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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.

9 Oct 2018: 4. Find the largest product of (some not necessarily distinct) natural numbers whose sum is 100. ... For each of claims (a)–(f) below, give aproof or counter-example. (a) f(AB) = f(A)f(B) (b) f1(C D) = f1(C)f1(D)(c) f(AB) =

13 Dec 2018: where the superscript denotes an X derivative of some de-gree, and h is any function. ... If we assume that the solution is of order a, then ak- a X(2kAo)a, where X is some constant that will vary withrealizations.

22 Jan 2018: EA(x,t) = cAtn/2 exp(x Bx/4t). and zero elsewhere, for some B Matn(C) and constant cA, both of which will obviously depend on A.For EA to ... Hence. sing supp F {. (t,x) : x p|p|. t = 0 for some p 6= 0}. =

20 Nov 2018: The square matrices A and B over the field F are congruent if B = PTAP for some invertible matrixP over F. ... for some λ,µ,ν R (which should turn out in this example to be integers).Does there exist a linear transformation which reduces the pair of

8 Oct 2018: 2. Show that the following sets of subsets of R all generate the same σ-algebra:(a) {(a,b) : a < b}, (b) {(a,b] : a < b}, (c) {(,b] : b R}. ... 10. Let (E,E,µ) be a measure space. Call a subset N E null if N B for some B E withµ(B) = 0.

16 Jul 2018: and poorly behaved. We give results showing that some of these issues may. ... A property P of some class F of finitely generated groups is said.

12 Nov 2018: Such a candidate shall offer the papers currently listed in Regulation 24(b). ... Paper A3 may include questions on literary, cultural, linguistic, or historical topics, some of which may be based on more extensive material in the foreign language, as

25 Sep 2018: 2. Consider the three equations in six unknowns given by Ax = b, where. ... that x Rn is an extreme point of the set{x Rn : x 0,Ax b} if and only if for some z Rm,.

26 Oct 2018: Observatory. 19 pages. [With some references not yet given above] Adam Perkins, 1996. ... B. E. J. Pagel, 1999. Some reminiscences of early days at the RGO, Herstmonceux.