Topologies on groups related to the theory of shape and generalized coverings

El presente trabajo consiste en dos partes diferenciadas: la principal de ellas (Cap tulos 1 y 2) est a dedicada a introducir estructura adicional en grupos que aparecen de manera natural en el contexto de la teor a de la forma. En la segunda parte (Cap tulo 3), se plantea c omo generalizar la teor a de espacios recubridores y, en particular, se propone una l nea de trabajo relacionada con la teor a de la forma. El punto de partida de esta tesis doctoral son los trabajos [25, 26, 68, 69, 70] en los que los autores introducen y utilizan algunas ultram etricas en el conjunto de los mor smos shape entre dos espacios topol ogicos punteados. En particular, si el dominio es (S1; 1); la construcci on realizada en [68] permite explicitar una ultram etrica en el grupo shape 1(X; x0) de un espacio m etrico compacto X; como ya fue observado en [69] y [80]. Si el espacio no es m etrico compacto, la construcci on nos lleva a utilizar el concepto de ultram etrica generalizada, en el sentido de Priess-Crampe y Ribenboim [78, 79]. En [7], D. K. Biss introduce la idea de topologizar el grupo fundamental de un espacio, de forma que la topolog a en 1(X; x0) sea una topolog a de grupo que permita detectar la (no) existencia de un recubridor universal para X: La forma de proceder sugerida es tomar en 1(X; x0)la toplog a cociente inducida por la topolog a compacto-abierta en el espacio de lazos (X; x0): Sin embargo, hay algunos errores en el art culo mencionado: en concreto, el error relacionado con el presente trabajo fue puesto de mani esto por P. Fabel en [33], mostrando que, en general, la operaci on de grupo en 1(X; x0)con la topolog a cociente no es continua. Utilizando un punto de vista similar, varios autores han tratado de dotar al grupo fundamental con una topolog a, de forma que 1(X; x0) sea un grupo topol ogico y la proyecci on q (X; x0){u100000} 1(X; x0)sea continua...

Groups, information theory, and Einstein's likelihood principle

We propose a unifying picture where the notion of generalized entropy is related to information theory by means of a group-theoretical approach. The group structure comes from the requirement that an entropy be well defined with respect to the composition of independent systems, in the context of a recently proposed generalization of the Shannon-Khinchin axioms. We associate to each member of a large class of entropies a generalized information measure, satisfying the additivity property on a set of independent systems as a consequence of the underlying group law. At the same time, we also show that Einstein's likelihood function naturally emerges as a byproduct of our informational interpretation of (generally nonadditive) entropies. These results confirm the adequacy of composable entropies both in physical and social science contexts.

A spectroscopy study of nearby late-type stars, possible members of stellar kinematic groups

Context. Nearby late-type stars are excellent targets for seeking young objects in stellar associations and moving groups. The origin of these structures is still misunderstood, and lists of moving group members often change with time and also from author to author. Most members of these groups have been identified by means of kinematic criteria, leading to an important contamination of previous lists by old field stars. Aims. We attempt to identify unambiguous moving group members among a sample of nearby-late type stars by studying their kinematics, lithium abundance, chromospheric activity, and other age-related properties. Methods. High-resolution echelle spectra (R ~ 57 000) of a sample of nearby late-type stars are used to derive accurate radial velocities that are combined with the precise Hipparcos parallaxes and proper motions to compute galactic-spatial velocity components. Stars are classified as possible members of the classical moving groups according to their kinematics. The spectra are also used to study several age-related properties for young late-type stars, i.e., the equivalent width of the lithium Li i 6707.8 Å line or the R'_HK index. Additional information like X-ray fluxes from the ROSAT All-Sky Survey or the presence of debris discs is also taken into account. The different age estimators are compared and the moving group membership of the kinematically selected candidates are discussed. Results. From a total list of 405 nearby stars, 102 have been classified as moving group candidates according to their kinematics. i.e., only ~25.2% of the sample. The number reduces when age estimates are considered, and only 26 moving group candidates (25.5% of the 102 candidates) have ages in agreement with the star having the same age as an MG member.