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17 May 2024: pdf]. I. Kontoyiannis. "Some information-theoretic computations related to the distribution of prime numbers." In Festschrift in Honor of Jorma Rissanen, (P. ... Mutual information, synergy, and some curious phenomena for simple channels." 2005 IEEE

8 Mar 2024: 2(1. 71). Find an element of OK with norm 2a 3b 5 for some. ... a,b 0. Hence prove that the class group of K is cyclic and find its order.

3 Apr 2024: Show further that if f : X Y is a continuous function between Polish spacesand f is injective on a Borel set B X, then f(B) is Borel. ... 14. Prove that a set A (in some Polish space) is analytic if and only if there exist closedsets An1,.,nk (indexed by

23 Feb 2024: 2(1. 71). Find an element of OK with norm 2a 3b 5 for some. ... a,b 0. Hence prove that the class group of K is cyclic and find its order.

31 Mar 2024: Show further that if f : X Y is a continuous function between Polish spacesand f is injective on a Borel set B X, then f(B) is Borel. ... 14. Prove that a set A (in some Polish space) is analytic if and only if there exist closedsets An1,.,nk (indexed by

11 Jan 2024: 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.

11 Jan 2024: 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}.

15 Jan 2024: 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.

7 Mar 2024: space. What if 𝑠 1/2? Exercise 6. a) Suppose 𝑠 = 𝑛2 𝛾 for some 0 < 𝛾 < 1. ... ANALYSIS OF FUNCTIONS EXAMPLE SHEET 4. b) Show that if 𝑠 = 𝑛2 𝑘 𝛾 for some 𝑘 Z>0, 0 < 𝛾 < 1, then.

15 May 2024: O. Randal-Williams, Some phenomena in tautological rings of manifolds.Selecta Mathematica 24 (4) (2018) 3835-3873. ... Abelianisations of the mapping class groups of some highly-connected manifolds – Rice,Houston, 17th February 2014.

11 Mar 2024: f(B) then y = f(a) for some a A andy = f(b) for some b B but then f(a) = y = f(b) so a = b A Band y ... If y f(A) f(B) then y = f(a) for some a A and y = f(b) for someb B.

17 May 2024: C93. “Combining information from different sources: Some surprising phenomena,” with B. ... C95. “Mutual information, synergy and some curious phenomena for simple channels,” with B.

17 May 2024: C93. “Combining information from different sources: Some surprising phenomena,” with B. ... C95. “Mutual information, synergy and some curious phenomena for simple channels,” with B.

8 Mar 2024: a1α1. adαdq. for some a1,. ,ad Z. Proof. We take. q =(disc(α1,. , ... In what follows, we will exhibit asuitable α, but this requires some preparations.

16 Jan 2024: γABA′B′ = φABεA′B′ ψA′B′εAB. for some symmetric spinors φ an ψ. ... for some complex Zα, α = 0, 1, 2, 3, where Z2 6= 0.

22 Jan 2024: 2 Background on quadratic formsWe collect together some standard definitions and results on quadratic forms. ... and henceto see that they satisfy some interesting symmetries (see Corollary 5.8).

31 Jan 2024: 223.4 The Galois correspondence. 25. 4 Some group theory 284.1 Free groups and presentations. ... H|{y0}[i/N,(i1)/N] : {y0} [iN ,. i1N ] X. lies entirely inside some Uα.

31 Jan 2024: If g2H = g′2H, then we know g. ′2 = g2 h for some h H. ... So we have to show p - ||. This is just some basic counting: we have. ||

5 Jun 2024: For injectivity, let τ NFalg(A)n be a cyclesuch that ψ[τ] = 0, i.e., τ(n1,⟨0⟩) = b (1)n1b for some b A. ... By the firststep in the proof of Proposition 3.18, τW = τb+(1)dn b for some b Wh(M).

29 May 2024: 0. for some valuation v. Then v(p q) = 1 and v(p r) = 0. ... Finally, assume that dom(f) 6= X. Then dom(f) = Ix for some x X.