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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.

5 Jun 2020: But Wn, the “waiting time” until the firstD-close match for Xn1 , is approximately equal to the reciprocalof the probability of finding such a match. ... Given such a code, we candefine a (sub-)probability distribution on Ân by.

10 Oct 2020: algebra. 1.2. Show that the following sets of subsets of R all generate the same σ-algebra:(a) {(a,b) : a < b}, (b) {(a,b] : a < b}, (c) {(,b] : b ... an,bn] such that the Lebesgue measure of A4B(= (Ac B) (ABc)) is less than ε.

28 Oct 2020: if any), and why? (b) By replacing A in the construction of B with a suitably chosen set, construct a setC N such that neither C nor N C are ... and such that Wn is recursive foreach n I but. nI Wn is not r.e.

5 Jun 2020: This means thatthere is aσ-finite measureψ on (X,B) such that,for any A B satisfying ψ(A) > 0 and any initialconditionx,. ... There exists an integern0 1, a constanta0 > 0, and a probability measure̺ on(X,B), such that,.

3 Mar 2020: Show that f is integrable. What is. 10f(x)dx? 4. Give an example of a continuous function f : [0,) [0,) such that0f(x)dx. ... Show. that for every > 0 and every closed interval I [a,b] of positive length, there isa closed interval J I such that J has

5 Jun 2020: This isformalized by requiring that{Xn} has a unique stationarydistribution π such that. ... Recall [13]that an irreducible, aperiodic chain{Xn} with values in thecountable alphabetA is geometrically ergodicif there existsa functionV : A [1, ), positive

5 Jun 2020: what happens when we look for longer and longer such initial patterns how. ... ii If then there exists a standard Brownian motion fBt t g such that.

5 Jun 2020: probability:. Doeblin Condition (DC). There exists an integer r 1 and a real numberβ > 0 such that,. ... This gives the first boot-strap sample. Then the whole process is repeated to generate a total of B such blocksL1, L2,. ,

5 Jun 2020: probability:. Doeblin Condition (DC). There exists an integer r 1 and a real number β > 0 such that,. ... This gives the first bootstrap sample. Then the whole process is repeated to generate a totalof B such blocks L1, L2,. ,

5 Jun 2020: Indeed, Comets [23] provides such anargument when and are both i.i.d. ... We assume the existence of aref-erence letter, i.e., an such that.