Generators for primary closures of Galois fields

AbstractWe continue to study the existence of (norm- and) trace-compatible sequences of primitive normal bases for prime power extensions of finite fields, introduced by the author in Hachenberger (Finite Fields Appl. 5 (1999) 378–385; in: D. Jungnickel, H. Niederreiter (Eds.), Proceedings of the Fifth International Conference on Finite Fields and Applications, Augsburg, August 1999, Springer, Heidelberg, 2001, pp. 208–223), and improve on some aspects of these papers

Exact Minkowski sums of polyhedra and exact and efficient decomposition of polyhedra in convex pieces

We present the first exact and robust implementation of the 3D Minkowski sum of two non-convex polyhedra. Our implementation decomposes the two polyhedra into convex pieces, performs pairwise Minkowski sums on the convex pieces, and constructs their union. We achieve exactness and the handling of all degeneracies by building upon 3D Nef polyhedra as provided by Cgal. The implementation also supports open and closed polyhedra. This allows the handling of degenerate scenarios like the tight passage problem in robot motion planning. The bottleneck of our approach is the union step. We address efficiency by optimizing this step by two means: we implement an efficient decomposition that yields a small amount of convex pieces, and develop, test and optimize multiple strategies for uniting the partial sums by consecutive binary union operations. The decomposition that we implemented as part of the Minkowski sum is interesting in its own right. It is the first robust implementation of a decomposition of polyhedra into convex pieces that yields at most O(r 2) pieces, where r is the number of edges whose adjacent facets comprise an angle of more than 180 degrees with respect to the interior of the polyhedron

Elation generalised quadrangles of order (s,p), where p is prime

We show that an elation generalised quadrangle which has p+1 lines on each point, for some prime p, is classical or arises from a flock of a quadratic cone (i.e., is a flock quadrangle).Comment: 14 page

Boolean operations on 3D selective Nef complexes : data structure, algorithms, optimized implementation, experiments and applications

Nef polyhedra in d-dimensional space are the closure of half-spaces under boolean set operations. Consequently, they can represent non-manifold situations, open and closed sets, mixed-dimensional complexes, and they are closed under all boolean and topological operations, such as complement and boundary. The generality of Nef complexes is essential for some applications. In this thesis, we present a new data structure for the boundary representation of three-dimensional Nef polyhedra and efficient algorithms for boolean operations. We use exact arithmetic to avoid well known problems with floating-point arithmetic and handle all degeneracies. Furthermore, we present important optimizations for the algorithms, and evaluate this optimized implementation with extensive experiments. The experiments supplement the theoretical runtime analysisNef-Polyeder sind d-dimensionale Punktmengen, die durch eine endliche Anzahl boolescher Operationen über Halbräumen generiert werden. Sie sind abgeschlossen hinsichtlich boolescher und topologischer Operationen. Als Konsequenz daraus können sie nicht-mannigfaltige Situationen, offene und geschlossene Mengen und gemischt-dimensionale Komplexe darstellen. Die Allgemeinheit von Nef-Komplexen ist unentbehrlich für einige Anwendungen. In dieser Doktorarbeit stellen wir eine neue Datenstruktur vor, die eine Randdarstellung von dreidimensionalen Nef-polyedern und Algorithmen für boolesche Operationen realisiert. Wir benutzen exakte Arithmetik um die bekannten Probleme mit Gleitkommaarithmetik und Degeneriertheiten zu vermeiden. Außerdem präsentieren wir wichtige Optimierungen der Algorithmen und bewerten die optimierte Implementierung an Hand umfassender Experimente. Weitere Experimente belegen die theoretische Laufzeitanalyse und vergleichen unsere Implementation mit dem kommerziellen CAD kernel ACIS. ACIS is meistens bis zu sechs mal schneller, aber es gibt auch Beispiele bei denen ACIS scheitert. Nef-Polyeder können bei einer Vielzahl von Anwendungen eingesetzt werden. Wir präsentieren einfache Implementationen zweier Anwendungen - von der visuellen Hülle und von der Minkowski-Summe zwei abgeschlossener Nef-Polyeder

Streaming algorithms for line simplification under the Fréchet distance

We study the following variant of the well-known linesimplification problem: we are getting a possibly infinite sequence of points p0, p1, p2, . . . defining a polygonal path, and as we receive the points we wish to maintain a simplification of the path seen so far. We study this problem in a streaming setting, where we only have a limited amount of storage so that we cannot store all the points. We analyze the competitive ratio of our algorithm, allowing resource augmentation: we let our algorithm maintain a simplification with 2k (internal) points, and compare the error of our simplification to the error of the optimal simplification with k points

The role of the context of physical activity on its association with affective well-being : an experience sampling study in young adults

Physical activity and being outdoors both improve affective well-being. However, little is known about the synergistic effects between them and the influences of contextual factors such as the life domain of physical activity (work-, chores-, leisure, or sports-related) or the type of the outdoor environment (green space, blue space, or city area) on mood. This study investigates the synergistic effects of physical activity and being outdoors as well as the potential role of contextual factors on mood. A total of 158 individuals aged 18−25 years (133 females) participated in a 14-day experience sampling study. Participants received seven prompts per day and answered questions about their physical activity, contextual factors, and affective well-being. Physical activity and being outdoors were associated with concurrent higher levels of positive and lower levels of negative affect compared to being physically inactive or being indoors, respectively. However, no synergistic effects were found. Being outdoors in a city area was associated with a less positive and more negative affect than being in nature. Work- and chores-related physical activity was associated with less positive affect and more negative affect compared to sports- or leisure-related physical activity. To foster positive affect, people should schedule leisure-related physical activity in nature