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Analysis II Example Sheet 1 Michaelmas 2006 MJW Corrections ...
https://www.dpmms.cam.ac.uk/study/IB/AnalysisII/2006-2007/ex1.pdf13 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,)? -
Analysis II Example Sheet 3 Michaelmas 2006 MJW Corrections ...
https://www.dpmms.cam.ac.uk/study/IB/AnalysisII/2006-2007/ex3.pdf10 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. -
Analysis II Example Sheet 4 Michaelmas 2006 MJW Corrections ...
https://www.dpmms.cam.ac.uk/study/IB/AnalysisII/2006-2007/ex4.pdf28 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. -
Representation Theory Sheet 4 4.1 Question Show that any ...
https://www.dpmms.cam.ac.uk/study/II/RepresentationTheory/2005-2006/sheet4.pdf13 Mar 2006: a) LetG be a compact group. Show that there is a group homomorphismρ : G On(R) if and only if G has ann dimensional representation overR.(b) Explicitly construct such a ... subset ofH given by matrices such thatb = 0, and letφ : Fp C be a non-trivial -
GEOMETRY AND GROUPS – Example Sheet 4TKC Michaelmas 2006 ...
https://www.dpmms.cam.ac.uk/study/II/Geometry%2BGroups/2006-2007/Exercise4.pdf24 Nov 2006: Show that, for such a map,. dimH f (M ) 61α. ... Prove that the limit set for such a group is totallydisconnected. -
Example Sheet 3, Geometry 2006 pmhw@dpmms.cam.ac.uk (1) Show the ...
https://www.dpmms.cam.ac.uk/study/IB/Geometry/2005-2006/Geom06.3.pdf7 Feb 2006: Prove that there exists such a reparametrization Γ̃ = Γffor which ‖dΓ̃/dt‖ is constant, namely l. ... 10) For a surface of revolution S, corresponding to a curve η : (a,b) R3 given byη(u) = (f(u),0,g(u)), where η is parametrized in such a way -
Michaelmas Term 2006 J. Saxl Linear Algebra: Example Sheet ...
https://www.dpmms.cam.ac.uk/study/IB/LinearAlgebra/2006-2007/sheet406.pdf30 Nov 2006: If A is the matrix of the first of these (say), find a non-singular matrix P such that P tAP is diagonalwith entries 1. ... 19. Let a1, a2,. , an be real numbers such that a1 an = 0 and a21 a. -
Michaelmas Term 2006 J. Saxl Linear Algebra: Example Sheet ...
https://www.dpmms.cam.ac.uk/study/IB/LinearAlgebra/2006-2007/sheet306.pdf13 Nov 2006: 1. 11. Show that if x 6= y are vectors in the finite dimensional vector space V , then there is a linear functionalθ V such that θ(x) 6= θ(y). ... If no such subspace exists,show that there exists a basis v,α(v),. -
COMPLEX ANALYSIS — Example Sheet 1TKC Lent 2006 The ...
https://www.dpmms.cam.ac.uk/study/IB/ComplexAnalysis/2005-2006/Exercise_1.pdf16 Feb 2006: for b1 6 y 6 b2.(b) Deduce that. R. f. x+ i. ... such piecewise continuously differentiable curves, andf : D C is an analytic function, then. -
1 Metric & Topological Spaces, sheet 2: (2006) 1. ...
https://www.dpmms.cam.ac.uk/study/IB/MetricTopologicalSpaces/2005-2006/2006sheet2.pdf4 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
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