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  2. Lent Term 2024 O. Randal-Williams IB Groups, Rings, and ...

    https://www.dpmms.cam.ac.uk/study/IB/GroupsRings%2BModules/2023-2024/Sheet3.pdf
    15 Jan 2024: Could there be some other Euclidean function φ making Z[3] into a Euclidean domain? ... iii) Show that if R is a PID, the gcd of elements a and b exists and can be written as rasbfor some r,s R.
  3. ANALYSIS II—EXAMPLES 3 Mich. 2018 Please email comments, corrections…

    https://www.dpmms.cam.ac.uk/study/IB/AnalysisII/2018-2019/18sheet3.pdf
    13 Nov 2018: Eo = {x X : B(x) E for some > 0}. ... 8. Use the Contraction Mapping Theorem to show that the equation cos x = x has a unique real solution.Find this solution to some reasonable accuracy using a calculator (remember to
  4. ANALYSIS II—EXAMPLES 4 Mich. 2018 Please email comments, corrections…

    https://www.dpmms.cam.ac.uk/study/IB/AnalysisII/2018-2019/18sheet4.pdf
    23 Nov 2018: Suppose thatsup. [a,b]BR(x0)‖F‖ R(b a)1 and that ‖F(t,x) F(t,y)‖ K‖x y‖ for some. ... 9. (Some useful properties of the operator norm). Recall that the operator norm on the.
  5. Metric and Topological Spaces Easter 2016 Example Sheet 2 ...

    https://www.dpmms.cam.ac.uk/study/IB/MetricTopologicalSpaces/2015-2016/MetTop_2016_Ex2.pdf
    16 May 2016: 7. (a) If f : [0, 1] [0, 1] is continuous, show there is some x [0, 1] withf(x) = x. ... For eachinteger n > 1, show that there is some x [0, 1] with f(x) = f(x 1.
  6. 1 Metric & Topological Spaces, sheet 2: (2006) 1. ...

    https://www.dpmms.cam.ac.uk/study/IB/MetricTopologicalSpaces/2005-2006/2006sheet2.pdf
    4 May 2006: Prove in this case thata map f from the unit interval [0, 1] to B may be “lifted” to E: there is some g : [0, 1] Esuch that p g = f. ... c) From above, we know (x, P ) Cl(Γ) for some x X.
  7. ANALYSIS II—EXAMPLES 3 Mich. 2015 Please email comments, corrections…

    https://www.dpmms.cam.ac.uk/study/IB/AnalysisII/2015-2016/15sheet3.pdf
    13 Nov 2015: Eo = {x X : B(x) E for some > 0}. ... 8. Use the Contraction Mapping Theorem to show that the equation cos x = x has a unique real solution.Find this solution to some reasonable accuracy using a calculator (remember to
  8. ANALYSIS II—EXAMPLES 3 Mich. 2014 The questions marked with ...

    https://www.dpmms.cam.ac.uk/study/IB/AnalysisII/2014-2015/14sheet3.pdf
    21 Nov 2014: Eo = {x X : B(x) E for some > 0}. ... and that. E = {x X : xn x for some sequence (xn) in E}.
  9. Metric and Topological Spaces Easter 2015 Example Sheet 2 ...

    https://www.dpmms.cam.ac.uk/study/IB/MetricTopologicalSpaces/2014-2015/MetTop_2015_Ex2.pdf
    31 Aug 2015: 7. (a) If f : [0, 1] [0, 1] is continuous, show there is some x [0, 1] withf(x) = x. ... For eachinteger n > 1, show that there is some x [0, 1] with f(x) = f(x 1.
  10. Erdös-Rado without choice Thomas ForsterCentre for Mathematical…

    https://www.dpmms.cam.ac.uk/~tef10/erdosrado.pdf
    11 Mar 2007: ordering <a—which is initialised to <K. We use a to discard some orderedpairs. ... abbreviate to <<a. We will weed out some ordered pairs from <<a to obtain<a.
  11. ANALYSIS II—EXAMPLES 4 Mich. 2014 The questions marked with ...

    https://www.dpmms.cam.ac.uk/study/IB/AnalysisII/2014-2015/14sheet4revised.pdf
    27 Feb 2015: for some constant Kand an open subset U of Rm containing a, |F(t,x) F(t,y)| K‖xy‖ for all t [0, 1],x,y U. ... E is of the form (a,b), [a,b), (a,b] or [a,b] for some a,b with a b.

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