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15 Oct 2020: The least such n iscalled the order of g.). (b) Show that there exists a positive integer n such that gn = e for all g G. ... Can such a group beinnite? 8. Let G be a group of even order.

5 Jun 2020: According to Occam’s razor, the simplest such explanation isthe one that should be favored. ... Given a string x, suppose wefocus on a special sub-class of programs p that produce x, namely, programs thatfirst describe a set B of strings such that x B,

5 Jun 2020: such thatlim supnn1 log Qn(B(Xn1 ,D)) R < a.s. ... The question then becomes how to find such a Q when P is unknown.

11 Feb 2020: Show that for any b C, there exists a sequence of points zn D(a,R)with zn 6= a such that zn a and f(zn) b as n. ... Find such a sequence when f(z) = e1/z, a = 0 and b = 2.[A much harder theorem of Picard says that in any neighbourhood of an essential

22 Oct 2020: Let B(G,F) be the (extended) Bruhat–Tits building of G(F). For any x B(G,F), there is asmooth affine group scheme Gx over OF such that ... a choice of basis b for Λ such that the embeddings of buildings (3.3.1) and (3.3.2) induceembeddings of the

20 Nov 2020: payout Y is called attainable if there exists a scalar a andportfolio b Rd such that Y = a b>S1. ... For the general case, show that there exists aportfolio θ Rd such that Y = θ S1 Z where Z and S1 are independent.] Verify thatY π(Y ) is concave.

5 Jun 2020: 15, 18]. This means that there is a σ-finite measure ψ on(X,B) such that, for any A B with ψ(A) > 0, and any x X,. ... There is a function V : X [1,], a small set C X,and constants δ > 0, b < , such that:.

20 Nov 2020: S aSbb | T, T bTaa | S | ǫ. (3) Give a CFG for each of the following CFL’s, and then transform each such CFG intoCNF (giving a justification for the transformation). ... b) Show that the language L := {anbncn | n 1} is not a CFL.

5 Jun 2020: π(f) :=. π(dx)f(x) <. (ii) There exists a closed, small set C B such that,. ... x X and every B B satisfying ψ(B) > 0, there is a constantcB < such that,.

12 Nov 2020: 5. Let A Rn be such that every continuous function f : A R is bounded. ... V such that A Uand B V.