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12 Oct 2021: The least such n iscalled the order of g.). (b) Show that there exists a positive integer n such that gn = e for all g G. ... θ(g) = e for all g D2n. Find all homomorphisms D2n Cnwhen n is even.

25 Aug 2021: of Jb(F) of maximal volume.(2) Show that all the stabilizers have the maximal volume. ... We now fix a choice of Haar measure on G(F) such that all Iwahori sub-groups of G(F) have volume 1.

29 Oct 2021: Let X be a Markov chain containing an absorbing state s to which all other states lead (i.e., j sfor all j). ... i < b, where 0 < λi, µi < 1 for all i, and λi µi = 1 for 1 i < b.Show that this process is time-reversible in equilibrium.

29 Nov 2021: 5. Show that T B(H) is normal iff ‖Tv‖ = ‖Tv‖ for all v H. ... 13. Let U B(H) unitary. Show that for all x H, the sequence n1n1.

30 Oct 2021: Assume that for some K > 0 we have. |ψ(t,x) ψ(t,y)|6 K‖xy‖ for all t [a,b] and all x,y BR(z). ... xn1) BR(z). Then ϕ is continuous and satisfies‖ϕ(t,x) ϕ(t,y)‖6 (K 1)‖xy‖ for all t [a,b] and all x,y BR(z).

16 Mar 2021: Homotopy Theory, Examples 4. Oscar Randal-Williams. Lent 2021. All cohomology is with Z/2-coefficients unless otherwise specified. ... i) Show that Sqi : Hni(M) Hn(M) = Z/2 is zero for all i > 0.

15 Oct 2021: a) f L1(Rn), g Ck(Rn) with supRn |Dαg| < for all |α| k. ... Deduce Hölder’s inequality:Rn|f(x)g(x)|dx ||f||p ||g||q , for all f L. p(Rn), g, Lq(Rn).

6 Aug 2021: Define τxφ by:. τxφ : C,y 7 φ(y x). (1.2). Then there exists > 0 such that τxφ Ckc () for all x B(0). ... for all x (a,b).Then: b.

8 Aug 2021: x. for all x so (λA)B = λ(AB). Again(. A(λB)). x = A(. ( ... for all x Rn so A B = B A).(iii) Observe that.

21 Nov 2021: Thus B(y,r) E for all r > 0 and E is not. ... as m. Thusa B(xj,δj) Uj. for all j 1 and. a j=1.