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15 Oct 2010: On the other hand, show that if fn does not converge uniformly to f, then we can find aconvergent sequence xm x in [a, b] such that fn(xn) does not ... Give an example of a sequence of uniformly continuousfunctions fn on R, such that fn converges

23 Jun 2012: 2) Let (Rn,d) denote Euclidean n-space. If P,Q,R are points in Rn such that. ... For any sequence (gn) Y such that gn g X, show that g Y.

21 May 2005: iii) Give an example of disjoint closed sets A and B in R2 with points an A and bn B such. ... 10. [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

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

18 Oct 2011: On the other hand, show that if fn does not converge uniformly to f, then we can find aconvergent sequence xm x in [a,b] such that fn(xn) does not ... Give an example of a sequence of uniformly continuousfunctions fn on R, such that fn converges

20 Oct 2014: metric relation R defined on X such that for any k elements x1,. ... Prove that there existu,v Zp such that. |A (uB v)| αβp αβp, where uB v denotes the set{ub v : b B}.

4 May 2006: Prove that f has a real root. 17. Prove there is no continuous function f : [0, 1] R such that x Q f (x) 6 Q (whereQ denotes the rational numbers). ... 18. A continuous surjective map p : E B is a covering if every point b B has an openneighbourhood V

11 Mar 2007: Then there is asubset B A of size β such that all the unordered γ-tuples from it are in thesame piece of the partition. ... an1, ν〉 such that 〈a1. an1, ν, c〉 and 〈a1. an1, ν, c′〉 are coloureddifferently by.

21 May 2005: n=1 = (x1,x2,x3, ) such that for each n, |xn| 1/n. ... In such circumstances, T is a bounded linear operator. (What is the moral of this equivalence?)Let B(E,F) be the space of bounded linear operators equipped with the operator

4 Nov 2005: n=1 = (x1,x2,x3, ) such that for each n, |xn| 1/n. ... In such circumstances, T is a bounded linear operator. (What is the moral of this equivalence?)Let B(E,F) be the space of bounded linear operators equipped with the usual

6 May 2011: c) Give a topology on R which is neither trivial nor discrete such that every open set isclosed (and vice-versa). ... Prove the following:. (a) G is homogeneous: given any x, y G there is a homeomorphism φ : G G such thatφ(x) = y.

29 Apr 2010: c) Give a topology on R which is neither trivial nor discrete such that every open set isclosed (and vice-versa). ... Prove the following:. (a) G is homogeneous: given any x, y G there is a homeomorphism φ : G G such thatφ(x) = y.

29 Jan 2010: The irreducibility of Sp(χ) for such a prime pin conjunction with a) implies that Φq,χ = 0. ... Since GẐ acts transitively on |MK |,. there exists a positive constant cn Q such that for every x |M|,.

27 Feb 2015: Does there exist c (a,b)such tat f(b) f(a) = f′(c)(ba)?(v) Let f: Rn Rn be a C1 map with ‖Df(x)I‖ 1/2 for ... continuous function g: Bε(I) Mn such that g(A)2 = A for all A Bε(I).

4 May 2006: b) Give a topology on R which is neither trivial nor discrete such that every open set isclosed (and vice-versa). ... a) Prove (R, TS ) is Hausdorff. (b) Prove (R, TS ) is not metrisable.

20 Oct 2014: Prove that there are. subsets A X and B G such that |η(A,B) δαβ| θ, where θ > 0 depends on γ andδ only. ... x1,. ,xk such that no xi xj with i,j distinct is a quadratic residue mod p.

16 Jul 2014: singularities is a real manifold Bwith an open subset B0 B, with := B B0 of codimension 2, such that B0 has the structure of an integral affine manifold.We also allow ... X Spec k[[t]]. such that the special fibre satisfies. X0 =. σPmax. Pσ. and the

25 Apr 2002: 3. dy. 2. eyf in y and finally (the most important part) cleverly choose u and v in such a way that d. ... d. 2. equal to 3e, or it may be that, although such a choice exists, the dependence of u and v on a, b and c is so complicated that we

20 Sep 2004: And yet there doesn't seem to be such a fact of the matter. ... Similarly, if B={x:Q(x)} then the statement "A cap B = emptyset" means nothing more than that there is no x such that P(x) and Q(x).

27 Oct 2023: 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) c log b d.

11 May 2001: 1/2. , so two such numbers must multiply to something of modulus at least 5. ... Analogues of Euclid's algorithm work in other contexts, such as the ring of all polynomials with rational coefficients.

14 Jun 2018: an〉 such that for all i < n 1, ai 6= ai1, and that〈a0,. ... C0 I is symmetrical. C1 For two distinct elements a and b there is a unique c such that I(a,c) andI(b,c).

5 Nov 2009: Then there is a subset B A of size β such that all theunordered γ-tuples from it are in the same piece of the partition. ... b) the property of relative largeness which plays such a central rôle inParis -Harrington is ill-typed in at least two senses.

19 Dec 2014: if B = {min(Xα) : α < κ} is stationary.(iii) If {Xα : α < κ} is pairwise disjoint, then there exists a [κ]κ such that. ... Prove that there exists a family F of2κ stationary subsets of S such that if A 6= B F , then A B and B A arestationary in

19 Dec 2014: i) Prove that 20 = 1 implies the following assertion SD :there exist sets A and B such that R2 = AB and Al and Bm are countablefor every horizontal line l and ... exist x1 and x2 such that x1 6 f(x2) and x2 6 f(x1).

7 Aug 2002: A formal statement of the result to be proved. Let X and Y be metric spaces and let f be a function from X to Y such that f. ... in U (for the reason I have just given) and such that [f(y) is in U] implies [d(f(x),f(y)) < a].

25 Nov 2002: We also know that CxB is a subset of AxB, and that the set-theoretic difference AxB-CxB consists of all ordered pairs (m,x) such that x is in B. ... me that if m> n then there is a unique positive integer t such that nt=m.

9 Apr 2012: Obviously we want I and II be the least fixedpoints for these two operations, and since P and b are both monotone functionson the complete poset 〈V,〉 there will be such ... More generally, butless idiomatically, we can write:. There is no -least set

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,)?

29 Mar 2007: The intuitionistic correctness of wffs such as. 4. A (B C). ... B (A C) reveals an infinite family of terms such as.

23 Mar 2007: Fix some subset B of [n]. How large a family A of subsets of [n] can we find such that the intersection of any two sets in A contains a cyclic ... Chung, Graham, Frankl and Shearer have proved that, in the case where B is a block of length t, we can do

12 Oct 2023: Mich. 2023 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 not converge uniformly to f thenwe can find a convergent sequence xm x in [a, b] such that fn(xn) 6

29 Jan 2010: Therefore there are constants a, b, c k such that if v, v′ Spand w IndGB δ1/2, then. ... π′′ = I(1)0, then we can choose w such that HomGw (πw.

29 May 2010: Let Γ Aut(X′) be a finite subgroup such that f factorsas X′ X′/Γ g X, where g is a radicial alteration. ... Then one knows that there exists a finite extension K′/K such that ρ(IK′) isunipotent.

16 Jul 2014: Mark Gross Mirror symmetry and the SYZ conjecture. Definition. A tropical affine manifold with singularities is a real (C 0) manifoldB along with an open subset B0 B such that := B ... For each integral point v of a two-face, we can choose anopen

23 May 2001: then construct a new sequence v in such a way that the first two terms of v mean that it is not a multiple of v. ... If the vector space V is infinite-dimensional, then this means that V contains a subset B such that every vector v in V is a linear

5 Nov 2003: AxB, the Cartesian product of A and B, is defined to be the set of all ordered pairs (x,y) such that x is an element of A and y is ... possible such that r is non-negative) with the same highest common factor.

9 Jun 2003: To give a simple example: if I take two subsets A and B, I can define their union to be the set of all n such that n belongs either to ... a. 1. ,b. 1. ] is any closed interval such that a.

10 Mar 2017: Provethat A contains an infinite set B such that no point of B is a convex combination of otherpoints of B. ... 6. Show that every graph G has a partition of its vertex-set as X Y such that the numberof edges from X to Y is at least 12e(G).

29 Jan 2021: 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.

23 May 2024: Tho15, Lemma 2.8]):. Lemma 2.11. There exists a non-zero N0 OF such that for all primes p and for all b B(Op) wehave N0 κb V (Op). ... There exists b′ B(Op) such that ordp (b tb′) = 1,where t is a uniformiser of Op.

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.

18 Feb 2015: ri 1 such thati ri = r. For example, in the quaternionic case H = B the plectic reflex Galois group isisomorphic to (St n ΓtF ) (Srt n Γ. ... The cup productof two such classes is an element g1 g2 H2(Y, 2).

29 Jan 2010: W2 E = G. Q(1). and GrWj E = 0 for j > 0 or j < 2, such that. ... The reader who is sceptical of the general finiteness of groups such as IIIwill be reassured that we do not require equality in (b).).

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.

7 Sep 2017: On the other hand, as the signature andEuler characteristic agree modulo 2, and the latter is multiplicative, it is true thatσ(E) σ(B) σ(F) mod 2 for all such ... More recently, Hambleton,Korzeniewski, and Ranicki [5, Theorem A] proved that σ(E) σ(B)

22 Oct 2006: iii) For any two subsets A, B V (G) the number of pairs (x, y) A B such thatxy E(G) differs from p|A||B| by at most c2n2. ... number of pairs (x, y) A B such.

11 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

24 Jan 2024: 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.

25 Jun 2015: Foreach open set A Ψ(U )cs and each M A, we shall show that there is a smalleropen neighbourhood M B A Ψ(U )cs such that. ... Then we may choose smooth functions ϕi : M [0, 1] such that supp(ϕi) int(Vi) and p(ν K) (. ϕi)1(1) and choose B =. B(A, ϕ1