We study the range S(A) := {xT Ay : x, y are orthonormal in Rn}, where A is an n×n complex skew symmetric matrix. It is a compact convex set. Power inequality s(A) ≤ s(A), k ∈ N, for the radius s(A) := maxξ∈S(A) |ξ| is proved. When n = 3, 4, 5, 6, relations between S(A) and the classical numerical range and the k-numerical range are given. Axiomatic characterization of S(A) is given. Sharp poin...

Notation. K will always denote a number eld. xI. Ranks of Families of Elliptic Curves For our purposes, a family of elliptic curves will be given by an equation E : y 2 = x 3 + A(T)x + B(T); A(T); B(T) 2 KT]; (T) = 4A(T) 3 + 27B(T) 2 6 = 0: We will always assume that E is non-split (i.e., is not obtained by base extension from a curve deened over K). The rank Generic Rank = Rank E(K(T)) is the ...