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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)

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.

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.

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 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.

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)?

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.

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

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