Finite Boolean Algebras for Solid Geometry using Julia's Sparse Arrays

The goal of this paper is to introduce a new method in computer-aided geometry of solid modeling. We put forth a novel algebraic technique to evaluate any variadic expression between polyhedral d-solids (d = 2, 3) with regularized operators of union, intersection, and difference, i.e., any CSG tree. The result is obtained in three steps: first, by computing an independent set of generators for the d-space partition induced by the input; then, by reducing the solid expression to an equivalent logical formula between Boolean terms made by zeros and ones; and, finally, by evaluating this expression using bitwise operators. This method is implemented in Julia using sparse arrays. The computational evaluation of every possible solid expression, usually denoted as CSG (Constructive Solid Geometry), is reduced to an equivalent logical expression of a finite set algebra over the cells of a space partition, and solved by native bitwise operators.Comment: revised version submitted to Computer-Aided Geometric Desig

An Efficient Representation for Filtrations of Simplicial Complexes

A filtration over a simplicial complex $K$ is an ordering of the simplices of $K$ such that all prefixes in the ordering are subcomplexes of $K$. Filtrations are at the core of Persistent Homology, a major tool in Topological Data Analysis. In order to represent the filtration of a simplicial complex, the entire filtration can be appended to any data structure that explicitly stores all the simplices of the complex such as the Hasse diagram or the recently introduced Simplex Tree [Algorithmica '14]. However, with the popularity of various computational methods that need to handle simplicial complexes, and with the rapidly increasing size of the complexes, the task of finding a compact data structure that can still support efficient queries is of great interest. In this paper, we propose a new data structure called the Critical Simplex Diagram (CSD) which is a variant of the Simplex Array List (SAL) [Algorithmica '17]. Our data structure allows one to store in a compact way the filtration of a simplicial complex, and allows for the efficient implementation of a large range of basic operations. Moreover, we prove that our data structure is essentially optimal with respect to the requisite storage space. Finally, we show that the CSD representation admits fast construction algorithms for Flag complexes and relaxed Delaunay complexes.Comment: A preliminary version appeared in SODA 201

Native vegetation of the southern forests : south-east highlands, Australian alps, south-west Slopes, and SE Corner bioregions

The Southern Forests study area covers an area of about six million hectares of south-eastern New South Wales, south of Oberon and Kiama and east of Albury and Boorowa (latitude 33° 02’–37 ° 06’ S; longitude 146° 56’ – 147° 06’ E). The total area of existing vegetation mapped was three million hectares (3 120 400 hectares) or about 50% of the study area. Terrestrial, wetland and estuarine vegetation of the Southern Forests region were classified into 206 vegetation groups and mapped at a scale between 1: 25 000 and 1: 100 000. The classification was based on a cluster analysis of detailed field surveys of vascular plants, as well as field knowledge in the absence of field survey data. The primary classification was based on 3740 vegetation samples with full floristics cover abundance data. Additional classifications of full floristics presence-absence and tree canopy data were carried out to guide mapping in areas with few full floristic samples. The mapping of extant vegetation was carried out by tagging vegetation polygons with vegetation codes, guided by expert knowledge, using field survey data classified into vegetation groups, remote sensing, and other environmental spatial data. The mapping of pre-1750 vegetation involved tagging of soils mapping with vegetation codes at 1: 100 000 scale, guided by spatial modelling of vegetation groups using generalised additive statistical models (GAMS), and expert knowledge. Profiles of each of the vegetation groups on the CD-ROM* provide key indicator species, descriptions, statistics and lists of informative plant species. The 206 vegetation groups cover the full range of natural vegetation, including rainforests, moist eucalypt forests, dry shrub forests, grassy forests, mallee low forests, heathlands, shrublands, grasslands and wetlands. There are 138 groups of Eucalyptus forests or woodlands, 12 rainforest groups, and 46 non-forest groups. Of the 206 groups, 193 were classified and mapped in the study area. Thirteen vegetation groups were not mapped because of their small size and lack of samples, or because they fell outside the study area. Updated regional extant and pre-1750 vegetation maps of southern New South Wales have been produced in 2005, based on those originally prepared in 2000 for the southern Regional Forest Agreement (RFA). Further validation and remapping of extant vegetation over 10% of the study area has subsequently improved the quality of the vegetation map, and removed some of the errors in the original version. The revised map provides a reasonable representation of native vegetation at a scale between 1: 25 000 and 1: 100 000 across the study area. In 2005 native vegetation covers 50% of the study area. Environmental pressures on the remaining vegetation include clearing, habitat degradation from weeds and nutrification, severe droughts, changing fire regimes, and urbanisation. Grassy woodlands and forests, temperate grasslands, and coastal and riparian vegetation have been the most reduced in areal extent. Over 90% of the grassy woodlands and temperate grasslands have been lost. Conservation of the remaining vegetation in these formations is problematic because of the small, discontinuous, and degraded nature of the remaining patches of vegetation

Geometric Reasoning with polymake

The mathematical software system polymake provides a wide range of functions for convex polytopes, simplicial complexes, and other objects. A large part of this paper is dedicated to a tutorial which exemplifies the usage. Later sections include a survey of research results obtained with the help of polymake so far and a short description of the technical background

Mining Maximal Cliques from an Uncertain Graph

We consider mining dense substructures (maximal cliques) from an uncertain graph, which is a probability distribution on a set of deterministic graphs. For parameter 0 < {\alpha} < 1, we present a precise definition of an {\alpha}-maximal clique in an uncertain graph. We present matching upper and lower bounds on the number of {\alpha}-maximal cliques possible within an uncertain graph. We present an algorithm to enumerate {\alpha}-maximal cliques in an uncertain graph whose worst-case runtime is near-optimal, and an experimental evaluation showing the practical utility of the algorithm.Comment: ICDE 201

Finding complex balanced and detailed balanced realizations of chemical reaction networks

Reversibility, weak reversibility and deficiency, detailed and complex balancing are generally not "encoded" in the kinetic differential equations but they are realization properties that may imply local or even global asymptotic stability of the underlying reaction kinetic system when further conditions are also fulfilled. In this paper, efficient numerical procedures are given for finding complex balanced or detailed balanced realizations of mass action type chemical reaction networks or kinetic dynamical systems in the framework of linear programming. The procedures are illustrated on numerical examples.Comment: submitted to J. Math. Che