Recent developments on absolute geometries and algebraization by K-loops

AbstractLet (P,L,α) be an ordered space. A spatial version of Pasch's assertion is proved, with that a short proof is given for the fact that (P,L) is an exchange space and the concepts h-parallel, one sided h-parallel and hyperbolic incidence structure are introduced (Section 2). An ordered space with hyperbolic incidence structure can be embedded in an ordered projective space (Pp,Lp,τ) of the same dimension such that P is projectively convex and projectively open (cf. Property 3.2). Then spaces with congruence (P,L,≡) are introduced and those are characterized in which point reflections do exist (Section 4). Incidence, congruence and order are joined together by assuming a compatibility axiom (ZK) (Section 5). If (P,L,α,≡) is an absolute space, if o∈P is fixed and if for x∈P,x′ denotes the midpoint of o and x and x̄ the point reflection in x then the map o: P→J; x→xo≔x̃′ satisfies the conditions (B1) and (B2) of Section 6, and if one sets a+b≔ao∘0o(b) then (P,+) becomes a K-loop (cf. Theorem 6.1) and the J of all lines through o forms an incidence fibrtion in the sense of Zizioli consisting of commutative subgroups of (P,+) (cf. Property 7.1). Therefore K-loops can be used for an algebraization of absolute spaces; in this way Ruoff's proportionality Theorem 8.4 for hyperbolic spaces is presented

Classification of general absolute geometries with Lambert-Saccheri quadrangles

Without claiming any kind of continuity we show that an absolute geometry has either a singular, a hyperbolic or an elliptic congruence

Lorentzian approach to noncommutative geometry

This thesis concerns the research on a Lorentzian generalization of Alain Connes' noncommutative geometry. In the first chapter, we present an introduction to noncommutative geometry within the context of unification theories. The second chapter is dedicated to the basic elements of noncommutative geometry as the noncommutative integral, the Riemannian distance function and spectral triples. In the last chapter, we investigate the problem of the generalization to Lorentzian manifolds. We present a first step of generalization of the distance function with the use of a global timelike eikonal condition. Then we set the first axioms of a temporal Lorentzian spectral triple as a generalization of a pseudo-Riemannian spectral triple together with a notion of global time in noncommutative geometry.Comment: PhD thesis, 200 pages, 9 figures, University of Namur FUNDP, Belgium, August 201

Spaces of dimension three with congruence

It is well known that every Euclidean plane (E, L, α, ≡) is isomorphic to an affine plane AG(2,K) over a Pythagorean ordered commutative field K .There is a corrisponding theorem for hyperbolic panes (cf. [2]). For both proofs one first considers the group of motions, in particular the line reflections

Introduction of measures for segments and angles in a general absolute plane

AbstractTo an absolute plane (E,L,≡,α) in the general sense of Karzel et al. [Einführung in die Geometrie, UTB 184, Vandenhoeck, Göttingen, 1973, Section 16] there will be associated an ordered commutative group (W,+,<) such that (W,+) is a subgroup of the corresponding K-loop (E,+) of the absolute plane and a cyclic ordered commutative group (E1,·,ζ) where (E1,·) is isomorphic to a rotation group fixing a point. (W,+,<), resp. (E1,·,ζ), will serve to introduce a distance λ describing the congruence and satisfying the triangular inequality or resp. a measure μ for angles describing the congruence and conjugacy of angles

Modular Theory, Non-Commutative Geometry and Quantum Gravity

This paper contains the first written exposition of some ideas (announced in a previous survey) on an approach to quantum gravity based on Tomita-Takesaki modular theory and A. Connes non-commutative geometry aiming at the reconstruction of spectral geometries from an operational formalism of states and categories of observables in a covariant theory. Care has been taken to provide a coverage of the relevant background on modular theory, its applications in non-commutative geometry and physics and to the detailed discussion of the main foundational issues raised by the proposal.Comment: Special Issue "Noncommutative Spaces and Fields

Numerical computation and avoidance of manipulator singularities

This thesis develops general solutions to two open problems of robot kinematics: the exhaustive computation of the singularity set of a manipulator, and the synthesis of singularity-free paths between given configurations. Obtaining proper solutions to these problems is crucial, because singularities generally pose problems to the normal operation of a robot and, thus, they should be taken into account before the actual construction of a prototype. The ability to compute the whole singularity set also provides rich information on the global motion capabilities of a manipulator. The projections onto the task and joint spaces delimit the working regions in such spaces, may inform on the various assembly modes of the manipulator, and highlight areas where control or dexterity losses can arise, among other anomalous behaviour. These projections also supply a fair view of the feasible movements of the system, but do not reveal all possible singularity-free motions. Automatic motion planners allowing to circumvent problematic singularities should thus be devised to assist the design and programming stages of a manipulator. The key role played by singular configurations has been thoroughly known for several years, but existing methods for singularity computation or avoidance still concentrate on specific classes of manipulators. The absence of methods able to tackle these problems on a sufficiently large class of manipulators is problematic because it hinders the analysis of more complex manipulators or the development of new robot topologies. A main reason for this absence has been the lack of computational tools suitable to the underlying mathematics that such problems conceal. However, recent advances in the field of numerical methods for polynomial system solving now permit to confront these issues with a very general intention in mind. The purpose of this thesis is to take advantage of this progress and to propose general robust methods for the computation and avoidance of singularities on non-redundant manipulators of arbitrary architecture. Overall, the work seeks to contribute to the general understanding on how the motions of complex multibody systems can be predicted, planned, or controlled in an efficient and reliable way.Aquesta tesi desenvolupa solucions generals per dos problemes oberts de la cinemàtica de robots: el càlcul exhaustiu del conjunt singular d'un manipulador, i la síntesi de camins lliures de singularitats entre configuracions donades. Obtenir solucions adequades per aquests problemes és crucial, ja que les singularitats plantegen problemes al funcionament normal del robot i, per tant, haurien de ser completament identificades abans de la construcció d'un prototipus. La habilitat de computar tot el conjunt singular també proporciona informació rica sobre les capacitats globals de moviment d'un manipulador. Les projeccions cap a l'espai de tasques o d'articulacions delimiten les regions de treball en aquests espais, poden informar sobre les diferents maneres de muntar el manipulador, i remarquen les àrees on poden sorgir pèrdues de control o destresa, entre d'altres comportaments anòmals. Aquestes projeccions també proporcionen una imatge fidel dels moviments factibles del sistema, però no revelen tots els possibles moviments lliures de singularitats. Planificadors de moviment automàtics que permetin evitar les singularitats problemàtiques haurien de ser ideats per tal d'assistir les etapes de disseny i programació d'un manipulador. El paper clau que juguen les configuracions singulars ha estat àmpliament conegut durant anys, però els mètodes existents pel càlcul o evitació de singularitats encara es concentren en classes específiques de manipuladors. L'absència de mètodes capaços de tractar aquests problemes en una classe suficientment gran de manipuladors és problemàtica, ja que dificulta l'anàlisi de manipuladors més complexes o el desenvolupament de noves topologies de robots. Una raó principal d'aquesta absència ha estat la manca d'eines computacionals adequades a les matemàtiques subjacents que aquests problemes amaguen. No obstant, avenços recents en el camp de mètodes numèrics per la solució de sistemes polinòmics permeten ara enfrontar-se a aquests temes amb una intenció molt general en ment. El propòsit d'aquesta tesi és aprofitar aquest progrés i proposar mètodes robustos i generals pel càlcul i evitació de singularitats per manipuladors no redundants d'arquitectura arbitrària. En global, el treball busca contribuir a la comprensió general sobre com els moviments de sistemes multicos complexos es poden predir, planificar o controlar d'una manera eficient i segur

Noncommutative Geometry and Gauge theories on AF algebras

Non-commutative geometry (NCG) is a mathematical discipline developed in the 1990s by Alain Connes. It is presented as a new generalization of usual geometry, both encompassing and going beyond the Riemannian framework, within a purely algebraic formalism. Like Riemannian geometry, NCG also has links with physics. Indeed, NCG provided a powerful framework for the reformulation of the Standard Model of Particle Physics (SMPP), taking into account General Relativity, in a single "geometric" representation, based on Non-Commutative Gauge Theories (NCGFT). Moreover, this accomplishment provides a convenient framework to study various possibilities to go beyond the SMPP, such as Grand Unified Theories (GUTs). This thesis intends to show an elegant method recently developed by Thierry Masson and myself, which proposes a general scheme to elaborate GUTs in the framework of NCGFTs. This concerns the study of NCGFTs based on approximately finite $C^*$-algebras (AF-algebras), using either derivations of the algebra or spectral triples to build up the underlying differential structure of the Gauge Theory. The inductive sequence defining the AF-algebra is used to allow the construction of a sequence of NCGFTs of Yang-Mills Higgs types, so that the rank $n+1$ can represent a grand unified theory of the rank $n$. The main advantage of this framework is that it controls, using appropriate conditions, the interaction of the degrees of freedom along the inductive sequence on the AF algebra. This suggests a way to obtain GUT-like models while offering many directions of theoretical investigation to go beyond the SMPP