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7 Oct 2021: ii) M R n N x S |fn(x)| 6 M. (b) In this case assume that S is a metric space. ... Prove that there is some subinterval [a,b] of [0, 1] with a < b on which f isbounded.

15 Oct 2021: b. a) Show that the sequences (f(xj. )). 1j=0. and (f(yj. )). ... sup. x2(a,b). f. 0i. (x).  M. for some M independent of i.

5 Jun 2020: iii) There exists a closed, small set C and an extended-valued non-negative function Vsatisfying V (x0) < for some x0 X, such that Condition (V3) holds. ... Theorem 5.3 of [9] implies that (1.3) will holdif f is subject to this additional bound: For some

20 Feb 2013: For every isomorphism ι : Ql = C, the image of the residual representation rι(π′) contains SL2(Flb),up to conjugation, for some b a. • ... This contradiction shows that rι(π′) must in fact have residual imagecontaining SL2(Flb), some b a, and

6 Feb 2021: Prove that f is continuous on (a,b). Must it be continuous at a andb too? ... 13. Let f : R R be a function which has the intermediate value property: Iff(a) < c < f(b), then f(x) = c for some x between a and b.

21 May 2005: for some x,y Z with (x,y) = 1, then there exist a,b Z such that p = a2 b2. ... subgroup of Q/Q2 generated by a,b. Determine whether the field Q(.

6 Mar 2023: By considering the singularity at or otherwise, show that any injective an-alytic map f C C has the form f(z) = az b, for some a C and b ... 9. Define an equivalence relation on C by z w iff z = 2sw for some s Z.Show that the quotient space R = C/ has

6 Mar 2023: By considering the singularity at or otherwise, show that any injective an-alytic map f C C has the form f(z) = az b, for some a C and b ... 9. Define an equivalence relation on C by z w iff z = 2sw for some s Z.Show that the quotient space R = C/ has

4 Oct 2017: 8. By considering the singularity at or otherwise, show that any injective analytic map f : C Chas the form f(z) = az b, for some a C and b C. ... conformally equivalent). 10. Define an equivalence relation on C by z w iff z = 2sw for some s Z.

23 Feb 2018: Could there be some other Euclidean function φ making Z[3] into a Euclidean domain? ... iii) Show that if R is a PID, the gcd of elements a and b exists and can be written as rasbfor some r,s R.

15 Nov 2019: a) All words w {0, 1} consisting of either the string 01 repeated some number oftimes (possibly none), or the string 010 repeated some number of times (possiblynone). ... b) All words w {a, b, c} consisting of some number of a’s (possibly none),

1 Feb 2010: COMPLEX ANALYSIS EXAMPLES 1. Lent 2010 J. M. E. Hyland. These questions are a mix of questions from recent versions of the course together with some specificto my own take on ... 16. Let f be an entire function such that for some a C and r > 0, f takes

7 Nov 2016: a) All words w {0, 1} consisting of either the string 01 repeated some number oftimes (possibly none), or the string 010 repeated some number of times (possiblynone). ... b) All words w {a,b,c} consisting of some number of a’s (possibly none), followed

25 Jul 2017: By the pigeonhole principle, we must haveca = cb for some a and b with 0 6 a < b 6 k. ... First, what if instead of simply requiring that X be infinite, we seek an Xof cardinality 1, say, or of some larger specified cardinality?

28 Feb 2022: some a,b 0. Hence prove that the class group of K is cyclic and find itsorder.

20 Feb 2011: 5. Let f be an entire function such that for some a C and r > 0, f takes no values in the discD(a, r). ... Show that |f (z)| as |z| ,and deduce that |f (z)| takes a minimum at some a C.

15 Oct 2021: Bj,γ,δ. zγxδ, j = 1,. ,n1,. andc(z,x′) =. γ,δ. cγ,δzγxδ,. where these power series converge if |z| |x′| < s for some small s > 0. ... cγ,δzγxδ,. 4. with these power series converging for |x| |x′| < s for some (possibly smaller) s.

23 Oct 2012: J : z 7 az bcz d. for some complex numbers a, b, c, d with ad bc = 1. ... For which choices of a, b, c, d is this map J aninvolution, that is J 2 = I?

28 Feb 2019: b) For some n > 2, let X1,. ,Xniid Exp(θ). Find a minimal sufficient statistic T, and. ... 4. For some n 2 let X1,. ,Xniid U[θ, 2θ], for some θ > 0.

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 ... 1.9. Let (E,E,µ) be a measure space. Call a subset N E null if N B for some B E withµ(B) = 0.

22 Oct 2020: 2) If µ is noncentral, then. π0(X({µ},b)K) = π1(G)σΓ0. 0.3. Now we discuss some applications. ... Our first major step is to show that every point in X({µ},b) is connected to a point in Xw(b)for some σ-straight element w Adm({µ}).

3 Jun 2007: The questions carry equal weight. This is a Mock examination, intended to give you some idea of the sortof questions you will face in the proper Part III examination. ... The function f : DR C is analytic on some disc DR = {z C : |z| < R} withR > 1 and

3 Jun 2007: u(z) = b log |z| Re a(z)for some b R and some analytic function a : A C. ... a) Show that Z(T) is a subgroup of Möb.(b) Find which groups (up to isomorphism) can arise as Z(T) for some Möbius transformation T.

13 Oct 2012: Let u =(. ac. ), v =. (bd. )for some integers a, b, c, d with ad bc = 1. ... Show that every vector. w Z Z can be written as mu nv for some integers m and n.

13 Nov 2011: Explain geometrically why this is thecase. 5. (i) Suppose that f : R2 R is such that f/x is continuous in some open ball around (a,b),and f/y exists ... 11. Show that there is a continuous square-root function on some neighbourhood of I in Mn;that is,

7 Oct 2021: ii) M R n N x S |fn(x)| 6 M. (b) In this case assume that S is a metric space. ... Prove that there is some subinterval [a,b] of [0, 1] with a < b on which f isbounded.

26 Oct 2021: Some more questions. 13. (a) A topological space is said to be second countable if there is a countable family Bof open sets such that every open set is a union ... of some members of B.

19 Jan 2017: Could there be some other Euclidean function φ making Z[3] into a Euclidean domain? ... iii) Show that if R is a PID, the gcd of elements a and b exists and can be written as rasbfor some r,s R.

23 Dec 2010: Byuniqueness, we can call x = t(R). Lemma 20. Any x V is t(R) for some R B. ... B,b] to an SPG [Q,q]. 28. By Lemma 23, there is a bijection θ : Dom Q ι5“H for some H M.

16 Aug 2011: larger family Aand then randomly discard some of the sets in A to form a subfamily B. ... Then A = vf (W )Xfor some W U , X X W2 YW and W X B.

21 May 2005: vary in difficulty but cover some instructive points. 1. For what values of a and b does the system of simultaneous linear equations. ... only if it is nilpotent, that is, αk = 0 for some natural number k.

18 Jan 2021: Proof. Let f : J0(N) A be the quotient map. There exists a mapg : A J0(N) such that fg = [n] for some n > 0. ... and for a given f, it’s possible that some Vf,p are of the form (trivial) (cyclotomic) while others aren’t.

25 Oct 2023: Proofs for some results inTopics in Analysis. T. W. Körner. October 25, 2023. ... Now P(z) = 0 if and only if z = ajfor some 1 j m.

5 Oct 2020: 4. If the map f of Question 3 is a conformal equivalence, show that F(z) = λz µfor some λ C. ... 5. Show that complex tori C/⟨1,τ1⟩ and C/⟨1,τ2⟩ are analytically isomorphic ifand only if τ2 = (aτ1 b)/(cτ1 d), for some matrix ( a bc d ) SL2

15 Aug 2011: 614.2 An example from criminology, and some exercises. 624.3 From recent Mathematical Tripos questions. ... equivalentlylog E(Nij) = const ai bj for some a,b. Thus, in terms of log E(Nij), testing H0 is equivalent to testing a hypothesis which islinear

15 Jan 2021: some a,b 0. Hence prove that the class group of K is cyclic and find itsorder.

12 Feb 2018: Xn be independent Poisson random variables, with Xi having parameteriθ for some θ > 0. ... b) For some n > 2, let X1,. ,Xniid Exp(θ). Find a minimal sufficient statistic T, and.

25 Dec 2010: Then the collection ofpairs (α, β) that can occur in quadrilaterals of the form A(w), B(w), C(w), wfor some w 6= 0 is exactly the zero set of ... It follows that the parameters (α, β) occur for some some (non-trivial)irreducible representation of G′

21 May 2005: Is ittrue that for some c in [a,b] we have. 1. ... i) Suppose that λ > 0. Show that there is some interval [c,d] in [a,b] with the property that.

7 Oct 2021: 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 ... 1.9. Let (E,E,µ) be a measure space. Call a subset N E null if N B for some B E withµ(B) = 0.

19 May 2007: The next lemma gives us some information under weaker hypotheses than those of. ... with some probability p, whereas a graph with the weaker intersections property looks like.

12 Apr 2005: intersection of A with (x,y)A has expected size cδ2N2 for some absolute constant c > 0,lives inside the grid [N,N]2, and has the property that B = ... It is straightforwardto show that B has density at least η > 0 for some η that depends on δ only.

2 Feb 2010: Show that f (x) = 0 for allx. 8. Let f : [a, b] R be bounded. ... 13. Let f : R R be a function which has the intermediate value property: If f (a) < c < f (b),then f (x) = c for some x between a and b.

11 Nov 2016: u(0,x) =1. 1 x2. 1. 5. (Cauchy-Kovalevskaya Theorem for systems of ODEs) Suppose b > 0 and F : u0 (b,b)d Rd bereal analytic in a neighbourhood of ... u′(t) = F(t,u(t)), u(0) = u0 Rd,. on (a,a) for some a > 0 with u((a,a)) u0 (b,b)d.

14 Feb 2017: 6. Let (An : n N) be a sequence of events in some probability space (,F,P). ... Set. A = {ω : ω An infinitely often}, B = {ω : ω An for all sufficiently large n}.

19 Oct 2020: ii) M R n N x S |fn(x)| 6 M. (b) In this case assume that S is a metric space. ... Prove that there is some subinterval [a,b] of [0, 1] with a < b on which f isbounded.

31 Jan 2024: If g2H = g′2H, then we know g. ′2 = g2 h for some h H. ... So we have to show p - ||. This is just some basic counting: we have. ||

5 Oct 2020: az b, for some a C and b C. ... conformally equivalent). 9. Define an equivalence relation on C by z w iff z = 2sw for some s Z.Show that the quotient space R = C/ has the natural structure of

17 Dec 2021: Explain ‘in-. variant’? 2.1 Wrapping up miscellaneous definitions. Finally we wrap up some definitions and notations. ... Let τbe some permutation that adds a Boffa atom and a Boffa antiatom, suchas (,B())(V,BV ). Then we set both yin and yang to be

14 Oct 2020: δ is abelian.(c) Assume that δ(g) = 1 for some g G. ... irreducible. 3 Find an example of a representation of some finite group over some field of charac-teristic p, which is not completely reducible.