An examination of the logical relationships between attitude statements suggests that attitudes can be ordered according to favourability, and can also stand in relationships of implication to one another. The traditional representation of attitudes, as points on a single dimension, is inadequate for representing both these relations but representing attitudes as intervals on a single dimension can incorporate both favourability and implication. An interval can be parameterised using its two endpoints or alternatively by its midpoint and latitude. Using this latter representation, the midpoint can be understood as the �favourability� of the attitude, while the latitude can be understood as its �generality�. It is argued that the generality of an attitude statement is akin to its latitude of acceptance, since a greater semantic range increases the likelihood of agreement. When Coombs� Theory of Unidimensional Unfolding is reformulated using the interval representation, the key question is how to measure the distance between two intervals on the dimension. There are innumerable ways to answer this question, but the present study restricts attention to eighteen possible �distance� measures. These measures are based on nine basic distances between intervals on a dimension, as well as two families of models, the Minkowski r-metric and the Generalised Hyperbolic Cosine Model (GHCM). Not all of these measures are distances in the strict sense as some of them fail to satisfy all the metric axioms. To distinguish between these eighteen �distance� measures two empirical tests, the triangle inequality test, and the aligned stimuli test, were developed and tested using two sets of attitude statements. The subject matter of the sets of statements differed but the underlying structure was the same. It is argued that this structure can be known a priori using the logical relationships between the statement�s predicates, and empirical tests confirm the underlying structure and the unidimensionality of the statements used in this study. Consequently, predictions of preference could be ascertained from each model and either confirmed or falsified by subjects� judgements. The results indicated that the triangle inequality failed in both stimulus sets. This suggests that the judgement space is not metric, contradicting a common assumption of attitude measurement. This result also falsified eleven of the eighteen �distance� measures because they predicted the satisfaction of the triangle inequality. The aligned stimuli test used stimuli that were aligned at the endpoint nearest to the ideal interval. The results indicated that subjects preferred the narrower of the two stimuli, contrary to the predictions of six of the measures. Since these six measures all passed the triangle inequality test, only one measure, the GHCM (item), satisfied both tests. However, the GHCM (item) only passes the aligned stimuli tests with additional constraints on its operational function. If it incorporates a strictly log-convex function, such as cosh, the GHCM (item) makes predictions that are satisfied in both tests. This is also evidence that the latitude of acceptance is an item rather than a subject or combined parameter.

La construction des courbes parallèles est fondamentale pour différentes applications en modélisation géométrique, telles que l'étude des trajectoires d'outils pour les machines à commande numérique ou pour la définition des zones de tolérance. En général, la courbe parallèle d'une courbe rationnelle n'est pas rationnelle, ce qui conduit à déterminer une approximation de cette courbe parallèle par une courbe spline. Récemment, J. C. Fiorot et T. Gensane et indépendamment H. Pottmann ont donné la forme générale de toutes les courbes rationnelles à parallèles rationnelles (courbes à hodographe pythagorien). Dans cette dernière famille figurent les quartiques de Tschirnhausen. Ces courbes ont même flexibilité que les coniques, leurs courbes parallèles sont rationnelles de degré quatre et sont exactement les développantes des cubiques de Tschirnhausen. En se basant sur cette caractérisation, nous présentons un algorithme d'approximation, avec un contact d'ordre deux, d'une courbe et de ses parallèles par des quartiques de Tschirnhausen préservant la variation de la courbure. Par ailleurs, le caractère judicieux de la représentation Bézier duale des courbes à hodographe pythagorien et de leurs parallèles, nous a permis de construire des ovales et des rosettes rationnelles à largeur constante qui jouent un rôle important en mécanique des cames. Enfin, suite aux travaux de H. Busemann et H. Guggenheimer sur la géométrie plane de Minkowski, nous généralisons la notion de courbes parallèles ainsi que les résultats de H. Pottmann (concernant la caractérisation Bézier duale et la caractérisation géométrique des courbes à hodographe pythagorien) au plan de Minkowski

This bachelor thesis shows a way to classify second order symmetric tensors in the Lorentz metric. Some basic prerequisite about indefinite and definite algebra is introduced, such as the Jordan form, indefinite inner products, the Segre type, and the Minkowski space. There are also some results concerning the invariant 2-spaces of a symmetric tensor and a different approach on how to classify second order symmetric tensor.

<p>One way of modeling human knowledge is by using multidimensional spaces, in which an object is represented as a point in the space, and the distances among the points reflect the similarities among the represented objects. The distances are measured with some metric, commonly some instance of the Minkowski metric. The instances differ with the magnitude of the so-called r-parameter. The instances most commonly mentioned in the literature are the ones where r equals 1, 2 and infinity.</p><p>Cognitive scientists have found out that different metrics are suited to describe different dimensional combinations. From these findings an important distinction between integral and separable dimensions has been stated (Garner, 1974). Separable dimensions, e.g. size and form, are best described by the city-block metric, where r equals 1, and integral dimensions, such as the color dimensions, are best described by the Euclidean metric, where r equals 2. Developmental psychologists have formulated a hypothesis saying that small children perceive many dimensional combinations as integral whereas adults perceive the same combinations as separable. Thus, there seems to be a shift towards increasing separability with age or maturity.</p><p>Earlier experiments show the same phenomenon in adult short-term learning with novel stimuli. In these experiments, the stimuli were first perceived as rather integral and were then turning more separable, indicated by the Minkowski-r. This indicates a shift towards increasing separability with familiarity or skill.</p><p>This dissertation aims at investigating the generality of this phenomenon. Five similarity-rating experiments are conducted, for which the best fitting metric for the first half of the session is compared to the last half of the session. If the Minkowski-r is lower for the last half compared to the first half, it is considered to indicate increasing separability.</p><p>The conclusion is that the phenomenon of increasing separability during short-term learning cannot be found in these experiments, at least not given the operational definition of increasing separability as a function of a decreasing Minkowski-r. An alternative definition of increasing separability is suggested, where an r-value ‘retreating’ 2.0 indicates increasing separability, i.e. when the r-value of the best fitting metric for the last half of a similarity-rating session is further away from 2.0 compared to the first half of the session.</p>

The objective of this dissertation is the application of Minkowskian cross-section measures (i.e., section and projection measures in finite-dimensional linear normed spaces over the real field) to various topics of geometric convexity in Minkowski spaces, such as bodies of constant Minkowskian width, Minkowskian geometry of simplices, geometric inequalities and the corresponding optimization problems for convex bodies. First we examine one-dimensional Minkowskian cross-section measures deriving (in a unified manner) various properties of these measures. Some of these properties are extensions of the corresponding Euclidean properties, while others are purely Minkowskian. Further on, we discover some new results on the geometry of a simplex in Minkowski spaces, involving descriptions of the so-called tangent Minkowskian balls and of simplices with equal Minkowskian heights. We also give some (characteristic) properties of bodies of constant width in Minkowski planes and in higher dimensional Minkowski spaces. This part of investigation has relations to the well known \emph{Borsuk problem} from the combinatorial geometry and to the widely used monotonicity lemma from the theory of Minkowski spaces. Finally, we study bodies of given Minkowskian thickness ($=$ minimal width) having least possible volume. In the planar case a complete description of this class of bodies is given, while in case of arbitrary dimension sharp estimates for the coefficient in the corresponding geometric inequality are found. / Die Dissertation befasst sich mit Problemen fuer spezielle konvexe Koerper in Minkowski-Raeumen (d.h. in endlich-dimensionalen Banach-Raeumen). Es wurden Klassen der Koerper mit verschiedenen metrischen Eigenschaften betrachtet (z.B., Koerper konstante Breite, reduzierte Koerper, Simplexe mit Inhaltsgleichen Facetten usw.) und einige kennzeichnende und andere Eigenschaften fuer diese Klassen herleitet.

The subject of this monograph can be described as the local properties of geometric Steiner minimal trees in finite-dimensional normed spaces. A Steiner minimal tree of a finite set of points is a shortest connected set interconnecting the points. For a quick introduction to this topic and an overview of all the results presented in this work, see Chapter 1. The relevant mathematical background knowledge needed to understand the results and their proofs are collected in Chapter 2. In Chapter 3 we introduce the Fermat-Torricelli problem, which is that of finding a point that minimizes the sum of distances to a finite set of given points. We only develop that part of the theory of Fermat-Torricelli points that is needed in later chapters. Steiner minimal trees in finite-dimensional normed spaces are introduced in Chapter 4, where the local Steiner problem is given an exact formulation. In Chapter 5 we solve the local Steiner problem for all two-dimensional spaces, and generalize this solution to a certain class of higher-dimensional spaces (CL spaces). The twodimensional solution is then applied to many specific norms in Chapter 6. Chapter 7 contains an abstract solution valid in any dimension, based on the subdifferential calculus. This solution is applied to two specific high-dimensional spaces in Chapter 8. In Chapter 9 we introduce an alternative approach to bounding the maximum degree of Steiner minimal trees from above, based on the illumination problem from combinatorial convexity. Finally, in Chapter 10 we consider the related k-Steiner minimal trees, which are shortest Steiner trees in which the number of Steiner points is restricted to be at most k. / Das Thema dieser Habilitationsschrift kann als die lokalen Eigenschaften der geometrischen minimalen Steiner-Bäume in endlich-dimensionalen normierten Räumen beschrieben werden. Ein minimaler Steiner-Baum einer endlichen Punktmenge ist eine kürzeste zusammenhängende Menge die die Punktmenge verbindet. Kapitel 1 enthält eine kurze Einführung zu diesem Thema und einen Überblick über alle Ergebnisse dieser Arbeit. Die entsprechenden mathematischen Vorkenntnisse mit ihren Beweisen, die erforderlich sind die Ergebnisse zu verstehen, erscheinen in Kapitel 2. In Kapitel 3 führen wir das Fermat-Torricelli-Problem ein, das heißt, die Suche nach einem Punkt, der die Summe der Entfernungen der Punkte einer endlichen Punktmenge minimiert. Wir entwickeln nur den Teil der Theorie der Fermat-Torricelli-Punkte, der in späteren Kapiteln benötigt wird. Minimale Steiner-Bäume in endlich-dimensionalen normierten Räumen werden in Kapitel 4 eingeführt, und eine exakte Formulierung wird für das lokale Steiner-Problem gegeben. In Kapitel 5 lösen wir das lokale Steiner-Problem für alle zwei-dimensionalen Räume, und diese Lösung wird für eine bestimmte Klasse von höher-dimensionalen Räumen (den sog. CL-Räumen) verallgemeinert. Die zweidimensionale Lösung wird dann auf mehrere bestimmte Normen in Kapitel 6 angewandt. Kapitel 7 enthält eine abstrakte Lösung die in jeder Dimension gilt, die auf der Analysis von Subdifferentialen basiert. Diese Lösung wird auf zwei bestimmte höher-dimensionale Räume in Kapitel 8 angewandt. In Kapitel 9 führen wir einen alternativen Ansatz zur oberen Schranke des maximalen Grads eines minimalen Steiner-Baums ein, der auf dem Beleuchtungsproblem der kombinatorischen Konvexität basiert ist. Schließlich betrachten wir in Kapitel 10 die verwandten minimalen k-Steiner-Bäume. Diese sind die kürzesten Steiner-Bäume, in denen die Anzahl der Steiner-Punkte auf höchstens k beschränkt wird.

The central objective of this work is to present an analysis of the asymptotic conformal Killing vectors in asymptotically-flat space-times of general relativity. This problem has been examined by two different methods; in Chapter 5 the asymptotic expansion technique originated by Newman and Unti [31] leads to a solution for asymptotically-flat spacetimes which admit an asymptotically shear-free congruence of null geodesics, and in Chapter 6 the conformal rescaling technique of Penrose [54] is used both to support the findings of the previous chapter and to set out a procedure for solution in the general case. It is pointed out that Penrose's conformal technique is preferable to the use of asymptotic expansion methods, since it can be established in a rigorous manner without leading to the possible convergence difficulties associated with asymptotic expansions. Since the asymptotic conformal symmetry groups of asymptotically flat space-times Are generalisations of the conformal group of Minkowski space-time we devote Chapters 3 and 4 to a study of the flat space case so that the results of later chapters may receive an interpretation in terms of familiar concepts. These chapters fulfil a second, equally important, role in establishing local isomorphisms between the Minkowski-space conformal group, 90(2,4) and SU(2,2). The SO(2,4) representation has been used by Kastrup [61] to give a physical interpretation using space-time gauge transformations. This appears as part of the survey of interpretative work in Chapter 7. The SU(2,2) representation of the conformal group has assumed a theoretical prominence in recent years. through the work of Penrose [9-11] on twistors. In Chapter 4 we establish contact with twistor ideas by showing that points in Minkowski space-time correspond to certain complex skew-symmetric rank two tensors on the SU(2,2) carrier space. These objects are, in Penrose's terminology [91, simple skew-symmetric twistors of valence [J. A particularly interesting aspect of conformal objects in space-time is explored in Chapter 8, where we extend the work of Geroch [16] on multipole moments of the Laplace equation in 3-space to the consideration. of Q tý =0 in Minkowski space-time. This development hinges upon the fact that multipole moment fields are also conformal Killing tensors. In the final chapter some elementary applications of the results of Chapters 3 and 5 are made to cosmological models which have conformal flatness or asymptotic conformal flatness. In the first class here we have 'models of the Robertson-Walker type and in the second class we have the asymptotically-Friedmann universes considered by Hawking [73].

An examination of the logical relationships between attitude statements suggests that attitudes can be ordered according to favourability, and can also stand in relationships of implication to one another. The traditional representation of attitudes, as points on a single dimension, is inadequate for representing both these relations but representing attitudes as intervals on a single dimension can incorporate both favourability and implication. An interval can be parameterised using its two endpoints or alternatively by its midpoint and latitude. Using this latter representation, the midpoint can be understood as the �favourability� of the attitude, while the latitude can be understood as its �generality�. It is argued that the generality of an attitude statement is akin to its latitude of acceptance, since a greater semantic range increases the likelihood of agreement. When Coombs� Theory of Unidimensional Unfolding is reformulated using the interval representation, the key question is how to measure the distance between two intervals on the dimension. There are innumerable ways to answer this question, but the present study restricts attention to eighteen possible �distance� measures. These measures are based on nine basic distances between intervals on a dimension, as well as two families of models, the Minkowski r-metric and the Generalised Hyperbolic Cosine Model (GHCM). Not all of these measures are distances in the strict sense as some of them fail to satisfy all the metric axioms. To distinguish between these eighteen �distance� measures two empirical tests, the triangle inequality test, and the aligned stimuli test, were developed and tested using two sets of attitude statements. The subject matter of the sets of statements differed but the underlying structure was the same. It is argued that this structure can be known a priori using the logical relationships between the statement�s predicates, and empirical tests confirm the underlying structure and the unidimensionality of the statements used in this study. Consequently, predictions of preference could be ascertained from each model and either confirmed or falsified by subjects� judgements. The results indicated that the triangle inequality failed in both stimulus sets. This suggests that the judgement space is not metric, contradicting a common assumption of attitude measurement. This result also falsified eleven of the eighteen �distance� measures because they predicted the satisfaction of the triangle inequality. The aligned stimuli test used stimuli that were aligned at the endpoint nearest to the ideal interval. The results indicated that subjects preferred the narrower of the two stimuli, contrary to the predictions of six of the measures. Since these six measures all passed the triangle inequality test, only one measure, the GHCM (item), satisfied both tests. However, the GHCM (item) only passes the aligned stimuli tests with additional constraints on its operational function. If it incorporates a strictly log-convex function, such as cosh, the GHCM (item) makes predictions that are satisfied in both tests. This is also evidence that the latitude of acceptance is an item rather than a subject or combined parameter.

Helmut-Schmidt-Univ./Univ. der Bundeswehr, Diss.--Hamburg, 2005.

Stuttgart, Univ., Diplomarb., 2005.