On differentiability and analyticity of positive definite functions

We derive a set of differential inequalities for positive definite functions based on previous results derived for positive definite kernels by purely algebraic methods. Our main results show that the global behavior of a smooth positive definite function is, to a large extent, determined solely by the sequence of even-order derivatives at the origin: if a single one of these vanishes then the function is constant; if they are all non-zero and satisfy a natural growth condition, the function is real-analytic and consequently extends holomorphically to a maximal horizontal strip of the complex plane.

Paleomagnetic study of the Great Foum Zguid dyke (southern Morocco): A positive contact test related to metasomatic processes

When a paleomagnetic pole is sought for in an igneous body, the host rocks should be subjected to a contact test to assure that the determined paleopole has the age of the intrusion. If the contact test is positive, it precludes the possibility that the measured magnetization is a later effect. Therefore, we investigated the variations of the remanent magnetization along cross-sections of rocks hosting the Foum Zguid dyke (southern Morocco) and the dyke itself. A positive contact test was obtained, but it is mainly related with Chemical/Crystalline Remanent Magnetization due to metasomatic processes in the host-rocks during magma intrusion and cooling, and not only with Thermo-Remanent Magnetization as ordinarily assumed in standard studies. Paleomagnetic data obtained within the dyke then reflect the Earth magnetic field during emplacement of this well-dated (196.9 +/- 1.8 Ma) intrusion.

Positive Solutions of the Dirichlet Problem for the One-dimensional Minkowski-Curvature Equation

We discuss existence and multiplicity of positive solutions of the Dirichlet problem for the quasilinear ordinary differential equation-(u' / root 1 - u'(2))' = f(t, u). Depending on the behaviour of f = f(t, s) near s = 0, we prove the existence of either one, or two, or three, or infinitely many positive solutions. In general, the positivity of f is not required. All results are obtained by reduction to an equivalent non-singular problem to which variational or topological methods apply in a classical fashion.

Positive solutions of fourth order problems with clamped beam boundary conditions

n this paper we make an exhaustive study of the fourth order linear operator u((4)) + M u coupled with the clamped beam conditions u(0) = u(1) = u'(0) = u'(1) = 0. We obtain the exact values on the real parameter M for which this operator satisfies an anti-maximum principle. Such a property is equivalent to the fact that the related Green's function is nonnegative in [0, 1] x [0, 1]. When M < 0 we obtain the best estimate by means of the spectral theory and for M > 0 we attain the optimal value by studying the oscillation properties of the solutions of the homogeneous equation u((4)) + M u = 0. By using the method of lower and upper solutions we deduce the existence of solutions for nonlinear problems coupled with this boundary conditions. (C) 2011 Elsevier Ltd. All rights reserved.

Relatório de estágio

I - As minhas expectativas eram elevadas pois este regresso à Escola Superior de Música de Lisboa permitia-me voltar a trabalhar com os professores que me formaram como músico e professor e com eles poder actualizar-me sobre vários temas ligados à pedagogia. Este aspecto é muito importante pois chego à conclusão que o tempo por vezes provoca-nos excesso de confiança que parece “cegar-nos” não nos deixando ver erros pedagógicos muitas vezes evitáveis. Quando ingressei neste estágio sentia-me confiante e seguro quanto às minhas capacidades como professor. O momento de viragem na minha perspectiva do estágio dá-se quando surgem as observações/gravações e respectivas análises e reflexões das aulas. Procurei trabalhar nessas aulas da forma mais natural possível pois o meu objectivo era observar o meu trabalho diário. A primeira observação das aulas permitiu-me anotar algumas coisas menos boas. Contudo, quando essa observação foi feita com o professor de didática os aspectos menos positivos ganharam uma enorme proporção: (1) falhas ao nível da instrução: demasiado longo, (2) feedback de pouca qualidade ou eficácia , (3) pouca percentagem de alunos que atingiam os objectivos., (4) ritmo de aula por vezes baixo devido a períodos longos de instrução ou devido a uma má gestão do espaço. Todos estes problemas eram mais visíveis quando as turmas eram maiores. Ao longo do estágio, e após a detecção destas falhas, fui procurando evitar estas práticas em todas as turmas onde leccionava. Senti que o ritmo de aula aumentou substancialmente não apenas à custa da energia do professor e de boas estratégias mas porque sobretudo se “falava menos e trabalhava-se mais”. Os erros dos alunos passaram a ser corrigos enquanto trabalhavam (feedback corretivo próximo do momento positivo ou negativo), o feedback positivo passou a ser mais destacado, a disposição da sala alterou-se de forma aos alunos estarem mais perto do professor, e este procurou ser menos “criativo” no momento de alterar o plano de aula devido a ideias momentâneas o que provocou mais tempo para cada estratégia e para que mais alunos fossem atingindo os objectivos. Apesar da evolução no sentido de proporcionar aos alunos aulas mais rentáveis e de ainda melhor qualidade, existe a consciência que alguns dos erros cometidos eram hábitos e como tal poderão levar algum tempo a ser corrigidos. Contudo, existe a consciência e a vontade em debelá-los da minha prática docente.

Práticas de avaliação formativa na aula de matemática: um estudo no 2º ciclo

Dissertação apresentada à Escola Superior de Educação de Lisboa para obtenção de grau de mestre em Educação Matemática na Educação Pré- Escolar e nos 1º e 2º Ciclo do Ensino Básico

Complex variable positive definite functions

In this paper we develop an appropriate theory of positive definite functions on the complex plane from first principles and show some consequences of positive definiteness for meromorphic functions.

Positive radial solutions of the Dirichlet problem for the Minkowski-Curvature equation in a ball

We study the existence and multiplicity of positive radial solutions of the Dirichlet problem for the Minkowski-curvature equation { -div(del upsilon/root 1-vertical bar del upsilon vertical bar(2)) in B-R, upsilon=0 on partial derivative B-R,B- where B-R is a ball in R-N (N >= 2). According to the behaviour off = f (r, s) near s = 0, we prove the existence of either one, two or three positive solutions. All results are obtained by reduction to an equivalent non-singular one-dimensional problem, to which variational methods can be applied in a standard way.

Estudo sobre as diferenças interativas e comunicativas das educadoras e das mães

Dissertação apresentada na Escola Superior de Educação de Lisboa para obtenção do grau de Mestre em Ciências da Educação - Especialidade Intervenção Precoce

Predictors of infant positive, negative and self-direct coping during face to face still-face in a Portuguese preterm sample

Past studies found three types of infant coping behaviour during Face-to-Face Still-Face paradigm (FFSF): a Positive Other-Directed Coping; a Negative Other-Directed Coping and a Self-Directed Coping. In the present study, we investigated whether those types of coping styles are predicted by: infants’ physiological responses; maternal representations of their infant’s temperament; maternal interactive behaviour in free play; and infant birth and medical status. The sample consisted of 46, healthy, prematurely born infants and their mothers. At one month, infant heart rate was collected in basal. At three months old (corrected age), infant heart-rate was registered during FFSF episodes. Mothers described their infants’ temperament using a validated Portuguese temperament scale, at infants three months of corrected age. As well, maternal interactive behaviour was evaluated during a free play situation using CARE-Index. Our findings indicate that positive coping behaviours were correlated with gestational birth weight, heart rate (HR), gestational age, and maternal sensitivity in free play. Gestational age and maternal sensitivity predicted Positive Other-Direct Coping behaviours. Moreover, Positive Other-Direct coping was negatively correlated with HR during Still-Face Episode. Self-directed behaviours were correlated with HR during Still-Face Episode and Recover Episode and with maternal controlling/intrusive behaviour. However, only maternal behaviour predicted Self-direct coping. Early social responses seem to be affected by infants’ birth status and by maternal interactive behaviour. Therefore, internal and external factors together contribute to infant ability to cope and to re-engage after stressful social events.