Querying for the Largest Empty Geometric Object in a Desired Location

We study new types of geometric query problems defined as follows: given a geometric set $P$, preprocess it such that given a query point $q$, the location of the largest circle that does not contain any member of $P$, but contains $q$ can be reported efficiently. The geometric sets we consider for $P$ are boundaries of convex and simple polygons, and point sets. While we primarily focus on circles as the desired shape, we also briefly discuss empty rectangles in the context of point sets.Comment: This version is a significant update of our earlier submission arXiv:1004.0558v1. Apart from new variants studied in Sections 3 and 4, the results have been improved in Section 5.Please note that the change in title and abstract indicate that we have expanded the scope of the problems we stud

One-dimensional itinerant interacting non-Abelian anyons

We construct models of interacting itinerant non-Abelian anyons moving along one-dimensional chains. We focus on itinerant Ising (Majorana) and Fibonacci anyons, which are, respectively, related to SU(2)_2 and SU(2)_3 anyons and also, respectively, describe quasiparticles of the Moore-Read and Z_3-Read-Rezayi fractional quantum Hall states. Following the derivation of the electronic large-U effective Hubbard model, we derive effective anyonic t-J models for the low-energy sectors. Solving these models by exact diagonalization, we find a fractionalization of the anyons into charge and (neutral) anyonic degrees of freedom -- a generalization of spin-charge separation of electrons which occurs in Luttinger liquids. A detailed description of the excitation spectrum can be performed by combining spectra for charge and anyonic sectors. The anyonic sector is the one of a squeezed chain of localized interacting anyons, and hence is described by the same conformal field theory (CFT), with central charge c=1/2 for Ising anyons and c=7/10 or c=4/5 for Fibonacci anyons with antiferromagnetic or ferromagnetic coupling, respectively. The charge sector is the spectrum of a chain of hardcore bosons subject to phase shifts which coincide with the momenta of the combined anyonic eigenstates, revealing a subtle coupling between charge and anyonic excitations at the microscopic level (which we also find to be present in Luttinger liquids), despite the macroscopic fractionalization. The combined central charge extracted from the entanglement entropy between segments of the chain is shown to be 1+c, where c is the central charge of the underlying CFT of the localized anyon (squeezed) chain.Comment: 19 pages, 18 figure

Enumerative Geometry of Del Pezzo Surfaces

We prove an equivalence between the superpotential defined via tropical geometry and Lagrangian Floer theory for special Lagrangian torus fibres in del Pezzo surfaces constructed by Collins-Jacob-Lin. We also include some explicit calculations for the projective plane, which confirm some folklore conjecture in this case.Comment: 42 pages, 1 figrure. Comments are welcom

Domain patterns in incommensurate systems with the uniaxial real order parameter

The basic Landau model for the incommensurate-commensurate transition to the uniform or dimerized uniaxial ordering is critically reexamined. The previous analyses identified only sinusoidal and homogeneous solutions as thermodynamically stable and proposed a simple phase diagram with the first-order phase transition between these configurations. By performing the numerical analysis of the free-energy and the Euler-Lagrange equation we show that the phase diagram is more complex. It also contains a set of metastable solutions present in the range of coexistence of homogeneous and sinusoidal solutions. These new configurations are periodic patterns of homogeneous domains connected by sinusoidal segments. They are Lyapunov unstable, very probably due to the nonintegrability of the free-energy functional. We also discuss some other mathematical aspects of the model and compare it with the essentially simpler sine-Gordon model for the transitions to the states with higher commensurabilities. We argue that the present results might be a basis for the explanation of phenomena such as thermal hystereses, cascades of phase transitions, and memory effects

C-9 and Other Microgravity Simulations

This document represents a summary of medical and scientific evaluations conducted aboard the C-9 or other NASA-sponsored aircraft from June 30, 2006, to June 30, 2007. Included is a general overview of investigations manifested and coordinated by the Human Adaptation and Countermeasures Office. A collection of brief reports that describe tests conducted aboard the NASA-sponsored aircraft follows the overview. Principal investigators and test engineers contributed significantly to the content of the report, describing their particular experiment or hardware evaluation. Although this document follows general guidelines, each report format may vary to accommodate differences in experiment design and procedures. This document concludes with an appendix that provides background information about the Reduced Gravity Program

Autonomous Robot Navigation through a Crowded and Dynamic Environment: Using A Novel form of Path Planning to Demonstrate Consideration towards Pedestrians and other Robots

This thesis presents a novel path planning algorithm for robotic crowd navigation through a pedestrian environment. The robot is designed to negotiate its way through the crowd using considerate movements. Unlike many other path planning algorithms, which assume cooperation with other pedestrians, this algorithm is completely independent and requires only observation. A considerate navigation strategy has been developed in this thesis, which utilises consideration as an directs an autonomous mobile robot. Using simple methods of predicting pedestrian movements, as well as simple relative distance and trajectory measurements between the robot and pedestrians, the robot can navigate through a crowd without causing disruption to pedestrian trajectories. Dynamic pedestrian positions are predicted using uncertainty ellipses. A novel Voronoi diagram-visibility graph hybrid roadmap is implemented so that the path planner can exploit any available gaps in between pedestrians, and plan considerate paths. The aim of the considerate path planner is to have the robot behave in specific ways when moving through the crowd. By predicting pedestrian trajectories, the robot can avoid interfering with them. Following preferences to move behind pedestrians, when cutting across their trajectories; to move in the same direction of the crowd when possible; and to slow down in crowded areas, will prevent any interference to individual pedestrians, as well as preventing an increase in congestion to the crowd as a whole. The effectiveness of the considerate navigation strategy is evaluated using simulated pedestrians, multiple mobile robots loaded with the path planning algorithm, as well as a real-life pedestrian dataset. The algorithm will highlight its ability to move with the aforementioned consideration towards each individual dynamic agent