Completion problems for real matrices, II

Thirty years ago, G.N. de Oliveira has proposed the following completion problems: Describe the possible characteristic polynomials of [C-ij], i,j is an element of {1, 2}, where C-1,C-1 and C-2,C-2 are square submatrices, when some of the blocks C-ij are fixed and the others vary. Several of these problems remain unsolved. This paper gives the solution, over the field of real numbers, of Oliveira's problem where the blocks C-1,C-1, C-2,C-2 are fixed and the others vary.

On the monoids of transformations that preserve the order and a uniform partition

In this article we consider the monoid O(mxn) of all order-preserving full transformations on a chain with mn elements that preserve a uniformm-partition and its submonoids O(mxn)(+) and O(mxn)(-) of all extensive transformations and of all co-extensive transformations, respectively. We determine their ranks and construct a bilateral semidirect product decomposition of O(mxn) in terms of O(mxn)(-) and O(mxn)(+).

Unified min-max and interlacing theorems for linear operators

There exist striking analogies in the behaviour of eigenvalues of Hermitian compact operators, singular values of compact operators and invariant factors of homomorphisms of modules over principal ideal domains, namely diagonalization theorems, interlacing inequalities and Courant-Fischer type formulae. Carlson and Sa [D. Carlson and E.M. Sa, Generalized minimax and interlacing inequalities, Linear Multilinear Algebra 15 (1984) pp. 77-103.] introduced an abstract structure, the s-space, where they proved unified versions of these theorems in the finite-dimensional case. We show that this unification can be done using modular lattices with Goldie dimension, which have a natural structure of s-space in the finite-dimensional case, and extend the unification to the countable-dimensional case.

A qualidade das salas de berçários nos concelhos de Setubal e de Palmela

Tendo por principal objetivo contribuir para o estudo da qualidade das creches dos Concelhos de Setúbal e de Palmela, com este trabalho pretende-se (1) avaliar e descrever a qualidade de 30 salas de berçário através da aplicação da Escala de Avaliação do Ambiente de Creche – Edição Revista (Infant/Toddler Environment Rating Scale – Revised Edition [ITERS-R], Harms, Cryer, & Clifford, 2006), (2) avaliar e descrever as práticas de socialização dos prestadores de cuidados responsáveis pelas 30 salas de berçário através da aplicação da Escala de Interação do Prestador de Cuidados (Caregiver Interaction Scale [CIS], Arnett, 1989), (3) estudar a associação entre as práticas de socialização dos profissionais e a qualidade das salas de berçário, (4) estudar associações entre a qualidade das salas de berçário e as práticas de socialização dos profissionais e algumas variáveis de estrutura (rácio adulto/criança, tamanho do grupo, educação, formação e experiência dos prestadores de cuidados). Tendo em conta que os estudos efetuados em Portugal revelam qualidade inadequada e mínima nas salas de creche, os resultados obtidos no presente estudo sugerem que as salas observadas possuem qualidade entre o mínimo e o bom. Embora o presente estudo apresente resultados superiores, verificou-se que os itens que apresentam maiores fragilidades (áreas deficitárias), assim como os itens que expressam maior qualidade (áreas fortes) parecem convergir com os resultados das investigações já efetuadas, com efeito os itens relacionados com as subescalas Atividades e Rotinas de cuidados pessoais são os que apresentam os resultados mais preocupantes. A subescala Interação foi a que apresentou resultados superiores e as práticas de socialização dos prestadores de cuidados revelaram ser excelentes. Verificou-se, igualmente, que as salas orientadas por educadores de infância, assim como, as salas em que o principal prestador de cuidados era educador de infância revelaram qualidade superior do que salas orientadas por assistentes operacionais (auxiliares de ação educativa). Este efeito chama a atenção para a importância da formação e qualificação dos profissionais de educação, assim como, para a necessidade de se alterarem as políticas educativas, nomeadamente no que diz respeito à obrigatoriedade de existirem educadores de infância nas salas de berçário. - ABSTRACT With the main objective of contributing to the study of quality of infant/toddler center-based child care in the Counties of Setúbal and Palmela, this work aims to (1) assess and describe the quality of 30 infant classrooms by applying the Infant/Toddler Environment Rating Scale-Revised Edition ([ITERS-R], Harms, Cryer, & Clifford, 2006), (2) assess and describe the socialization practices of caregivers responsible for 30 infant classrooms by applying the Caregiver Interaction Scale ([CIS], Arnett, 1989), (3) study the association between the socialization practices of professionals and the quality of infant classrooms, (4) to study associations between the quality of infant classrooms and the socialization practices of professionals and some structure variables (ratio adult/child group size, education, training, and experience of carers). Considering that the studies carried out in Portugal reveal minimum or inadequate quality in infant/toddler center-based child care, the results of this study suggest that, in accordance with ITERS-R, the quality of the infant classrooms observed varies between minimum and good. Although the present study reports higher quality scores, it was found that items which present the biggest weaknesses (deficit areas) as well as the items which express higher quality (strengths) appear to converge with the results of the investigations already carried out, since the items related with subscales Activities and Personal Care Routines are those with the most worrisome results. The Interaction subscale from ITERS-R proved to be the one with higher scores, and socialization practices of caregivers proved to be excellent. It was also found that the rooms oriented by early childhood teachers, as well as the rooms in which the primary care provider was also guided by childhood teachers showed higher quality than when guided by operational assistants (educational activities assistants). This effect draws attention to the importance of the education professionals training and qualification, as well as the need for altering the educational policies, in particular as regards the obligation to ensure the existence of early childhood teachers in infant classrooms.

Avaliação e promoção da qualidade dos cuidados prestados em creche familiar: primeiros resultados de um estudo de investigação/acção

Neste artigo apresentam-se os resultados do projecto de investigação/acção Promoção da qualidade dos cuidados prestados em amas e creche familiar (resposta enquadrada pelo Decreto-Lei n.º158/84). É um estudo exploratório, de carácter longitudinal, que teve como principais objectivos: (1) avaliar a qualidade da resposta creche familiar em duas instituições do Distrito de Lisboa; (2) analisar se variáveis como a idade, nível de escolaridade e tempo de experiência das amas, o rácio TE/ama e a idade das crianças estavam associadas à qualidade de cuidados prestada; (3) sistematizar os passos de um programa de promoção da qualidade baseado numa metodologia de consultoria; e (4) determinar os efeitos do programa de consultoria na promoção da qualidade. Participaram neste projecto 10 amas enquadradas na creche familiar de um centro infantil do Centro Distrital de Segurança Social de Lisboa, 21 amas de centros infantis da Santa Casa da Misericórdia de Lisboa e os Técnicos de Enquadramento responsáveis pelo apoio técnico. A Family Child Care Environment Rating Scale, Revised Edition (FCCERS-R) de Harms, Cryer e Clifford (2007), traduzida pela equipa de investigação, foi o principal instrumento utilizado. A recolha de dados decorreu em três momentos: (1) avaliação inicial; (2) reunião de consultoria; e (3) avaliação final.Os resultados encontrados mostram que a qualidade dos cuidados prestados pelas amas é adequada/mínima, com excepção dos itens relacionados com a interacção ama-criança onde a qualidade é boa, sendo a subescala actividades aquela onde foram encontrados valores mais baixos. Não foram encontradas associações significativas entre a qualidade e o tempo experiência das amas. Relativamente às restantes variáveis – idade das amas, rácio TE/ama e idade das crianças – encontraram-se algumas associações com a qualidade global e em algumas das subescalas da FCCERS-R. Foram encontradas diferenças entre os dados da 1ª e da 2ª observação que parecem poder associar-se ao processo de consultoria implementado.

O pensamento algébrico e a descoberta de padrões na formação de professores

Este texto incide sobre o tema do pensamento algébrico nos primeiros anos de escolaridade ao nível da formação contínua de professores. Na abordagem apresentada o tema dos padrões surge como contexto estruturante para o desenvolvimento do pensamento algébrico. Começa-se por fazer um enquadramento dos referenciais teóricos que fundamentam a adoção de uma proposta didática desenvolvida nesse âmbito, recorrendo em seguida a um estudo empírico que foi desenvolvido, do qual se relata aqui uma parte centrada nas práticas de sala de aula de uma professora do primeiro ciclo em formação, de modo a poder confirmar a eficácia da proposta no ensino e desenvolvimento do pensamento algébrico dos alunos. Os resultados permitem concluir que a professora interiorizou e projetou na sua prática os aspetos essenciais que conduziram ao desenvolvimento do pensamento algébrico nos alunos. O texto termina com algumas conclusões e implicações para a formação de professores.

Modeling for control design of a multi-reach water canal

This article addresses the problem of obtaining reduced complexity models of multi-reach water delivery canals that are suitable for robust and linear parameter varying (LPV) control design. In the first stage, by applying a method known from the literature, a finite dimensional rational transfer function of a priori defined order is obtained for each canal reach by linearizing the Saint-Venant equations. Then, by using block diagrams algebra, these different models are combined with linearized gate models in order to obtain the overall canal model. In what concerns the control design objectives, this approach has the advantages of providing a model with prescribed order and to quantify the high frequency uncertainty due to model approximation. A case study with a 3-reach canal is presented, and the resulting model is compared with experimental data. © 2014 IEEE.

Modeling for control design of a multi-reach water canal

This article addresses the problem of obtaining reduced complexity models of multi-reach water delivery canals that are suitable for robust and linear parameter varying (LPV) control design. In the first stage, by applying a method known from the literature, a finite dimensional rational transfer function of a priori defined order is obtained for each canal reach by linearizing the Saint-Venant equations. Then, by using block diagrams algebra, these different models are combined with linearized gate models in order to obtain the overall canal model. In what concerns the control design objectives, this approach has the advantages of providing a model with prescribed order and to quantify the high frequency uncertainty due to model approximation. A case study with a 3-reach canal is presented, and the resulting model is compared with experimental data. © 2014 IEEE.

On the ranks of certain monoids of transformations that preserve a uniform partition

The rank of a semigroup, an important and relevant concept in Semigroup Theory, is the cardinality of a least-size generating set. Semigroups of transformations that preserve or reverse the order or the orientation as well as semigroups of transformations preserving an equivalence relation have been widely studied over the past decades by many authors. The purpose of this article is to compute the ranks of the monoid

Spectral and weak polynomial completeness for the porduct of nonsingular matrices

Let F be a field with at least four elements. In this paper, we identify all the pairs (A, B) of n x n nonsingular matrices over F , satisfying the following property: for every monic polynomial f(x) = xn + an-1xn-1 + … +a1x + aο over F, with a root in F and aο = (-1)n det(AB), there are nonsingular matrices X, Y ϵ Fnxn such that X A X-1 Y BY-1 has characteristic polynomial f (x). © 2014 © 2014 Taylor & Francis.

The K-Th derivatives of the immanant and the X-symmetric power of an operator

In recent papers, formulas are obtained for directional derivatives, of all orders, of the determinant, the permanent, the m-th compound map and the m-th induced power map. This paper generalizes these results for immanants and for other symmetric powers of a matrix.

The norm of the K- Th derivative of the X-symmetric power of an operator

In this paper, the exact value for the norm of directional derivatives, of all orders, for symmetric tensor powers of operators on finite dimensional vector spaces is presented. Using this result, an upper bound for the norm of all directional derivatives of immanants is obtained.

Crossing balanced and stair nested desings

Balanced nesting is the most usual form of nesting and originates, when used singly or with crossing of such sub-models, orthogonal models. In balanced nesting we are forced to divide repeatedly the plots and we have few degrees of freedom for the first levels. If we apply stair nesting we will have plots all of the same size rendering the designs easier to apply. The stair nested designs are a valid alternative for the balanced nested designs because we can work with fewer observations, the amount of information for the different factors is more evenly distributed and we obtain good results. The inference for models with balanced nesting is already well studied. For models with stair nesting it is easy to carry out inference because it is very similar to that for balanced nesting. Furthermore stair nested designs being unbalanced have an orthogonal structure. Other alternative to the balanced nesting is the staggered nesting that is the most popular unbalanced nested design which also has the advantage of requiring fewer observations. However staggered nested designs are not orthogonal, unlike the stair nested designs. In this work we start with the algebraic structure of the balanced, the stair and the staggered nested designs and we finish with the structure of the cross between balanced and stair nested designs.

Periodic paths on nonautonomous graphs

We define nonautonomous graphs as a class of dynamic graphs in discrete time whose time-dependence consists in connecting or disconnecting edges. We study periodic paths in these graphs, and the associated zeta functions. Based on the analytic properties of these zeta functions we obtain explicit formulae for the number of n-periodic paths, as the sum of the nth powers of some specific algebraic numbers.

On derivatives and norms of generalized matrix functions and respective symmetric powers

In recent papers, the authors obtained formulas for directional derivatives of all orders, of the immanant and of the m-th xi-symmetric tensor power of an operator and a matrix, when xi is a character of the full symmetric group. The operator norm of these derivatives was also calculated. In this paper, similar results are established for generalized matrix functions and for every symmetric tensor power.