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20 Jan 2022: 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 called acovering map.]. (b) Show that T2 also admits a covering map p̂ : T2 K to

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)|

4 Oct 2022: b) Now let X and Y be jointly Gaussian random vectors such that Cov(X) is positivedefinite. ... Let Qbe the equivalent probability measure such that dQ. dP eX.

4 Oct 2022: 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.

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.

12 Aug 2022: Theorem 4.2. If f : [a,b] [a,b] is a continuous function, then we canfind a w [a,b] such that f(w) = w. ... Supposethat we can find a a0 < a1 < < an b such that, writing σ = ‖fP‖we have either.

1 Feb 2022: b) Find a condition on s such that x 2 Hs(Rn). ... 2u u = f. b) Show further that there exists C > 0 such that kukHs4.

21 Nov 2022: Show that the following sets are computablyenumerable:. (a) {w ; there are three distinct words v such that f(w, v)};(b) {w ; there is a word v of even length such ... g(u, v) :=. {ffu,1(u)(v) if u K and. otherwiseis computable. (b) By the s-m-n theorem,

24 Oct 2022: similar to any of them? If so, which? Find a basis such that it is in Jordan Normal Form. ... Withoutappealing to Jordan Normal Form, show that there is a vector y such that (y,α(y),α2(y),. ,

12 Oct 2022: Mich. 2022 ANALYSIS AND TOPOLOGY—EXAMPLES 1 PAR. 1. Let (zn) be a sequence in R2 such that zn a and zn b. ... On the other hand, show that if fn does notconverge uniformly to f then we can find a convergent sequence xm x in [a,b] such that fn(xn) 6

6 Jul 2022: vd of V such that v1,. ,vr is a basis of W and the. ... vd of V such that v1,. ,vr is a basis of W the matrices.

29 Nov 2022: vd of V such that v1,. ,vr is a basis of W and the. ... vd of V such that v1,. ,vr is a basis of W the matrices.

12 Dec 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.

12 Dec 2022: Theorem 4.2. If f : [a,b] [a,b] is a continuous function, then we canfind a w [a,b] such that f(w) = w. ... Supposethat we can find a a0 < a1 < < an b such that, writing σ = ‖fP‖we have either.

5 Oct 2022: algebra. 1.2. Show that the following sets of subsets of R all generate the same σ-algebra:(a) {(a,b) : a < b}, (b) {(a,b] : a < b}, (c) {(,b] : b ... an,bn] such that the Lebesgue measure of A4B(= (Ac B) (ABc)) is less than ε.

30 Mar 2022: In 7 we prove that such orbits admit integral representatives away from smallprimes. ... We have a Gm-action on Vgiven by λ v = λv and there is a unique Gm-action on B such that π is Gm-equivariant.

30 Aug 2022: and PrV then we can define a prob-ability Pr on = U V W in such a way that. ... 76. Exercise 2.4.23. Suppose such a die exists. We must have.

11 Dec 2022: There are lots of such books (because lots of different com-munities need to do first-year mathematics: engineers, students doing psyschology,science, economics, computer science, medicine. ) ... Typically,for a programming language, numbers such as

24 May 2022: Whether or not there are such things depend on whetheror not countable choice holds. ... every y such that R(y, x),and obeys the recursion wherever it is defined.

7 Oct 2022: Choose then a and b such that. 1. a. def=. 1. ... 29. Proof of Lemma 3.1.1. x K , rx > 0 such that B(x,rx).