Parallel connections over symmetric spaces

Let M be a simply connected Riemannian symmetric space, with at most one flat direction. We show that every Riemannian (or unitary) vector bundle with parallel curvature over M is an associated vector bundle of a canonical principal bundle, with the connection inherited from the principal bundle. The problem of finding Riemannian (or unitary) vector bundles with parallel curvature then reduces to finding representations of the structure group of the canonical principal bundle.     

Local connection forms revisited

Local connection forms provide a very useful tool for handling connections on principal bundles, because, essentially, they involve only the adjoint representation and the left (logarithmic) differential of the structure group, thus overcoming any complexities of the total space. The main results here characterize connections related together by bundle morphisms. A few applications refer to connections on (Banach) associated bundles and connections on projective limit bundles (in the Fréchet framework). The role of local connection forms is further illustrated by their sheaf-theoretic globalization, resulting in a sheaf-theoretic approach to principal connections. The latter point of view is naturally leading to a theory of connections on abstract principal sheaves.     

State space collapse and stability of queueing networks

We study the stability of subcritical multi-class queueing networks with feedback allowed and a work-conserving head-of-the-line service discipline. Assuming that the fluid limit model associated to the queueing network satisfies a state space collapse condition, we show that the queueing network is stable provided that any solution of an associated linear Skorokhod problem is attracted to the origin in finite time. We also give sufficient conditions ensuring this attraction in terms of the reflection matrix of the Skorokhod problem, by using an adequate Lyapunov function. State space collapse establishes that the fluid limit of the queue process can be expressed in terms of the fluid limit of the workload process by means of a lifting matrix.     

On the geometry of the space of smooth fibrations

We study geometrical aspects of the space of smooth fibrations between two given manifolds M and B, from the point of view of Fréchet geometry. As a first result, we show that any connected component of this space is the base space of a Fréchet-smooth principal bundle with the identity component of the group of diffeomorphisms of M as total space. Second, we prove that the space of fibrations is also itself the total space of a smooth Fréchet principal bundle with structure group the group of diffeomorphisms of the base B.     

On the stability problem for functional differential equations with infinite delay

We study the stability of functional differential equations with infinite delay, using the Lyapunov functional of constant sign with a derivative of constant sign. Limit equations are constructed in a special phase space. We establish a theorem on localization of a positive limit set and theorems on the stability and the asymptotic stability. The results are illustrated by examples.     

Boundedness of orbits and stability of closed sets

This paper explores fundamental connections between boundedness of orbits, and stability and attractivity of closed sets. For this purpose, the paper considers topological notions of stability and attractivity which do not depend on a metric. The notions considered are characterized in terms of restricted prolongations and positive limit sets, and connections with boundedness are studied. Finally, the notion of nontangency is used to give a Lyapunov result for Lyapunov stability, attractivity and boundedness of orbits. Unlike previous sufficient conditions for boundedness, our result does not require the Lyapunov function to be proper or weakly proper. Examples are provided to illustrate the results.     

Some Contributions to Stochastic Asymptotic Stability and Boundedness via Multiple Lyapunov Functions

Most of the existing results on stochastic stability use a single Lyapunov function, but we shall instead use multiple Lyapunov functions in this paper to establish some sufficient criteria for locating the limit sets of solutions of stochastic differential equations. From them follow many useful results on stochastic asymptotic stability and boundedness, which enable us to construct the Lyapunov functions much more easily in applications. In particular, the well-known classical theorem on stochastic asymptotic stability is a special case of our more general results. These show clearly the power of our new results.     

The Stiefel bundle of a Banach algebra

We introduce the Stiefel bundle associated to a given Banachable algebra and study the properties of this analytic principal fiber bundle over the Grassmannian of equivalence classes of idempotents in the algebra. Our main application concerns the bounded linear operators of a Banach space. In particular, the problem of smooth parametrization of subspaces can then be reduced to one involving the smooth extension of sections.     

Asymptotic Stability and Boundedness of Stochastic Differential Equations with Respect to Semimartingales

In this paper, we shall use multiple Lyapunov functions to establish some sufficient criteria for locating the limit sets of solutions of stochastic differential equations with respect to semimartingales. From them follow many useful results on stochastic asymptotic stability and boundedness, including some classical results as special cases. In particular, our new asymptotic stability criteria do not require the diffusion operator associated with the underlying stochastic differential equation be negative definite, while most of the existing results do require this negative definite property essentially.     

Localization of the limit set of trajectories of the Euler-Bernoulli equation with control

We investigate a differential equation in a Hilbert space that describes vibrations of the Euler-Bernoulli elastic beam with feedback control. The relative compactness of positive semitrajectories of the considered equation is proved. Constructing a Lyapunov functional in explicit form and using the invariance principle, we obtain representations of limit sets. __________ Translated from Ukrains'kyi Matematychnyi Zhurnal, Vol. 60, No. 2, pp. 173–182, February, 2008.     

Smooth almost Δ-bundles over polyhedrons

In this paper, we construct and study a category of principal fiber bundles with the following properties: 1) The base of a bundle is a polyhedron and the structure group is a k-dimensional torus. 2) The transition functions of the bundle atlas are smooth on simplexes of the base. 3) On the base, a simplicial action of a finite group Δ is given which has a multivalued lifting to the total space of the bundle. We study invariant connections and construct integer-valued realizable characteristic classes.     

Geometry and Topology of Some Fibered Riemannian Manifolds

We investigate a principal G-bundle with G-invariant Riemannian metric on its total space. We derive formulas describing the Levi-Civita connection and curvatures in two-dimensional directions. We obtain estimates of the influence of properties of sectional curvatures to topological invariants of the bundle.     

Spin (7)-structures in principal fibre bundles over Riemannian manifolds withG 2-structure

We consider principal fibre bundles with one-dimensional fiber over manifolds withG ₂-structure. We define twoSpin(7)-structures on the total bundle space and find relations between the respective structures on the total space and the base. Finally we construct examples ofSpin(7)-structures using the results previously proved.     

Lyapunov direct method for semidynamical systems

The stability of closed invariant sets of semidynamical systems defined on an arbitrary metric space is analyzed. The main theorems of Lyapunov’s second method for the uniform stability and uniform asymptotic stability (local and global) are stated. Illustrative examples are given.     

Integrable almost complex structures in principal bundles and holomorphic curves

We consider almost complex structures that arise naturally in a particular class of principal fibre bundles, where the choice of a connection can be used to determine equivariant isomorphisms between the vertical and horizontal tangent bundles of the total space. For instance, such data always exist on the frame bundle of a 3-manifold, but also in many other situations. We study the integrability condition to a complex structure, obtaining a system of gauge invariant coupled first order partial differential equations. This yields to a few correspondences between complex-geometric properties on the total space and metric properties on the base.     

The Moduli Space of Flat Connections on Oriented Surfaces with Boundary

A 2-form is constructed on the space of connections on a principal bundle over an oriented surface with boundary. This induces a symplectic structure for the moduli space of flat connections with boundary holonomies lying in prescribed conjugacy classes. The Yang-Mills quantum field measure is described for this situation. This measure converges to the normalized symplectic volume measure in the “classical” limit.     

Reductions of the object and the affine torsion tensor of a centroprojective connection on a distribution of planes

The projective group is represented as a bundle of centroprojective frames. This bundle is endowed with a centroprojective connection and becomes the space of this centroprojective connection. Structure equations of this space are found, which include the affine torsion tensor and the centroprojective curvature tensor containing the affine curvature subtensor. A distribution of planes in projective space and its associated principal bundle (which has two simplest and two simple (in the sense of [1]) quotient principal bundles) are considered. On the associated bundle, a group connection is defined. The object of the centroprojective connection is reduced to the object of the group connection. The object of the group connection contains the objects of the flat and normal linear connections, the centroprojective subconnection, and the affine-group connection as subobjects. The torsion object of the affine-group connection is determined. It is proved that it forms a tensor, which contains the torsion tensor of the normal linear connection as a subtensor. It is shown that the affine torsion tensor of the centroprojective connection reduces to the torsion tensor of the affine-group connection.     

Asymptotic self invariant sets and functional differential equations in Banach spaces

Summary The stability properties of asymptotically self-invariant sets of functional differential equations in a Banach space are discussed using a nonlinear variation of constants formula analogous to Alekseev's formula in Euclidean spaces and Lyapunov functions. Entrata in Redazione il 14 aprile 1972.     

On the nonlinear stability of an epidemic SEIR reaction-diffusion model

This paper deals with a reaction-diffusion SEIR model for infections. The longtime behaviour of the solutions is analyzed and, in particular, absorbing sets in the phase space are determined. By using a peculiar Lyapunov function, the nonlinear asymptotic stability of endemic equilibrium is investigated.     

On the Stability of a SEIR Reaction Diffusion Model for Infections Under Neumann Boundary Conditions

This paper deals with a reaction-diffusion SEIR model for infections under homogeneous Neumann boundary conditions. The longtime behaviour of the solutions is analyzed and, in particular, absorbing sets in the phase space are determined. By using a peculiar Lyapunov function, the nonlinear asymptotic stability of endemic equilibrium is investigated.