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25 Jan 2024: random variables, independent of N. Show that if g(s,x) is a function and Tj are thejump times of N then. ... E[exp{θNtj=1. g(Tj,Xj)}] = exp{λ t0. (E(eθg(s,X)) 1)ds}. This is called Campbell’s theorem.(b) Cars arrive at the beginning of a long

9 Nov 2023: closed: if tj B with tj tfin then tfin B. Relies on higher orderCk estimates for all the quantities and Arzela-Ascoli theorem to showconvergence. ... the gauge assumptions need to be constructed for the extendedspacetime. closed: if tj B with tj tfin

16 Jan 2024: The Ward Correspondence and StationaryAxisymmetric Spacetimes. Grigalius Taujanskas. Mathematical InstituteOxford University. Radcliffe Observatory QuarterOxford OX2 6GG, UK. Contents. 1 Introduction 2. 2 Mathematical Background 32.1 Setting. 32.2

31 Jan 2024: Groups, Rings, and ModulesOscar Randal-Williams. Based on notes taken by Dexter Chua. https://www.dpmms.cam.ac.uk/or257/teaching/notes/grm.pdf. 1 Groups 11.1 Basic concepts. 11.2 Normal subgroups, quotients, homomorphisms, isomorphisms. 21.3 Actions

20 Dec 2023: Wedefine. A〈/r,s/〉 :=. jZ. ajj. jZ. Aj : lim|j|. |aj|tj = 0 if s 6 t 6 r. ... and give it the norm jZ ajj. := sups6t6r supjZ |aj|tj.Of course this norm can be written in the somewhat less symmetric form.

25 Oct 2023: Indeed,. pn(t) =nj=0. (n. j. )f(j/n)tj(1 t)nj. (ii) ‖pn f‖ 0 as n.

25 Oct 2023: n. j. )f(j/n)tj(1 t)nj. (ii) Automatically,. EYn = EX1 X2 Xn.