Global Solvability of the Cauchy Problem for the Landau-Lifshitz-Gilbert Equation in Higher Dimensions

We prove existence, uniqueness and asymptotics of global smooth solutions for the Landau-Lifshitz-Gilbert equation in dimension $n \ge 3$, valid under a smallness condition of initial gradients in the $L^n$ norm. The argument is based on the method of moving frames that produces a covariant complex Ginzburg-Landau equation, and a priori estimates that we obtain by the method of weighted-in-time norms as introduced by Fujita and Kato

Analogues of Non-Gibbsianness in Joint Measures of Disordered Mean Field Models

It is known that the joint measures on the product of spin-space and disorder space are very often non-Gibbsian measures, for lattice systems with quenched disorder, at low temperature. Are there reflections of this non-Gibbsianness in the corresponding mean-field models? We study the continuity properties of the conditional probabilities in finite volume of the following mean field models: (a) joint measures of random field Ising, (b) joint measures of dilute Ising, (c) decimation of ferromagnetic Ising. The conditional probabilities are functions of the empirical mean of the conditionings; so we look at the large volume behavior of these functions to discover non-trivial limiting objects. For (a) we find (1) discontinuous dependence for almost any realization and (2) dependence of the conditional probabilities on the phase. In contrast to that we see continuous behavior for (b) and (c), for almost any realization. This is in complete analogy to the behavior of the corresponding lattice models in high dimensions. It shows that non-Gibbsian behavior which seems a genuine lattice phenomenon can be partially understood already on the level of mean-field models.

Decision Problems for Deterministic Pushdown Automata on Infinite Words

The article surveys some decidability results for DPDAs on infinite words (omega-DPDA). We summarize some recent results on the decidability of the regularity and the equivalence problem for the class of weak omega-DPDAs. Furthermore, we present some new results on the parity index problem for omega-DPDAs. For the specification of a parity condition, the states of the omega-DPDA are assigned priorities (natural numbers), and a run is accepting if the highest priority that appears infinitely often during a run is even. The basic simplification question asks whether one can determine the minimal number of priorities that are needed to accept the language of a given omega-DPDA. We provide some decidability results on variations of this question for some classes of omega-DPDAs.Comment: In Proceedings AFL 2014, arXiv:1405.527