Extremal problems on sum-free sets and coverings in tridimensional spaces

Given a prime power q, define c (q) as the minimum cardinality of a subset H of F 3 q which satisfies the following property: every vector in this space di ff ers in at most 1 coordinate from a multiple of a vector in H. In this work, we introduce two extremal problems in combinatorial number theory aiming to discuss a known connection between the corresponding coverings and sum-free sets. Also, we provide several bounds on these maps which yield new classes of coverings, improving the previous upper bound on c (q)

Decomposability problem on branched coverings

Given a branched covering of degree d between closed surfaces, it determines a collection of partitions of d, the branch data. In this work we show that any branch data are realized by an indecomposable primitive branched covering on a connected closed surface N with chi(N) <= 0. This shows that decomposable and indecomposable realizations may coexist. Moreover, we characterize the branch data of a decomposable primitive branched covering. Bibliography: 20 titles.

In-shoe plantar pressure distribution during running on natural grass and asphalt in recreational runners

The type of surface used for running can influence the load that the locomotor apparatus will absorb and the load distribution could be related to the incidence of chronic injuries. As there is no consensus on how the locomotor apparatus adapts to loads originating from running Surfaces with different compliance, the objective of this study was to investigate how loads are distributed over the plantar surface while running on natural grass and on a rigid surface-asphalt. Forty-four adult runners with 4 3 years of running experience were evaluated while running at 12 km/h for 40 m wearing standardised running shoes and Pedar insoles (Novel). Peak pressure, contact time and contact area were measured in six regions: lateral, central and medial rearfoot, midfoot, lateral and media] forefoot. The Surfaces and regions were compared by three ANOVAS (2 x 6). Asphalt and natural grass were statistically different in all variables. Higher peak pressures were observed on asphalt at the central (p < 0.001) [grass: 303.8(66.7) kPa; asphalt: 342.3(76.3) kPa] and lateral rearfoot (p < 0.001) [grass: 312.7(75.8) kPa: asphalt: 350.9(98.3) kPa] and lateral forefoot (p < 0.001) [grass: 221.5(42.9) kPa asphalt: 245.3(55.5) kPa]. For natural grass, contact time and contact area were significantly greater at the central rearfoot (p < 0.001). These results suggest that natural grass may be a Surface that provokes lighter loads on the rearfoot and forefoot in recreational runners. (C) 2008 Sports Medicine Australia. Published by Elsevier Ltd. All rights reserved.