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8 Jan 2020: iii) Let H be a subgroup of G and suppose there exists a homomorphismρ: G H such that ρ(h) = h for all h H. ... ii) Let X G, H G be subsets. Show that H X if and only ifpi(H) pi(X) for all i I.

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. ... ϕ(g) = e for all g D2n. Find all homomorphisms D2n Cnwhen n is even.

5 Jun 2020: π(n) 12. log n for all n 2Kolmogorov complexity arguments [Bergman, Tropp, Li & Vitanyi] show. ... of all these primes also divides B. In particular, their product must be Bn1<p2n1.

16 Jan 2020: Equivalently, it hopes that. Hd(GLn(k), GLn1(k)) = 0 for all d f (n). ... with the two-sided bar construction. (Something similar can be done for all Ek.).

9 Nov 2020: b) All words w {a, b, c} consisting of some number of a’s (possibly none), followed bysome number of b’s (at least one), followed by some number of ... If |w| < 4,then w must contain a 1 somewhere. (d) All words w {a,. ,

5 Jun 2020: Then for all xn1 :. Ln(xn1 ) = Ln(qn(x. n1 )) log Qn(qn(xn1 )) log Qn(B(xn1, D)) bits. () ... a) The generalized AEP holds for all Qθ in Θ. (b) The rate function R(P, Qθ, D) of the AEP is lower semicont’s in θ.

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 ... Show that. (a) C is uncountable and has Lebesgue measure 0,(b) for all x [0, 1], the limit F(x) =

9 Nov 2020: pn = e2 and pni = 0 for all i > 0. (!). ... Using these relations to contract all internal edges, this is the sameas partitions of {1, 2,.

5 Jun 2020: as knowingthe values of all theXi; the last inequality comes from (3).Therefore,. ... An. loge n π(n) B n. loge n, for all n n0.

13 Nov 2020: b) Show that S1 S1 is homeomorphic to a graph.(c) Draw all the covering spaces of S1 of degree 2 or 3. ... d) Draw all the covering spaces of S1 S1 of degree 2 or 3.2.

5 Jun 2020: log pp. log n,. where the sum is over all primes p not exceeding n. ... that:. An. loge n π(n) B n. loge n, for all n n0.

14 May 2020: Assuming that ϕ(b) > for all b R, show that ϕ is convex. ... Which of the basic solutions arefeasible? Are all the basic solutions non-degenerate?

5 Jun 2020: offinite-length binary strings to the set of all finite- or infinite-length binary strings. ... The idea behind this definition is that the description of B contains all the “struc-ture” in x, while the remaining log2 |B| bits describe the

20 Nov 2020: Show that(a) E[eθXf(X)] = eµθθ2σ2/2E[f(X θσ2)] for all θ R and suitable f,(b) E[f(X)(X µ)] = σ2E[f ′(X)] for suitable f,(c) ... payout Y is called attainable if there exists a scalar a andportfolio b Rd such that Y = a b>S1.

5 Jun 2020: n1 ),. where H(Xn1 ) is the entropy of Xn1. (b) For all n and all K 1, the probability. ... b) For all n and all K 1, the probability. Pr{n beats (S)n by K bits or more}= Pr{n(Xn1 ) (S)n (Xn1 ) K} 2K.

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 ... f(A) = trA for all A Matn,n(F).(b) Now let V be a non-zero finite dimensional real vector space.

5 Jun 2020: Q N M L (x) =Qθ̂ (x )(x). all y Qθ̂ (y)(y). ... The idea behind this definition is that the description of B contains all the “structure”in x , while the remaining log2 |B| bits describe the “noise” in x.

12 Nov 2020: 11. Let M be a non-empty compact metric space and f : M M be a function.(a) Show that if d(f(x),f(y)) < d(x,y) for all ... x 6= y in M, then f has a unique fixed point.(b) Show that if f is isometric, i.e., d(f(x),f(y)) = d(x,y) for all

16 Nov 2020: b) Find all 1-dimensional representations of G.(c) Let ψ : Fp C be a non-trivial 1-dimensional representation of the cyclic group. ... d) Prove that the collection of representations constructed in (b) and (c) gives a com-plete list of all irreducible

5 Jun 2020: then the union of the balls B(yn1 , D) centered at the points yn1 Cncovers all of An except for a set of probability no greater than. ... isthat, since all the elements yn1 of the covering set B are drawn from the set T.