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GEOMETRY AND GROUPS – Example Sheet 2TKC Michaelmas 2006 ...
https://www.dpmms.cam.ac.uk/study/II/Geometry%2BGroups/2006-2007/Exercise2.pdf8 Nov 2006: J : z 7 az bcz d. for some complex numbers a, b, c, d with ad bc = 1. ... For which choices of a, b, c, d is this map J aninvolution, that is J 2 = I? -
Geometric group theoryExample sheet 1 Lent 2023 Questions marked ...
https://www.dpmms.cam.ac.uk/files/study/III/GeometricGroupTheory/2022-2023/2023%20GGT%20Sheet%201.pdf16 Feb 2023: 5. Consider the (2,3)-Baumslag–Solitar group BS(2, 3) = 〈a,b | ba2b1a3〉. ... You may use the simplicial approximation theoremwithout proof.]. (b) By considering the preimage of the midpoint of I in the disc, provethat some term ai or bi of g is -
ANALYSIS II—EXAMPLES 1 Mich. 2014 Please email comments, corrections…
https://www.dpmms.cam.ac.uk/study/IB/AnalysisII/2014-2015/14sheet1.pdf20 Oct 2014: 4. Let (fn) be a sequence of real-valued continuous functions on a closed, bounded interval [a,b], and. ... a) If Dn is the set of discontinuities of fn and D is the set of discontinuities of f, show that D n=1j=nDj.(b) Suppose that for some finite -
GEOMETRY AND GROUPS – Example Sheet 1TKC Lent 2010 ...
https://www.dpmms.cam.ac.uk/study/II/Geometry%2BGroups/2009-2010/Exercise1.pdf4 Feb 2010: Let u =(. ac. ), v =. (bd. )for some integers a, b, c, d with ad bc = 1. ... Show that every vector. v Z Z can be written as mu nv for some integers m and n. -
GEOMETRY AND GROUPS — Example Sheet 1TKC Michaelmas 2006 ...
https://www.dpmms.cam.ac.uk/study/II/Geometry%2BGroups/2006-2007/Exercise1.pdf8 Nov 2006: Let u =(. ac. ), v =. (bd. )for some integers a, b, c, d with ad bc = 1. ... Show that every vector. v Z Z can be written as mu nv for some integers m and n. -
STATISTICAL MODELLING Part IIC / Michaelmas 2023Example Sheet 4 ...
https://www.dpmms.cam.ac.uk/study/II/StatisticalModelling/2023-2024/ExampleSheet4.pdf27 Nov 2023: ni=1. µ̂i =. ni=1. Yi. 2. Suppose that for some strictly increasing function f, we have. ... for some β that you should specify. (b) How can we estimate β from the data (Yi, xi)ni=1? -
ANALYSIS II (Michaelmas 2010): EXAMPLES 3 The questions are ...
https://www.dpmms.cam.ac.uk/study/IB/AnalysisII/2010-2011/anII_ex_2010_3.pdf19 Nov 2010: 5. (i) Suppose that f : R2 R is such that f/x is continuous in some open ball around (a,b),and f/y exists at (a,b). ... 11. Show that there is a continuous square-root function on some neighbourhood of I in Mn;that is, show that there is an open ball B(I; -
ANALYSIS II—EXAMPLES 4 Mich. 2015 Please email comments, corrections…
https://www.dpmms.cam.ac.uk/study/IB/AnalysisII/2015-2016/15sheet4.pdf30 Nov 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. ... for some K and all t [a,b], x,y BR(x0).We showed in lecture that for each t0 [a,b], there is a unique f C([a,b]; -
ANALYSIS II—EXAMPLES 1 Mich. 2015 Please email comments, corrections…
https://www.dpmms.cam.ac.uk/study/IB/AnalysisII/2015-2016/15sheet1.pdf16 Oct 2015: 5. Let (fn) be a sequence of real-valued continuous functions on a closed, bounded interval [a,b], and. ... a) If Dn is the set of discontinuities of fn and D is the set of discontinuities of f, show that D n=1j=nDj.(b) Suppose that for some finite -
Why do we need the real numbers?
https://www.dpmms.cam.ac.uk/~wtg10/reals.html3 Sep 2003: What does that mean? Well, if f(x)=u and f(y)=v and z=axby for some a,b> 0 with ab=1 then f(z) ought to be close ... there must be some c between a and b such that f(c)=w. -
Lent Term 2024 O. Randal-Williams IB Groups, Rings, and ...
https://www.dpmms.cam.ac.uk/study/IB/GroupsRings%2BModules/2023-2024/Sheet3.pdf15 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. -
ANALYSIS II—EXAMPLES 3 Mich. 2018 Please email comments, corrections…
https://www.dpmms.cam.ac.uk/study/IB/AnalysisII/2018-2019/18sheet3.pdf13 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 -
ANALYSIS II—EXAMPLES 4 Mich. 2018 Please email comments, corrections…
https://www.dpmms.cam.ac.uk/study/IB/AnalysisII/2018-2019/18sheet4.pdf23 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. -
Metric and Topological Spaces Easter 2016 Example Sheet 2 ...
https://www.dpmms.cam.ac.uk/study/IB/MetricTopologicalSpaces/2015-2016/MetTop_2016_Ex2.pdf16 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. -
1 Metric & Topological Spaces, sheet 2: (2006) 1. ...
https://www.dpmms.cam.ac.uk/study/IB/MetricTopologicalSpaces/2005-2006/2006sheet2.pdf4 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. -
ANALYSIS II—EXAMPLES 3 Mich. 2015 Please email comments, corrections…
https://www.dpmms.cam.ac.uk/study/IB/AnalysisII/2015-2016/15sheet3.pdf13 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 -
ANALYSIS II—EXAMPLES 3 Mich. 2014 The questions marked with ...
https://www.dpmms.cam.ac.uk/study/IB/AnalysisII/2014-2015/14sheet3.pdf21 Nov 2014: Eo = {x X : B(x) E for some > 0}. ... and that. E = {x X : xn x for some sequence (xn) in E}. -
Metric and Topological Spaces Easter 2015 Example Sheet 2 ...
https://www.dpmms.cam.ac.uk/study/IB/MetricTopologicalSpaces/2014-2015/MetTop_2015_Ex2.pdf31 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. -
Erdös-Rado without choice Thomas ForsterCentre for Mathematical…
https://www.dpmms.cam.ac.uk/~tef10/erdosrado.pdf11 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. -
ANALYSIS II—EXAMPLES 4 Mich. 2014 The questions marked with ...
https://www.dpmms.cam.ac.uk/study/IB/AnalysisII/2014-2015/14sheet4revised.pdf27 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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