Ordered metric geometry

After a brief biography of David Hilbert, we look at how he came to take up elementary geometry and to write his Grundlagen der Geometrie (1899). In this book he gave a careful exposition of axiom systems for elementary geometry that, in Hurwitz’s opinion, created the subject of axiomatics. Hilbert was able to show how different geometries may be studied and found to be inter-related, and how they may be established rigorously in terms of arithmetic. His book ran to 10 editions and opened the way to the axiomatic method in other branches of mathematics. The place of Desargues’ theorem in projective geometry is considered in some detail, using Forest Ray Moulton’s presentation (adopted by Hilbert in the 2nd and subsequent editions).More extracts from Hilbert’s Grundlagen der Geometrie are given at the end of Chapter 28.KeywordsEuclidean GeometryProjective GeometryAxiom SystemWinter SemesterParallel AxiomThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Targets are matroids that arise from a nested sequence of flats in a projective geometry. This class of matroids was introduced by Nelson and Nomoto, who found the forbidden induced restrictions for binary targets. This paper generalizes their result to targets arising from projective geometries over $GF(q)$. We also consider targets arising from nested sequences of affine flats and determine the forbidden induced restrictions for affine targets.

Chapter 2 - Projective and Affine Geometry over Division Rings

Lattice-Theoretic Approach to Projective and Affine Geometry

Structural Properties of Models of N1-Categorical Theories

A cap set in a projective or affine geometry over a finite field is a set of points no three of which are collinear.We construct complete cap sets with sizes 274432, 13991936, and 30294016 in affine geometries (, ), (, ), and (, ), respectively.

It was about 125 years ago that the light bulb was commercialized by Thomas Edison. No doubt a brilliant invention at the time, today its low power conversion efficiency is one of the reasons why lighting in the western world has such high energy consumption. Thus, the potential for saving energy is enormous in this area. The introduction of halogen, discharge and fluorescent lamps has lead to certain efficiency improvements, however more than half of the energy is still lost as heat. Light-emitting diodes (LEDs) are very promising candidates for high efficiency light sources, with modern devices showing internal quantum efficiencies of virtually 100 %. However, due to the high refractive index of the commonly used semiconductor materials it is very difficult to have a large extraction efficiency; in a standard cubic geometry most of the internally emitted light is trapped inside the device due to total internal reflection. Several methods have been developed in order to circumvent this problem, either by optimizing the device geometry in order to increase the escape cone or by incorporating a resonant structure in order to force the internal emission into the existing escape cone. The latter approach is called microcavity LED (MCLED) or resonant cavity LED (RCLED). In a MCLED the spontaneous internal emission is controlled by placing the emitter inside an optical cavity with a thickness of the order of its emitting wavelength. The resulting interference effects increase the part of the emission that can be extracted. Contrary to the other approaches this is possible without changing the device geometry and thus without additional costly back-end processing steps. The control of the farfield radiation pattern makes these devices particularly interesting for high brightness applications, which demand highly directional emitters, such as for printing, bar code reading, large area displays and optical communication. The extraction efficiency of a MCLED is inversely proportional to the effective cavity length. An ideal cavity, allowing an extraction efficiency close to unity, consists of a low refractive index material and has an optical length of λ/2. In contrast to this, to obtain high internal quantum efficiencies it is necessary to use high index cavities with an optical length of at least λ. It should be noted, that the large penetration depth of the optical field in the semiconductor-based distributed Bragg reflectors (DBRs) leads to a significant increase of the effective cavity length and thus further reduces the achievable extraction efficiencies. In this thesis novel concepts to reduce effective cavity lengths and therefore increase extraction efficiencies are implemented into standard MCLED structures. The phaseshift cavity principle whilst maintaining the electrical properties of a standard A cavity achieves optical properties approaching that of a λ/2 cavity. The use of AlOx instead of AlAs as the, low refractive index component in the DBRs leads to smaller penetration depths and a concomitant reduction of the effective cavity length. A similar effect can be obtained by combining a resonant cavity with a thin-film structure. Thanks to these design improvements, the external quantum efficiency of different types of MCLEDs was increased. Near infrared emitting InGaAs/GaAs MCLEDs including a phase-shift cavity were realized, as both bottom and top emitting structures. The external quantum efficiencies achieved for emission into air were 18 and 19 %, respectively. With the additional incorporation of an oxide based bottom DBR, the efficiency of top emitting near infrared MCLEDs was further increased to 28 %. Red emitting AlGaInP-based structures are not compatible with the phase-shift cavity principle. However the beneficial effect of the implementation of an oxide DBR is greater at their wavelength rather than in the near infrared. Thus with preliminary red emitting GaInP/AlGaInP MCLEDs containing a bottom oxide DBR external quantum efficiencies of 12% could be achieved. Unfortunately, the incorporation of an oxide DBR significantly complicates the device design and the device fabrication. These problems can be avoided by combining the resonant cavity with a thin-film structure instead. Initial non-optimized red emitting thin-film MCLEDs were realized by OSRAM Opto Semiconductors and characterized in this work. They show external quantum efficiencies of 23 % and 18 % with and without encapsulation, respectively. It is assumed that a significant fraction of the high external quantum efficiency is due to a strong photon recycling effect in these devices. Simulations presented in this thesis show that the theoretical limits for the MCLEDs discussed above are slightly higher than the values obtained, encouraging further device optimization. The thin-film MCLEDs seem to hold the biggest potential for high efficiency emission from MCLEDs, independent of the wavelength range of emission.

Projective geometry is that area of geometry that treats geometric figures from the point of view of the perspective and the horizon line, figures that are considered invariable by projection. The origins are found in the Pappus works of Alexandria (4th century AD), which, referring to the results of Apollonius in Perga, introduces the concept of an anarchic report. The study of projective geometry is later resumed by mathematicians like Pascal or architects like Gerard Desargues in the seventeenth century, to be theorized and taught in the late 18th century by Gaspard Monge. Jean-Victor Poncelet, through his work, the Traite des proprietes geometriques des figures, gives a strong leap to this science, but this is based on Euclidean geometry. However, the affine geometry excluded the possibility of an intersection of parallel lines, an essential notion in the projective geometry. But the 19th-century discoveries of August Ferdinand Mobius, Julius Plucker and especially Felix Klein’s 1900s, definitively separate the projective geometry from the Euclidean one. There is also a conceptual revolution: If until then geometry was a science of figures, now the focus is on geometric transformations, the associated internal compositional laws, the structures of the various transformation groups.

From the analytic viewpoint of Klein’s definition of geometry, projective geometry is the logical generalization of the affine geometry introduced in Chapter 3. Just as we were able to generalize the isometries of the Euclidean plane to similarities, and these in turn to affinities, we will now be able to generalize affinities to collineations, the transformations that define projective geometry. There is, however, one new ingredient required in this last generalization. The set of points contained in the Euclidean plane must be enlarged to include points on one additional line, a line often referred to as the ideal line. Rather than complicating the geometry, these new ideal points simplify projective geometry and give it the highly desirable property of duality.KeywordsProjective PlaneIdeal PointProjective GeometryInvariant PointPoint ConicThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

In this article we extend Alon’s Nullstellensatz to functions which have multiple zeros at the common zeros of some polynomials g 1,g 2, …, g n , that are the product of linear factors. We then prove a punctured version which states, for simple zeros, that if f vanishes at nearly all, but not all, of the common zeros of g 1(X 1), …,g n (X n ) then every residue of f modulo the ideal generated by g 1, …, g n , has a large degree. This punctured Nullstellensatz is used to prove a blocking theorem for projective and affine geometries over an arbitrary field. This theorem has as corollaries a theorem of Alon and Furedi which gives a lower bound on the number of hyperplanes needed to cover all but one of the points of a hypercube and theorems of Bruen, Jamison and Brouwer and Schrijver which provides lower bounds on the number of points needed to block the hyperplanes of an affine space over a finite field.

Cayley algebra and bracket algebra are important approaches to invariant computing in projective and affine geometries, but there are some difficulties in doing algebraic computation. In this paper we show how the principle “breefs” – bracket-oriented representation, elimination and expansion for factored and shortest results, can significantly simplify algebraic computations. We present several typical examples on automated theorem proving in conics and make detailed discussions on the procedure of applying the principle to automated geometric theorem proving.

Accurate regulation and measurement of a load parameters can be difficult when there are significant voltage drops between the power supply and the load. In particular, some methods of the load voltage management or regulation are known. These methods have these or those disadvantages. A new power supply method is offered for the remote load. This method is based on correction of the power supply output voltage by using an invariant relationship of the line. The affine ratio of three points, known in projective and affine geometry, is such invariant. Therefore, three samples of the output voltage and the corresponding samples of the input current are used. The affine ratio value does not depend from the line parameters and the measurement accuracy.

A new definition for the dimension of a combinatorial t- (v,k,\lambda) design over a finite field is proposed. The complementary designs of the hyperplanes in a finite projective or affine geometry, and the finite Desarguesian planes in particular, are characterized as the unique (up to isomorphism) designs with the given parameters and minimum dimension. This generalizes a well-known characterization of the binary hyperplane designs in terms of their minimum 2-rank. The proof utilizes the q-ary analogue of the Hamming code, and a group-theoretic characterization of the classical designs.

Constructing a Class of Symmetric Graphs

Projective geometry became regarded by the mid 19th century as the fundamental geometry. This was very much the view of the English mathematicians Arthur Cayley, James Joseph Sylvester, and Henry Smith, and of George Salmon in Ireland, as it was of the Italian mathematician Luigi Cremona, whose book Elementi di geometria projettiva may be the first to have given the subject its present name. We look briefly at this book, concentrating on Cremona’s treatment of duality. Then we look at the disquiet over the foundations of geometry that were addressed by Moritz Pasch in his Vorlesungen uber neuere Geometrie (1882), who attempted to build elementary geometry axiomatically from ideas laid down empirically by Hermann von Helmholtz. Helmholtz, whose acceptance of non-Euclidean geometry was influential, based his approach on an analysis of what is required for a geometry to capture the idea of the free mobility of figures in space.