Social polarization: introducing distances between and within groups

The measurement of social polarization has received little attention from the literature. The only social polarization index that has been used to measure religious or ethnic polarization (the RQ index) has several shortcomings that are critically discussed in the paper. In particular, that index is not taking into account the existing distance between and within different groups. A couple of axiomatically characterized social polarization indices that overcome these limitations are presented. In the empirical section we show that the rankings of countries according to the levels of polarization change to a great extent when we replace the RQ index by the indices presented in this paper.

Musical actions of dihedral groups

The sequence of pitches which form a musical melody can be transposed or inverted. Since the 1970s, music theorists have modeled musical transposition and inversion in terms of an action of the dihedral group of order 24. More recently music theorists have found an intriguing second way that the dihedral group of order 24 acts on the set of major and minor triads. We illustrate both geometrically and algebraically how these two actions are dual. Both actions and their duality have been used to analyze works of music as diverse as Hindemith and the Beatles.

Conjugacy in Thompson's groups

We give a unified solution the conjugacy problem in Thompson’s groups F, V , and T using strand diagrams, and we analyze the complexity of the resulting algorithms.

Mather invariants in groups of piecewise-linear homeomorphisms

We describe the relation between two characterizations of conjugacy in groups of piecewise-linear homeomorphisms, discovered by Brin and Squier in [2] and Kassabov and Matucci in [5]. Thanks to the interplay between the techniques, we produce a simplified point of view of conjugacy that allows ua to easily recover centralizers and lends itself to generalization.

The simultaneous conjugacy problem in groups of piecewise linear functions

Guba and Sapir asked, in their joint paper [8], if the simultaneous conjugacy problem was solvable in Diagram Groups or, at least, for Thompson's group F. We give an elementary proof for the solution of the latter question. This relies purely on the description of F as the group of piecewise linear orientation-preserving homeomorphisms of the unit. The techniques we develop allow us also to solve the ordinary conjugacy problem as well, and we can compute roots and centralizers. Moreover, these techniques can be generalized to solve the same questions in larger groups of piecewise-linear homeomorphisms.

On uniform conjugators in torsion-free hyperbolic groups

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Structure theorems for subgroups of homeomorphism groups

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Bounding the residual finiteness of free groups

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Statistical properties of subgroups of free groups

The usual way to investigate the statistical properties of finitely generated subgroups of free groups, and of finite presentations of groups, is based on the so-called word-based distribution: subgroups are generated (finite presentations are determined) by randomly chosen k-tuples of reduced words, whose maximal length is allowed to tend to infinity. In this paper we adopt a different, though equally natural point of view: we investigate the statistical properties of the same objects, but with respect to the so-called graph-based distribution, recently introduced by Bassino, Nicaud and Weil. Here, subgroups (and finite presentations) are determined by randomly chosen Stallings graphs whose number of vertices tends to infinity. Our results show that these two distributions behave quite differently from each other, shedding a new light on which properties of finitely generated subgroups can be considered frequent or rare. For example, we show that malnormal subgroups of a free group are negligible in the raph-based distribution, while they are exponentially generic in the word-based distribution. Quite surprisingly, a random finite presentation generically presents the trivial group in this new distribution, while in the classical one it is known to generically present an infinite hyperbolic group.

Higher arithmetic Chow groups

We give a new construction of higher arithmetic Chow groups for quasi-projective arithmetic varieties over a field. Our definition agrees with the higher arithmetic Chow groups defined by Goncharov for projective arithmetic varieties over a field. These groups are the analogue, in the Arakelov context, of the higher algebraic Chow groups defined by Bloch. The degree zero group agrees with the arithmetic Chow groups of Burgos. Our new construction is shown to be a contravariant functor and is endowed with a product structure, which is commutative and associative.

The first steps of a secondary school teacher: experiences and studies of a student-teacher

This dissertation aims at fostering the professional development of the EFL teacher. This document compiles two small scale empirical studies carried out during the practicum periods of the TED's course. The first one is based on the role of the teacher's talk in the EFL classroom and the second one focuses on students’ small group talk, analysing the impact of cooperative learning in the EFL classroom by examining students' conversation. The following section gathers the teacher's personal reflections upon the process of professionalization. The paper concludes with a summary of the major findings and further professional improvement proposals

Adolescència i preferències vocacionals: anàlisi comparativa per sexes, de les preferències vocacionals expressades per estudiants de 4t. d'Educació Secundària Obligatòria, a través d'un disseny longitudinal amb mostratges de subjectes diferents

Aquest treball de recerca consisteix en un estudi sobre les preferències vocacionals expressades per adolescents que cursen 4t d’Educació Secundària Obligatòria, amb una anàlisi comparativa per sexes. Es tracta d’un estudi de cas realitzat en un centre del Vallès Oriental, de caràcter longitudinal, en el que s’ha recollit durant 7 cursos acadèmics consecutius les seves preferències vocacionals a través de la prova d’interessos Kuder-C. L'objectiu ha estat comparar les preferències expressades pels nois i per les noies, les influències dels models de gènere estereotipats, així com l’existència o no de diferències entre ambdós col·lectius amb el pas del temps.

Twisted conjugacy in braid groups

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