Questions about linear spaces

AbstractWe present three themes of interest for future research that require the cooperation of fairly large teams: 1.linear spaces as building blocks;2.data for an Atlas of linear spaces;3.morphisms of linear spaces

Spherical subcomplexes of spherical buildings

Let B be a thick spherical building equipped with its natural CAT(1) metric and let M be a proper, convex subset of B. If M is open or if M is a closed ball of radius pi/2, then the maximal subcomplex supported by the complement of M is spherical and non contractible.Comment: 33 pages, 7 figure

Classification of flag-transitive Steiner quadruple systems

A Steiner quadruple system of order v is a 3-(v,4,1) design, and will be denoted SQS(v). Using the classification of finite 2-transitive permutation groups all SQS(v) with a flag-transitive automorphism group are completely classified, thus solving the "still open and longstanding problem of classifying all flag-transitive 3-(v,k,1) designs" for the smallest value of k. Moreover, a generalization of a result of H. Lueneburg (1965, Math. Z. 89, 82-90) is achieved.Comment: 11 page

Working memory assessment in older adults: validation and norming studies of the month ordering task

Dissertação de mestrado em Psicologia Clínica e da Saúde (Psicogerontologia Clínica), apresentada à Faculdade de Psicologia e de Ciências da Educação da Universidade de CoimbraThe present work aimed to obtain construct-related evidence for the Month Ordering Task and to examine its reliability (test-retest and internal consistency). We equally sought to establish normative data for use with speakers of European Portuguese. More specifically, we intended to: 1) attain convergent and discriminant validity evidence for the Month Ordering Task, studying the correlations between this task and other working memory tasks (Reading Span Task and Digit Span Backward) and between the Month Ordering Task and measures of constructs less related to working memory (Digit Span Forward, and measures of inhibition and processing speed derived from the Stroop Neuropsychological Screening Test); 2) analyze the Month Ordering Task’s reliability; test-retest reliability was investigated by assessing 40 participants on two different occasions (with a 12-14 week interval) and internal consistency was checked by estimating Cronbach’s alpha; 3) establish regression-based norms for use with speakers of European Portuguese. The Month Ordering Task revealed to possess sound psychometric properties, strongly correlating with other measures of working memory and presenting non-significant to moderate correlations with measures of less related constructs (except for the Digit Span Forward, with which a strong correlation was unveiled). A high retest coefficient was obtained, suggesting that the Month Ordering Task is a temporally stable instrument. The internal consistency study revealed that the Month Ordering Task is an internally consistent and homogeneous scale. Linear regression analyses showed that Month Ordering Task performance is influenced by age, gender and years of formal education. In order to control for these influences, a regression-based algorithm was obtained, enabling the user to transform Month Ordering Task scores into standardized Z scores.Este trabalho teve como objetivo a análise das propriedades psicométricas da Tarefa de Ordenação de Meses e o estabelecimento de normas baseadas na regressão para uso com indivíduos cuja língua materna seja o português europeu. Mais especificamente, os objetivos desta tese são os seguintes: 1) análise da validade convergente e discriminante da Tarefa de Ordenação de Meses, através do estudo das correlações com outros instrumentos de avaliação da memória de trabalho (Tarefa de Amplitude de Leitura e Memória de Dígitos em sentido inverso) e com instrumentos que avaliam construtos menos relacionados com a memória de trabalho (Memória de Dígitos em sentido direto, e medidas de inibição e de velocidade de processamento obtidas através do Teste Stroop Neuropsicológico); 2) estudo da fiabilidade da Tarefa de Ordenação de Meses, nomeadamente a fiabilidade teste-reteste e a consistência interna; de forma a examinar a fiabilidade teste-reteste, uma amostra de 40 participantes foi avaliada duas vezes em momentos distintos (separados por um intervalo de 12-14 semanas), e a consistência interna foi analisada recorrendo ao índice alfa de Cronbach; 3) por último, procuramos obter dados normativos para falantes do português europeu. Os resultados obtidos sugerem que a Tarefa de Ordenação de Meses possui boas propriedades psicométricas, apresentando fortes correlações com outras medidas de memória de trabalho e correlações fracas a moderadas com instrumentos que visam a avaliação de construtos distintos (com exceção da Memória de Dígitos em sentido direto, tendo sido obtido um coeficiente de correlação elevado). A Tarefa de Ordenação de Meses mostra ter uma boa estabilidade temporal, demonstrando também ser uma escala homogénea com boa consistência interna. Análises de regressão lineares revelaram que o desempenho na Tarefa de Ordenação de Meses é influenciado por diversas variáveis demográficas (idade, género, anos de escolaridade). De forma a poder controlar o impacto destas variáveis demográficas na avaliação da memória de trabalho através do desempenho na Tarefa de Ordenação de Meses, foram obtidas normas baseadas na regressão. Esse algoritmo permite a transformação de resultados brutos em resultados standardizados

Regular Incidence Complexes, Polytopes, and C-Groups

Regular incidence complexes are combinatorial incidence structures generalizing regular convex polytopes, regular complex polytopes, various types of incidence geometries, and many other highly symmetric objects. The special case of abstract regular polytopes has been well-studied. The paper describes the combinatorial structure of a regular incidence complex in terms of a system of distinguished generating subgroups of its automorphism group or a flag-transitive subgroup. Then the groups admitting a flag-transitive action on an incidence complex are characterized as generalized string C-groups. Further, extensions of regular incidence complexes are studied, and certain incidence complexes particularly close to abstract polytopes, called abstract polytope complexes, are investigated.Comment: 24 pages; to appear in "Discrete Geometry and Symmetry", M. Conder, A. Deza, and A. Ivic Weiss (eds), Springe

A Quasi Curtis-Tits-Phan theorem for the symplectic group

We obtain the symplectic group \SP(V) as the universal completion of an amalgam of low rank subgroups akin to Levi components. We let \SP(V) act flag-transitively on the geometry of maximal rank subspaces of $V$. We show that this geometry and its rank $\ge 3$ residues are simply connected with few exceptions. The main exceptional residue is described in some detail. The amalgamation result is then obtained by applying Tits' lemma. This provides a new way of recognizing the symplectic groups from a small collection of small subgroups

A Census Of Highly Symmetric Combinatorial Designs

As a consequence of the classification of the finite simple groups, it has been possible in recent years to characterize Steiner t-designs, that is t-(v,k,1) designs, mainly for t = 2, admitting groups of automorphisms with sufficiently strong symmetry properties. However, despite the finite simple group classification, for Steiner t-designs with t > 2 most of these characterizations have remained longstanding challenging problems. Especially, the determination of all flag-transitive Steiner t-designs with 2 < t < 7 is of particular interest and has been open for about 40 years (cf. [11, p. 147] and [12, p. 273], but presumably dating back to 1965). The present paper continues the author's work [20, 21, 22] of classifying all flag-transitive Steiner 3-designs and 4-designs. We give a complete classification of all flag-transitive Steiner 5-designs and prove furthermore that there are no non-trivial flag-transitive Steiner 6-designs. Both results rely on the classification of the finite 3-homogeneous permutation groups. Moreover, we survey some of the most general results on highly symmetric Steiner t-designs.Comment: 26 pages; to appear in: "Journal of Algebraic Combinatorics

An alternative construction of B-M and B-T unitals in Desarguesian planes

We present a new construction of non-classical unitals from a classical unital $U$ in $PG(2,q^2)$. The resulting non-classical unitals are B-M unitals. The idea is to find a non-standard model $\Pi$ of $PG(2,q^2)$ with the following three properties: 1. points of $\Pi$ are those of $PG(2,q^2)$; 2. lines of $\Pi$ are certain lines and conics of $PG(2,q^2)$; 3. the points in $U$ form a non-classical B-M unital in $\Pi$. Our construction also works for the B-T unital, provided that conics are replaced by certain algebraic curves of higher degree.Comment: Keywords: unital, desarguesian plane 11 pages; ISSN: 0012-365

Representing some non-representable matroids

We extend the notion of representation of a matroid to algebraic structures that we call skew partial fields. Our definition of such representations extends Tutte's definition, using chain groups. We show how such representations behave under duality and minors, we extend Tutte's representability criterion to this new class, and we study the generator matrices of the chain groups. An example shows that the class of matroids representable over a skew partial field properly contains the class of matroids representable over a skew field. Next, we show that every multilinear representation of a matroid can be seen as a representation over a skew partial field. Finally we study a class of matroids called quaternionic unimodular. We prove a generalization of the Matrix Tree theorem for this class.Comment: 29 pages, 2 figure