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10 May 2002: A formal definition could be something like this. A multivalued function from A to B is a subset H of AxB such that for each a in A there is at ... least one b in B such that (a,b) is in H.

19 Feb 2024: 8. Find two totally ordered sets such that neither is isomorphic to a subset of the other.Can you find three such sets? ... Is there a countable ordinal α such thatα is the αth fixed point?

24 Nov 2001: 8. A real number is a partition of the rational numbers into two sets A and B such that every element of A is less than every element of B. ... B such that (a,b) is in F.

11 Oct 2023: Let A and B be subspaces of a vector space V such that V = A B, and let ‖.‖1and. ... Construct two normed spaces X and Y such that d(X, Y ) = 1 but X and Y are not.

8 May 2003: n. ) with certain properties, such that a. n. tends to infinity, then one can first try to get every a. ... n. ) such that b. n. /a. n. tends to zero and Sum.

8 Feb 2024: 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.

4 Oct 2001: Of course, we could say something like there exists a constant lambda such that a. ... x[y]-y[x]]. 2. =0. I put the square in because one always likes to expand such an expression in terms of inner products.

16 Nov 2023: 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?

26 Mar 2003: Let A be a set that is NOT. Then we can find a subset B of {1,2,.,N} with |B|> cN and a function g:B-->C such that ... Zeilberger should not be disappointed if attempts to find such a proof eventually succeed.

8 Feb 2024: 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 Dec 2001: Then there is some point c in (a,b) such that f'(c) equals (f(b)-f(a))/(b-a). ... x. 2. <. x. n. =b such that the intervals corresponding to neighbouring x.

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

7 Nov 2001: Because of irritating difficulties such as the need to carry digits and to identify 0.999999. ... all y(n) such that y is in B.

20 Sep 2023: Provethat A contains an infinite set B such that no point of B is a convex combination of otherpoints of B. ... Show also that there is a partitionof the vertex-set as X Y such that each of e(X) and e(Y ) is at most 13e(G).

6 Mar 2002: b in B (that is, some non-zero b (mod p)) such that the set of all bx. ... If, on the other hand, we cannot find such a b, then A=B and the theorem is proved.

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

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}.

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

29 Jan 2002: Once again let us consider a particular example, such as the equation x. ... What we then discovered was that if we take a new number b, given as a root of a polynomial with its coefficients in Q(a) (in such a way that

29 Oct 2018: Fix j, let > 0 and choose a partition P = {a = a0 < a1 <. < an = b} such thatU(P,f) L(P,f) < j1. Let K = {k : Ej (ak,ak1) 6= }. Then Ej ... an = b} be any partition of [a,b] such thataj1 aj < δ, and let J = {j : [aj,aj1] F 6= }. Show that sup[aj,aj1] f

24 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.

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

12 Oct 2018: x). On the other hand, show that if fn does not convergeuniformly to f then we can find a convergent sequence xn x in [a,b] such that fn(xn) 6 ... a) Must there exist a sequence (fn) of continuous functions on [0, 1] such that fn f uniformly on [0, 1]?(b)

10 Nov 2006: Does it follow that every open set must be. closed? Give an example of such a metric space. ... Show that thereexists a norm on C(I) such that the map sending f to y0.

28 Nov 2006: ii) Suppose that f : R2 R is such that D1f = f/x exists and is bounded near (a,b), and. ... is a function l : U M such that exp(l(A)) = A for all A U.

27 Nov 2023: X̃1, that is, β̃1 = (X̃T1 X̃1). 1X̃T1 Ỹ. (c) Find a condition on X0 and X1 such that β̃0 = β̂0. ... b) The model is to be reparametrised in such a way that it can be specialised to (i) two parallellines for the two groups, (ii) two lines with

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.

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.

20 Oct 2014: 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 ... a) Must there exist a sequence (fn) of continuous functions on [0, 1] such that fn f uniformly on [0, 1]?(b)?

13 Nov 2018: Supposethat there is a constant K such that ‖F(t,x) F(t,y)‖ K‖xy‖ for all t [0, 1] and all x,y B. ... Letx1,x2 Rn. By the Picard–Lindelöf theorem, we know that there is (0, 1] and differentiable functionsf1,f2 : [0,] B such that dfjdt = F(t,fj

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.

23 Oct 2017: 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

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.

21 May 2005: 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,

21 May 2005: 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,

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

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.

2 Nov 2015: Fix j, let > 0 and choose a partition P = {a = a0 < a1 <. < an = b} such thatU(P,f) L(P,f) < j1. Let K = {k : Ej (ak,ak1) 6= }. Then Ej ... an = b} be any partition of [a,b] such thataj1 aj < δ, and let J = {j : [aj,aj1] F 6= }. Show that sup[aj,aj1] f

16 Oct 2015: 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 ... a) Must there exist a sequence (fn) of continuous functions on [0, 1] such that fn f uniformly on [0, 1]?(b)?

29 Oct 2012: On the other hand, show that if fn does not. converge uniformly to f then we can find a convergent sequence xm x in [a, b] such that fn(xn) 6 ... valued functions (fn) on R such that fn converges pointwise to a function f which is continuous but not.

14 Oct 2021: y2= 0 (3). Construct a sequence of real analytic functions (uk)k=1 with uk : R2 R solving (4) such. ... u1. 1We write A B to mean that there exists a universal constant C such that A CB.

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

23 Oct 2023: Show that:. (a) There is a nondeterministic automaton N with n 1 states such that L(N) = Ln and(b) if D is a deterministic automaton with fewer than 2n states, ... Prove that there is a proper excerpt e of v such that xey L.

14 Oct 2021: Show thatS1M = 0 if and only if there exists an s S such that sM = 0. ... ii) There exists nonzero elements e1,e2 A such that e1e2 = 0, e21 = e1, e22 = e2 and e1 e2 = 1.iii) A is isomorphic to A1 A2 for some rings

24 Jan 2024: 3. Let f,g : R R be such that f(x) as x a and g(y) k as y. ... Construct a function f : R R that takes every value on every interval – in otherwords for every a < b and every c there is an x with a < x < b such

23 Nov 2018: Does it follow that thereexists c (a,b) such that f(b) f(a) = f′(c)(ba)? ... continuous function g: Bε(I) Mn such that g(A)2 = A for all A Bε(I).

17 Oct 2008: 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 f

3 Sep 2003: n. ) and (b. n. ) tending to root two from below and above respectively in such a way that f(b. ... there must be some c between a and b such that f(c)=w.

24 Nov 2006: Show that, for such a map,. dimH f (M ) 61α. ... Prove that the limit set for such a group is totallydisconnected.

8 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),