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21 Nov 2023: 10. Let f: R2 R and (a,b) R2. (a) Suppose that D1f exists and is continuous in some open ball around (a,b), and that. ... b) Suppose instead that D1f exists and is bounded on some open ball around (a,b), and.

8 Mar 2024: 2(1. 71). Find an element of OK with norm 2a 3b 5 for some. ... a,b 0. Hence prove that the class group of K is cyclic and find its order.

3 Apr 2024: Show further that if f : X Y is a continuous function between Polish spacesand f is injective on a Borel set B X, then f(B) is Borel. ... 14. Prove that a set A (in some Polish space) is analytic if and only if there exist closedsets An1,.,nk (indexed by

17 May 2024: pdf]. I. Kontoyiannis. "Some information-theoretic computations related to the distribution of prime numbers." In Festschrift in Honor of Jorma Rissanen, (P. ... Mutual information, synergy, and some curious phenomena for simple channels." 2005 IEEE

12 Oct 2023: a, b], and suppose that fn converges pointwise to a continuous function f. ... Show that there is some subinterval [a, b] of [0, 1] with a < b on which (fn) is uniformly.

24 Nov 2023: 10. Let f: R2 R and (a,b) R2. (a) Suppose that D1f exists and is continuous in some open ball around (a,b), and that. ... b) Suppose instead that D1f exists and is bounded on some open ball around (a,b), and.

21 Nov 2023: 4. Let K be a field containing a primitive mth root of unity for some m > 1. ... Showthat f and g have the same splitting field if and only if b = cmar for some c Kand r N with gcd(r,m) = 1.

23 Feb 2024: 2(1. 71). Find an element of OK with norm 2a 3b 5 for some. ... a,b 0. Hence prove that the class group of K is cyclic and find its order.

11 Jan 2024: 6. Let (An : n N) be a sequence of events in some probability space (, F, P). ... Set. A = {ω : ω An infinitely often}, B = {ω : ω An for all sufficiently large n}.

12 Oct 2023: 6. By considering the singularity at or otherwise, show that any injective holo-morphic map f C C has the form f(z) = az b, for some a C and b ... 0,1,a} to be conformallyequivalent to C{0,1,b}, where a,b C{0,1}.[Don’t forget the cross ratio from IB -