Academic literature on the topic 'Non-Orientability'

Loop averages and partition functions in the U(N) gauge theory are calculated for loops without intersections on arbitrary two-dimensional manifolds including nonorientable ones. The physical quantities are directly expressed through geometrical characteristics of a manifold (areas enclosed by loops and the genus) and gauge group parameters (Casimir eigenvalues and dimensions of the irreducible representations). It is shown that, from the physical quantities’ point of view, non-orientability of the manifold is equivalent to its non-compactness.

We recall that the “Sagnac Effect” can be viewed as a clock converting rotation (torsion) into a lapse of time. It is shown that the two available helicities can be used to show that PT invariance can be experimentally confirmed within the framework of General Relativity. These clocks are rather similar to those proposed by de Broglie (the particle-clocks) to convert (right or left) rotation into a quantum of energy. The Quantum analogue of this PT invariance is analyzed for zero rest-mass fields and related Mass and Spin Casimir operators. We suggest that Quantum PT invariance could have been violated in the monopole phase, due to mixing of homotopy classes of the Lorentz Lie Group along a spacetime track, appearance of superselected Mass and Spin Casimir operators, and spontaneous breaking of their proportionality. We suggest that this could be viewed as an evidence of the asymmetric flow of time, (arrow of time) in the Early Universe. Existence (non existence) of spinor structure and orientability of space, is discussed.

Let X be a bordered Klein surface, by which we mean a Klein surface with non-empty boundary. X is characterized topologically by its orientability, the number k of its boundary components and the genus p of the closed surface obtained by filling in all the holes. The algebraic genus g of X is defined by.If g≥2 it is known that if G is a group of automorphisms of X then |G|≤12(g-l) and that the upper bound is attained for infinitely many values of g ([4], [5]). A bordered Klein surface for which this upper bound is attained is said to have maximal symmetry. A group of 12(g-l) automorphisms of a bordered Klein surface of algebraic genus g is called an M*-group and it is known that a finite group G is an M*-group if and only if it is generated by 3 non-trivial elements T1, T2, T3 which obey the relations([4]).

A closed Riemann surface $X$ which can be realized as a 3-sheeted covering of the Riemann sphere is called trigonal, and such a covering will be called a trigonal morphism. A trigonal Riemann surface $X$ is called real trigonal if there is an anticonformal involution (symmetry) $\sigma$ of $X$ commuting with the trigonal morphism. If the trigonal morphism is a cyclic regular covering the Riemann surface is called real cyclic trigonal. The species of the symmetry $\sigma $ is the number of connected components of the fixed point set $\mathrm{Fix}(\sigma)$ and the orientability of the Klein surface $X/\langle\sigma\rangle$. We characterize real trigonality by means of Fuchsian and NEC groups. Using this approach we obtain all possible species for the symmetry of real cyclic trigonal and real non-cyclic trigonal Riemann surfaces.

The scope of this work is Geometric Modeling. We study the representation and the construction of subdivisions of quasi-manifolds, using a topology-based approach. Quasi-manifolds are here defined as a subset of pseudo-manifolds, which are well-known objects in Algebraic Topology. N-dimensional generalized maps (resp. n-dimensional maps) are combinatorial models defined for representing the topology of subdivisions of orientable or non-orientable quasi-manifolds with or without boundaries (resp. orientable quasi-manifolds without boundaries). In this paper, we define the models, main related notions and properties as cells, boundaries, duality, orientability, Euler characteristic. Basic operations are proposed for handling the models. We also show the correspondence between the combinatorial models and combinatorial cellular quasi-manifolds, here defined as combinatorial simplicial quasi-manifolds to which a structure into cells is added. This correspondence establishes that the models are rigorous and unambiguous ones. Moreover, the definitions of the models and the related notions and operations are quite simple, and they can be easily implemented and used in geometric modellers.

Abstract Fiber-reinforced composites are progressively more used in a variety of industrial applications. In recent years, carbon fiber-reinforced plastics have become increasingly popular, particularly in the aerospace sector because they offer outstanding mechanical properties combined with low weight. However, the orientation and distribution of the fibers have a significant effect on the mechanical and physical properties of the composite materials. Using conventional manufacturing technologies, it is not always technologically possible to adjust the fiber orientation to the load direction. One possible approach to targeted fiber alignment is the combination of classical manufacturing processes with a superimposed alignment mechanism so that the fibers can be oriented according to the load during component manufacturing. In this context, the orientation and distribution of short and long fibers through an external magnetic field seem to be well-suited to be integrated into the conventional manufacturing process of fiber-reinforced composites. Therefore, the generally non-magnetic reinforcement fibers, e.g. carbon or glass fibers, need to be modified or coated with magnetic materials. In this paper, carbon fibers coated with an iron-cobalt alloy are prepared by electrodeposition for the validation of simulation models developed in previous studies. Furthermore, numerical studies are presented in regard to the orientation of such fibers in polymeric matrices. Thus, simulative investigations of the orientability of coated carbon fibers in polymeric materials are shown and the works provide an important reference for future studies of fiber orientation and alignment using magnetic fields.

Fiber-reinforced polymers are increasingly being used, especially in lightweight structures. Here, the effective adaptation of mechanical or physical properties to the necessary application or manufacturing requirements plays an important role. In this context, the alignment of reinforcing fibers is often hindered by manufacturing aspects. To achieve graded or locally adjusted alignment of different fiber lengths, common manufacturing technologies such as injection molding or compression molding need to be supported by the external non-mechanical process. Magnetic or electrostatic fields seem to be particularly suitable for this purpose. The present work shows a first simulation study of the alignment of magnetic particles in polymer matrices as a function of different parameters. The parameters studied are the viscosity of the surrounding polymer as a function of the focused processing methods, the fiber length, the thickness and permeability of the magnetic fiber coatings, and the magnetic flux density. The novelty of the presented works is in the development of an advanced simulation model that allows the simulative representation and reveal of the fluid–structure interaction, the influences of these parameters on the inducible magnetic torque and fiber alignment of a single fiber. Accordingly, the greatest influence on fiber alignment is caused by the magnetic flux density and the coating material.

AbstractOrientability is an important global topological property of spacetime manifolds. It is often assumed that a test for spatial orientability requires a global journey across the whole 3-space to check for orientation-reversing paths. Since such a global expedition is not feasible, theoretical arguments that combine universality of physical experiments with local arrow of time, CP violation and CPT invariance are usually offered to support the choosing of time- and space-orientable spacetime manifolds. Another theoretical argument also offered to support this choice comes from the impossibility of having globally defined spinor fields on non-orientable spacetime manifolds. In this paper, we argue that it is possible to locally access spatial orientability of Minkowski empty spacetime through physical effects involving quantum vacuum electromagnetic fluctuations. We study the motions of a charged particle and a point electric dipole subject to these electromagnetic fluctuations in Minkowski spacetime with orientable and non-orientable spatial topologies. We derive analytic expressions for a statistical orientability indicator for both of these point-like particles in two inequivalent spatially flat topologies. For the charged particle, we show that it is possible to distinguish the orientable from the non-orientable topology by contrasting the time evolution of the orientability indicators. This result reveals that it is possible to access orientability through electromagnetic quantum vacuum fluctuations. However, the answer to the central question of the paper, namely how to locally probe the orientability of Minkowski 3-space intrinsically, comes about only in the study of the motions of an electric dipole. For this point-like particle, we find that a characteristic inversion pattern exhibited by the curves of the orientability statistical indicator is a signature of non-orientability. This result makes it clear that it is possible to locally unveil spatial non-orientability through the inversion pattern of curves of our orientability indicator for a point electric dipole under quantum vacuum electromagnetic fluctuations. Our findings might open the way to a conceivable experiment involving quantum vacuum electromagnetic fluctuations to locally probe the spatial orientability of Minkowski empty spacetime.

On s’intéresse à des liens entre les fonctions symétriques et l’énumération des cartes, qui sont des graphes dessinés sur des surfaces, pas nécessairement orientables. On considère des séries génératrices de certaines familles de cartes avec des sommets colorés, notamment des cartes bipartites et des constellations. Dans ces séries génératrices, certaines propriétés de la structure combinatoire de la carte sont contrôlées, et chaque carte est comptée avec un poids corrélé à sa "non-orientabilité". On se concentre sur deux familles de conjectures reliant ces séries aux polynômes de Jack, une déformation à un paramètre des fonctions de Schur. La conjecture Matching-Jack, introduite par Goulden et Jackson en 1996, suggère que le développement d’une certaine série de Jack à plusieurs alphabets de variables dans la base des sommes de puissances a des coefficients entiers et positifs. De plus, ces coefficients compteraient des cartes biparties avec contrôle des degrés de tous les sommets et de toutes les faces. En utilisant des techniques d’opérateurs différentiels récemment introduites par Chapuy et Dołęga, on prouve la conjecture Matching-Jack pour une spécialisation particulière de la série génératrice. On utilise ensuite ce résultat et un nouveau lien avec l’algèbre de Farahat–Higman pour prouver la "partie intégralité" de la conjecture. Dans une autre direction, on établit une formule combinatoire pour le développement en sommes de puissances des polynômes de Jack, en utilisant les cartes à niveaux, une famille de cartes biparties décorées introduite dans cette thèse. En fait, cette formule découle d’une formule plus générale qu’on prouve pour les caractères de Jack. Ce résultat généralise une formule conjecturée par Stanley et prouvée par Féray en 2010 pour les caractères du groupe symétrique. En combinant cette formule avec une approche basée sur une famille d’opérateurs introduite par Nazarov et Sklyanin, on prouve une conjecture de Lassalle de 2008 sur la positivité et l’intégralité des caractères de Jack dans les coordonnées de Stanley. Finalement, nous utilisons la formule combinatoire obtenue pour les caractères de Jack afin de prouver que la série génératrice des cartes biparties avec contrôle des degrés de sommets et des faces satisfait une famille d’équations différentielles qui la caractérise. Ce résultat s’étend également aux séries des constellations
We are interested in connections between symmetric functions and the enumeration of maps, which are graphs drawn on surfaces, not necessarily orientable. We consider generating series of some families of maps with colored vertices, including bipartite maps and constellations. In these generating series, some properties of the combinatorial structure of the map are controlled, and each map is counted with a weight correlated to its "non-orientability". We focus on two families of conjectures connecting these series to Jack polynomials, a one parameter deformation of Schur symmetric functions. The Matching-Jack conjecture, introduced by Goulden and Jackson in 1996, suggests that the expansion of a mutliparametric Jack series in the power-sum symmetric functions has non-negative integer coefficients. Moreover, these coefficients count bipartite maps with controlled degrees of all vertices and faces. Using techniques of differential operators recently introduced by Chapuy and Dołęga, we prove the Matching-Jack conjecture for a particular specialization of the generating series. We use this result and a new connection with the Farahat-Higman algebra to prove the "integrality part" in the conjecture. In another direction, we establish a combinatorial formula for the power-sum expansion of Jack polynomials using layered maps, a family of decorated bipartite maps introduced in this thesis. We deduce this formula from a more general one that we provide for Jack characters. Actually, this result generalizes a formula conjectured by Stanley and proved by Féray in 2010 for the characters of the symmetric group. We combine this formula withan approach based on a family of operators introduced by Nazarov and Sklyanin in orderto prove a conjecture of Lassalle from 2008 about the positivity and the integrality of Jack characters in Stanley’s coordinates. Finally, we use the map expansion of Jack characters in order to prove that the generating series of bipartite maps with controlled vertex and face degrees satisfies a family of differential equations that completely characterizes it. Similar differential equations are alsoprovided for the series of constellations

Abstract This chapter offers an overview of the geometry of elliptic de Sitter spacetime. Elliptic de Sitter spacetime is de Sitter spacetime’s only rival in symmetry and elegance among solutions of Einstein’s equations with a positive cosmological constant. It is also, in many ways, the most natural example of a temporally non-orientable spacetime. The chapter concludes with a discussion of some reasons that have motivated physicists to take elliptic de Sitter spacetime seriously as a physical model, despite its temporal non-orientability.

‘Making surfaces’ considers the shape of surfaces and discusses the work of some of the early topologists, Möbius, Klein, and Riemann. It introduces the torus shape and shows how its Euler number can be calculated along with that of a sphere. It discusses closed surfaces—ones without a boundary—and how they can be divided up into vertices, edges, and faces. It then introduces one-sided surfaces such as the Möbius strip and Klein bottle, which are examples of non-orientable surfaces. The Euler number goes a long way to separating out different surfaces, with the only missing ingredient in the classification the notion of orientability.