Geometric gradient-flow dynamics with singular solutions

The gradient-flow dynamics of an arbitrary geometric quantity is derived using a generalization of Darcy's Law. We consider flows in both Lagrangian and Eulerian formulations. The Lagrangian formulation includes a dissipative modification of fluid mechanics. Eulerian equations for self-organization of scalars, 1-forms and 2-forms are shown to reduce to nonlocal characteristic equations. We identify singular solutions of these equations corresponding to collapsed (clumped) states and discuss their evolution.Comment: 28 pages, 1 figure, to appear on Physica

Ising Spins on a Gravitating Sphere

We investigated numerically an Ising model coupled to two-dimensional Euclidean gravity with spherical topology, using Regge calculus with the $dl/l$ path-integral measure to discretize the gravitational interaction. Previous studies of this system with toroidal topology have shown that the critical behavior of the Ising model remains in the flat-space Onsager universality class, contrary to the predictions of conformal field theory and matrix models. Implementing the spherical topology as triangulated surfaces of three-dimensional cubes, we find again strong evidence that the critical exponents of the Ising transition are consistent with the Onsager values, and that KPZ exponents are definitely excluded.Comment: 13 pages, self unpacking uuencoded PostScript file, including all the figures. Paper also available at http://www.physik.fu-berlin.de/~holm

Vlasov moments, integrable systems and singular solutions

The Vlasov equation for the collisionless evolution of the single-particle probability distribution function (PDF) is a well-known Lie-Poisson Hamiltonian system. Remarkably, the operation of taking the moments of the Vlasov PDF preserves the Lie-Poisson structure. The individual particle motions correspond to singular solutions of the Vlasov equation. The paper focuses on singular solutions of the problem of geodesic motion of the Vlasov moments. These singular solutions recover geodesic motion of the individual particles.Comment: 16 pages, no figures. Submitted to Phys. Lett.

Construction of totally reflexive modules from an exact pair of zero divisors

Let A be a local ring which admits an exact pair x,y of zero divisors as defined by Henriques and Sega. Assuming that this pair is regular and that there exists a regular element on the A-module A/(x,y), we explicitly construct an infinite family of non-isomorphic indecomposable totally reflexive A-modules. In this setting, our construction provides an answer to a question raised by Christensen, Piepmeyer, Striuli, and Takahashi. Furthermore, we compute the module of homomorphisms between any two given modules from the infinite family mentioned above.Comment: 15 page

Formation and Evolution of Singularities in Anisotropic Geometric Continua

Evolutionary PDEs for geometric order parameters that admit propagating singular solutions are introduced and discussed. These singular solutions arise as a result of the competition between nonlinear and nonlocal processes in various familiar vector spaces. Several examples are given. The motivating example is the directed self assembly of a large number of particles for technological purposes such as nano-science processes, in which the particle interactions are anisotropic. This application leads to the derivation and analysis of gradient flow equations on Lie algebras. The Riemann structure of these gradient flow equations is also discussed.Comment: 38 pages, 4 figures. Physica D, submitte

Modules with cosupport and injective functors

Several authors have studied the filtered colimit closure lim(B) of a class B of finitely presented modules. Lenzing called lim(B) the category of modules with support in B, and proved that it is equivalent to the category of flat objects in the functor category (B^{op},Ab). In this paper, we study the category (Mod-R)^B of modules with cosupport in B. We show that (Mod-R)^B is equivalent to the category of injective objects in (B,Ab), and thus recover a classical result by Jensen-Lenzing on pure injective modules. Works of Angeleri-Hugel, Enochs, Krause, Rada, and Saorin make it easy to discuss covering and enveloping properties of (Mod-R)^B, and furthermore we compare the naturally associated notions of B-coherence and B-noetherianness. Finally, we prove a number of stability results for lim(B) and (Mod-R)^B. Our applications include a generalization of a result by Gruson-Jensen and Enochs on pure injective envelopes of flat modules.Comment: 16 page