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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?

13 Feb 2017: Such a c is called a fixed point of g.Give an example of a bijection h: [0, 1] [0, 1] with no fixed point. ... Thatis, for every a < b and every y there exists x (a,b) such that f(x) = y.Can such a function be continuous at any point?

27 Nov 2013: 12. Let S be a collection of subsets of N such that for every A, B S we have A B orB A. ... otherwords, for every a < b and every c there is an x with a < x < b such that f(x) = c.

23 Nov 2016: Mich. 2016 NUMBERS AND SETS—EXAMPLES 4 PAR. 1. The symmetric difference A4B of two sets A and B is defined to be the set of elements x such that xis ... 11. Let S be a collection of subsets of N such that for all A, B S we have A B or B A.

15 Jan 2024: 6. Let R be an integral domain. The greatest common divisor (gcd) of non-zero elements a andb in R is an element d in R such that d divides both ... b) Show that may be identified with the set Fq{} in such a way that GL2(Fq) acts on as the group of

13 Oct 2006: b) Suppose that f is bounded. Prove that there exists M and n0 such that |fn(x)| < M for all. ... a,b] such that fn(xn) does not converge to f(x). 7. Which of the following functions f are (a) uniformly continuous, (b) bounded on [0,)?

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).

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.

8 Feb 2017: 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 2020: The least such n iscalled the order of g.). (b) Show that there exists a positive integer n such that gn = e for all g G. ... Can such a group beinnite? 8. Let G be a group of even order.

19 Jan 2021: Show that φis isotopic to the identity. 8. Let α,β be a pair of simple closed curves on a surface S, such thati(α,β) = 1. ... Let α,β be astandard pair of simple closed curves on S such that i(α,β) = 1, andlet γ be the boundary curve.

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}

7 Mar 2024: b) Find a condition on 𝑠 such that 𝛿𝑥 𝐻𝑠(R𝑛). c) Show that 𝐻𝑡 (R𝑛) is continuously embedded in 𝐻𝑠(R𝑛) for 𝑠 < 𝑡. ... Δ2𝑢 𝑢 = 𝑓. b) Show further that there exists 𝐶 > 0 such that ‖𝑢‖𝐻 𝑠4

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

14 Oct 2016: c) Let x be a non-zero rational number. Show that there is a rational number y such thatxy = 1. ... d) Show that if a and b are natural numbers such that a is a factor of b then a 6 b.

17 Nov 2011: NB: maybexn < 0. ). 14. Let S PN be such that ifA, B S then A B or B A. ... Can S be uncountable?Is there an uncountable familyT PN such thatA B is finite for all distinctA, B T?

21 Oct 2015: a) Let a G. Show that there is a positive integer n such that an = e, the identity element.(The least such positive n is the order of a.). ... b) Show that there exists a positive integer n such that an = e for all a G.(The least such positive n is the

9 Oct 2018: B. Prove that (A4B)4C = A4(B4C). 7. Let A1, A2, A3,. be sets such that A1 A2. ... c there is an x with a < x < b such that f(x) = c.

12 Nov 2019: Show that φis isotopic to the identity. 8. Let α,β be a pair of simple closed curves on a surface S, such thati(α,β) = 1. ... Let α,β be astandard pair of simple closed curves on S such that i(α,β) = 1, andlet γ be the boundary curve.

26 Nov 2018: 13. Let S PN be such that if A,B S then A B or B A. ... Can S be uncountable?Is there an uncountable family T PN such that AB is finite for all distinct A,B T?

25 Nov 2014: n=1x3n? 14. Let S PN be such that if A, B S then A B or B A. ... Can S be uncountable?Is there an uncountable family T PN such that AB is finite for all distinct A, B T?

10 Mar 2017: Show that for every. closed subinterval I [a,b] of positive length and every ε > 0 there existsa closed subinterval J I of positive length such that supJ f infJ f < ... 0. sinxx. dx = π2. 12. A function f : [a,b] R is said to have bounded variation if

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.

10 Oct 2014: a) Let a G. Show that there is a positive integer n such that an = e, the identity element.(The least such positive n is the order of a.). ... b) Show that there exists a positive integer n such that an = e for all a G.(The least such positive n is the

8 Mar 2023: Does 10f exist? 3. Give an example of a continuous function f : [0,) [0,), such that0f. ... Must f vanishidentically?[We say g vanishes near a and b if there exists > 0 such that g(x) = 0 for x 6(a ,b ).]. 1. 2 ANALYSIS I EXAMPLES 4.

9 Feb 2015: That is, for everya < b and every t there should exist x (a,b) such that f(x) = t. ... 14. Let f : R R be a function that has the intermediate value property: that is, iff(a) < c < f(b) then there exists x (a,b) such that f(x) =

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

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.

17 May 2019: Show. there exist disjoint open sets U1 and U2 such that Ci Ui.8. ... continuous path. f : [0, 1] Xconnecting a and b such that f is a homeomorphism onto its image.

8 Nov 2023: 12. (a) Show that if x ̸= y are vectors in the finite dimensional vector space V , then there is a linear func-tional θ V such that θ(x) ̸= ... Let A, B V. Prove that A B if and only if Ao Bo.Show that A = V if and only if Ao = {0}.

17 Nov 2017: Can S be uncountable?Is there an uncountable family T PN such that AB is finite for all distinct A, B T? ... You are asked to supply them with astrategy such that, should they follow it, only finitely many of them will guess wrongly.

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

27 Nov 2019: 12. Let S be a collection of subsets of N such that for every A, B S we have A B orB A. ... otherwords, for every a < b and every c there is an x with a < x < b such that f(x) = c.

18 Nov 2010: 14. Is there an uncountable family S PN such that A B is finite for all distinct A, B S? ... such that. (qn qn1)2 converges? A.G.Thomason@dpmms.cam.ac.uk - 1 - 18 November 2010.

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

5 Oct 2023: Theleast such n is called the order of g.). (b) Show from first principles that there is a positive integer N such that gN = e for all g G. ... 9. Let H and G be groups and let X G such that 〈X〉 = G.

8 Nov 2017: Show that there is a divergentseries. n=1 yn with yn > 0 for all n, such that yn/xn 0. ... a < x < b such that f(x) = c.

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.

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.

7 Feb 2014: That is, for everya < b and every t there should exist x (a,b) such that f(x) = t. ... 14. Let f : R R be a function that has the intermediate value property: that is, iff(a) < c < f(b) then there exists x (a,b) such that f(x) =

25 Oct 2018: Find all such solutions. 2. Let a, b, c, d N. ... 3. Find integers x, y and z such that 56x 63y 72z = 1.

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.

12 Oct 2007: 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 f ... b) Give an example of a sequence of uniformly continuous, real-valued functions (fn) on R such thatfn

1 Nov 2021: Fix such an n and prove the theorem as follows. (a) Show directly that a nowhere-zero vector field on Sn induces a homotopy from the identityto the antipodal map. ... b) Use de Rham cohomology to prove that no such homotopy exists.

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.

20 Nov 2015: Can S be uncountable?Is there an uncountable family T PN such that AB is finite for all distinct A, B T? ... You are asked to supply them with a strategy such that,should they follow it, only finitely many of them will guess wrongly.

6 Feb 2019: Show that for any b C, there exists a sequence of pointszn 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.

23 Oct 2013: 11 fn 6 0. Is it possible to find such a sequence with |fn(x)| 1 for all n and for all x? ... 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

13 Oct 2021: foreach γ C there exist elements α(γ) A and β(γ) B, such that for all γ we have:. ... multiplication. An R-linear derivation OX,p R is an R-linearmap d : OX,p R such that d(fg) = d(f)g(p) f(p)d(g) for all f

15 Apr 2024: b) Consider a polygonal decomposition of Σ such that at least three edges meet at each vertex andeach face has at least three edges. ... Deduce. that there exists δ > 0 such that in any hyperbolic triangle the union of the δ-neighbourhoods oftwo of