Some geometric applications of Dilworth’s theorem

A geometric graph is a graph drawn in the plane such that its edges are closed line segments and no three vertices are collinear. We settle an old question of Avital, Hanani, Erdős, Kupitz, and Perles by showing that every geometric graph withn vertices andm>k ⁴ n edges containsk+1 pairwise disjoint edges. We also prove that, given a set of pointsV and a set of axis-parallel rectangles in the plane, then either there arek+1 rectangles such that no point ofV belongs to more than one of them, or we can find an at most 2·10⁵ k ⁸ element subset ofV meeting all rectangles. This improves a result of Ding, Seymour, and Winkler. Both proofs are based on Dilworth’s theorem on partially ordered sets. The research by János Pach was supported by Hungarian National Foundation for Scientific Research Grant OTKA-4269 and NSF Grant CCR-91-22103.     

Semiovals contained in the union of three concurrent lines

Semiovals which are contained in the union of three concurrent lines are studied. The notion of a strong semioval is introduced, and a complete classification of these objects in PG(2,p) and PG(2,p²), p an odd prime, is given. © 2007 Wiley Periodicals, Inc. J Combin Designs 15: 491–501, 2007     

Some Global Optimization Problems on Stiefel Manifolds

Optimization on Stiefel manifolds was discussed by Rapcsák in earlier papers, and some global optimization methods were considered and tested on Stiefel manifolds. In the paper, test functions are given with known global optimum points and their optimal function values. A restriction, which leads to a discretization of the problem is suggested, which results in a problem equivalent to the well-known assignment problem.     

Monotone drawings of planar graphs

Let G be a graph drawn in the plane so that its edges are represented by x‐monotone curves, any pair of which cross an even number of times. We show that G can be redrawn in such a way that the x‐coordinates of the vertices remain unchanged and the edges become non‐crossing straight‐line segments. © 2004 Wiley Periodicals, Inc. J Graph Theory 46: 39–47, 2004     

Digital compensation of pressure sensors in the time domain

This contribution presents an innovative technique to determine the transfer function of pneumatic and fast-response pressure probes. The dynamic response is determined experimentally with pressure step-tests. In the case of conventional instrumentation fast-opening valves or balloon explosions are used. For the fast-response pressure sensors, shock tube tests are performed. The response of the probe is fitted in the time domain with the response of an m-order linear system. This numerical system is then used to correct the lag and dynamic error of the measurement chain. Received: 27 November 2000 / Accepted: 4 July 2001     

Fast calculation of DNMR spectra on CUDA-enabled graphics card

During the past few years, general-purpose graphics processing units (GPGPUs) have become rather popular in the high performance computing community. In this study, we present an implementation of the simulation of dynamic nuclear magnetic resonance (DNMR) spectra. The algorithm is based on the kinetic Monte Carlo method and therefore can benefit from the multithreaded architecture of the GPGPU. By careful optimization of the algorithm a 30-100-fold speed increase could be achieved.     

Indirect voltammetric detection of fluoride ions in toothpaste on a comb-shaped interdigitated microelectrode array

A novel technique based on dynamic electrochemistry for the detection of fluoride ions was developed. It is based on its strong complexation with ferric ion. Formed fluoroferric complex is cathodically inactive at the potential of the reduction of free ferric aquo ion. The voltammetric and amperometric response of platinum comb-shaped interdigitated microelectrode array is decreased after fluoride addition. This decrease serves for the quantification of fluoride ions added to the solution. The detection limit of 4.5 × 10⁻⁵ mol dm⁻³ was achieved when one of the segments of interdigitated microelectrode array (IDA) was used as an indicating electrode. The detection limit is about one order of magnitude lower than in the case of conventional platinum macroelectrode. In comparison with ISE electrodes this method is faster and also avoiding large error resulting from the antilogarithmization of ISE Nerstian response. The method was applied to the analysis of toothpaste.