Genericity of Nondegenerate Critical Points and Morse Geodesic Functionals

We consider a family of variational problems on a Hilbert manifold parameterized by an open subset of a Banach manifold, and we discuss the genericity of the nondegeneracy condition for the critical points. Using classical techniques, we prove an abstract genericity result that employs the infinite dimensional Sard-Smale theorem, along the lines of an analogous result of B. White [29]. Applications are given by proving the genericity of metrics without degenerate geodesics between fixed endpoints in general (non compact) semi-Riemannian manifolds, in orthogonally split semi-Riemannian manifolds and in globally hyperbolic Lorentzian manifolds. We discuss the genericity property also in stationary Lorentzian manifolds.

Iteration of closed geodesics in stationary Lorentzian manifolds

Following the lines of Bott in (Commun Pure Appl Math 9:171-206, 1956), we study the Morse index of the iterates of a closed geodesic in stationary Lorentzian manifolds, or, more generally, of a closed Lorentzian geodesic that admits a timelike periodic Jacobi field. Given one such closed geodesic gamma, we prove the existence of a locally constant integer valued map Lambda(gamma) on the unit circle with the property that the Morse index of the iterated gamma(N) is equal, up to a correction term epsilon(gamma) is an element of {0,1}, to the sum of the values of Lambda(gamma) at the N-th roots of unity. The discontinuities of Lambda(gamma) occur at a finite number of points of the unit circle, that are special eigenvalues of the linearized Poincare map of gamma. We discuss some applications of the theory.

Dually discrete spaces

A neighbourhood assignment in a space X is a family O = {O-x: x is an element of X} of open subsets of X such that X is an element of O-x for any x is an element of X. A set Y subset of X is a kernel of O if O(Y) = U{O-x: x is an element of Y} = X. We obtain some new results concerning dually discrete spaces, being those spaces for which every neighbourhood assignment has a discrete kernel. This is a strictly larger class than the class of D-spaces of [E.K. van Douwen, W.F. Pfeffer, Some properties of the Sorgenfrey line and related spaces, Pacific J. Math. 81 (2) (1979) 371-377]. (c) 2008 Elsevier B.V. All rights reserved.

An existence theorem for G-structure preserving affine immersions

We prove an existence result for local and global G-structure preserving affine immersions between affine manifolds. Several examples are discussed in the context of Riemannian and semi-Riemannian geometry, including the case of isometric immersions into Lie groups endowed with a left-invariant metric, and the case of isometric immersions into products of space forms.

Abelian torsion groups with a countably compact group topology

Comfort and Remus [W.W. Comfort, D. Remus, Abelian torsion groups with a pseudo-compact group topology, Forum Math. 6 (3) (1994) 323-337] characterized algebraically the Abelian torsion groups that admit a pseudocompact group topology using the Ulm-Kaplansky invariants. We show, under a condition weaker than the Generalized Continuum Hypothesis, that an Abelian torsion group (of any cardinality) admits a pseudocompact group topology if and only if it admits a countably compact group topology. Dikranjan and Tkachenko [D. Dikranjan. M. Tkachenko, Algebraic structure of small countably compact Abelian groups, Forum Math. 15 (6) (2003) 811-837], and Dikranjan and Shakhmatov [D. Dikranjan. D. Shakhmatov, Forcing hereditarily separable compact-like group topologies on Abelian groups, Topology Appl. 151 (1-3) (2005) 2-54] showed this equivalence for groups of cardinality not greater than 2(c). We also show, from the existence of a selective ultrafilter, that there are countably compact groups without non-trivial convergent sequences of cardinality kappa(omega), for any infinite cardinal kappa. In particular, it is consistent that for every cardinal kappa there are countably compact groups without non-trivial convergent sequences whose weight lambda has countable cofinality and lambda > kappa. (C) 2009 Elsevier B.V. All rights reserved.

Generalized solutions of a nonlinear parabolic equation with generalized functions as initial data

In [H. Brezis, A. Friedman, Nonlinear parabolic equations involving measures as initial conditions, J. Math. Pure Appl. (9) (1983) 73-97.] Brezis and Friedman prove that certain nonlinear parabolic equations, with the delta-measure as initial data, have no solution. However in [J.F. Colombeau, M. Langlais, Generalized solutions of nonlinear parabolic equations with distributions as initial conditions, J. Math. Anal. Appl (1990) 186-196.] Colombeau and Langlais prove that these equations have a unique solution even if the delta-measure is substituted by any Colombeau generalized function of compact support. Here we generalize Colombeau and Langlais` result proving that we may take any generalized function as the initial data. Our approach relies on recent algebraic and topological developments of the theory of Colombeau generalized functions and results from [J. Aragona, Colombeau generalized functions on quasi-regular sets, Publ. Math. Debrecen (2006) 371-399.]. (C) 2009 Elsevier Ltd. All rights reserved.

Group identities on symmetric units

Let F be an infinite field of characteristic different from 2, G a group and * an involution of G extended by linearity to an involution of the group algebra FG. Here we completely characterize the torsion groups G for which the *-symmetric units of FG satisfy a group identity. When * is the classical involution induced from g -> g(-1), g is an element of G, this result was obtained in [ A. Giambruno, S. K. Sehgal, A. Valenti, Symmetric units and group identities, Manuscripta Math. 96 (1998) 443-461]. (C) 2009 Elsevier Inc. All rights reserved.

Olika föreställningar om samband mellan musik och matematik

Mitt syfte är att beskriva olika föreställningar om samverkan mellan musik och matematik samt erfarenheter och resultat av sådan samverkan i pedagogisk verksamhet. Studien är en litteratur-studie över ST Math + Music. Jag har även intervjuat två lärare på en svensk skola. Jag har be-skrivit samverkan genom överspridningseffekter, musikens inneboende matematik och undervis-ning i matematik med hjälp av musik. Resultaten tyder på att det finns såväl föreställningar om som erfarenheter av att integrering av musik och matematik kan ha en gynnsam inverkan när det gäller matematisk förståelse.

A Genetic Algorithm for Solar Boat

Genetic algorithm has been widely used in different areas of optimization problems. Ithas been combined with renewable energy domain, photovoltaic system, in this thesis.To participate and win the solar boat race, a control program is needed and C++ hasbeen chosen for programming. To implement the program, the mathematic model hasbeen built. Besides, the approaches to calculate the boundaries related to conditionhave been explained. Afterward, the processing of the prediction and real time controlfunction are offered. The program has been simulated and the results proved thatgenetic algorithm is helpful to get the good results but it does not improve the resultstoo much since the particularity of the solar driven boat project such as the limitationof energy production

Matematikläxans fördelar och nackdelar : En intervjustudie av lärares syn på matematikläxor i grundskolans tidigare år

Syftet med denna studie är att få kunskap om lärares syn på matematikläxor i grundskolans tidigare år. Förhoppningen var att få förståelse för lärares syfte med matematikläxor och vilka förmågor eleverna förväntas utveckla med matematikläxors hjälp samt att jämföra vilka för- och nackdelar matematikläxan har. För att undersöka detta har sju intervjuer gjorts med lärare från olika skolor och olika årskurser från 1-6. Samtliga lärare som intervjuats undervisar i matematik. Lärares erfarenheter av läxornas betydelse för matematikundervisningen är ett viktigt område att få kunskap om. De lärare som deltog i intervjuerna uttryckte olika synpunkter på matematikläxan. Något lärarna var överens om var att läxa är en skoluppgift som eleven tar med hem efter skolans slut och arbetar med hemma. Lärarna uttryckte samstämmigt att eleverna inte utvecklar några förmågor med matematikläxans hjälp utan förmågorna utvecklar eleverna i skolan med hjälp av lärarna. Det framgick att samtliga lärare upplevde att matematikläxan var tidskrävande och för att matematikläxan ska nå syftet på bästa möjliga sätt behövs mer tid. Att läxan kopplas till undervisning och uppföljning är centralt för att syftet med matematikläxan ska uppnås enligt de intervjuade lärarna. Av resultaten framgår att några lärare är negativt inställda till matematikläxor för att uppföljning och koppling till undervisningen inte finns med och därför förlorar matematikläxan sitt syfte. Föräldrarnas förutsättningar är väldigt olika som exempelvis att de har svårigheter med det svenska språket eller inte är engagerade i barnets skolgång. Andra lärare är positivt inställda till matematikläxor för att färdighetsträningen och repetitionen är viktig och tiden som behövs för det inte räcker till i skolan.

PANA V : Evaluation of a Literacy Project

PANA V Evaluation of a Literacy ProjectSUMMARY AND CONCLUSIONSThis evaluation set out to explore the impact of the literacy work carried out through PANA V. It focussed on clarifying effects such as empowerment and poverty reduction in relation to the civil society. Two specific objectives were to evaluate the methodological approach and the didactic materials and to evaluate the sustainability of the project.Although the focus of the evaluation has been PANA V, the project has been evaluated in its context, as one in a series of five projects located in Rwanda ten years after the war and genocide. The conclusion will consider future plans in this field.The evaluator has striven to create a holistic picture of the effects of the project, although the given time for the evaluation was short. Only three weeks were spent in the field study and only ten days in the actual field. Although there were some organisational and logistic problems, as is common when carrying out a study like this in a poor country, many literacy sites were visited and quite many participators were interviewed. The overall impression from the study is overwhelmingly positive. So many people commit themselves in this task of teaching Rwandans reading, writing and numeracy. Despite harsh conditions learners strive to learn and group leaders devote themselves to the task. Many leaders on different levels try their very best to manage their difficult and demanding task. The main objective was to explore the impact of the project on poverty reduction, particularly on empowerment and strategies for everyday life. Women were to be regarded particularly. From the results it is clear that the project has a strong, positive impact both on poverty reduction and empowerment of marginalised groups. Among those who have benefited from the alphabetisation are mainly women. Unfortunately, when it comes to leaders in PANA, who may also be said to have benefited from the project, only a small minority is women. This is something that is recommended that it be reconsidered inside the organisation. As a majority of the targeted learners are women, and as the economic and social situation of women in Rwanda is generally weak, this is a question that I recommend the Pentesostal church and ADEPR to look particularly into. With many women being single breadwinners of their households, it is important that also women get access to positions that may bring benefits of different kind.It is also clear that the project has positive effects for the civil society. In the present situation in Rwanda, during the process of reconciliation and rapid progress, basic education for the poor majority is a democratic issue. In a country with a plethora of internet-cafés in the capital and a small minority that use cars and mobile-telephones to communicate nation-wide, it is of outmost importance that the majority acquires basic education, of which literacy is a central part. To strengthen the civil society in Rwanda literacy is important. One central issue is then that Rwanda develops toward becoming a country where literacy is used for the benefit of the citizens and it is a democratic issue that all citizens get an opportunity to participate. Crucial for this is that strong efforts are put into primary schools nation-wide. Literacy projects for adults, like PANA, may only complement these efforts, but they constitute important and necessary complements. Other relevant ways to promote literacy are campaigns in Radio and TV and through cultural events such as festivals, music and theatre. News papers, magazines and books are natural parts of such campaigns as well as adult education. As stated under the results not much can be said about the didactics in this evaluation. On the whole the methodology and the materials fill their function well and receive a high reputation. As people learn to read and write under very simple conditions, obviously the approach is appropriate. A few suggestions may be given from the study:•Focus groups leaders’ attention on clearness, that they show very clearly what is to be read. Good structuring is probably of great importance for many learners.•Make clear what is tested in the tests and consider the possibility to use a holistic test that would be more congruent with the methodology. The possibility to use only one grade, pass, would enable a more practical test, such as reading a short, relevant text, writing something relevant and solving practical mathematic problems. Avoid tests that demand school knowledge.•Avoid using methaphors such as “fight against illiteracy” and connections between illiteracy/literacy and darkness/light. It is not true that illiteracy causes bad things and that literacy only brings good. •Be prepared that it may be more difficult in the future to achieve the goals as it may be the case that the early learners where the ones who achieved easily. The goal of “literacy in six month” in PANA will probably hold only for some learners but also those who do not manage in six months need literacy skills.A third objective was to secure sustainability. As for sustainability of the project in itself, and of the literacy process, the main conclusion is that there is a good potential. The commitment and devotedness among many involved in PANA proves good. One weakness is individual leaders in ADEPR who do not see this as an important task for the Pentecostal church in Rwanda. Other weaknesses are the unwillingness to mention explicitly the wish, for example among group leaders, to get some kind of incentive and the fear of loosing believers by cooperation with other organisations. A higher degree of transparency in this issue would probably solve some irritations and tensions.As for the sustainability of the literacy skills much may be done to improve. The acquired skills seem to be comparably relevant. The level achieved, and the level tested, may be defined as basic literacy skills, consisting of basic reading, writing and numeracy skills. However, these skills are very restricted and there is a high risk that the skills will decline, which means that there is a high risk that people will forget how to read and write because of lack of exercising. From these conclusions a few suggestions for future development will be given.

Interview with Jane West

Abstract: In her July 15, 2013 interview with Martha Manning, Jane West details her time at Winthrop as a Biology major in the 1960s. Briefly, West discusses student life and Winthrop traditions, but she provides insight into her life student teaching at Florence High School. Additionally, West includes her experience in required courses like English and Math, as well as experiments conducted in her Biology courses. West concludes the interview with her overall perceptions of Winthrop. This interview was conducted for inclusion into the Louise Pettus Archives and Special Collections Oral History Program.

Medindo valor em comunidades de prática: caso multicurso

Communities of Practice are groups of people who share a concern, problem or passion about a topic, and who deepen their knowledge and expertise by interacting regularly. It¿s use as a tool for planning and implementation of programs for continuous training of teachers in Brazil is not widespread, but used occasionally. This dissertation seeks to demonstrate the use of the communities of practice theory in continuous teachers training in the Brazilian reality. It presents some models of continuous formation currently in process in Brazil, points to the principles of knowledge management taken by this work, presents the communities of practice theory and the criticism of this in its theoretical part. For the case study was used the methodology described by Robert Yin (2005) of single case study to examine the Multicurso, program to continuously train math teachers in the state of Goiás, Brazil. The community of practice formed in this program was examined on its form, as to the cultivation of communities of practice, as to the development and about the challenges of geographical distribution. Going beyond, this work has as its main objective to propose a methodology for measuring the value of communities of practice from the case studied, without the intention to make generalizations about the measurement of value in projects of various kinds. The model proposed indicators separated into two categories, the outcome and impact, to facilitate the measurement of value and prevent that communities of practice are to be criticized or super estimated by assignment of value to those without the unequivocal proof of unique causality at the same time it searchs to attest that the contribution of communities of practice in improvements in the systems in which the CoP are in.

Exames curriculares e resultados educacionais: uma análise do exame nacional do ensino médio

Esta dissertação avalia os efeitos do Exame Nacional do Ensino Médio (ENEM) sobre o ensino médio brasileiro. Esta avaliação baseia-se na teoria de que exames curriculares com a qualidade de gerar conseqüências diretas sobre os alunos, como no caso do ENEM, são capazes de influenciar positivamente o aprendizado dos alunos por dois canais: por aumento de seu esforço e pela pressão que exercem sobre professores e diretores. O presente trabalho avalia especificamente o segundo canal, sobre o qual o ENEM deve atuar por meio da divulgação ao público das notas médias por escola em todo o país. Neste sentido, estima-se o efeito da divulgação das notas do ENEM sobre os seguintes resultados educacionais: proficiência em Matemática e Língua Portuguesa no Saeb, alocação de insumos escolares e comportamento dos professores. Para tanto, dois métodos são propostos: i) diferenças em diferenças, utilizando dados dos Saeb de 2005 – um ano antes do início da divulgação – e 2007 – um ano após a primeira publicação – e ii) regressão com descontinuidade, explorando o fato de haver uma regra estrita para entrada da escola na divulgação – ter ao menos 10 concluintes do ensino médio participando do ENEM. Os resultados mostram que o fato uma escola ter seu resultado médio no ENEM divulgado não parece influenciar o desempenho de seus alunos nem a quantidade de seus insumos escolares, por nenhum dos métodos propostos. O método de diferenças em diferenças aponta ainda que os professores de escolas submetidas ao tratamento dessa divulgação parecem ter maior enfoque em atividades voltadas a resolução de exercícios e fixação de regras gramaticais, reação que é interpretada por parte da literatura como adversa. Estes últimos resultados, no entanto, podem estar refletindo apenas tendências temporais preexistentes e não um efeito de tratamento legítimo. Conclui-se então que o ENEM não deve estar servindo como indutor de melhorias dos resultados educacionais via pressão dos alunos do ensino médio. Porém, é possível levantar a hipótese de que este tipo de efeito ainda pode ser captado em prazos mais longos ou para grupos de escolas diferentes das usadas na análise.