Vector fields can arise in the cosmological context in different ways, and we discuss both abelian and nonabelian sector. In the abelian sector vector fields of the geometrical origin (from dimensional reduction and Einstein–Eddington modification of gravity) can provide a very non-trivial dynamics, which can be expressed in terms of the effective dilaton-scalar gravity with the specific potent...

In this paper we will address the question of how many nonvanishing, linearly independent tangent vector fields can exist on a sphere Sn−1 ⊆ R. By this we mean the following, a tangent vector field on Sn−1 = {x ∈ R : ‖x‖ = 1} is a map v : Sn−1 → R such that v(x) ⊥ x for all x ∈ Sn−1. However, by assumption v is nonvanishing, so we can normalize such that ‖v(x)‖ = 1 and we obtain a map v : Sn−1 ...

We propose statistically self-similar and rotation-invariant models for vector fields, study some of the more significant properties of these models, and suggest algorithms and methods for reconstructing vector fields from numerical observations, using the same notions of self-similarity and invariance that give rise to our stochastic models. We illustrate the efficacy of the proposed schemes b...

We present alternative ways of looking at vector elds that complement existing ow visualization methods. These techniques are based on bump mapping and are simple, easy to compute and fast to render. For example, in one of the techniques a surface from the vector eld is extracted and the directions of the vector values on the surface are normalized and used as surface normals. The shading of th...

This paper presents a solution to the problem of finding the maximum number of linearly independent vector fields that can be placed on a sphere. To produce the correct upper bound, we make use of K-theory. After briefly recapitulating the basics of K-theory, we introduce Adams operations and compute the K-theory of the complex and real projective spaces. We then define the characteristic class...

We introduce a scheme of control polygons to design topological skeletons for vector fields of arbitrary topology. Based on this we construct piecewise linear vector fields of exactly the topology specified by the control polygons. This way a controlled construction of vector fields of any topology is possible. Finally we apply this method for topology-preserving compression of vector fields co...

These notes cover some of the recent developments in the renormalization of quasiperiodic flows. This includes skew flows over tori, Hamiltonian flows, and other flows on T×R. After stating some of the problems and describing alternative approaches, we focus on the definition and basic properties of a single renormalization step. A second part deals with the construction of conjugacies and inva...

This work puts forward an extended definition of vector fractional Brownian motion (fBm) using a distribution theoretic formulation in the spirit of Gel’fand and Vilenkin’s stochastic analysis. We introduce random vector fields that share the statistical invariances of standard vector fBm (self-similarity and rotation invariance) but which, in contrast, have dependent vector components in the g...

For a smooth fibre bundle F i −→ E p −→ B where F is a compact manifold with or without boundary, a vertical vector field V gives rise to a transfer τ V as an S-map. Our goal is to show these transfers satisfy an equation analogous to one that the index of vector fields satisfy. This equation gives results involving equivariant vector fields as well as a characterization of those transfers defi...