Counting a black hole in Lorentzian product triangulations

We take a step toward a nonperturbative gravitational path integral for black-hole geometries by deriving an expression for the expansion rate of null geodesic congruences in the approach of causal dynamical triangulations. We propose to use the integrated expansion rate in building a quantum horizon finder in the sum over spacetime geometries. It takes the form of a counting formula for various types of discrete building blocks which differ in how they focus and defocus light rays. In the course of the derivation, we introduce the concept of a Lorentzian dynamical triangulation of product type, whose applicability goes beyond that of describing black-hole configurations.Comment: 42 pages, 11 figure

Dehn Twists in Heegaard Floer Homology

We derive a new exact sequence in the hat-version of Heegaard Floer homology. As a consequence we see a functorial connection between the invariant of Legendrian knots and the contact element. As an application we derive two vanishing results of the contact element making it possible to easily read off its vanishing out of a surgery presentation in suitable situations.Comment: 61 pages, 39 figures; added details to several proofs. Version published by Algebr. Geom. Topol. 10 (2010), 465--52

Localized mirror functor for Lagrangian immersions, and homological mirror symmetry for P 1 a,b,c

This paper gives a new way of constructing Landau–Ginzburg mirrors usingdeformation theory of Lagrangian immersions motivated by the works of Seidel,Strominger –Yau–Zaslow and Fukaya–Oh–Ohta–Ono. Moreover, we construct acanonical functor from the Fukaya category to the mirror category of matrixfactorizations. This functor derives homological mirror symmetry under someexplicit assumptions.As an application, the construction is applied to spheres with three orbifoldpoints to produce their quantum-corrected mirrors and derive homological mirrorsymmetry. Furthermore, we discover an enumerative meaning of the (inverse)mirror map for elliptic curve quotients

A nodal domain theorem for integrable billiards in two dimensions

Eigenfunctions of integrable planar billiards are studied - in particular, the number of nodal domains, $\nu$, of the eigenfunctions are considered. The billiards for which the time-independent Schr\"odinger equation (Helmholtz equation) is separable admit trivial expressions for the number of domains. Here, we discover that for all separable and non-separable integrable billiards, $\nu$ satisfies certain difference equations. This has been possible because the eigenfunctions can be classified in families labelled by the same value of $m\mod kn$, given a particular $k$, for a set of quantum numbers, $m, n$. Further, we observe that the patterns in a family are similar and the algebraic representation of the geometrical nodal patterns is found. Instances of this representation are explained in detail to understand the beauty of the patterns. This paper therefore presents a mathematical connection between integrable systems and difference equations.Comment: 13 pages, 5 figure

Sweep-Line Extensions to the Multiple Object Intersection Problem: Methods and Applications in Graph Mining

Identifying and quantifying the size of multiple overlapping axis-aligned geometric objects is an essential computational geometry problem. The ability to solve this problem can effectively inform a number of spatial data mining methods and can provide support in decision making for a variety of critical applications. The state-of-the-art approach for addressing such problems resorts to an algorithmic paradigm, collectively known as the sweep-line or plane sweep algorithm. However, its application inherits a number of limitations including lack of versatility and lack of support for ad hoc intersection queries. With these limitations in mind, we design and implement a novel, exact, fast and scalable yet versatile, sweep-line based algorithm, named SLIG. The key idea of our algorithm lies in constructing an auxiliary data structure when the sweep line algorithm is applied, an intersection graph. This graph can effectively be used to provide connectivity properties among overlapping objects and to inform answers to ad hoc intersection queries. It can also be employed to find the location and size of the common area of multiple overlapping objects. SLIG performs significantly faster than classic sweep-line based algorithms, it is more versatile, and provides a suite of powerful querying capabilities. To demonstrate the versatility of our SLIG algorithm we show how it can be utilized for evaluating the importance of nodes in a trajectory network - a type of dynamic network where the nodes are moving objects (cars, pedestrians, etc.) and the edges represent interactions (contacts) between objects as defined by a proximity threshold. The key observation to address the problem is that the time intervals of these interactions can be represented as 1-dimensional axis-aligned geometric objects. Then, a variant of our SLIG algorithm, named SLOT, is utilized that effectively computes the metrics of interest, including node degree, triangle membership and connected components for each node, over time

Post-processing partitions to identify domains of modularity optimization

We introduce the Convex Hull of Admissible Modularity Partitions (CHAMP) algorithm to prune and prioritize different network community structures identified across multiple runs of possibly various computational heuristics. Given a set of partitions, CHAMP identifies the domain of modularity optimization for each partition ---i.e., the parameter-space domain where it has the largest modularity relative to the input set---discarding partitions with empty domains to obtain the subset of partitions that are "admissible" candidate community structures that remain potentially optimal over indicated parameter domains. Importantly, CHAMP can be used for multi-dimensional parameter spaces, such as those for multilayer networks where one includes a resolution parameter and interlayer coupling. Using the results from CHAMP, a user can more appropriately select robust community structures by observing the sizes of domains of optimization and the pairwise comparisons between partitions in the admissible subset. We demonstrate the utility of CHAMP with several example networks. In these examples, CHAMP focuses attention onto pruned subsets of admissible partitions that are 20-to-1785 times smaller than the sets of unique partitions obtained by community detection heuristics that were input into CHAMP.Comment: http://www.mdpi.com/1999-4893/10/3/9