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12 Apr 2005: The pair is -regular if |d(A′,B′) d(A,B)| 6 for all subsets A′ Aand B′ B such that |A′|> |A| and |B′|> |B|. ... Proof. Define B [N]2 to be the set of all (x,y) such that x2y A.

22 Oct 2020: Let B(G,F) be the (extended) Bruhat–Tits building of G(F). For any x B(G,F), there is asmooth affine group scheme Gx over OF such that ... a choice of basis b for Λ such that the embeddings of buildings (3.3.1) and (3.3.2) induceembeddings of the

11 Feb 2014: It follows that for such q, gq λI is positive definite. ... The set of all such t is closed andcontained in [a,b], so it has a minimum value t0.

5 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).

13 Nov 2002: else prove that no such relation exists. 1. 11. Define a binary operation on Z2 by the formula (a,b) (c,d) = (ac,ad bc). ... a1,b1) (a2,b2). (ak,bk). 12. Let A1,A2,. be sets such that for every positive integer n we have A1.

9 Feb 2015: Show that for any b C, there exists a sequence of points zn D(a,R) with zn 6= a such thatzn a and f(zn) b as n. ... Find such a sequence when f(z) = e1/z, a = 0 and b = 2.

15 Oct 2021: Define τ by:. U τ for all x U, there exists B β such that x B and B U. ... b) Fix a local base β̇. Show that a sequence (xn)n=1 is τ-Cauchy if and only if for anyB β̇ we can find an integer N such that.

29 Jan 2024: 8. Let t1, t2,. be propositions such that, for every valuation v, there exists n N withv(tn) = 1. ... Show that there is a proposition b, inwhich the only primitive propositions appearing are those that appear in both a and c,such that a b and b c.

19 Jan 2021: Show that E has a subgroup isomorphic toE/M, and show that all such subgroups are conjugate. ... such that iApA iBpB = idM , pAiA = idA, pBiB = idB, pAiB = 0 andpBiA = 0.

5 Jun 2020: 15, 18]. This means that there is a σ-finite measure ψ on(X,B) such that, for any A B with ψ(A) > 0, and any x X,. ... There is a function V : X [1,], a small set C X,and constants δ > 0, b < , such that:.

6 Aug 2021: b) Find a sequence fi R([a,b]) such that (fi)iN is Cauchy with respect tothe L2 norm, but for which there is no f R([a,b]) such that ... Proof. Since is open, there exists ,x such that the ball B(x) = {y Rn : |y x| < }is contained in.

7 May 2013: Remark 2.2. — There exist coefficient systems B M such that. ... Let β H(M; (m )B) be a class such that (m )(β) = 0 H(M; B).

25 Oct 2023: 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.

19 May 2007: 1 log(α1)m such that |N (x) N (x′)| > αn for all but at most γ|B|2. ... Then there must exist for this r a set B such thatg(B) γ1b(B) > αrm2, which implies that |B| > αr/2m and b(B) 6 γ|B|2.

5 Apr 2007: It follows that vi is in particular bounded,i.e. there exists an R such that vi B(R). ... One direction is obvious. For the other, note that if U is an arbitraryopen set, and q = T (p) for p U, then there exists a δ such that p B(

5 Jun 2020: Ân such that Ln(xn1 ) log Qn(B(xn1,D)) for allxn1 An. ... Conversely, for any sequence of “admissible” distributions {Qn}, there is a sequenceof codes {Cn,Ln} operating at distortion level D, such that, for (almost) any realization ofthe source,

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

4 Feb 2018: Show that for any b C, there exists a sequence of points zn D(a,R)with zn 6= a such that zn a and f(zn) b as n. ... Find such a sequence when f(z) = e1/z, a = 0 and b = 2.[A much harder theorem of Picard says that in any neighbourhood of an essential

15 Oct 2009: Then there is N1 such that whenever n m N1 we have n. ... k=1 uk(x0). Also there is N2 such thatwhenever n m N2 we have n.

7 Dec 2022: Use (ii) to show that there exists b K such that det(σiσjg1(b)) 6=0. ... Deduce that if y = g1(b) then {σ(y) | σ G} is a K-basis for L.Such a basis {σ(y)} is said to be a normal basis for

12 Nov 2005: Let a be an integer with 1 a < p, and let σ Aut(L) be theunique automorphism such that σ(X) = aX. ... Use (ii) to show that there exists b K such that det(σiσj g1(b)) 6= 0.Deduce that if y = g1(b) then {σ(y) | σ G} is a

26 Oct 2021: of B such thatπ1(Uα) is a disjoint union. ... H(B) H(E) such that π π! is multiplication by N.

5 Jun 2020: π(f) :=. π(dx)f(x) <. (ii) There exists a closed, small set C B such that,. ... x X and every B B satisfying ψ(B) > 0, there is a constantcB < such that,.

18 Feb 2009: Showthat a diffeomorphism f : X X extends to X if and only if f([b]) = [b]. ... Show that if f ′ is anyother such f , the resulting manifold M ′ is diffeomorphic to M.

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.

27 Oct 2016: b) If (fn) does not converge uniformly, show that there is a convergent sequence (xm) x [0, 1] such that (fn(xn)) does not converge to f(x). ... a) The space Cb(R) of bounded continuous functions on R.(b) The space C0(R) of continuous functions f : R R

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.

25 Feb 2024: c) Give an example of a sequence {𝜙 𝑗}𝑗=1 𝐶𝑐 (R𝑛) such that. ... Deduce thatR𝑚𝜓(𝑦)𝑑𝑦 = 𝑢[Ψ], where Ψ(𝑥) =. R𝑚𝜙(𝑥, 𝑦)𝑑𝑦. b) Show that there exists a sequence of smooth functions 𝑓𝑛 𝐶𝑐

31 Jan 2024: 8. Let t1, t2,. be propositions such that, for every valuation v, there exists n N withv(tn) = 1. ... Show that there is a proposition b, inwhich the only primitive propositions appearing are those that appear in both a and c,such that a b and b c.

13 Jan 2022: Prove directly from the definition thatAX has an m-variate normal distribution AX Nm(Aµ,AΣAT ). 2. Let X and Y be N(0, 1) random variables such that ... a random set C(Y ) such that Pβ,σ2 (b C(Y )) = 1 α for all β,σ2.

17 Jan 2024: b) We say that a measurable 𝑓 : R𝑛 C is in weak-𝐿𝑝(R𝑛), written 𝑓 𝐿𝑝,𝑤 (R𝑛) if there exists aconstant 𝐶 such that. |{ ... Deduce 𝑁𝑖=1. 𝐵𝑖. 6 3𝑛 𝑘︁𝑗=1. 𝐵𝑖𝑗. b) () Suppose {𝐵 𝑗 : 𝑗

21 Nov 2023: Let a,b K such that the polynomials f = Xm a, g = Xm b are irreducible. ... Use asuitable Lagrange resolvent to find a Q(ζ3) such that K = Q(ζ3, 3.

16 Mar 2021: 7. Let R Q be a ring. If X is a 1-connected space such that H(X; R) = H(Sn; R),show that there is a map f : Sn X ... Hint: a) Show that such rings have Tor(,R) = 0;b) Consider the class of abelian groups A such that AR = 0.].

28 Nov 2003: 1. Does there exist an uncountable family B of subsets of N such that for every A,B B(distinct) the intersection of A with B is finite? ... a1,a0,a1,a2,. be a doubly infinite sequence of points in [0, 1] such that for.

19 Dec 2014: Let M be a CTM such that (T is an ever-branching Suslin tree )M. ... λ < κ. Prove that if | A |= κ and x Aimplies | x |< λ, then there exists B A such that | B |= κ and (r)(x B)(x 6=y xy = r).

5 Jun 2020: This means thatthere is aσ-finite measureψ on (X,B) such that,for any A B satisfying ψ(A) > 0 and any initialconditionx,. ... There exists an integern0 1, a constanta0 > 0, and a probability measure̺ on(X,B), such that,.

11 Oct 2023: 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.

18 Oct 2021: In each case, describe the subsetof maximal ideals. 2. Given an example of a homomorphism of rings ϕ : A B such that the preimage of a maximal ideal is notmaximal. ... the topological space consisting of two points{p1,p2} such that {p1} is dense and {p2}

17 Jul 2023: 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)| L for all x. ... Given constants b1, b2 such that b1 eb2 0 use the Lagrangian method to.

3 Nov 2014: b) Let C1c ([0, 1]) be the set of functions f C1([0, 1]) such that f(x) = 0 for x in some neighborhood ofthe end points 0 and 1. ... If Sis a subset of C([a,b]) such that every sequence in S has a subsequence converging uniformly to a functionin S (i.e.

21 May 2005: Tripos IB 96401(b), modified] (i) Let (X,d) be a nonempty complete metric space, and let f :X X be a continuous map such that, for any two points ... xn} is a finite set of points inX such that the balls B(xi,) cover X, consider the 2n 1 points of HX

24 Nov 2023: If A is the matrix of the first of these (say), find a non-singular matrix P such that PT AP is diagonalwith entries 1. ... an be real numbers such that a1 an = 0 and a21 a2n = 1.

10 Oct 2023: Suppose that α has minimal polynomial f over Q, and let β = α4.Find a, b, c Q such that β = a bα cα2. ... 9. Let K and L be subfields of a field M such that M/K is finite.

10 Oct 2023: Suppose that α has minimal polynomial f over Q, and let β = α4.Find a, b, c Q such that β = a bα cα2. ... 9. Let K and L be subfields of a field M such that M/K is finite.

1 Oct 2012: Let Bun(V,θγd)denote the space of such maps. Recall from [1] that the cobordism category Cθ isa topological category roughly defined as follows. ... In dimension two, this can be improved. Theorem B. Let θ : X BO(2) be a tangential structure such

27 Jan 2020: 8. Let t1, t2,. be propositions such that, for every valuation v, there exists n withv(tn) = 1. ... Show that there is a proposition b, inwhich the only primitive propositions appearing are those that appear in both a and c,such that a b and b c.

5 Jun 2020: such thatlim supnn1 log Qn(B(Xn1 ,D)) R < a.s. ... The question then becomes how to find such a Q when P is unknown.

5 Jun 2020: Doeblintype condition there exists r such that ess inf PXnr j Bn Here. ... Lemma There exists a constant such that for any positive integers s and.

10 Jan 2018: 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).

21 May 2005: Why would such a thesis be unlikely to be consideredworth a Ph.D? ... xn} is a finite set of points inX such that the balls B(xi, ) cover X, consider the 2n 1 points of HX which are the nonemptysubsets of X0.].