On the Euler characteristic of spherical polyhedra and the Euler relation

Let En+1, for some integer n ≥ 0, be the (n + 1)-dimensional Euclidean space, and denote by Sn the standard n-sphere in En+1, . It is convenient to introduce the (-1)-dimensional sphere , where denotes the empty set. By an i-dimensional subsphere T of Sn, i = 0 n, we understand the intersection of Sn with some (i+1)-dimensional subspace of En+1. The affine hull of T always contains, with this definition, the origin of En+1. is the unique (-1)-dimensional subsphere of Sn. By the spherical hull, sph X, of a set , we understand the intersection of all subspheres of Sn containing X. Further we set dim X: = dim sph X. The interior, the boundary and the complement of an arbitrary set , with respect to Sn, shall be denoted by int X, bd X and cpl X. Finally we define the relative interior rel int X to be the interior of with respect to the usual topology sphZ . For each (n-1)-dimensional subsphere of Sn defines two closed hemispheres of Sn, whose common boundary it is. The two hemispheres of the sphere Sº are denned to be the two one-pointed subsets of Sº. A subset is called a closed (spherical) polytope, if it is the intersection of finitely many closed hemispheres, and, if, in addition, it does not contain a subsphere of Sn. is called an i-dimensional, relatively open polytope, , or shortly an i-open polytope, if there exists a closed polytope such that dim P = i and Q = rel int P. is called a closed polyhedron, if it is a finite union of closed polytopes P1 ..., Pr. The empty set is the only (-1)-dimensional closed polyhedron of Sn. We denote by the set of all closed polyhedra of Sn. is called an i-open polyhedron, for some , if there are finitely many i-open polytopes Q1 ..., Qr in Sn such that , and dim . By we denote the set of all i-open polyhedra. Clearly for all , and each i-dimensional subsphere of Sn, , belongs to and to , For each i-dimensional subsphere T of Sn, set . A map is defined by , for all , and, for al

Thomassen's Choosability Argument Revisited

Thomassen (1994) proved that every planar graph is 5-choosable. This result was generalised by {\v{S}}krekovski (1998) and He et al. (2008), who proved that every $K_5$-minor-free graph is 5-choosable. Both proofs rely on the characterisation of $K_5$-minor-free graphs due to Wagner (1937). This paper proves the same result without using Wagner's structure theorem or even planar embeddings. Given that there is no structure theorem for graphs with no $K_6$-minor, we argue that this proof suggests a possible approach for attacking the Hadwiger Conjecture

Eine Erweiterung der Croftonschen Formeln für konvexe Körper

§0. Einführung. Durch die bekannten Croftonschen Integrale können bekanntlich die Minkowskischen Quermaβintegrale konvexer Körper dargestellt werden. In der vorliegenden Note betrachten wir gewisse Erweiterungen dieser klassischen integralgeometrischen Formeln, durch die allgemeinere invariante Eikörperfunktionale gegeben sind. Es handelt sich hierbei um kinematische Integrale mit beweglichen unterdimensionalen Teilräumen, wobei passend gewahlte Funktionen ihrer Abstände vom Eikörper eingehe

Integral Geometry of Linearly Combined Gaussian and Student-t, and Skew Student's t Random Fields

International audienceThe integral geometry of random fields has been investigated since the 1980s, and the analytic formulae of the Minkowski functionals (also called Lipschitz-Killing curvatures, shortly denoted LKCs) of their excursion sets on a compact subset S in the n-dimensional Euclidean space have been reported in the specialized literature for Gaussian and student-t random fields. Very recently, explicit analytical formulae of the Minkowski functionals of their excursion sets in the bi-dimensional case (n = 2) have been defined on more sophisticated random fields, namely: the Linearly Combined Gaussian and Student-t, and the Skew Student-t random fields. This paper presents the theoretical background, and gives the explicit analytic formulae of the three Minkowski functionals. Simulation results are also presented both for illustration and validation, together with a real application example on an worn engineered surface

Morphological Thermodynamics of Fluids: Shape Dependence of Free Energies

We examine the dependence of a thermodynamic potential of a fluid on the geometry of its container. If motion invariance, continuity, and additivity of the potential are fulfilled, only four morphometric measures are needed to describe fully the influence of an arbitrarily shaped container on the fluid. These three constraints can be understood as a more precise definition for the conventional term "extensive" and have as a consequence that the surface tension and other thermodynamic quantities contain, beside a constant term, only contributions linear in the mean and Gaussian curvature of the container and not an infinite number of curvatures as generally assumed before. We verify this numerically in the entropic system of hard spheres bounded by a curved wall.Comment: 4 pages, 3 figures, accepted for publication in PR

Lines pinning lines

A line g is a transversal to a family F of convex polytopes in 3-dimensional space if it intersects every member of F. If, in addition, g is an isolated point of the space of line transversals to F, we say that F is a pinning of g. We show that any minimal pinning of a line by convex polytopes such that no face of a polytope is coplanar with the line has size at most eight. If, in addition, the polytopes are disjoint, then it has size at most six. We completely characterize configurations of disjoint polytopes that form minimal pinnings of a line.Comment: 27 pages, 10 figure

Domótica e a requalificação de edifícios

Para a elaboração deste artigo técnico foi necessário adoptar uma estrutura que possibilitasse fornecer um estudo teóricoprático, transversal e equilibrado, das diferentes tecnologias domóticas. Inicialmente realizou-se um pequeno estudo teórico das tecnologias domóticas mais relevantes, de uma forma transversal e resumida (Capítulo 2). Em função do estudo teórico do capítulo anterior, no Capítulo 3 realizou-se uma análise mais prática, em que ao invés de abordar um caso prático existente, de grandes instalações com o seu valor emblemático, optou-se por utilizar como modelo o edifício F do Instituto Superior de Engenharia do Porto e apresentar uma das soluções possíveis de implementação de tecnologias domóticas em edifícios já existentes (aplicação do conceito de requalificação de edifícios). Depois da exposição do caso prático, expôs-se o futuro e oportunidades de mercado da domótica ou sistema de gestão técnica centralizada, mais focalizado para o mundo académico (Capítulo 4). Por fim, são tecidas as conclusões e considerações finais do artigo (capitulo 5). Esse artigo foi elaborado sob o ponto de vista de integrador. Por outras palavras, procurou-se realizar uma aproximação da realidade prática a nível de implementação das tecnologias domóticas em edifícios, ao dar uma linha de conhecimento abrangente e ao mesmo acessível aos leitores, que muitas das vezes esse tema acaba por transmitir conceitos errados

Eutactic quantum codes

We consider sets of quantum observables corresponding to eutactic stars. Eutactic stars are systems of vectors which are the lower dimensional ``shadow'' image, the orthogonal view, of higher dimensional orthonormal bases. Although these vector systems are not comeasurable, they represent redundant coordinate bases with remarkable properties. One application is quantum secret sharing.Comment: 6 page