Growth of positive words and lower bounds of the growth rate for Thompson's group (F)p

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Computational explorations in Thompson's group F

Here we describe the results of some computational explorations in Thompson's group F. We describe experiments to estimate the cogrowth of F with respect to its standard finite generating set, designed to address the subtle and difficult question whether or not Thompson's group is amenable. We also describe experiments to estimate the exponential growth rate of F and the rate of escape of symmetric random walks with respect to the standard generating set.

Combinatorial and metric properties of Thompson's group T

We discuss metric and combinatorial properties of Thompson's group T, such as the normal forms for elements and uniqueness of tree pair diagrams. We relate these properties to those of Thompson's group F when possible, and highlight combinatorial differences between the two groups. We define a set of unique normal forms for elements of T arising from minimal factorizations of elements into convenient pieces. We show that the number of carets in a reduced representative of T estimates the word length, that F is undistorted in T, and that cyclic subgroups of T are undistorted. We show that every element of T has a power which is conjugate to an element of F and describe how to recognize torsion elements in T.

Pure braid subgroups of braided Thompson's groups

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Minimal lenght elements of Thompson's groups F(p)

We describe a method for determining the minimal length of elements in the generalized Thompson's groups F(p). We compute the length of an element by constructing a tree pair diagram for the element, classifying the nodes of the tree and summing associated weights from the pairs of node classifications. We use this method to effectively find minimal length representatives of an element.

Metric properties of the braided Thompson's groups

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Tree automorphisms and quasi-isometries of Thompson's group F

We prove that automorphisms of the infinite binary rooted tree T2 do not yield quasi-isometries of Thompson's group F, except for the map which reverses orientation on the unit interval, a natural outer automorphism of F. This map, together with the identity map, forms a subgroup of Aut(T2) consisting of 2-adic automorphisms, following standard terminology used in the study of branch groups. However, for more general p, we show that the analgous groups of p-adic tree automorphisms do not give rise to quasiisometries of F(p).

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.

Dynamics in Thompson's group F

We describe an explicit relationship between strand diagrams and piecewise-linear functions for elements of Thompson’s group F. Using this correspondence, we investigate the dynamics of elements of F, and we show that conjugacy of one-bump functions can be described by a Mather-type invariant.

Bounding right-arm rotation distances

Rotation distance quantifies the difference in shape between two rooted binary trees of the same size by counting the minimum number of elementary changes needed to transform one tree to the other. We describe several types of rotation distance, and provide upper bounds on distances between trees with a fixed number of nodes with respect to each type. These bounds are obtained by relating each restricted rotation distance to the word length of elements of Thompson's group F with respect to different generating sets, including both finite and infinite generating sets.

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.

The Lorenz-maximal core allocations and the kernel in some classes of assignment games

En aquest treball demostrem que en la classe de jocs d'assignació amb diagonal dominant (Solymosi i Raghavan, 2001), el repartiment de Thompson (que coincideix amb el valor tau) és l'únic punt del core que és maximal respecte de la relació de dominància de Lorenz, i a més coincideix amb la solucié de Dutta i Ray (1989), també coneguda com solució igualitària. En segon lloc, mitjançant una condició més forta que la de diagonal dominant, introduïm una nova classe de jocs d'assignació on cada agent obté amb la seva parella òptima almenys el doble que amb qualsevol altra parella. Per aquests jocs d'assignació amb diagonal 2-dominant, el repartiment de Thompson és l'únic punt del kernel, i per tant el nucleolo.

The Lorenz-maximal core allocations and the kernel in some classes of assignment games

En aquest treball demostrem que en la classe de jocs d'assignació amb diagonal dominant (Solymosi i Raghavan, 2001), el repartiment de Thompson (que coincideix amb el valor tau) és l'únic punt del core que és maximal respecte de la relació de dominància de Lorenz, i a més coincideix amb la solucié de Dutta i Ray (1989), també coneguda com solució igualitària. En segon lloc, mitjançant una condició més forta que la de diagonal dominant, introduïm una nova classe de jocs d'assignació on cada agent obté amb la seva parella òptima almenys el doble que amb qualsevol altra parella. Per aquests jocs d'assignació amb diagonal 2-dominant, el repartiment de Thompson és l'únic punt del kernel, i per tant el nucleolo.

RTS,S/AS02A malaria vaccine does not induce parasite CSP T cell epitope selection and reduces multiplicity of infection

Objective: The candidate malaria vaccine RTS,S/AS02A is a recombinant protein containing part of the circumsporozoite protein (CSP) sequence of Plasmodium falciparum, linked to the hepatitis B surface antigen and formulated in the proprietary adjuvant system AS02A. In a recent trial conducted in children younger than age five in southern Mozambique, the vaccinedemonstrated significant and sustained efficacy against both infection and clinical disease. In a follow-up study to the main trial, breakthrough infections identified in the trial were examined to determine whether the distribution of csp sequences was affected by the vaccine and to measure the multiplicity of infecting parasite genotypes. Design: P. falciparum DNA from isolates collected during the trial was used for genotype studies. Setting: The main trial was carried out in the Manhiça district, Maputo province, Mozambique, between April 2003 and May 2004. Participants: Children from the two cohorts of the main trial provided parasite isolates as follows: children from Cohort 1 who were admitted to hospital with clinical malaria; children from Cohort 1 who were parasite-positive in a cross-sectional survey at study month 8.5; children from Cohort 2 identified as parasite-positive during follow-up by active detection of infection. Outcome: Divergence of DNA sequence encoding the CSP T cell-epitope region sequence from that of the vaccine sequence was measured in 521 isolates. The number of distinct P. falciparum genotypes was also determined. Results: We found no evidence that parasite genotypes from children in the RTS,S/AS02A arm were more divergent than those receiving control vaccines. For Cohort 1 (survey at studymonth 8.5) and Cohort 2, infections in the vaccine group contained significantly fewer genotypes than those in the control group, (p 1/4 0.035, p 1/4 0.006), respectively, for the two cohorts. This was not the case for children in Cohort 1 who were admitted to hospital (p 1/4 0.478). Conclusions: RTS,S/AS02A did not select for genotypes encoding divergent T cell epitopes in the C-terminal region of CSP in this trial. In both cohorts, there was a modest reduction in the mean number of parasite genotypes harboured by vaccinated children compared with controls, but only among those with asymptomatic infections.

Impacte de la Gimnàstica artística a Vic: la relació gimnasta-entrenadora dins del Club Gimnàstic Osona (CGO)

L’objectiu d’aquest estudi és comprovar si una millor relació entre gimnasta i entrenadora és un factor determinant per a poder obtenir més resultats en les competicions. S’estudien tres grups dividits per tres nivells diferents: Escolar (8 gimnastes), Promogym (11 gimnastes) i Tecnificació (28 gimnastes). A partir d’una adaptació feta dels qüestionaris Cart-Q de Jowett i Ntoumanis (2004) i l’ SF-PANAS de Thompson (2007) es realitzarà l’anàlisi de la relació gimnasta-entrenadora abans de dues competicions, on s’analitzaran els podis. Observant l’estudi, una baixada de mitjanes dels aspectes de les 3 C’s (proximitat, compromís i complementarietat) estava acompanyat d’una baixada de resultats i una pujada de les mitjanes de les 3 C’s estava relacionat amb l’obtenció de més podis. Per tant, a l’hora de valorar els resultats en competició la relació gimnasta-entrenadora és un factor important a tenir en compte.