Fast Fourier Transforms for Finite Inverse Semigroups

We extend the theory of fast Fourier transforms on finite groups to finite inverse semigroups. We use a general method for constructing the irreducible representations of a finite inverse semigroup to reduce the problem of computing its Fourier transform to the problems of computing Fourier transforms on its maximal subgroups and a fast zeta transform on its poset structure. We then exhibit explicit fast algorithms for particular inverse semigroups of interest--specifically, for the rook monoid and its wreath products by arbitrary finite groups.Comment: ver 3: Added improved upper and lower bounds for the memory required by the fast zeta transform on the rook monoid. ver 2: Corrected typos and (naive) bounds on memory requirements. 30 pages, 0 figure

Fast Fourier Transforms for the Rook Monoid

We define the notion of the Fourier transform for the rook monoid (also called the symmetric inverse semigroup) and provide two efficient divide-and-conquer algorithms (fast Fourier transforms, or FFTs) for computing it. This paper marks the first extension of group FFTs to non-group semigroups

Algebraic and combinatorial aspects of sandpile monoids on directed graphs

The sandpile group of a graph is a well-studied object that combines ideas from algebraic graph theory, group theory, dynamical systems, and statistical physics. A graph's sandpile group is part of a larger algebraic structure on the graph, known as its sandpile monoid. Most of the work on sandpiles so far has focused on the sandpile group rather than the sandpile monoid of a graph, and has also assumed the underlying graph to be undirected. A notable exception is the recent work of Babai and Toumpakari, which builds up the theory of sandpile monoids on directed graphs from scratch and provides many connections between the combinatorics of a graph and the algebraic aspects of its sandpile monoid. In this paper we primarily consider sandpile monoids on directed graphs, and we extend the existing theory in four main ways. First, we give a combinatorial classification of the maximal subgroups of a sandpile monoid on a directed graph in terms of the sandpile groups of certain easily-identifiable subgraphs. Second, we point out certain sandpile results for undirected graphs that are really results for sandpile monoids on directed graphs that contain exactly two idempotents. Third, we give a new algebraic constraint that sandpile monoids must satisfy and exhibit two infinite families of monoids that cannot be realized as sandpile monoids on any graph. Finally, we give an explicit combinatorial description of the sandpile group identity for every graph in a family of directed graphs which generalizes the family of (undirected) distance-regular graphs. This family includes many other graphs of interest, including iterated wheels, regular trees, and regular tournaments.Comment: v2: Cleaner presentation, new results in final section. Accepted for publication in J. Combin. Theory Ser. A. 21 pages, 5 figure

Inverse semigroup spectral analysis for partially ranked data

Motivated by the notion of symmetric group spectral analysis developed by Diaconis, we introduce the notion of spectral analysis on the rook monoid (also called the symmetric inverse semigroup), characterize its output in terms of symmetric group spectral analysis, and provide an application to the statistical analysis of partially ranked (voting) data. We also discuss generalizations to arbitrary finite inverse semigroups. This paper marks the first non-group semigroup development of spectral analysis.Comment: v3: Significant changes in terms of organization and presentation. Accepted for publication in Appl. Comput. Harmon. Anal. 25 pages, 5 tables. v2: Some reformatting, minor typos correcte

A Potential Energy Landscape Study of the Amorphous-Amorphous Transformation in H2_2O

We study the potential energy landscape explored during a compression-decompression cycle for the SPC/E (extended simple point charge) model of water. During the cycle, the system changes from low density amorphous ice (LDA) to high density amorphous ice (HDA). After the cycle, the system does not return to the same region of the landscape, supporting the interesting possibility that more than one significantly different configuration corresponds to LDA. We find that the regions of the landscape explored during this transition have properties remarkably different from those explored in thermal equilibrium in the liquid phase

Mass transport in a strongly sheared binary mixture of Maxwell molecules

Transport coefficients associated with the mass flux of a binary mixture of Maxwell molecules under uniform shear flow are exactly determined from the Boltzmann kinetic equation. A normal solution is obtained via a Chapman--Enskog-like expansion around a local shear flow distribution that retains all the hydrodynamics orders in the shear rate. In the first order of the expansion the mass flux is proportional to the gradients of mole fraction, pressure, and temperature but, due to the anisotropy induced in the system by the shear flow, mutual diffusion, pressure diffusion and thermal diffusion tensors are identified instead of the conventional scalar coefficients. These tensors are obtained in terms of the shear rate and the parameters of the mixture (particle masses, concentrations, and force constants). The description is made both in the absence and in the presence of an external thermostat introduced in computer simulations to compensate for the viscous heating. As expected, the analysis shows that there is not a simple relationship between the results with and without the thermostat. The dependence of the three diffusion tensors on the shear rate is illustrated in the tracer limit case, the results showing that the deviation of the generalized transport coefficients from their equilibrium forms is in general quite important. Finally, the generalized transport coefficients associated with the momentum and heat transport are evaluated from a model kinetic equation of the Boltzmann equation.Comment: 6 figure

On the critical nature of plastic flow: one and two dimensional models

Steady state plastic flows have been compared to developed turbulence because the two phenomena share the inherent complexity of particle trajectories, the scale free spatial patterns and the power law statistics of fluctuations. The origin of the apparently chaotic and at the same time highly correlated microscopic response in plasticity remains hidden behind conventional engineering models which are based on smooth fitting functions. To regain access to fluctuations, we study in this paper a minimal mesoscopic model whose goal is to elucidate the origin of scale free behavior in plasticity. We limit our description to fcc type crystals and leave out both temperature and rate effects. We provide simple illustrations of the fact that complexity in rate independent athermal plastic flows is due to marginal stability of the underlying elastic system. Our conclusions are based on a reduction of an over-damped visco-elasticity problem for a system with a rugged elastic energy landscape to an integer valued automaton. We start with an overdamped one dimensional model and show that it reproduces the main macroscopic phenomenology of rate independent plastic behavior but falls short of generating self similar structure of fluctuations. We then provide evidence that a two dimensional model is already adequate for describing power law statistics of avalanches and fractal character of dislocation patterning. In addition to capturing experimentally measured critical exponents, the proposed minimal model shows finite size scaling collapse and generates realistic shape functions in the scaling laws.Comment: 72 pages, 40 Figures, International Journal of Engineering Science for the special issue in honor of Victor Berdichevsky, 201