Cyclic sieving and cluster multicomplexes

Reiner, Stanton, and White \cite{RSWCSP} proved results regarding the enumeration of polygon dissections up to rotational symmetry. Eu and Fu \cite{EuFu} generalized these results to Cartan-Killing types other than A by means of actions of deformed Coxeter elements on cluster complexes of Fomin and Zelevinsky \cite{FZY}. The Reiner-Stanton-White and Eu-Fu results were proven using direct counting arguments. We give representation theoretic proofs of closely related results using the notion of noncrossing and semi-noncrossing tableaux due to Pylyavskyy \cite{PN} as well as some geometric realizations of finite type cluster algebras due to Fomin and Zelevinsky \cite{FZClusterII}.Comment: To appear in Adv. Appl. Mat

AT-GIS: highly parallel spatial query processing with associative transducers

Users in many domains, including urban planning, transportation, and environmental science want to execute analytical queries over continuously updated spatial datasets. Current solutions for largescale spatial query processing either rely on extensions to RDBMS, which entails expensive loading and indexing phases when the data changes, or distributed map/reduce frameworks, running on resource-hungry compute clusters. Both solutions struggle with the sequential bottleneck of parsing complex, hierarchical spatial data formats, which frequently dominates query execution time. Our goal is to fully exploit the parallelism offered by modern multicore CPUs for parsing and query execution, thus providing the performance of a cluster with the resources of a single machine. We describe AT-GIS, a highly-parallel spatial query processing system that scales linearly to a large number of CPU cores. ATGIS integrates the parsing and querying of spatial data using a new computational abstraction called associative transducers(ATs). ATs can form a single data-parallel pipeline for computation without requiring the spatial input data to be split into logically independent blocks. Using ATs, AT-GIS can execute, in parallel, spatial query operators on the raw input data in multiple formats, without any pre-processing. On a single 64-core machine, AT-GIS provides 3× the performance of an 8-node Hadoop cluster with 192 cores for containment queries, and 10× for aggregation queries

Molecular dynamics simulation: a tool for exploration and discovery using simple models

Emergent phenomena share the fascinating property of not being obvious consequences of the design of the system in which they appear. This characteristic is no less relevant when attempting to simulate such phenomena, given that the outcome is not always a foregone conclusion. The present survey focuses on several simple model systems that exhibit surprisingly rich emergent behavior, all studied by MD simulation. The examples are taken from the disparate fields of fluid dynamics, granular matter and supramolecular self-assembly. In studies of fluids modeled at the detailed microscopic level using discrete particles, the simulations demonstrate that complex hydrodynamic phenomena in rotating and convecting fluids, the Taylor-Couette and Rayleigh-B\'enard instabilities, can not only be observed within the limited length and time scales accessible to MD, but even quantitative agreement can be achieved. Simulation of highly counterintuitive segregation phenomena in granular mixtures, again using MD methods, but now augmented by forces producing damping and friction, leads to results that resemble experimentally observed axial and radial segregation in the case of a rotating cylinder, and to a novel form of horizontal segregation in a vertically vibrated layer. Finally, when modeling self-assembly processes analogous to the formation of the polyhedral shells that package spherical viruses, simulation of suitably shaped particles reveals the ability to produce complete, error-free assembly, and leads to the important general observation that reversible growth steps contribute to the high yield. While there are limitations to the MD approach, both computational and conceptual, the results offer a tantalizing hint of the kinds of phenomena that can be explored, and what might be discovered when sufficient resources are brought to bear on a problem.Comment: 21 pages, 20 figures (v2 - minor text addition