Point sets that minimize (≤k)(\le k)-edges, 3-decomposable drawings, and the rectilinear crossing number of K30K_{30}

There are two properties shared by all known crossing-minimizing geometric drawings of $K_n$, for $n$ a multiple of 3. First, the underlying $n$-point set of these drawings has exactly $3\binom{k+2}{2}$ $(\le k)$-edges, for all $0\le k < n/3$. Second, all such drawings have the $n$ points divided into three groups of equal size; this last property is captured under the concept of 3-decomposability. In this paper we show that these properties are tightly related: every $n$-point set with exactly $3\binom{k+2}{2}$ $(\le k)$-edges for all $0\le k < n/3$, is 3-decomposable. As an application, we prove that the rectilinear crossing number of $K_{30}$ is 9726.Comment: 14 page

Quasi-infra-red fixed points and renormalisation group invariant trajectories for non-holomorphic soft supersymmetry breaking

In the MSSM the quasi-infra-red fixed point for the top-quark Yukawa coupling gives rise to specific predictions for the soft-breaking parameters. We discuss the extent to which these predictions are modified by the introduction of additional ``non-holomorphic'' soft-breaking terms. We also show that in a specific class of theories there exists an RG-invariant trajectory for the ``non-holomorphic'' terms, which can be understood using a holomorphic spurion term.Comment: 24 pages, TeX, two figures. Uses Harvmac (big) and epsf. Minor errors corrected, and the RG trajectory explained in terms of a holomorphic spurion ter

On the number of simple arrangements of five double pseudolines

We describe an incremental algorithm to enumerate the isomorphism classes of double pseudoline arrangements. The correction of our algorithm is based on the connectedness under mutations of the spaces of one-extensions of double pseudoline arrangements, proved in this paper. Counting results derived from an implementation of our algorithm are also reported.Comment: 24 pages, 16 figures, 6 table

Embedding Four-directional Paths on Convex Point Sets

A directed path whose edges are assigned labels "up", "down", "right", or "left" is called \emph{four-directional}, and \emph{three-directional} if at most three out of the four labels are used. A \emph{direction-consistent embedding} of an \mbox{$n$-vertex} four-directional path $P$ on a set $S$ of $n$ points in the plane is a straight-line drawing of $P$ where each vertex of $P$ is mapped to a distinct point of $S$ and every edge points to the direction specified by its label. We study planar direction-consistent embeddings of three- and four-directional paths and provide a complete picture of the problem for convex point sets.Comment: 11 pages, full conference version including all proof

Quadratic-time, linear-space algorithms for generating orthogonal polygons with a given number of vertices

Programa de Financiamento Plurianual, Fundação para a Ciéncia e TecnologiaPrograma POSIPrograma POCTI, FCTFondo Europeo de Desarrollo Regiona

Matching Points with Things

Given an ordered set of points and an ordered set of geometric objects in the plane, we are interested in finding a non-crossing matching between point-object pairs. We show that when the objects we match the points to are finite point sets, the problem is NP-complete in general, and polynomial when the objects are on a line or when their number is at most 2. When the objects are line segments, we show that the problem is NP-complete in general, and polynomial when the segments form a convex polygon or are all on a line. Finally, for objects that are straight lines, we show that the problem of finding a min-max non-crossing matching is NP-complete

BigSUR: Large-scale Structured Urban Reconstruction

The creation of high-quality semantically parsed 3D models for dense metropolitan areas is a fundamental urban modeling problem. Although recent advances in acquisition techniques and processing algorithms have resulted in large-scale imagery or 3D polygonal reconstructions, such data-sources are typically noisy, and incomplete, with no semantic structure. In this paper, we present an automatic data fusion technique that produces high-quality structured models of city blocks. From coarse polygonal meshes, street-level imagery, and GIS footprints, we formulate a binary integer program that globally balances sources of error to produce semantically parsed mass models with associated facade elements. We demonstrate our system on four city regions of varying complexity; our examples typically contain densely built urban blocks spanning hundreds of buildings. In our largest example, we produce a structured model of 37 city blocks spanning a total of 1, 011 buildings at a scale and quality previously impossible to achieve automatically

Intra-operative real time intracranial subarachnoid haemorrhage during glial tumour resection: A case report

Glial tumours associated with subarachnoid haemorrhage are very rare. A 64-year-old woman admitted with a history of 3 weeks seizures and a left sided hemiparesis and dysphasia. The magnetic resonance disclosed heterogeneously enhancing a right temporal mass. During surgery, suddenly an abrupt and extensive swelling had occurred both in tumour and the brain tissue. The surgery was completed with a gross total tumour resection together with a partial temporal lobectomy. Postoperative computerized tomography demonstrated a massive subarachnoid hemorrhage (SAH). A cerebral Magnetic Resonance (MR) angiography showed neither an aneurysm nor arteriovenous malformation. Coincidence of an intracerebral tumour and subarachnoid haemorrhage would be devastating