Josep Ferrer

Differentiable families of subspaces

Abstract We show how the differentiable families of subspaces can be studied, from a geometrical point of view, by means of their representation as a differentiable mapping from the manifold of parameters M into the Grassmann manifold Gr k,n of all k -dimensional subspaces of K n , and the consideration of principal bundles over Gr k,n and M . In order to achieve this, first we present a survey of the topology, the differentiable structure and the bundle structure of Gr k,n . In particular, we show that the topology induced by the gap metric is equivalent to the usual quotient topology. Second, we use these structures to prove two local characterizations of the differentiability of a family of subspaces, and we state that these characterizations can be globalized if M is contractible. Finally, we apply these properties to study the existence of global Jordan basis of class C r .

Modelling Advanced Transport Telematic Applications with Microscopic Simulators: The Case of AIMSUN2

The simulation of Advanced Transport Telematic Applications requires specific modelling features, which have not been usually taken into account in the design of microscopic traffic simulation models. This paper discusses the general requirements of some of these applications, and describes how have they been implemented in the microscopic traffic simulator AIMSUN2.

Brunowsky local form of a holomorphic family of pairs of matrices

Following Arnold’s techniques, we obtain a local canonical form of a holomorphic family of pairs of matrices ( A( A), B(h)) ac t e d on by the state feedback group. We obtain an explicit formula to compute the dimension of the base space of any miniversal deformation of ( A(O), B(O)). W e m ak e some applications to local perturbations of a pair of matrices.

Differentiable structure of the set of controllable (A,B)t-invariant subspaces☆

Abstract Given (A,B) t ∈ Hom( C n , C n+m ) observable, we prove that the set of ( A,B ) t -invariant subspaces having a fixed Brunovsky-Kronecker structure is a connected manifold, and we compute its dimension. Also, we include some applications of these results.

Similarity of non-everywhere defined linear maps☆

Abstract The equivalence between linear maps defined only on a subspace (or, by duality, defined modulo a subspace) is studied, and applied to block similarity of rectangular matrices. In that way, we find the complete families of invariants obtained by Brunovsky and Zaballa, and the results of Gohberg and others for this particular case. Also, we describe explicit methods for obtaining the reduced matrix and the corresponding bases.

Client-driven load balancing through association control in IEEE 802.11 WLANs

The growth of IEEE 802.11 wireless local area networks (WLANs) (Wi-Fi) brings new possibilities of getting connected in public spaces, known as Hot Spots. Current client-access point associations are an interesting research topic because in these scenarios, users tend to be ‘gregarious’ and essentially static. Under IEEE 802.11 standards, association and roaming decisions are made by client devices and most implementations are based on signal strength measurements; i.e. a client station selects the access point (AP) that provides the strongest signal, which leads to an uneven distribution of clients and load between neighbouring APs. As it can be observed in practical scenarios, the default AP-client association scheme followed in IEEE WLANs, produces unfair situations. This work provides means to effectively alleviate this performance issue and also gives details for a feasible implementation. In this paper we analyse how new IEEE 802.11 standards will allow new radio measurements to provide more efficient association decisions. We propose a new load metric that will produce client-driven associations that ensure greater fairness and throughput. Copyright

On (A, B)t-invariant subspaces having extendible Brunovsky bases

We consider (A, BY-invariant subspaces having a Brunovsky basis which can be extended to a Brunovsky basis of the whole space. We obtain a geometrical characterization of this class of (A, BY-invariant subspaces, and a complete family of numerical invariants to classify them. 0 Elsevier Science Inc., 1997

GEOMETRIC CHARACTERIZATION AND CLASSIFICATION OF MARKED SUBSPACES

Abstract Given an endomorphism of a finite dimensional C -vector space, we obtain geometric conditions in order to characterize and to classify the marked invariant subspaces W ⊂ E (that is to say, the invariant subspaces having some Jordan basis which can be extended to a Jordan basis of E ). Both conditions are expressed in terms of the double family of subspaces (Ker f h ∩ Im f d ). The starting point is a new criterion for extensibility of Jordan bases.

Versal deformations of invariant subspaces

We describe a miniversal deformation of invariant subspaces (with regard to a fixed endomorphism) by means of a technique which can be applied to obtain explicitly miniversal deformations in a general orbit space. In addition, we present an application to the problem of classifying invariant subspaces.

Regularity of the Brunovsky--Kronecker Stratification

We study the partition of the set of pairs of matrices according to the Brunovsky--Kronecker type. We show that it is a constructible stratification, and that it is Whitney regular when the second matrix is a column matrix. We give an application to the obtainment of bifurcation diagrams for few-parameter generic families of linear systems.