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5 Jun 2020: 15, 18]. This means that there is a σ-finite measure ψ on(X,B) such that, for any A B with ψ(A) > 0, and any x X,. ... There is a function V : X [1,], a small set C X,and constants δ > 0, b < , such that:.

7 May 2013: Remark 2.2. — There exist coefficient systems B M such that. ... Let β H(M; (m )B) be a class such that (m )(β) = 0 H(M; B).

6 Aug 2021: b) Find a sequence fi R([a,b]) such that (fi)iN is Cauchy with respect tothe L2 norm, but for which there is no f R([a,b]) such that ... Proof. Since is open, there exists ,x such that the ball B(x) = {y Rn : |y x| < }is contained in.

5 Apr 2007: It follows that vi is in particular bounded,i.e. there exists an R such that vi B(R). ... One direction is obvious. For the other, note that if U is an arbitraryopen set, and q = T (p) for p U, then there exists a δ such that p B(

19 May 2007: 1 log(α1)m such that |N (x) N (x′)| > αn for all but at most γ|B|2. ... Then there must exist for this r a set B such thatg(B) γ1b(B) > αrm2, which implies that |B| > αr/2m and b(B) 6 γ|B|2.

5 Jun 2020: Ân such that Ln(xn1 ) log Qn(B(xn1,D)) for allxn1 An. ... Conversely, for any sequence of “admissible” distributions {Qn}, there is a sequenceof codes {Cn,Ln} operating at distortion level D, such that, for (almost) any realization ofthe source,

4 Feb 2018: Show that for any b C, there exists a sequence of points zn D(a,R)with zn 6= a such that zn a and f(zn) b as n. ... Find such a sequence when f(z) = e1/z, a = 0 and b = 2.[A much harder theorem of Picard says that in any neighbourhood of an essential

25 Oct 2023: Theorem 4.2. If f : [a,b] [a,b] is a continuous function, then we canfind a w [a,b] such that f(w) = w. ... Supposethat we can find a a0 < a1 < < an b such that, writing σ = ‖fP‖we have either.

4 Feb 2023: a) Construct a continuous surjection p : T2 T2 from the torus to itself such that for every x T2, p1(x)consists of exactly two points, and such that p is ... Such a map is calleda covering map.]. (b) Show that T2 also admits a covering map p̂ : T2 K to

7 Dec 2022: Use (ii) to show that there exists b K such that det(σiσjg1(b)) 6=0. ... Deduce that if y = g1(b) then {σ(y) | σ G} is a K-basis for L.Such a basis {σ(y)} is said to be a normal basis for