A História da Alimentação: balizas historiográficas

Os M. pretenderam traçar um quadro da História da Alimentação, não como um novo ramo epistemológico da disciplina, mas como um campo em desenvolvimento de práticas e atividades especializadas, incluindo pesquisa, formação, publicações, associações, encontros acadêmicos, etc. Um breve relato das condições em que tal campo se assentou faz-se preceder de um panorama dos estudos de alimentação e temas correia tos, em geral, segundo cinco abardagens Ia biológica, a econômica, a social, a cultural e a filosófica!, assim como da identificação das contribuições mais relevantes da Antropologia, Arqueologia, Sociologia e Geografia. A fim de comentar a multiforme e volumosa bibliografia histórica, foi ela organizada segundo critérios morfológicos. A seguir, alguns tópicos importantes mereceram tratamento à parte: a fome, o alimento e o domínio religioso, as descobertas européias e a difusão mundial de alimentos, gosto e gastronomia. O artigo se encerra com um rápido balanço crítico da historiografia brasileira sobre o tema

On Incidences Between Points and Hyperplanes

We show that if the number I of incidences between m points and n planes in R³ is sufficiently large, then the incidence graph (that connects points to their incident planes) contains a large complete bipartite subgraph involving r points and s planes, so that rs ≥ I2 mn −a(m+n), for some constant a> 0. This is shown to be almost tight in the worst case because there are examples of arbitrarily large sets of points and planes where the largest complete bipartite incidence subgraph records only I2 m+n mn − 16 incidences. We also make some steps towards generalizing this result to higher dimensions. On the way, we slightly improve upon a result of Brass and Knauer [BK] about the representation complexity of incidences between m points and n hyperplanes in Rd, and get rid of the logarithmic factor in their upper bound

Repeated angles in three and four dimensions

Abstract. We show that the maximum number of occurrences of a given angle in a set of n points in � 3 is O(n 7/3), and that a right angle can actually occur Ω(n 7/3) times. We then show that the maximum number of occurrences of any angle different from π/2 in a set of n points in � 4 is O(n 5/2 β(n)), where β(n) = 2 O(α(n)2) and α(n) is the inverse Ackermann function

Similar Simplices in a d-Dimensional Point Set

We consider the problem of bounding the maximum possible number fk,d(n) of ksimplices that are spanned by a set of n points in R d and are similar to a given simplex. We first show that f2,3(n) = O(n 13/6), and then tackle the general case, and show that fd−2,d(n) = O(n d−8/5) and 1 fd−1,d(n) = O ∗ (n d−72/55), for any d. Our technique extends to derive bounds for other values of k and d, and we illustrate this by showing that f2,5(n) = O(n 8/3)

Points with large quadrant depth

Given a set P of points in the plane we are interested in points that are `deep' in the set in the sense that they have two opposite quadrants both containing many points of P. We deal with an extremal version of this problem. A pair (a,b) of numbers is admissible if every point set P contains a point p in P that determines a pair (Q,Qop) of opposite quadrants, such thatQ contains at least an a-fraction and Qop contains at least a b-fraction of the points of P. We provide a complete description of the set F of all admissible pairs (a,b). This amounts to identifying three line segments and a point on the boundary of F. In higher dimensions we study the maximum a, such that (a,a) is opposite-orthant admissible. In dimension d we show that 1/(2γ)≤a≤1/γ for γ=22d-12d-1.Finally we deal with a variant of the problem where the opposite pairs of orthants need not be determined by a point in P. Again we are interested in values a, such that all subsets P inRd admit a pair (O,Oop) of opposite orthants both ontaining at least an a-fraction of the points. The maximum such value is a=1/2d. Generalizations of the problem are also disussed