Search

22 Sep 2014: homotopy equivalences f : B B such that. θ f = θ. ... Homology stability with respect to higher handles. A further immediateconsequence of Theorem B is an analogous homological stability theorem for gluingon to W a θ-cobordism K : W P such that

1 Mar 2023: i) f(z)/z 0 as |z|.(ii) There exists b C and > 0 such that for every z C, |f(z) b| >. (iii) f = u iv and |u(z)| > ... 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.

30 Nov 2021: ϕα : E|Uα := π1(Uα) Uα M,. such that each. ϕβ ϕ1α : (Uα Uβ) M (Uα Uβ) M. ... π. : Hid(E) Ei,d Ei,d2 = Hi(B; b 7 Hd(π1(b))) = Hi(B).

20 Jan 2007: Show that the curve joining A and B with length and such that together with AB bounds the largest possible area is an arc of a circle passingthrough A and B. ... Compute the Gaussian curvature and the mean curvature. 10. (i) Determine an equation for the

7 Oct 2023: g : C such that |g| = 1 |ν|-a.e. and |ν|(A) =Ag dν for all A F. ... Recall thatµ and ν2 are said to be orthogonal if there is a measurable partition = AB such thatµ(B) = |ν2|(A) = 0.].

25 Jan 2019: Can such a map be a Möbiustransformation? 10. Calculateγ z sin z dz when γ is the straight line joining 0 to i. ... functions of which theyare derivatives) on the domains indicated:. (a)1. z, (0 < |z| < ); (b) z.

8 Oct 2021: Choose an element δ DS such that δ = δ as in [TX16, Lemma 3.8]. ... τ. We define the integer nτ to be the smallest integer. such that σnττ Σ S.

15 Oct 2021: b) Show that f is smooth and periodic. c) Show that there exists a constant cn depending only on n such that:. ... 2q. ) such that 0 andX. g2g = 1. b) Show that if u 2 D0(Rn) is periodic and , 0 are both as in part a), then.

20 Nov 2015: If A is the matrix of the first of these (say), find a non-singular matrix P such that PTAP is diagonalwith entries 1. ... 14. Let a1, a2,. , an be real numbers such that a1 an = 0 and a21 a2n = 1.

6 Mar 2023: Logarithmic Gromov-Witten theory defines the notion of a logarithmic stable mapto such targets X B; more generally, this morphism just needs to be toroidal ratherthan normal crossings, i.e., log ... The type of such a family consistsof the data τ := (G,

15 Oct 2021: b) Suppose that v V , with components vi and ω V with components ωj. ... Please send any corrections to c.warnick@warwick.ac.uk. 2. b) Fix x U and suppose that V ,W TxU are two vectors such that for any f

11 Nov 2019: vector space isomorphic to V ;. (ii) any point p B has a neighbourhood U, such that there is a diffeomorphism. ... A local section of the principal bundle P is a smooth map s : U P defined on aneighbourhood U B and such that π s = idU.

15 Oct 2021: supxA||fi(x)f(x)|| <. b) For all x A, for all i N there exists > 0 such that:. ... xK} [a,b] Q such that [a,b] Km=1(xm δ,xm δ). Since gi converges ateach rational point, for each m = 1,.

11 Oct 2022: Find a, b, c Q such thaty = a bx cx2. ... 15. Let K and L be subfields of a field M such that M/K is finite.

15 Feb 2016: 7. 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

28 Oct 2023: functionf : (X,w) [0, 1] such that f(p) = 1 and f = 0 on C. ... Showthat (K,M,µ1) is the completion of (K,B,µ). 19. Let X be a real or complex vector space with a topology such that addition, scalarmultiplication are continuous, and

29 Sep 2023: The discriminant = (a, b)=16(4a327b2)is non-zero. Conversely, for any pair (a, b) O2K such that (a, b) , 0, the equation(2.1) defines an elliptic curve. ... ap bp = cp. to the Fermat equation in exponent p; thus a, b, c Z are coprime non-zero

11 Oct 2022: 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. ... 10. Let H and G be groups and let X G such that 〈X〉 = G.

19 Nov 2022: 13. Let S be a collection of subsets of N such that for every A,B S we have A B or B A.Can S be uncountable? ... Is there an uncountable collection T of subsets of N such that A B isfinite for all distinct A,B T?

15 Oct 2021: Exercise 3.1. a) Suppose that {j. }1j=1 E () is a sequence such that j! ... f 2 L1(Rn). Fix > 0. Show thatthere exists f. 2 C10(Rn) such that.

6 Mar 2014: Show that for any Hecke character χ, thereexist continuous χv : Kv C (v ΣK) such that. ... πv7. )= 1. Show that there is a unique Hecke character χ of K such that. •

29 Sep 2003: 1. 2 ANALYSIS II EXAMPLES 1. (ii) If fn does not converge uniformly, show that we can find a convergent sequence xn x in[a,b] such that fn(xn) does ... derivative’ limh0+ ((f(h) f(0))/h),and similarly for f′(1)), but such that f′ is unbounded on [0,

14 Mar 2005: Proof. Define B [N]2 to be the set of all (x,y) such that xy A. ... It follows that there are functions a, b and c. taking values in [0, 1] such that.

8 Feb 2012: 14. Exhibit a subset X of R2 such that (a) any two points x, y X can be joined bya continuous path γ : [0, 1] X and (b) for x 6= ... A regular polyhedron is one in which all two-dimensional faces are congruent regularpolygons, and such that for each

11 Nov 2016: Let x0 and 0 < ρ < ρ̄ be such that the ball B(x0, ρ̄). ... where b Rd and c R. Show that there exists a constant C such thatB(x0,ρ).

26 Oct 2015: fn for V such that. ψ.  mi=1. xiei,. nj=1. yjfj. ... Suppose thatα: V W is a linear map such that α(Y ) Z.

23 Nov 2021: 13. Let S be a collection of subsets of N such that for every A,B S we have A B or B A.Can S be uncountable? ... Is there an uncountable collection T of subsets of N such that A B isfinite for all distinct A,B T?

1 Oct 2012: Suppose W is a (n1)-connected 2n-manifold such that πn(W) πn(BO) is surjective. ... Theorems A and B follow fromtwo observations about the monoid E: firstly that the submonoid E′ consisting ofthose path components represented by manifolds W such that

4 Dec 2014: If A is the matrix of the first of these (say), find a non-singular matrix P such that PTAP is diagonalwith entries 1. ... 14. Let a1, a2,. , an be real numbers such that a1 an = 0 and a21 a2n = 1.

17 Nov 2013: 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 L/K, and the result just proved is the NormalBasis Theorem.

9 Nov 2023: SC =2π. κ. C. [Q[ξ] ξ BH]. However, it is not possible for θ to be of the form of a totalvariation δBH on H in general because such a ... M =. [Q[t] t B]. and there is no difficulty in obtaining an M that satisfies thedesired relation.

5 Jun 2020: According to Occam’s razor, the simplest such explanation is theone that should be favored. ... h x (α) = min{. log2 |B|, for sets B such that x B and K (B) α}.

6 Oct 2019: map(B,M) = {families of (.) over B}. Then. H(M) = {characteristic classes of such families}. ... This “resolution” point of view applies to many further examples,such as Confn(Rk) or BDi (W).

11 Mar 2024: Proof 2: more words. Let ϵ > 0. Choose N N such that N > 1/ϵ2. ... f1 When f : A B is a function, and C is a subset of B, thesymbol f1(C) denotes the preimage of the set C: the set ofelements a A such

22 Nov 2022: Let a, b K such thatthe polynomials f = Xm a, g = Xm b are irreducible. ... Use a suitable Lagrange resolvent to finda Q(ζ3) such that K = Q(ζ3, 3.

15 Oct 2021: Questions marked () are optional. 2. b) For all x 2 A, for all i 2 N there exists > 0 such that:. ... R such that f(x) = ˜f(x) for all x 2 (a, b).

28 Nov 2012: An immersion is a smooth map f : X Y such that dfx is injective forall x X. ... A submersion is a smooth map f : X Y such that dfx is surjectivefor all x X.

10 Feb 2023: g :(D2,S1. ) (M,B). such that[g|S1. ]6 N. We first explain why the Loop Theorem implies Dehn’s Lemma:. ... g :(D2,S1. ) (M,B). such that[g|S1. ]6= 1. Set c =. [g|S1. ] π1B, and write c = ka b, with. k, Z.

19 Mar 2008: iii) Give an example of disjoint closed sets A and B in R2 with points an A and bn B such. ... 9. [Tripos IB 96401(b), modified] (i) Suppose that (X, d) is a nonempty complete metric space, andf : X X a continuous map such that, for any x, y X, the

13 May 2016: If φ : (Sn,) (B,B) , show that we can find a mapΦ : X E such that p Φ = φ π and Φ(Sn1 0) =. If we define φ =Φ|Sn11 : Sn1 ... Show that there are maps Ap : TpM TpM such that ifv,w TpM, ω(v,w) = 〈Apv,w〉.

7 Dec 2009: 5. (i) Suppose that f : R2 R is such that D1f = f/x is continuous in some open ball around(a,b), and D2f = f/y exists at (a,b). ... Show that f is differentiable at (a,b). (ii) Suppose that f : R2 R is such that D1f = f/x exists and is bounded near (a,b),

24 Oct 2023: Vn such that V ' V1 Vn. Theorem 4.12. Suppose that G is finite and |G| 6= 0 in F. ... central character, such that z v = λ(z) v forall z Z(G) and v V.

12 Oct 2021: 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. ... 10. Let H and G be groups and let X G such that 〈X〉 = G.

26 Jan 2024: a2. b2. ak. bk. Example. Given a, b G,. sclG ([a, b]) 12χ. ... such thatD(φ) := sup.

23 Feb 2018: 7. 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

25 Jul 2017: By Lemma 1, we can find A B with|A| = 1 and colours ca (0 6 a 6 k) such that if x is in the span of A andhas pattern ... By Lemma 1, there exist a subset A B with |A| = 1 and colours cπ(π Π′) such that if x is in the span of A and has pattern π

13 Oct 2017: Calculate d = (a,b) and find integers r and s such that ra sb = d when. ... iii) Find constants c and d such that λ(a,b) 6 c log b d.

15 Oct 2021: 1t t > 0. a) Show that χ C(R). b) Show that there exists a function ψ C0 (Rn) such that. ... c) Give an example of a sequence {φj}j=1 C0 (Rn) such that.

13 May 2016: p. k, where k is the smallest value ofn such that xn 6= yn. ... Hint:first find a subsequence (fni ) such that fni (x) converges for all x Q [0, 1].)Deduce that the closure of B in (C[0, 1], d) is compact.

10 Nov 2014: linearfunctional θ V such that θ(x) 6= θ(y).(b) Suppose that V is finite dimensional. ... If no such subspaceexists, show that there exists a basis 〈v,α(v),.