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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.

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.

5 Jun 2020: log Px{ n1. i=0. Fj (Xi) ncj for all j, and Xn1 B}. ... i=0 Fj (Xi) ncj for all j, and Xn1 B}, and letµn.

4 Feb 2020: Showthat f1 : f(I) I is continuous. 8. Suppose that f : R R satisfies |f(x) f(y)| |x y|2 for all x,y, R. ... a) If f is increasing, then f ′(x) 0 for every x (a,b).(b) If f ′(x) 0 for every x (a,b), then f is increasing.(c) If f

5 Jun 2020: As it turns out,. l(x) log Q(B(x,D)) bits, (1.2)where B(x,D) denotes the distortion-ball of all reproduction strings that are withindistortion D or ... xn) Sn, we define the distortion-ball. B(xn1,D) :=. {yn1 Tn :. 1. n. nk=1. ρ(xk,yk) D. }. to be the

5 Jun 2020: We assume, without loss of generality, thatP(a) > 0 for all a A, and also that for each a A thereexists a b A with ρ(a,b) = 0 (otherwise we ... a) For all n 1, β enε(α).(b) If αn 0, then.

10 Jan 2020: tβn. are joined by an edge if, after renumbering:. 2. (a) αi is isotopic to βi for all i > 1;. ... b) if Sα2,.,αn is a one-holed torus then i(α1,β1) = 1;.

31 Jul 2020: fs(x) f(x) ft(x) for all f B(l,u). It follows that. 16. • ... all f B(l,u) are negative definite if and only if both fu and ft are negative definite;. •

5 Jun 2020: Then, , for all. Proof: Note that, for all. so that, in particular, , and,. ... the have finite means and variances,and that:. a) They are “uniformly ultra bounded,” in that,for all , with a common ;. b) Their means satisfy as ;c) Their covariances

5 Jun 2020: For part (ii) recall the assumption that for all a A there is b = b(a) such that ρ(a,b) = 0.If we let W (a,b) = ... b=b(a)}, then W M(P,D) for any D 0 and from (47), R(D) EWY [log M(Y )] < for all D 0.

5 Jun 2020: A B| |AB|3. |A| |B|, (3). all of which hold for arbitrary subsets A,B,C of the integers or any other discrete abelian group,and where the difference set ... h(X1 X2,Y ) = h(X1 X2|Y ) h(Y ). Finally combining all the above estimates yields,.