections of the generalized Kawaguchi space of order two is constructed. It is proved that the linear connection of this space incorporates intrinsic antiquaternional structures, the conditions of their complete integrability are found, and the affine connections associated with the above-mentioned structures are constructed. Translated from Lietuvos Matematikos Rinkinys, Vol. 40, No. 3, pp. 321–334, July–September, 2000. Translated by R. Lapinskas     

Some Properties of Quasi-keep-range Operators and Their Applications

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Twistors and Cartan Connections

We give a simple, general approach for constructing a twistorspace Z with a CR structure for a manifold Mwith a geometric structure. The starting point is a principalG-bundle : P M with a Cartanconnection. The twistor space is an appropriate subgroup ofG. The integrability condition for the CR structure isdescribed in terms of the curvature of the Cartan connection.     

Hidden linearity in systems for competition with evolution in ecology and finance

In [J.C. Sprott, Competition with evolution in ecology and finance, Phys. Lett. A 325 (2004) 329-333] a variant of the multispecies Lotka-Volterra model was proposed in which species competing for fixed finite resources are replaced by a new randomly chosen species whenever they become extinct. We applied Lie group analysis to this system and found that it can be reduced to a linear equation if certain values are given to the parameters involved, which rules out any chaotic behavior in these instances. Also different routes to linearization are shown.     

On lattices of congruences of relational systems and universal algebras

Let \(\mathfrak{X}\) =〈X;R〉 be a relational system.X is a non-empty set andR is a collection of subsets ofX ^α, α an ordinal. The system of equivalence relations onX having the substitution property with respect to members ofR form a complete latticeC( \(\mathfrak{X}\) ) containing the identity but not necessarilyX×X. It is shown that for any relational system (X;R) there is a groupoid definable onX whose congruence lattice isC( \(\mathfrak{X}\) )U{X×X} . Theorem 2 and Corollary 2 contain some interesting combinatorial pecularities associated with oriented complete graphs and simple groupoids.     

Cartan connection and its geodesics on a manifold of nondegenerate affinor fields

We consider the quotient manifold of the manifold of nondegenerate affinor fields on a compact manifold with respect to the action of the group of nowhere vanishing functions. On this manifold, we construct a Cartan connection and find its torsion tensor. We also find the geodesics of the Cartan connection.     

Oscillation theorems for linear Hamiltonian systems with nonlinear dependence on the spectral parameter and the comparative index

We consider linear Hamiltonian differential systems which depend in general nonlinearly on the spectral parameter and with Dirichlet boundary conditions. In our consideration we do not impose any controllability and strict normality assumptions and omit the Legendre condition for the Hamiltonian. We adapt the theory developed by A.A. Abramov for the Hamiltonian spectral problems based on piecewise constant transformations of their conjoined bases using the notion of the comparative index. We introduce the concept of oscillation numbers which generalize the notion of proper focal points and prove the oscillation theorem relating the number of finite eigenvalues in the given interval with the values of the oscillation numbers at the end points of this interval.     

Necessary and sufficient conditions for quadratic stabilizability of switched systems on non-uniform time domains

In this paper, we consider the quadratic stabilizability via state feedback for a particular class of switched systems that evolve on a non-uniform time domain by introducing time scales theory. The system considered switches between a continuous-time subsystem with variable lengths and a discrete-time subsystem with variable discrete step sizes. Necessary and sufficient conditions are derived to guarantee the quadratic stability of this class of switched systems via a switching state feedback law based on the existence of a common positive definite matrix satisfying the quadratic stabilizability condition by considering that the two subsystems are unstable. By state feedback, we mean that the switching among subsystems depends on the system states. Current results for this kind of state switching feedback control are derived only for switched systems evolving on a continuous time domain or a discrete time domain with fixed step’s size. These results are not applicable for the particular class of switched systems where there is a mixing between the continuous and discrete dynamics. This motivates the derivation of a new and more general state feedback control law for switched systems in this work. A numerical example illustrating the results is presented.     

Hilbertian Jamison sequences and rigid dynamical systems

A strictly increasing sequence (nk)_k?0 of positive integers is said to be a Hilbertian Jamison sequence if for any bounded operator T on a separable Hilbert space such that supk?0‖Tnk‖<+∞, the set of eigenvalues of modulus 1 of T is at most countable. We first give a complete characterization of such sequences. We then turn to the study of rigidity sequences (nk)_k?0 for weakly mixing dynamical systems on measure spaces, and give various conditions, some of which are closely related to the Jamison condition, for a sequence to be a rigidity sequence. We obtain on our way a complete characterization of topological rigidity and uniform rigidity sequences for linear dynamical systems, and we construct in this framework examples of dynamical systems which are both weakly mixing in the measure-theoretic sense and uniformly rigid.     

New stability criteria for a class of neutral systems with discrete and distributed time-delays: an LMI approach

In this paper, a problem of the asymptotic stability for a class of neutral systems with multiple discrete and distributed time-delays is considered. Lyapunov stability theory is applied to guarantee the stability for the systems. New discrete-delay-independent and discrete-delay-dependent stability conditions are derived in terms of the spectral radius and linear matrix inequality. By mathematical analysis, the stability criteria are proved to be less conservative than the ones reported in the current literatures. A numerical example is given to illustrate the availability of the proposed results.