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6 May 2020: d. (p, Im c|[b′,b]. ) D0. Therefore there exist s [a,a′] and t [b′,b] such that d (p,c(s)) ,d (p,c(t)) D0. ... By definition, form,n 1, there exists bm,n B such that amRnbm,n.

28 Jan 2020: Regarding T as a map from C into C by identifyingR2 with C in the usual way, show that there exist unique complex numbers A, B such thatfor every z C, ... Can such a map everbe a Möbius transformation? 9. Let U C be open and let f = uiv : U C.

2 Dec 2020: identity). (G3) For each a G there exists a b G such that a b = e. ... ii) If a G then e a = a. (iii) If a,b,b′ G are such that a b = e and a b′ = e, then b = b′.

30 May 2020: Abigon for α,β is an embedded (closed) disc D S such that D(αβ) = D = ab where a αand b β are arcs. ... Therefore, after iterating, we eventually findψ Zd 6 Mod(S) such that φαi ψ αi.

6 May 2020: By definition, form,n 1, there exists bm,n B such that amRnbm,n. ... R such that dX (ai,b) =dA (ai,αi) ε2 dB (βi,b).

8 Jun 2020: A representation ofthis functor, if it exists, is called a product of A and B: it consists of an object AB together withmorphisms AB π1 A and AB π2 B such ... iI of objects of (A G) indexed by a set I, such that every B ob (A G) has amorphism Bi B in

19 Oct 2020: 5. (a) Suppose that f Matn,n(F) is such that f(AB) = f(BA) for all A,B Matn,n(F) and f(I) = n.Show that f is the ... fn for V such that. ψ.  mi=1. xiei,. nj=1. yjfj.

28 May 2020: F :X I Y (where I = [0, 1]) such that F(, 0) = f0 and F(, 1) = f1. ... Let f : Sn Sn and q Sn be such that f1 (f(q)) = {q1,.

30 Nov 2020: f. There is a chart (φ,U) about p withφ(p) = 0 such that. ... of isolated zeroes, there is an analytic functiong with g(0) 0 such that.

5 Jun 2020: Q(B(Xn1 ;D)) that such a match indeed occurs. In the case when no distortion is allowed, Q(B(Xn1 ;D)). ... the supremum is taken over all integers r 1 and all pairs of events A and B such that B 2 (Y 1rk),.