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Lent 2017 ANALYSIS I – EXAMPLES 3 AZ 1. ...
https://www.dpmms.cam.ac.uk/study/IA/AnalysisI/2016-2017/sheet3.pdf10 Mar 2017: 3. Prove Cauchy’s mean value theorem: let f,g : [a,b] R be continuous functionswhich are differentiable on the open interval (a,b); show that for some c ... Show thatif f ′(a) < y < f ′(b) for some a < b in I and y R, then there exists x I witha < -
Lent Term 2017 O. Randal-Williams IB Groups, Rings, and ...
https://www.dpmms.cam.ac.uk/~or257/teaching/IBGRM/2017/Sheet3.pdf19 Jan 2017: 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. -
RIEMANN SURFACES EXAMPLES 2 Michaelmas 2017 Comments on and/or ...
https://www.dpmms.cam.ac.uk/study/II/Riemann/2017-2018/es2.pdf4 Oct 2017: C/Λ2 are conformally equivalent if and only if thelattices are related by Λ2 = λΛ1 for some λ C. ... 5. Show that complex tori C/〈1,τ1〉 and C/〈1,τ2〉 are analytically isomorphic if and only if τ2 =. (aτ1 b)/(cτ1 d), for some matrix(a bc d -
RIEMANN SURFACES EXAMPLES 1 Michaelmas 2017 Comments on and/or ...
https://www.dpmms.cam.ac.uk/study/II/Riemann/2017-2018/es1.pdf4 Oct 2017: 8. By considering the singularity at or otherwise, show that any injective analytic map f : C Chas the form f(z) = az b, for some a C and b C. ... conformally equivalent). 10. Define an equivalence relation on C by z w iff z = 2sw for some s Z. -
MATHEMATICAL TRIPOS: PART IA Lent 2017 PROBABILITY JRNExample Sheet…
https://www.dpmms.cam.ac.uk/study/IA/Probability/2016-2017/pex1.pdf14 Feb 2017: 6. Let (An : n N) be a sequence of events in some probability space (,F,P). ... Set. A = {ω : ω An infinitely often}, B = {ω : ω An for all sufficiently large n}. -
RIEMANN SURFACES EXAMPLES 2 H. Krieger Lent 2017 Comments ...
https://www.dpmms.cam.ac.uk/study/II/Riemann/2016-2017/rs_ex2_17.pdf18 Jan 2017: C/Λ2 are conformally equivalent if and only if thelattices are related by Λ2 = λΛ1 for some λ C. ... 5. Show that complex tori C/〈1,τ1〉 and C/〈1,τ2〉 are analytically isomorphic if and only if τ2 =. (aτ1 b)/(cτ1 d), for some matrix(a bc d -
Numbers and Sets (2017–18) Example Sheet 1 of 4 ...
https://www.dpmms.cam.ac.uk/study/IA/Numbers%2BSets/2017-2018/examples-NS-17-1.pdf6 Oct 2017: ii) m n a b (n m) [(a = 1) (b = 1) (ab 6= n)]Is (ii) true? ... 5. How large can the product of some (not necessarily distinct) natural numbers be, if their sumis 100? -
PART II REPRESENTATION THEORYSHEET 1 Unless otherwise stated, all ...
https://www.dpmms.cam.ac.uk/study/II/RepresentationTheory/2016-2017/repex1.pdf10 Jan 2017: δ is abelian.(c) Assume that δ(g) = 1 for some g G. ... irreducible. 3 Find an example of a representation of some finite group over some field of charac-teristic p, which is not completely reducible. -
RIEMANN SURFACES EXAMPLES 1 H. Krieger Lent 2017 Comments ...
https://www.dpmms.cam.ac.uk/study/II/Riemann/2016-2017/rs_ex1_17.pdf18 Jan 2017: 8. By considering the singularity at or otherwise, show that any injective analytic map f : C Chas the form f(z) = az b, for some a C and b C. ... conformally equivalent). 10. Define an equivalence relation on C by z w iff z = 2sw for some s Z. -
Lent Term 2017 O. Randal-Williams IB Groups, Rings, and ...
https://www.dpmms.cam.ac.uk/study/IB/GroupsRings%2BModules/2016-2017/Sheet3.pdf23 Feb 2017: 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.
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