Computing the diameter of Coxeter groups

This thesis addresses the problem of computing the minimal and maximal diameter of the Cayley graph of Coxeter groups. We first present and assert relevant parts of polytope theory and related Coxeter theory. After this, a method of contracting the orthogonal projections of a polytope from Rd onto R2 and R3, d ¸ 3 is presented. This method is the Equality Set Projection algorithm that requires a constant number of linearprogramming problems per facet of the projection in the absence of degeneracy. The ESP algorithm allows us to compute also projected geometric diameters of high-dimensional polytopes. A representation set of projected polytopes is presented to illustrate the methods adopted in this thesis.

Fundamental domains for proper affine actions of Coxeter groups in three dimensions

We study proper actions of groups $G \cong \Z/2\Z \ast \Z/2\Z \ast \Z/2\Z$ on affine space of three real dimensions. Since $G$ is nonsolvable, work of Fried and Goldman implies that it preserves a Lorentzian metric. A subgroup $\Gamma < G$ of index two acts freely, and $\R^3/\Gamma$ is a Margulis spacetime associated to a hyperbolic surface $\Sigma$. When $\Sigma$ is convex cocompact, work of Danciger, Gu{\'e}ritaud, and Kassel shows that the action of $\Gamma$ admits a polyhedral fundamental domain bounded by crooked planes. We consider under what circumstances the action of $G$ also admits a crooked fundamental domain. We show that it is possible to construct actions of $G$ that fail to admit crooked fundamental domains exactly when the extended mapping class group of $\Sigma$ fails to act transitively on the top-dimensional simplices of the arc complex of $\Sigma$. We also provide explicit descriptions of the moduli space of $G$ actions that admit crooked fundamental domains.

A Geometrical Construction for the Polynomial Invariants of some Reflection Groups

2000 Mathematics Subject Classification: Primary 20F55, 13F20; Secondary 14L30.

Brisure de symétrie par la réduction des groupes de Lie simples à leurs sous-groupes de Lie réductifs maximaux

Dans ce travail, nous exploitons des propriétés déjà connues pour les systèmes de poids des représentations afin de les définir pour les orbites des groupes de Weyl des algèbres de Lie simples, traitées individuellement, et nous étendons certaines de ces propriétés aux orbites des groupes de Coxeter non cristallographiques. D'abord, nous considérons les points d'une orbite d'un groupe de Coxeter fini G comme les sommets d'un polytope (G-polytope) centré à l'origine d'un espace euclidien réel à n dimensions. Nous introduisons les produits et les puissances symétrisées de G-polytopes et nous en décrivons la décomposition en des sommes de G-polytopes. Plusieurs invariants des G-polytopes sont présentés. Ensuite, les orbites des groupes de Weyl des algèbres de Lie simples de tous types sont réduites en l'union d'orbites des groupes de Weyl des sous-algèbres réductives maximales de l'algèbre. Nous listons les matrices qui transforment les points des orbites de l'algèbre en des points des orbites des sous-algèbres pour tous les cas n<=8 ainsi que pour plusieurs séries infinies des paires d'algèbre-sous-algèbre. De nombreux exemples de règles de branchement sont présentés. Finalement, nous fournissons une nouvelle description, uniforme et complète, des centralisateurs des sous-groupes réguliers maximaux des groupes de Lie simples de tous types et de tous rangs. Nous présentons des formules explicites pour l'action de tels centralisateurs sur les représentations irréductibles des algèbres de Lie simples et montrons qu'elles peuvent être utilisées dans le calcul des règles de branchement impliquant ces sous-algèbres.

ON CLOSED GEODESICS IN THE LEAF SPACE OF SINGULAR RIEMANNIAN FOLIATIONS

In this paper we prove the existence of closed geodesics in the leaf space of some classes of singular Riemannian foliations (s.r.f.), namely s.r.fs. that admit sections or have no horizontal conjugate points. We also investigate the shortening process with respect to Riemannian foliations.

Braid groups, mapping class groups, and Torelli groups

This thesis discusses subgroups of mapping class groups of particular surfaces. First, we study the Torelli group, that is, the subgroup of the mapping class group that acts trivially on the first homology. We investigate generators of the Torelli group, and we give an algorithm that factorizes elements of the Torelli group into products of particular generators. Furthermore, we investigate normal closures of powers of standard generators of the mapping class group of a punctured sphere. By using the Jones representation, we prove that in most cases these normal closures have infinite index in the mapping class group. We prove a similar result for the hyperelliptic mapping class group, that is, the group that consists of mapping classes that commute with a fixed hyperelliptic involution. As a corollary, we recover an older theorem of Coxeter (with 2 exceptional cases), which states that the normal closure of the m-th power of standard generators of the braid group has infinite index in the braid group. Finally, we study finite index subgroups of braid groups, namely, congruence subgroups of braid groups. We discuss presentations of these groups and we provide a topological interpretation of their generating sets.

Design, Synthesis And Biological Evaluation Of Hybrid Bioisoster Derivatives Of N-acylhydrazone And Furoxan Groups With Potential And Selective Anti-trypanosoma Cruzi Activity.

Hybrid bioisoster derivatives from N-acylhydrazones and furoxan groups were designed with the objective of obtaining at least a dual mechanism of action: cruzain inhibition and nitric oxide (NO) releasing activity. Fifteen designed compounds were synthesized varying the substitution in N-acylhydrazone and in furoxan group as well. They had its anti-Trypanosoma cruzi activity in amastigotes forms, NO releasing potential and inhibitory cruzain activity evaluated. The two most active compounds (6, 14) both in the parasite amastigotes and in the enzyme contain the nitro group in para position of the aromatic ring. The permeability screening in Caco-2 cell and cytotoxicity assay in human cells were performed for those most active compounds and both showed to be less cytotoxic than the reference drug, benznidazole. Compound 6 was the most promising, since besides activity it showed good permeability and selectivity index, higher than the reference drug. Thereby the compound 6 was considered as a possible candidate for additional studies.

Comparative Cytogenetics Of Physalaemus albifrons And Physalaemus Cuvieri Species Groups (anura, leptodactylidae).

Recently, Physalaemus albifrons (Spix, 1824) was relocated from the Physalaemus cuvieri group to the same group as Physalaemus biligonigerus (Cope, 1861), Physalaemus marmoratus (Reinhardt & Lütken, 1862) and Physalaemus santafecinus Barrio, 1965. To contribute to the analysis of this proposition, we studied the karyotypes of Physalaemus albifrons, Physalaemus santafecinus and three species of the Physalaemus cuvieri group. The karyotype of Physalaemus santafecinus was found to be very similar to those of Physalaemus biligonigerus and Physalaemus marmoratus, which were previously described. A remarkable characteristic that these three species share is a conspicuous C-band that extends from the pericentromeric region almost to the telomere in the short arm of chromosome 3. This characteristic is not present in the Physalaemus albifrons karyotype and could be a synapomorphy of Physalaemus biligonigerus, Physalaemus marmoratus and Physalaemus santafecinus. The karyotype of Physalaemus santafecinus is also similar to those of Physalaemus marmoratus and Physalaemus biligonigerus owing to the presence of several terminal C-bands and the distal localization of the NOR in a small metacentric chromosome. In contrast, the Physalaemus albifrons karyotype has no terminal C-bands and its NOR is located interstitially in the long arm of submetacentric chromosome 8. The NOR-bearing chromosome of Physalaemus albifrons very closely resembles those found in Physalaemus albonotatus (Steindachner, 1864), Physalaemus cuqui Lobo, 1993 and some populations of Physalaemus cuvieri Fitzinger, 1826. Additionally, the Physalaemus albifrons karyotype has an interstitial C-band in chromosome 5 that has been exclusively observed in species of the Physalaemus cuvieri group. Therefore, we were not able to identify any chromosomal feature that supports the reallocation of Physalaemus albifrons.

Metacommunity Versus Biogeography: A Case Study Of Two Groups Of Neotropical Vegetation-dwelling Arthropods.

Biogeography and metacommunity ecology provide two different perspectives on species diversity. Both are spatial in nature but their spatial scales do not necessarily match. With recent boom of metacommunity studies, we see an increasing need for clear discrimination of spatial scales relevant for both perspectives. This discrimination is a necessary prerequisite for improved understanding of ecological phenomena across scales. Here we provide a case study to illustrate some spatial scale-dependent concepts in recent metacommunity studies and identify potential pitfalls. We presented here the diversity patterns of Neotropical lepidopterans and spiders viewed both from metacommunity and biogeographical perspectives. Specifically, we investigated how the relative importance of niche- and dispersal-based processes for community assembly change at two spatial scales: metacommunity scale, i.e. within a locality, and biogeographical scale, i.e. among localities widely scattered along a macroclimatic gradient. As expected, niche-based processes dominated the community assembly at metacommunity scale, while dispersal-based processes played a major role at biogeographical scale for both taxonomical groups. However, we also observed small but significant spatial effects at metacommunity scale and environmental effects at biogeographical scale. We also observed differences in diversity patterns between the two taxonomical groups corresponding to differences in their dispersal modes. Our results thus support the idea of continuity of processes interactively shaping diversity patterns across scales and emphasize the necessity of integration of metacommunity and biogeographical perspectives.

Cephalometric And Anthropometric Data Of Obstructive Apnea In Different Age Groups.

Patients with obstructive sleep apnea syndrome usually present with changes in upper airway morphology and/or body fat distribution, which may occur throughout life and increase the severity of obstructive sleep apnea syndrome with age. To correlate cephalometric and anthropometric measures with obstructive sleep apnea syndrome severity in different age groups. A retrospective study of cephalometric and anthropometric measures of 102 patients with obstructive sleep apnea syndrome was analyzed. Patients were divided into three age groups (≥20 and <40 years, ≥40 and <60 years, and ≥60 years). Pearson's correlation was performed for these measures with the apnea-hypopnea index in the full sample, and subsequently by age group. The cephalometric measures MP-H (distance between the mandibular plane and the hyoid bone) and PNS-P (distance between the posterior nasal spine and the tip of the soft palate) and the neck and waist circumferences showed a statistically significant correlation with apnea-hypopnea index in both the full sample and in the ≥40 and <60 years age group. These variables did not show any significant correlation with the other two age groups (<40 and ≥60 years). Cephalometric measurements MP-H and PNS-P and cervical and waist circumferences correlated with obstructive sleep apnea syndrome severity in patients in the ≥40 and <60 age group.

Quality Of Life On Arterial Hypertension: Validity Of Known Groups Of Minichal.

Introductions: In the care of hypertension, it is important that health professionals possess available tools that allow evaluating the impairment of the health-related quality of life, according to the severity of hypertension and the risk for cardiovascular events. Among the instruments developed for the assessment of health-related quality of life, there is the Mini-Cuestionario of Calidad de Vida en la Hipertensión Arterial (MINICHAL) recently adapted to the Brazilian culture. Objective: To estimate the validity of known groups of the Brazilian version of the MINICHAL regarding the classification of risk for cardiovascular events, symptoms, severity of dyspnea and target-organ damage. Methods: Data of 200 hypertensive outpatients concerning sociodemographic and clinical information and health-related quality of life were gathered by consulting the medical charts and the application of the Brazilian version of MINICHAL. The Mann-Whitney test was used to compare health-related quality of life in relation to symptoms and target-organ damage. The Kruskal-Wallis test and ANOVA with ranks transformation were used to compare health-related quality of life in relation to the classification of risk for cardiovascular events and intensity of dyspnea, respectively. Results: The MINICHAL was able to discriminate health-related quality of life in relation to symptoms and kidney damage, but did not discriminate health-related quality of life in relation to the classification of risk for cardiovascular events. Conclusion: The Brazilian version of the MINICHAL is a questionnaire capable of discriminating differences on the health‑related quality of life regarding dyspnea, chest pain, palpitation, lipothymy, cephalea and renal damage.Fundamento: No cuidado ao hipertenso, é importante que o profissional de saúde disponha de ferramentas que possibilitem avaliar o comprometimento da qualidade de vida relacionada à saúde, de acordo com a gravidade da hipertensão e o risco para eventos cardiovasculares. Dentre os instrumentos criados para avaliação da qualidade de vida relacionada à saúde, destaca-se o Mini-Cuestionario de Calidad de Vida en la Hipertensión Arterial (MINICHAL), recentemente adaptado para a cultura brasileira. Objetivo: Estimar a validade de grupos conhecidos da versão brasileira do MINICHAL em relação à classificação de risco para eventos cardiovasculares, sintomas, intensidade da dispneia e lesões de órgãos-alvo. Métodos: Foram investigados 200 hipertensos em seguimento ambulatorial, cujos dados sociodemográficos, clínicos e de qualidade de vida relacionada à saúde foram obtidos por meio de consulta ao prontuário e da aplicação da versão brasileira do MINICHAL. O teste de Mann-Whitney foi utilizado para comparar qualidade de vida relacionada à saúde em relação aos sintomas e às lesões de órgãos-alvo. Teste de Kruskal-Wallis e ANOVA com transformação nos ranks foram empregados para comparar qualidade de vida relacionada à saúde em relação à classificação de risco para eventos cardiovasculares e intensidade da dispneia, respectivamente. Resultados: O MINICHAL discriminou qualidade de vida relacionada à saúde em relação aos sintomas e dano renal (lesões de órgãos-alvo), porém não discriminou qualidade de vida relacionada à saúde em relação à classificação de risco para eventos cardiovasculares. Conclusão: A versão brasileira do MINICHAL é um instrumento capaz de discriminar diferenças na qualidade de vida relacionada à saúde em relação aos sintomas de dispneia, precordialgia, palpitação, lipotímia, cefaleia e presença de dano renal.

Identification of family variables in parents' groups of children with epilepsy

OBJECTIVE: To verify the effectiveness of the support group in the identification of family variables linked to epilepsy. METHOD: Pre-test were applied to parents of 21 children with benign epilepsy of childhood recently diagnosed, from 5 to 15 years, who participated in the groups at HC/Unicamp. There was a presentation of an educational video, discussion and application of the post-test 1. After six months, the post-test 2 was applied. RESULTS: The beliefs were: fear of swallowing the tongue during the seizures (76.19%) and of a future mental disease (66.67%). Facing the epilepsy, fear and sadness appeared. 76.19% of the parents presented overprotection and 90.48%, expected a new seizure. In the post-test 1, the parents affirmed that the information offered had modified the beliefs. In the post-test 2, 80.95% didn't report great doubts about epilepsy and 90.48% considered their relationship with their children better. CONCLUSIONS: The demystification of beliefs supplied from the groups influenced the family positively, prevented behavior alterations and guaranteed effective care in the attendance to the child with epilepsy.

Permeability of different groups of maxillary teeth after 38% hydrogen peroxide internal bleaching

This study evaluated the influence of internal tooth bleaching with 38% hydrogen peroxide (H2O2) on the permeability of the coronal dentin in maxillary anterior teeth and premolars. Seventy teeth (14 per group) were used: central incisors (CI), lateral incisor (LI), canines (C), first premolars (1PM) and second premolars (2PM). Pulp chamber access and transversal sectioning at 2 mm from the cementoenamel junction were performed and the specimens were divided into 2 groups (n= 7): a) no treatment and b) bleaching with 38% H2O2. The bleaching agent was applied to the buccal surface and to the pulp chamber for 10 min. This procedure was repeated 3 times. The specimens were processed histochemically with copper sulfate and rubeanic acid, sectioned longitudinally, and digitalized in a scanner. The area of stained dentin was measured using Image Tool software. Data were analyzed statistically by ANOVA and Tukey's HSD test (?=0.05). There was statistically significant difference (p<0.001) among the untreated groups, CI (0.23 ± 0.26) having the lowest permeability and LI (10.14 ± 1.89) the highest permeability. Among the bleached groups, dentin permeability was increased in all groups of teeth except for 2PM. It may be concluded that bleaching with 38% H2O2 affected dentin permeability near the pulp chamber in maxillary anterior teeth and in first and second premolars.

Dentin permeability of the apical third in different groups of teeth

This ex vivo study evaluated dentin permeability of the root canal in the apical third of different human groups of teeth. Eighty teeth were used, 8 from each dental group: maxillary and mandibular central incisors, lateral incisors and canines, maxillary first premolars (buccal and palatal roots), mandibular first premolars, and maxillary and mandibular second premolars, totalizing 88 roots that were distributed in 11 groups. The root canals were instrumented, irrigated with 1% NaOCl and 15% EDTA. Roots were immersed in 10% copper sulfate for 30 min and then in 1% rubeanic acid alcohol solution for the same period; this chemical reaction reveals dentin permeability by the formation of copper rubeanate, which is a dark-colored compound. Semi-serial 100-µm-thick cross-sections were obtained from the apical third of the roots. Five sections of each apical third were washed, dehydrated, cleared and mounted on glass slides for examination under optical microscopy. The percentage of copper ion infiltration and the amount of tubular dentin were quantified by morphometric analysis. The penetration of copper ions in the apical third ranged from 4.60 to 16.66%. The mandibular central and lateral incisors presented the highest dentin permeability (16.66%), while the maxillary canines and mandibular second and first premolars presented the lowest dentin permeability (4.60%, 4.80% and 5.71%, respectively; p<0.001). The other teeth presented intermediate permeability. In conclusion, dye penetration into dentin tubules at the apical region is strongly dependent on the group of teeth evaluated.

Population dynamics: seasonal variation of phytoplankton functional groups in brazilian reservoirs (Billings and Guarapiranga, São Paulo)

Phytoplankton may function as a "sensor" of changes in aquatic environment and responds rapidly to such changes. In freshwaters, coexistence of species that have similar ecological requirements and show the same environmental requirements frequently occurs; such species groups are named functional groups. The use of phytoplankton functional groups to evaluate these changes has proven to be very useful and effective. Thus, the aim of this study was to evaluate the occurrence of functional groups of phytoplankton in two reservoirs (Billings and Guarapiranga) that supply water to millions of people in São Paulo city Metropolitan Area, southeastern Brazil. Surface water samples were collected monthly and physical, chemical and biological (quantitative and qualitative analyses of the phytoplankton) were performed. The highest biovolume (mm³.L-1) of the descriptor species and functional groups were represented respectively by Anabaena circinalis Rabenh. (H1), Microcystis aeruginosa (Kützing) Kützing (L M/M) and Mougeotia sp. (T) in the Guarapiranga reservoir and Cylindrospermopsis raciborskii (Wolosz.) Seen. and Subba Raju (S N), Microcystis aeruginosa and M. panniformis Komárek et al. (L M/M), Planktothrix agardhii (Gom.) Anagn. and Komárek and P. cf. clathrata (Skuja) Anagn. and Komárek (S1) in the Billings reservoir. The environmental factors that most influenced the phytoplankton dynamics were water temperature, euphotic zone, turbidity, conductivity, pH, dissolved oxygen, nitrate and total phosphorous.