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27 Oct 2023: Calculate d = (a, b) and find integers r and s such that ra sb = d when. ... iii) Find constants c and d such that λ(a, b) c log b d.

31 Jan 2023: Regarding T as a map from C into C, show that there existunique complex numbers A, B such that for every z C, T(z) = Az Bz. ... Can such a map be the restriction tothe strip of a Möbius map?

4 Feb 2023: a) Construct a continuous surjection p : T2 T2 from the torus to itself such that for every x T2, p1(x)consists of exactly two points, and such that p is ... Such a map is calleda covering map.]. (b) Show that T2 also admits a covering map p̂ : T2 K to

24 Nov 2023: 45) Show that there is a w such that Ww = {w} and a w such that |Ww| = |w|. ... Show that the following sets arecomputably enumerable:. (a) {w ; there are three distinct words v such that f(w, v)};(b) {w ; there is a word v of even length such

3 Feb 2023: 3. Let f,g : R R be such that f(x) as x a and g(y) k as y. ... Construct a function f : R R that takes every value on every interval – in otherwords for every a < b and every c there is an x with a < x < b such

10 Oct 2023: b) Now let X and Y be jointly Gaussian random vectors such that Cov(X) is positivedefinite. ... Let Q bethe equivalent probability measure such that dQ. dP eX.

28 Oct 2023: functionf : (X,w) [0, 1] such that f(p) = 1 and f = 0 on C. ... Showthat (K,M,µ1) is the completion of (K,B,µ). 19. Let X be a real or complex vector space with a topology such that addition, scalarmultiplication are continuous, and

10 Oct 2023: n such that Ax = 0.• There exists a y Rm with (AT y)i 0 for all i = 1,. , ... Make this assumption for the rest of the problem.(b) Why must there exist an equivalent measure Q such that 1.

7 Oct 2023: g : C such that |g| = 1 |ν|-a.e. and |ν|(A) =Ag dν for all A F. ... Recall thatµ and ν2 are said to be orthogonal if there is a measurable partition = AB such thatµ(B) = |ν2|(A) = 0.].

14 Nov 2023: Show that spanwB =. (B). for B X.Deduce that a w-closed subspace Y of X is a dual space: there is a normedspace Z such that Y = Z. ... Note that such a linear map is bounded.) Show that forT B(X,Y ) the following are all equivalent to T being weakly