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Here we give a necessary and sufficient condition for a Banach space to be separable.
For the construction of a phylogenetic network, it is essential to know if the new sequence that is being inserted is the result of mutations or due to events like recombinations. Recombination is a process of formation of new genetic sequences by piecing together segments of previously existing sequences . Once it is determined that it is a… (More)
It is observed that during the evolution of species, reticulation is an important event and is very prevalent in several organisms. However, the analytical tools for the representation of these events are still under development. This is primarily because of the difficulty involved in detecting these reticulation events that have taken place during the… (More)
It is shown that (1) if a good set has finitely many related components, then they are full, (2) loops correspond one-to-one to extreme points of a convex set. Some other properties of good sets are discussed.
Department of Mathematics, Periyar University, Salem 636011, India E-mail: firstname.lastname@example.org 2000 Mathematics Subject Classification: primary 60A05, 47A35, secondary 28D05, 37Axx Abstract:. We show that as in the case of nfold Cartesian product for n ≥ 4, even in 3-fold Cartesian product, a related component need not be full component.
The purpose of this note is to answer two questions about good sets raised in  and . Let X1,X2, . . . ,Xn be nonempty sets and let Ω = X1 ×X2 × ·· · ×Xn be their cartesian product. We will write ~x to denote a point (x1,x2, . . . ,xn) ∈ Ω. For each 1 ≤ i ≤ n,Πi denotes the canonical projection of Ω onto Xi. A subset S ⊂ Ω is said to be good, if every… (More)
Let X and Y be finite sets with |X| = m and |Y | = n. Let G be a group acting on X and Y. Let G act on X × Y by g(x, y) = (g(x), g(y)) for all g ∈ G and (x, y) ∈ X × Y. Let X/G be the set of G orbits of X. Let |X/G| = m1. Write n1 = |Y/G| and m12 = |(X × Y )/G| . Let π1 and π2 denote the projection maps from X × Y to X and Y respectively. The sets G(x),… (More)