The disturbing function for polar Centaurs and transneptunian objects

The classical disturbing function of the three-body problem is based on an expansion of the gravitational interaction in the vicinity of nearly coplanar orbits. Consequently, it is not suitable for the identification and study of resonances of the Centaurs and transneptunian objects on nearly polar orbits with the solar system planets. Here, we provide a series expansion algorithm of the gravitational interaction in the vicinity of polar orbits and produce explicitly the disturbing function to fourth order in eccentricity and inclination cosine. The properties of the polar series differ significantly from those of the classical disturbing function: the polar series can model any resonance as the expansion order is not related to the resonance order. The powers of eccentricity and inclination of the force amplitude of a $p$:$q$ resonance do not depend on the value of the resonance order $|p-q|$ but only on its parity. Thus all even resonance order eccentricity amplitudes are $\propto e^2$ and odd ones $\propto e$ to lowest order in eccentricity $e$. With the new findings on the structure of the polar disturbing function and the possible resonant critical arguments, we illustrate the dynamics of the polar resonances 1:3, 3:1, 2:9 and 7:9 where transneptunian object 471325 could currently be locked.Comment: 18 pages, 9 figures, 7 tables. Accepted for publication in Monthly Notices of the Royal Astronomical Societ

A numerical investigation of coorbital stability and libration in three dimensions

Motivated by the dynamics of resonance capture, we study numerically the coorbital resonance for inclination180 >=I>=0 in the circular restricted three-body problem. We examine the similarities and differences between planar and three dimensional coorbital resonance capture and seek their origin in the stability of coorbital motion at arbitrary inclination. After we present stability maps of the planar prograde and retrograde coorbital resonances, we characterize the new coorbital modes in three dimensions. We see that retrograde mode I (R1) and mode II (R2) persist as we change the relative inclination, while retrograde mode III (R3) seems to exist only in the planar problem. A new coorbital mode (R4) appears in 3D which is a retrograde analogue to an horseshoe-orbit. The Kozai-Lidov resonance is active for retrograde orbits as well as prograde orbits and plays a key role in coorbital resonance capture. Stable coorbital modes exist at all inclinations, including retrograde and polar obits. This result confirms the robustness the coorbital resonance at large inclination and encourages the search for retrograde coorbital companions of the solar system's planets.Comment: accepted for publication in Celestial Mechanics and Dynamical Astronom

Communication in the classroom: Practice and reflection of a mathematics teacher

This paper discusses the conceptions, practices and reflections about practices of a mathematics teacher, Maria, with respect to classroom communication and their change during the activity of a collaborative project involving a researcher and two other mathematics teachers. The case study of this teacher, who teaches at grades 5-6, draws on interviews and participant observation of the collaborative project meetings. The results show the relevance of the project to develop the teacher’s understanding of communication issues in her classroom, putting her practices under scrutiny, and developing richer communication processes between her and her students

Neotropical Cerambycidae (Coleoptera) primarily in the Canadian Museum of Nature, Ottawa. I. Falsamblesthiini (Lamiinae)

The following new species are described: Nyctonympha andersoni, sp. n., and N. howdenarum, sp. n., both from Colombia; N. genieri, sp. n., from Ecuador; N. taeniata, sp. n., from Trinidad; Falsamblesthis microps, sp. n., from Venezuela; Bactriola circundata, sp. n., from Brazil (Rio de Janeiro); B. maculata, sp. n., from Venezuela and Ecuador; and B. falsa, sp. n., from Brazil (Minas Gerais to Rio Grande do Sul). A redescription of Bactriola vittulata Bates, 1886, herein designated as the type species of the genus, is provided. Accurate data on the occurrence of Saepiseuthes chilensis Thomson, 1868, in Chile are given. Keys to the species of Bactriola Bates, 1886 and Nyctonympha Thomson, 1868 are added

A utilização do geoplano no ensino e aprendizagem da geometria. Uma experiência com alunos do 4.º ano do Ensino Básico

O presente artigo retrata parte de uma investigação levada a cabo numa turma de 4.º ano do 1.º ciclo do ensino básico, a propósito da utilização de materiais manipuláveis no ensino e aprendizagem da matemática. Esta investigação, de caráter qualitativo, foi desenvolvida segundo algumas caraterísticas da metodologia de investigação-ação. Ao longo do artigo apresenta-se os resultados obtidos em duas sessões concretizadas em torno dos conceitos de área e perímetro com recurso ao geoplano. A análise das sessões em causa terá por base as seguintes questões de investigação: Como foi utilizado o geoplano ao longo da realização das atividades propostas? Quais as dificuldades sentidas pelos alunos quando resolviam a tarefa com recurso ao geoplano? De um modo geral, a utilização do geoplano funcionou como uma base concreta e dinâmica para a (re)construção dos conceitos de área e perímetro, assim como para o desenvolvimento de capacidades de visualização espacial. As dificuldades mais notórias foram na transferência das figuras do geoplano para o papel ponteado e vice-versa, na distinção entre os comprimentos do lado e da diagonal de um quadrado formado no geoplano, bem como na determinação das medidas dos lados de uma figura como sendo a distância entre os pregos do geoplano.Este trabalho é financiado por Fundos FEDER através do Programa Operacional Factores de Competitividade – COMPETE e por Fundos Nacionais através da FCT-Fundação para a Ciência e a Tecnologia no âmbito do projeto “FCOMP-01-0124-FEDER-041405 (Ref. FCT, EXPL/MHCCED/0645/2013)”.info:eu-repo/semantics/publishedVersio

Diatom assemblages in Portuguese temporary rivers

During a survey of benthic diatoms in watercourses from Portugal in spring and summer 2006 and 2007, 76 sites in temporary streams (according to the National Water Institute classification) have been studied. These streams belong to the Ribeiras do Algarve (10 sites), Guadiana (19 sites), Mira (7 sites), Sado (16 sites), Tejo (20 sites), and Douro (4 sites) watersheds. Among these 76 sites, the National Water Institute classified only 48 as reference sites, and only these were therefore considered in the statistical analysis performed. A total of 276 diatom taxa have been identified in the 48 reference sites, from which 112 were present in abundance above 2% in at least one inventory. The most frequent taxa, present in more than 50% of the studied sites, were Achnanthidium minutissimum (Kützing) Czarnecki, Amphora pediculus (Kützing) Grunow, Cocconeis euglypta Ehrenberg, Eolimna minima (Grunow) Lange-Bertalot, Gomphonema parvulum Kützing, G. rosenstockianum Lange-Bertalot & E. Reichardt, Navicula gregaria Donkin, N. veneta Kützing, Nitzschia inconspicua Grunow and Planothidium frequentissimum (Lange-Bertalot) Lange-Bertalot. These are neutrophilous to alcaliphilous, mesosaprobous to -meso-saprobous, eutraphentic to indifferent taxa. Furthermore, a multivariate analysis relating diatom assemblages with environmental parameters is presented and the ecological preferences of not yet well known taxa, such as G. rosenstockianum, are provided

‘Managing pieces of a personal puzzle’ — older people’s experiences of self-management falls prevention exercise guided by a digital program or a booklet

Background: Exercise is effective in order to prevent falls in community-dwelling older people. Self-management programs have the potential to increase access and reduce costs related to exercise-based fall prevention. However, information regarding older people's views of participating in such programs is needed to support implementation. The aim of this study was to explore older people's experiences of a self-management fall prevention exercise routine guided either by a digital program (web-based or mobile) or a paper booklet. Methods: This qualitative study was part of a feasibility study exploring two completely self-managed exercise interventions in which the participants tailored their own program, guided either by a digital program or a paper booklet. Individual face-to-face semi-structured interviews were conducted with a purposeful sample of 28 participants (18 women), mean age 76yrs. Qualitative content analysis was used to analyse the data. Results: Self-managing and self-tailoring these exercise programs was experienced as Managing pieces of a personal puzzle'. To independently being able to create a program and manage exercise was described in the categories Finding my own level' and Programming it into my life'. The participants experienced the flexibility and independence provided by completely self-managed exercise as positive and constructive although it required discipline. Furthermore, different needs and preferences when managing their exercise were described, as well as varying sources of motivation for doing the exercise, as highlighted in the category Defining my source of motivation'. The category Evolving my acquired knowledge' captures the participants' views of building their competence and strategies for maintenance of the exercise. It describes a combined process of learning the program and developing reflection, which was more clearly articulated by participants using the digital program. Conclusions: This study provides new knowledge regarding experiences, preferences and motivations of older people to engage in home-based self-managed fall prevention exercise. They expressed both a capability and willingness to independently manage their exercise. A digital program seems to have strengthened the feeling of support while creating their own exercise program and tailoring it to their preferences and circumstances, which might therefore create better opportunities for adoption and adherence in the long term

Critical dialogues: slow readings of English literary texts

The reader will find gathered together in this volume a selection of articles and essays that have been separately published over the past three decades as a result of my teaching practice and research activity. In chronological terms, they span a period from 1985 until May 2008. These texts were chosen because they are considered to be representative of the type of work I usually do in my classes, undoubtedly as a result of my own academic background, based on a deliberate and specific form of approach to literary texts, my own choice of English authors and, lastly, my most recent interest in inter-art studies.Fundação para a Ciência e a Tecnologi