Area preservation in computational fluid dynamics

Incompressible two-dimensional flows such as the advection (Liouville) equation and the Euler equations have a large family of conservation laws related to conservation of area. We present two Eulerian numerical methods which preserve a discrete analog of area. The first is a fully discrete model based on a rearrangement of cells; the second is more conventional, but still preserves the area within each contour of the vorticity field. Initial tests indicate that both methods suppress the formation of spurious oscillations in the field.Comment: 14 pages incl. 3 figure

The supermembrane with central charges:(2+1)-D NCSYM, confinement and phase transition

The spectrum of the bosonic sector of the D=11 supermembrane with central charges is shown to be discrete and with finite multiplicities, hence containing a mass gap. The result extends to the exact theory our previous proof of the similar property for the SU(N) regularised model and strongly suggest discreteness of the spectrum for the complete Hamiltonian of the supermembrane with central charges. This theory is a quantum equivalent to a symplectic non-commutative super-Yang-Mills in 2+1 dimensions, where the space-like sector is a Riemann surface of positive genus. In this context, it is argued how the theory in 4D exhibits confinement in the N=1 supermembrane with central charges phase and how the theory enters in the quark-gluon plasma phase through the spontaneous breaking of the centre. This phase is interpreted in terms of the compactified supermembrane without central charges.Comment: 33 pages, Latex. In this new version, several changes have been made and various typos were correcte

Coincidence free pairs of maps

This paper centers around two basic problems of topological coincidence theory. First, try to measure (with help of Nielsen and minimum numbers) how far a given pair of maps is from being loose, i.e. from being homotopic to a pair of coincidence free maps. Secondly, describe the set of loose pairs of homotopy classes. We give a brief (and necessarily very incomplete) survey of some old and new advances concerning the first problem. Then we attack the second problem mainly in the setting of homotopy groups. This leads also to a very natural filtration of all homotopy sets. Explicit calculations are carried out for maps into spheres and projective spaces

Structure-Preserving Discretization of Incompressible Fluids

The geometric nature of Euler fluids has been clearly identified and extensively studied over the years, culminating with Lagrangian and Hamiltonian descriptions of fluid dynamics where the configuration space is defined as the volume-preserving diffeomorphisms, and Kelvin's circulation theorem is viewed as a consequence of Noether's theorem associated with the particle relabeling symmetry of fluid mechanics. However computational approaches to fluid mechanics have been largely derived from a numerical-analytic point of view, and are rarely designed with structure preservation in mind, and often suffer from spurious numerical artifacts such as energy and circulation drift. In contrast, this paper geometrically derives discrete equations of motion for fluid dynamics from first principles in a purely Eulerian form. Our approach approximates the group of volume-preserving diffeomorphisms using a finite dimensional Lie group, and associated discrete Euler equations are derived from a variational principle with non-holonomic constraints. The resulting discrete equations of motion yield a structure-preserving time integrator with good long-term energy behavior and for which an exact discrete Kelvin's circulation theorem holds

Discrete and continuum third quantization of Gravity

We give a brief introduction to matrix models and the group field theory (GFT) formalism as realizations of the idea of a third quantization of gravity, and present in some more detail the idea and basic features of a continuum third quantization formalism in terms of a field theory on the space of connections, building up on the results of loop quantum gravity that allow to make the idea slightly more concrete. We explore to what extent one can rigorously define such a field theory. Concrete examples are given for the simple case of Riemannian GR in 3 spacetime dimensions. We discuss the relation between GFT and this formal continuum third quantized gravity, and what it can teach us about the continuum limit of GFTs.Comment: 21 pages, 5 eps figures; submitted as a contribution to the proceedings of the conference "Quantum Field Theory and Gravity Conference Regensburg 2010" (28 September - 1 October 2010, Regensburg/Bavaria); v2: preprint number include

On volume-preserving vector fields and finite type invariants of knots

We consider the general nonvanishing, divergence-free vector fields defined on a domain in three space and tangent to its boundary. Based on the theory of finite type invariants, we define a family of invariants for such fields, in the style of Arnold's asymptotic linking number. Our approach is based on the configuration space integrals due to Bott and Taubes.Comment: 30 pages, 6 figures, exposition improve