Exact results for extensions of twofold nested anova for variance components

In this paper we study a problem related to the classical Erdos--Szekeres Theorem on finding points in convex position in planar point sets. We study for which n and k there exists a number h(n,k) such that in every planar point set X of size h(n,k) or larger, no three points on a line, we can find n points forming a vertex set of a convex n-gon with at most k points of X in its interior. Recall that h(n,0) does not exist for n = 7 by a result of Horton. In this paper we prove the following results. First, using Horton's construction with no empty 7-gon we obtain that h(n,k) does not exist for k = 2(n+6)/4-n-3. Then we give some exact results for convex hexagons: every point set containing a convex hexagon contains a convex hexagon with at most seven points inside it, and any such set of at least 19 points contains a convex hexagon with at most five points inside it.

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Function of coprophagous beetles (Coleoptera: Scarabaeidae, Geotrupidae, Hydrophilidae) in cattle pastures inferred from pitfall trapping data

Journal of Agrobiology ◽

10.2478/s10146-009-0011-2 ◽

2010 ◽

Vol 27 (2) ◽

pp. 85-91 ◽

Cited By ~ 1

Author(s):

Martin Šlachta ◽

Jan Frelich ◽

Tomáš Tonka

Keyword(s):

Seasonal Variation ◽

Spatial Variation ◽

Dry Weight ◽

Seasonal Distribution ◽

The Other ◽

Pitfall Trapping ◽

Other Hand ◽

Nested Anova ◽

Species Groups ◽

Dung Decomposition

Function of coprophagous beetles (Coleoptera: Scarabaeidae, Geotrupidae, Hydrophilidae) in cattle pastures inferred from pitfall trapping dataAn analysis of data on the dry weight biomass of coprophagous beetles in standardized dung (4.5 l) was conducted in order to characterize the spatial and the seasonal distribution of the beetles' biomass in cattle pastures and to elucidate their function in dung decomposition. Nested Anova with factors of farm, site (nested in farm), seasonal period and year was used to evaluate the effect of these factors on the biomass of four functional species groups: the dung dwellers ofScarabaeidae(subfamilyAphodiinae), the dung dwellers ofHydrophilidae, the small tunnellers ofScarabaeidae(subfamilyCoprinae) and the large tunnellers ofGeotrupidae. The spatial variation of biomass (between the sites and the farms) was insignificant (P>0.05) in the two dung-dweller groups and in the large-tunnellers group. On the other hand, a significant (P<0.05) seasonal variation of biomass was found in all but the large tunneller group. In dung dwellers, the spring biomass was formed mainly by two species,Aphodius prodromusandA. sphacelatus. In summer, most of the biomass was accounted for bySphaeridium lunatum, S. scarabaeoidesandA. rufipes. In the two tunneller groups,Onthophagus fracticornis, Geotrupes stercorariusandG. spinigerformed a majority of the biomass in dung.

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