Boundary of the rauzy fractal sets in R x C generated by P(x)=x(4)-x(3)-x(2)-x-1

Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)

Kinematics of a spacetime with an infinite cosmological constant

A solution of the sourceless Einstein's equation with an infinite value for the cosmological constant L is discussed by using Inonu-Wigner contractions of the de Sitter groups and spaces. When Lambda --> infinity, spacetime becomes a four-dimensional cone, dual to Minkowski space by a spacetime inversion. This inversion relates the four-cone vertex to the infinity of Minkowski space, and the four-cone infinity to the Minkowski light-cone. The non-relativistic limit c --> infinity. is further considered, the kinematical group in this case being a modified Galilei group in which the space and time translations are replaced by the non-relativistic limits of the corresponding proper conformal transformations. This group presents the same abstract Lie algebra as the Galilei group and can be named the conformal Galilei group. The results may be of interest to the early Universe Cosmology.

The nonlinear essence of gravitational waves

A critical review of gravitational wave theory is made. It is pointed out that the usual linear approach to the gravitational wave theory is neither conceptually consistent nor mathematically justified. Relying upon that analysis it is argued that-analogously to a Yang-Mills propagating field, which must be nonlinear to carry its gauge charge-a gravitational wave must necessarily be nonlinear to transport its own charge-that is, energy-momentum.

Free actions of abelian p-groups on the n-torus

In this work we make some contributions to the theory of actions of abelian p-groups on the n-Torus T-n. Set congruent to Z(pk1)(h1) x Z(pk2)(h2) x...x Z(pkr)(hr), r >= 1, k(1) >= k(2) >=...>= k(r) >= 1, p prime. Suppose that the group H acts freely on T-n and the induced representation on pi(1)(T-n) congruent to Z(n) is faithful and has first Betti number b. We show that the numbers n, p, b, k(i) and h(i) (i = 1,..,r) satisfy some relation. In particular, when H congruent to Z(p)(h), the minimum value of n is phi(p) + b when b >= 1. Also when H congruent to Z(pk1) x Z(p) the minimum value of n is phi(p(k1)) + p - 1 + b for b >= 1. Here phi denotes the Euler function.

On the hardy space H¹ on products of half-spaces

We show that the Hardy space H¹ anal (R2+ x R2+) can be identified with the class of functions f such that f and all its double and partial Hubert transforms Hk f belong to L¹ (R2). A basic tool used in the proof is the bisubharmonicity of |F|q, where F is a vector field that satisfies a generalized conjugate system of Cauchy-Riemann type.

COMPARATIVE COLONY PRODUCTIVITY OF POLISTES-SIMILLIMUS AND POLISTES-VERSICOLOR (HYMENOPTERA, VESPIDAE) AND THE EVOLUTION OF PARAGYNY IN THE POLISTINAE

Over a 3-year period, all colony foundations of the social wasps Polistes versicolor and Polistes simillimus were registered, and the fate and growth of all colonies were followed. P. simillimus exhibited a greater number of colony-founding attempts, while P. versicolor had a larger number of adult colonies. P. simillimus had greater cell numbers and number of adults produced per colony. P. simillimus reutilized only a small percentage of brood cells for adult production for up to 2 generations, while P. versicolor reutilized a large percentage of brood cells for up to 3 generations. Consequently, cell production was higher in P. simillimus. Because of a high rate of adult production and extensive cell production, we suggest that P. simillimus may demonstrate paragynous social organization, and may demonstrate an intermediate form between polygynous and monogynous Vespidae. Paragynous associations may lead to lower predation pressures and a relative independence of adult production on time.

DIFFERENT MEDIA, DIFFERENT TYPES OF COLLECTIVE WORK IN ONLINE CONTINUING TEACHER EDUCATION: WOULD YOU PASS THE PEN, PLEASE?

This paper presents some findings regarding the interaction between different computer interfaces and different types of collective work. We want to claim that design in online learning environments has a paramount role in the type of collaboration that happens among participants. In this paper, we report on data that illustrate how teachers can collaborate online in order to learn how to use geometry software in teaching activities. A virtual environment which allows that construction to be carried out collectively, even if the participants are not sharing a classroom, is the setting for the research presented in this paper.

Phenomenological perspective on mathematics education

This is a philosophical essay on a phenomenological way to understand and to work out Mathematics Education. Its philosophical grounding is the Husserlian work, focusing on its key word "going to the things themselves" in order to keep us away from the theoretical educational truth, took as the unique one. We assume the attitude of being on the life-world with the students and Mathematics as a field of research and practice that show and express themselves through lived experiences and through language. We assume to be in search of understanding of education, learning and Mathematics, as we take care, consciously, of what we are doing and saying in the same movement of saying and doing it.

O modelo de Romberg e o percurso metodológico de uma pesquisa qualitativa em educação de uma pesquisa qualitativa em educação

The objective of this study is to present the trajectory of a research project (ALLEVATO, 2005) whose phenomenon of interest is the teaching of mathematics using problem solving with computers. The text is an attempt to portray this trajectory, from the point of view of the methodological route followed by the researcher, which was based on two main axes: the guidance of the educator Thomas A. Romberg (1992), and the guidelines provided by the foundations of qualitative research. The study was developed during a doctoral course offered by the Graduate Program in Mathematics Education at the State University of São Paulo (UNESP), Rio Claro campus.

On the injectivity of C1 maps of the real plane

Let X : ℝ2 → ℝ2 be a C1 map. Denote by Spec(X) the set of (complex) eigenvalues of DXp when p varies in ℝ2. If there exists ε > 0 such that Spec(X) ∩ (-ε, ε) = ∅, then X is injective. Some applications of this result to the real Keller Jacobian conjecture are discussed.