Packing triangles in low degree graphs and indifference graphs

We consider the problems of finding the maximum number of vertex-disjoint triangles (VTP) and edge-disjoint triangles (ETP) in a simple graph. Both problems are NP-hard. The algorithm with the best approximation ratio known so far for these problems has ratio 3/2 + epsilon, a result that follows from a more general algorithm for set packing obtained by Hurkens and Schrijver [On the size of systems of sets every t of which have an SDR, with an application to the worst-case ratio of heuristics for packing problems, SIAM J. Discrete Math. 2(1) (1989) 68-72]. We present improvements on the approximation ratio for restricted cases of VTP and ETP that are known to be APX-hard: we give an approximation algorithm for VTP on graphs with maximum degree 4 with ratio slightly less than 1.2, and for ETP on graphs with maximum degree 5 with ratio 4/3. We also present an exact linear-time algorithm for VTP on the class of indifference graphs. (C) 2007 Elsevier B.V. All rights reserved.

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.

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.

Changes in test scores distribution for students of the fourth grade in Brazil: a relative distribution analysis for the years 1997 to 2005

To assess the quality of school education, much of educational research is concerned with comparisons of test scores means or medians. In this paper, we shift this focus and explore test scores data by addressing some often neglected questions. In the case of Brazil, the mean of test scores in Math for students of the fourth grade has declined approximately 0,2 standard deviation in the late 1990s. But what about changes in the distribution of scores? It is unclear whether the decline was caused by deterioration in student performance in upper and/or lower tails of the distribution. To answer this question, we propose the use of the relative distribution method developed by Handcock and Morris (1999). The advantage of this methodology is that it compares two distributions of test scores data through a single distribution and synthesizes all the differences between them. Moreover, it is possible to decompose the total difference between two distributions in a level effect (changes in median) and shape effect (changes in shape of the distribution). We find that the decline of average-test scores is mainly caused by a worsening in the position of all students throughout the distribution of scores and is not only specific to any quantile of distribution.

The impact of structured teaching methods on the quality of education

This paper estimates the impact of the use of structured methods on the quality of education of the students in primary public school in Brazil. Structure methods encompass a range of pedagogical and managerial instruments applied to the education system. In recent years, several municipalities in the State of São Paulo have contracted out private educational providers to implement these structured methods in their schooling system. Their pedagogical proposal involves structuring curriculum contents, elaboration and use of teachers and students textbooks, and training and supervision of the teachers and instructors. Using a difference in differences estimation strategy, we find that the fourth and eighth grader students in the municipalities with structured methods performed better in Portuguese and Math than students in municipalities not exposed to the methods. We find no differences in approval rates. However, a robustness check is not able to discard the possibility that unobserved municipal characteristics may affect the results.