2. Let E be measurable, and suppose that E is ν-null but |ν|(E) 6= 0. Then ν+(E) − ν−(E) = ν(E) = 0 and ν+(E) + ν−(E) = |ν|(E) > 0, so ν+(E) = ν−(E) > 0. Since ν+ ⊥ ν−, there exist disjoint measurable sets A and B covering X such that A is ν+-null and B is ν−-null. In particular ν+(E ∩ B) = ν+(E ∩ B) + ν+(E ∩ A) = ν+(E) > 0 but ν−(E∩B) ≤ ν−(B) = 0. This implies that ν(E∩B) > 0, which is a contr...

Let G be a group. We denote by ν(G) certain extension of the non-abelian tensor square G⊗G G×G. In this paper we prove that derived subgroup ν(G)′ is central product three normal subgroups ν(G), all isomorphic to G⊗G. As consequence, describe structure each term and lower series group ν(G).

1. Prove the results about the smoothness of the members of the Matèrn family. Proof. From the class we know that for small values of τ , we have K ν (τ) ≈ Γ(ν)2 ν−1 τ −ν. Also, for the derivatives we have that d dτ (τ ν K ν (τ)) = −τ ν K ν−1 (τ). If the Matèrn family has the correlation function ρ(τ) = 1 Γ(ν)2 ν−1 τ ν K ν (τ), then ρ (τ) = −1 Γ(ν)2 ν−1 τ ν K ν−1 (τ) Note that K ν (τ) = K −ν (τ...

1 The Weyl dimension formula and its q-analogue This material is taken from notes for a joint paper with Zajj Daugherty and Rahbar Virk. Everything here is " well known ". Proposition 1.1. Let g be a finite dimensional complex semisimple Lie algebra etc.etc. ...???? Let ν be a dominant integral weight so that the irreducible module L(ν) of highest weight ν is finite dimensional. Letˆκ 1 = ν, ν ...

1. Proof Proposition 1. Ψ y ν (x) is a surrogate function of ψ ν (x). Proof. A function g y (x) is said to a surrogate function of f (x) provided ⇢ f (x) ≤ g y (x), ∀x, y ∈ [0, ∞) f (y)=g y (y), ∀y ∈ [0, ∞). (1) Let us consider the following function: f ν (x)=(1− exp(−νx))/ν. (2) Since f 00 ν (x) < 0, this function is strictly concave, ∀x, y, f ν (x) ≤ f ν (y)+(x − y)f 0 ν (y), (3) with equalit...