Maslov index, Lagrangians, Mapping Class Groups and TQFT

Given a mapping class f of an oriented surface Sigma and a lagrangian lambda in the first homology of Sigma, we define an integer n_{lambda}(f). We use n_{lambda}(f) (mod 4) to describe a universal central extension of the mapping class group of Sigma as an index-four subgroup of the extension constructed from the Maslov index of triples of lagrangian subspaces in the homology of the surface. We give two descriptions of this subgroup. One is topological using surgery, the other is homological and builds on work of Turaev and work of Walker. Some applications to TQFT are discussed. They are based on the fact that our construction allows one to precisely describe how the phase factors that arise in the skein theory approach to TQFT-representations of the mapping class group depend on the choice of a lagrangian on the surface.Comment: 31 pages, 11 Figures. to appear in Forum Mathematicu

Effects of Lateral Diffusion on the Dynamics of Desorption

The adsorbate dynamics during simultaneous action of desorption and lateral adsorbate diffusion is studied in a simple lattice-gas model by kinetic Monte Carlo simulations. It is found that the action of the coverage-conserving diffusion process during the course of the desorption has two distinct, competing effects: a general acceleration of the desorption process, and a coarsening of the adsorbate configuration through Ostwald ripening. The balance between these two effects is governed by the structure of the adsorbate layer at the beginning of the desorption process

Microstructure and velocity of field-driven solid-on-solid interfaces moving under stochastic dynamics with local energy barriers

We study the microscopic structure and the stationary propagation velocity of (1+1)-dimensional solid-on-solid interfaces in an Ising lattice-gas model, which are driven far from equilibrium by an applied force, such as a magnetic field or a difference in (electro)chemical potential. We use an analytic nonlinear-response approximation [P.A. Rikvold and M. Kolesik, J. Stat. Phys. 100, 377 (2000)] together with kinetic Monte Carlo simulations. Here we consider interfaces that move under Arrhenius dynamics, which include a microscopic energy barrier between the allowed Ising/lattice-gas states. Two different dynamics are studied: the standard one-step dynamic (OSD) [H.C. Kang and W. Weinberg, J. Chem. Phys. 90, 2824 (1992)] and the two-step transition-dynamics approximation (TDA) [T. Ala-Nissila, J. Kjoll, and S.C. Ying, Phys. Rev. B 46, 846 (1992)]. In the OSD the effects of the applied force and the interaction energies in the model factorize in the transition rates (a soft dynamic), while in the TDA such factorization is not possible (a hard dynamic). In full agreement with previous general theoretical results we find that the local interface width under the TDA increases dramatically with the applied force. In contrast, the interface structure with the OSD is only weakly influenced by the force, in qualitative agreement with the theoretical expectations. Results are also obtained for the force-dependence and anisotropy of the interface velocity, which also show differences in good agreement with the theoretical expectations for the differences between soft and hard dynamics. Our results confirm that different stochastic interface dynamics that all obey detailed balance and the same conservation laws nevertheless can lead to radically different interface responses to an applied force.Comment: 18 pages RevTex. Minor revisions. Phys. Rev. B, in pres

Integral closure of rings of integer-valued polynomials on algebras

Let $D$ be an integrally closed domain with quotient field $K$. Let $A$ be a torsion-free $D$-algebra that is finitely generated as a $D$-module. For every $a$ in $A$ we consider its minimal polynomial $\mu_a(X)\in D[X]$, i.e. the monic polynomial of least degree such that $\mu_a(a)=0$. The ring ${\rm Int}_K(A)$ consists of polynomials in $K[X]$ that send elements of $A$ back to $A$ under evaluation. If $D$ has finite residue rings, we show that the integral closure of ${\rm Int}_K(A)$ is the ring of polynomials in $K[X]$ which map the roots in an algebraic closure of $K$ of all the $\mu_a(X)$, $a\in A$, into elements that are integral over $D$. The result is obtained by identifying $A$ with a $D$-subalgebra of the matrix algebra $M_n(K)$ for some $n$ and then considering polynomials which map a matrix to a matrix integral over $D$. We also obtain information about polynomially dense subsets of these rings of polynomials.Comment: Keywords: Integer-valued polynomial, matrix, triangular matrix, integral closure, pullback, polynomially dense set. accepted for publication in the volume "Commutative rings, integer-valued polynomials and polynomial functions", M. Fontana, S. Frisch and S. Glaz (editors), Springer 201

Monte Carlo with Absorbing Markov Chains: Fast Local Algorithms for Slow Dynamics

A class of Monte Carlo algorithms which incorporate absorbing Markov chains is presented. In a particular limit, the lowest-order of these algorithms reduces to the $n$-fold way algorithm. These algorithms are applied to study the escape from the metastable state in the two-dimensional square-lattice nearest-neighbor Ising ferromagnet in an unfavorable applied field, and the agreement with theoretical predictions is very good. It is demonstrated that the higher-order algorithms can be many orders of magnitude faster than either the traditional Monte Carlo or $n$-fold way algorithms.Comment: ReVTeX, Request 3 figures from [email protected]

Unconventional MBE Strategies from Computer Simulations for Optimized Growth Conditions

We investigate the influence of step edge diffusion (SED) and desorption on Molecular Beam Epitaxy (MBE) using kinetic Monte-Carlo simulations of the solid-on-solid (SOS) model. Based on these investigations we propose two strategies to optimize MBE growth. The strategies are applicable in different growth regimes: During layer-by-layer growth one can exploit the presence of desorption in order to achieve smooth surfaces. By additional short high flux pulses of particles one can increase the growth rate and assist layer-by-layer growth. If, however, mounds are formed (non-layer-by-layer growth) the SED can be used to control size and shape of the three-dimensional structures. By controlled reduction of the flux with time we achieve a fast coarsening together with smooth step edges.Comment: 19 pages, 7 figures, submitted to Phys. Rev.

New distinguished classes of spectral spaces: a survey

In the present survey paper, we present several new classes of Hochster's spectral spaces "occurring in nature", actually in multiplicative ideal theory, and not linked to or realized in an explicit way by prime spectra of rings. The general setting is the space of the semistar operations (of finite type), endowed with a Zariski-like topology, which turns out to be a natural topological extension of the space of the overrings of an integral domain, endowed with a topology introduced by Zariski. One of the key tool is a recent characterization of spectral spaces, based on the ultrafilter topology, given in a paper by C. Finocchiaro in Comm. Algebra 2014. Several applications are also discussed