he study of coherent structures in turbu- lent boundary layers is an active area of research. Driven by applications including the reduc- tion of turbulent skin-friction drag over aircraft, researchers have long been interested in developing a deeper understanding of the key physical features with- in turbulent flows. One application of interest is the analysis of multiple scalar and vector distributions and how the interaction of variables contributes to theories of drag and the formation of vortex packets within a turbulent boundary layer. A second application involves examining the correlation between different layers within a numerical simulation of a turbulent channel flow. The analysis of these applica- tions can be greatly facilitated by a better understanding of the compli- cated interactions that occur between the vortices that make up the motion. Important research conducted in the computational physics field focuses on modeling 3D magneto- hydrodynamic light supersonic jets in the context of astrophysical jets in galaxy clusters. These high-speed jets propagate distances of over 650,000 light years from their sources, transporting energy and magnetic fields to their surrounding environments. The jet magnetic field is advected along with the flow and is expected to reflect properties of the evolving velocity field. Of particular interest is the extent to which the magnetic and velocity vector fields are spa- tially aligned and/or orthogonal to one another and the interplay between magnetic field strength and the cor- responding velocity structures. Our research explores strategies for developing effec- tive methods for the visual representation of multiple colocated vector and scalar fields to allow understand- ing and analyses of each field, both individually and in the context of the other. Mining the knowledge base of previous visualization research yields important find- ings and insights to assist the research of our specific applications. Using these insights, we can reconstruct, manipulate, and expand upon the existing state of the art to further the knowledge-discovery process related to specific tasks and conditions. Our present efforts focus primarily on the investigation of methods for effective- ly visualizing multiple fields defined over a 2D domain; the problem of effectively visualizing multiple fields defined over a 3D domain is even more challenging, and an important area for future work. The "Classifying and Combining Visualization Techniques" sidebar presents previous and related work. Dual vector fields

In this paper we obtain results for the systematic study of reversible-equivariant vector fields - namely, in the simultaneous presence of symmetries and reversing symmetries - by employing algebraic techniques from invariant theory for compact Lie groups. We introduce the Hilbert-Poincare series and their associated Molien formula and we prove the character formulas for the computation of dimensions of spaces of homogeneous anti-invariant polynomial functions and reversible- equivariant polynomial mappings. Two symbolic algorithms are also obtained,

The aim of this paper is to introduce a concrete notion of multiplicity for invariant algebraic curves in polynomial vector fields. In fact, we give several natural definitions and show that they are all equivalent to our main definition, under some 'generic' assumptions. In particular, we show that there is a natural equivalence between the algebraic viewpoint (multiplicities defined by extactic curves or exponential factors) and the geometric viewpoint (multiplicities defined by the number of algebraic curves which can appear under bifurcation or by the holonomy group of the curve). Furthermore, via the extactic, we can give an eective method for calculating the multiplicity of a given curve. As applications of our results, we give a solution to the inverse problem of describing the module of vector fields with prescribed algebraic curves with their multiplicities; we also give a completed version of the Darboux theory of integration which takes the multiplicities of the curves into account. In this paper, we have concentrated mainly on the multiplicity of a single irreducible and reduced curve. We hope, however, that the range of equivalent definitions given here already demonstrates that this notion of multiplicity is both natural and useful for applications.