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7 Jan 2023: Suppose that b′ > b. Then we know by the rules for N+ that wecan find a c N+ such that b′ = b c. ... If c C then, since g is surjective, we can find b B such that g(b) = cand, since f is surjective we can find a A such that f (a) =

12 Aug 2022: Then there is a point y / E such. that B(y,r) E 6= whenever r > 0. ... Since Uj1 isopen we can find a δj1 > 0 with δj1 δj/4 such that B(xj1,δj1) Uj1.

8 Feb 2017: Show that there exist unique complexnumbers A, B such that for every z C, T(z) = Az Bz̄. ... Can such a map everbe a Möbius transformation? 9. Let f : U C be a holomorphic function where U is an open set (you may assume f isalso C1).

23 Sep 2013: Hence. UTV U = diag(λ1,. ,λp). Thus, given V , we can always construct an orthogonal matrix U such that ifY = UTX, then Y1,. ... yn), in the following sense:take k orthonormal vectors a1,. ,ak (ie such that a.

11 Feb 2015: If X is a space such that H(X; R) = H(Sn; R), show thatthere is a map f : Sn X inducing an isomorphism on H(; R), and hence thatπi(X) ... Hint: a) Show that such rings have Tor(,R) = 0; b)Consider the class of abelian groups A such that AR = 0.].

14 Oct 2019: 1. Let E be a Lebesgue measurable subset of the real line with positive Lebesgue measure m(E).Show that for every > 0 there exists an open interval (a,b) such ... A Dirac mass at x X is themeasure δx on B such that δx(A) = 1 or 0 according as x A or x /

10 Dec 2012: encourage mathematical investigation without any examination emphasis. 1. Calculate d = (a, b) and find integers x and y such that d = ax by when. ... 11. Let A be the group (Z/65520Z). Determine the least positive integer n such thatgn = 1 for all g in A

26 Apr 2022: Show that ϕ is convex. 5. Suppose f : X R is twice-differentiable and such that 2f(x) is positive definite for allx X. ... a) Show that the unique minimizer is x = 0. (b) Find constants 0 < m < M and L > 0 such that m f ′′(x) M and |f ′′′(x)|

15 Oct 2021: Fix > 0. Then thereexist partitions PL of [a,c] and PR of [c,b] such that:. ... b) R such that fi f uniformly, then f : (a,b) R is uniformlycontinuous.

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