In this paper, we study Mannheim partner curves in three dimensional space. We obtain the necessary and sufficient conditions for the Mannheim partner curves in Euclidean space E 3 and Minkowski space E 3 1 , respectively. Some examples are also given. (2000): 53A04, 53B30.
Mathematics Subject Classification
We consider diploid bi-parental analogues of Cannings models: in a population of fixed size N the next generation is composed of Vi,j offspring from parents i and j, where V = (Vi,j) 1≤i =j≤N is a (jointly) exchangeable (symmetric) array. Every individual carries two chromosome copies, each of which is inherited from one of its parents. We obtain general conditions, formulated in terms of the vector of the total number of offspring to each individual, for the convergence of the properly scaled ancestral process for an n-sample of genes towards a (Ξ-)coalescent. This complements Möhle and Sagitov's (2001) result for the haploid case and sharpens the profile of Möhle and Sagitov's (2003) study of the diploid case, which focused on fixed couples, where each row of V has at most one non-zero entry.We apply the convergence result to several examples, in particular to two diploid variations of Schweinsberg's (2003) model, leading to Beta-coalescents with two-fold and with four-fold mergers, respectively.
Date
Using the lookdown construction of Donnelly and Kurtz we prove that, at any fixed positive time, the Λ-Fleming-Viot process with underlying Brownian motion has a compact support provided that the corresponding Λ-coalescent comes down from infinity not too slowly. We also find both upper bound and lower bound on the Hausdorff dimension for the support.
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