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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.

20 Nov 2020: 4. Find all bases for which 39 is an Euler pseudoprime. ... More generally, showthat if p and 2p1 are both prime numbers, then N = p(2p1) is a pseudoprimefor precisely half of all bases.