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The Saros cycle has been known since antiquity and refers to the periodicity of eclipses. It is the least common multiple of three periods: the synodic, the draconic, and the anomalistic months. We show how to obtain these periods from Newton's laws with a precision greater than 0.02% using only the sidereal month and year as references. O ciclo do Saros é conhecido desde a antiguidade e se refere a periodicidade dos eclipses. Ele é o mínimo múltiplo comum de três períodos: os meses sinódico, dracônico e anomalístico. Mostramos neste trabalho como obter estes três períodos a partir das leis de Newton com uma precisão superior a 0.02% usando apenas os mês e ano siderais como referência.

We study the evolution of cooperation under indirect reciprocity, believed to constitute the biological basis of morality. We employ an evolutionary game theoretical model of multilevel selection, and show that natural selection and mutation lead to the emergence of a robust and simple social norm, which we call stern-judging. Under stern-judging, helping a good individual or refusing help to a bad individual leads to a good reputation, whereas refusing help to a good individual or helping a bad one leads to a bad reputation. Similarly for tit-for-tat and win-stay-lose-shift, the simplest ubiquitous strategies in direct reciprocity, the lack of ambiguity of stern-judging, where implacable punishment is compensated by prompt forgiving, supports the idea that simplicity is often associated with evolutionary success., Synopsis Humans, unlike other animal species, form large social groups in which cooperation among non-kin is widespread. This contrasts with the general assumption that the strong and selfish individuals are the ones who benefit most from natural selection. Among the different mechanisms invoked to explain the evolution of cooperation, indirect reciprocity is associated with cooperation supported by reputation: I help you and someone else helps me. However, how did reputation evolve and which type of moral is encapsulated in those social norms that are evolutionary successful? Suggesting a simple scenario for the evolution of social norms, Pacheco, Santos, and Chalub propose a reputation-based multilevel selection model, where individual behaviour and moral systems co-evolve, governed by competition and natural selection. Evolution leads to the emergence of a simple and robust social norm, which the authors call stern-judging, where implacable punishment goes side-by-side with prompt forgiving. The low level of complexity of this norm, which is supported by empirical observations in e-trade, conveys the idea that simplicity is often associated with evolutionary success.

Sem PDF conforme despacho. For a given graph, there is a natural question of the possible lists of multiplicities for the eigenvalues among the spectra of Hermitian matrices with that graph (no constraint is placed upon the diagonal entries of the matrices by the graph). Here, we survey some of what is known about this question and include some new information about it. There is a natural focus upon the case in which the graph is a tree. In this event, there is remarkable structure to the possible lists. Both the general theory and a summary of specific results is given. At the end, this allows to give, in compact tabular form, all lists for trees on fewer than 11 vertices (a potentially valuable tool for further work). There is a brief discussion of non-trees. publishersversion published

Work partially supported by the Sub-programa Ciência e Tecnologia do 2o Quadro Comunitário de Apoio (grant number BD/18012/98) and project POCTI/32440/MAT/2000 (CAUL) of FCT and FEDER. A transformation semigroup over a set X with N elements is said to be a near permutation semigroup if it is generated by a group of permutations on N elements and by a set of transformations of rank N −1. In this paper we make a summary of the author’s thesis, in which are exhibited ways of checking whether those specific transformation semigroups belong to various structural classes of semigroups, just by analysis of the generators and without first having to generate them. proof published

In industry, when flexible manufacturing systems are designed within a group technology approach, numerous decision-taking processes emerge requiring control of the multiple characteristics of the system. In this context, several grouping problems are identified within the scope of combinatorial optimisation. Such is the case of the part families with precedence constraints problem, which is defined in order to set up families where the total dissimilarity among the parts placed in the same family is minimal and precedence constraints, as well as capacity constraints arise when grouping parts. The present paper describes the use of an improved genetic heuristic to tackle this problem. It comprises a standard genetic heuristic with appropriate operators, improved through specific local search. In order to study the performance of the improved genetic approach, a special purpose constructive heuristic plus an earlier version of the genetic heuristic were implemented. CPLEX software was used from a binary linear formulation for this problem. Computational results are given from the experiment performed using test instances partly taken from the literature while others were semi-randomly generated. The improved genetic heuristic produced optimal solutions for most of the shortest dimension test instances and acted positively in relation to the constructive heuristic results, over almost all the instances. As for the CPLEX it found optimal solutions only for the small instances, besides which for the higher dimensioned instances CPLEX failed to obtain any integer solutions at all, in 10h running time. Therefore, these experiments demonstrate that the improved genetic is a good tool to tackle high dimensioned test instances, when one does not expect an exact method to find an optimal solution in reasonable computing time. (C) 2004 Elsevier Ltd. All rights reserved. publishersversion published

An important theorem about the existence of principal submatrices of a Hermitian matrixwhose graph is a tree, in which the multiplicity of an eigenvalue increases, was largely developed in separate papers by Parter and Wiener. Here, the prior work is fully stated, then generalized with a self-contained proof. The more complete result is then used to better understand the eigenvalue possibilities of reducible principal submatrices of Hermitian tridiagonal matrices. Sets of vertices, for which the multiplicity increases, are also studied. publishersversion published

Research supported in part by Fundação para a Ciência e a Tecnologia, Portugal, through the research grant SFRH/BD/899/2000. Part of the research was done while visiting the College of William and Mary. The maximum multiplicity among eigenvalues of matrices with a given graph cannot generally be expressed in terms of the degrees of the vertices (even when the graph is a tree). Given are best possible lower and upper bounds, and characterization of the cases of equality in these bounds. A by-product is a sequential algorithm to calculate the exact maximum multiplicity by simple counting. publishersversion published

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Among those real symmetric matrices whose graph is a given tree $T$, the maximum multiplicity is known to be the path cover number of $T$. An explicit characterization is given for those trees for which whenever the maximum multiplicity is attained, all other multiplicities are $1$. publishersversion published

There is remarkable and distinctive structure among Hermitian matrices, whose graph is a given tree T and that have an eigenvalue of multiplicity that is a maximum for T. Among such structure, we give several new results: (1) no vertex of T may be "neutral"; (2) neutral vertices may occur if the largest multiplicity is less than the maximum; (3) every Parter vertex has at least two downer branches; (4) removal of a Parter vertex changes the status of no other vertex; and (5) every set of Parter vertices forms a Parter set. Statements (3), (4) and (5) are also not generally true when the multiplicity is less than the maximum. Some of our results are used to give further insights into prior results, and both the review of necessary background and the development of new structural lemmas may be of independent interest. http://www.sciencedirect.com/science/article/B6V0R-4SMF2K9-2/1/e80ee40f33898e52f517c5d58dbfa5bc

Esta comunicação está inserida no desenvolvimento de um projecto de investigação que procura compreender a forma como os professores de matemática podem integrar o uso de materiais tecnológicos em benefício da aprendizagem dos alunos. O projecto centra-se essencialmente nos materiais electrónicos que acompanham os manuais escolares, CD-Roms, eBooks, portais, filmes e conjuntos de outras actividades que apelam ao uso do computador. Procura-se compreender o papel que estes materiais desempenham no processo de ensino aprendizagem, nomeadamente na forma como os professores se apropriam desses materiais e o uso que fazem dos mesmos na sala de aula. Procurar-se-á apresentar nesta comunicação um breve enquadramento teórico do tema em estudo indicando as principais opções assumidas pelos autores. publishersversion published

Sem PDF In this paper we are concerned with the numerical solution of a diffusion equation in which the time derivative is of non-integer order, i.e., in the interval (0, 1). An implicit numerical method is presented and its unconditional stability and convergence are proved. Two numerical examples are provided to illustrate the obtained theoretical results. publishersversion published

Nikolai V. Chemetov was supported by the project no. 2021/03758-8 of Regular Research Grant-FAPESP, Brazil, and by the Program of "Docentes Novos", Process no. 20.1.4175.1.0 of Universidade de Sao Paulo. F. Cipriano was partially supported by the Fundacao para a Ciencia e a Tec-nologia. We study an optimal control problem with a quadratic cost functional for non-Newtonian fluids of differential type. More precisely, we consider the system governing the evolution of a second grade fluid filling a two-dimensional bounded domain, supplemented with a Navier slip boundary con-dition. Under certain assumptions on the size of the initial data and pa-rameters of the model, we prove second-order sufficient optimality conditions. Furthermore, we establish a global uniqueness result for the solutions of the first-order optimality system. publishersversion published

In this work we compute the probabilities of obtaining the values associated with the throwing of four rods in the Egyptian game Senet. Our purpose is to, among several sets of rules obtained by some historians and archeologists, deduce which ones provide the most balanced game. In the course of this study it was also possible to infer the most likely format for the rod’s cross section. publishersversion published