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14 Sep 2023: The 2-Selmergroup is. (20) S(2)(J/k) = ker. (H1(k,J [2]). v. H1(kv,J). ). ... The fake Selmer group is S(2)fake(J/k) = S(2)(J/k)/〈c〉. The analogue of (20) is.

25 Oct 2023: 7Some other proofs are given in Exercises 20.6, 20.7 and 20.8. ... The proof is givenin Exercise 20.10 but is not part of the course.

10 Feb 2023: simple descriptions in terms of the homologyand Laplacian of a generalized reduced dual graph (Corollary 1.20). ... There has been considerable interest in “Jacobians of graphs” — for example, Lorenzini[19, 20], Bacher–de la Harpe–Nagnibeda [1]

31 Aug 2023: 20. Chapitre 7Dipôles Linéaires Passifs en RégimeVariable. I Relations courant-tension en convention récepteur. ... G dB = 20 log |H|. Son déphasage φ est défini parφ = arg H.

22 Aug 2023: Theorem 2.1.20 ([Hag08]). If ϕ : Y X is a local isometry, then:. ... Proof. One direction follows from Theorem 2.1.36, together with the π1-injectivity of localisometries stated in Theorem 2.1.20.

24 Oct 2023: Lemma 6.20. 1. If χ is an (irreducible) character, then so is χ̄. ... 20. 6.4 First projection formulaLecture 6starts here. We will now work our way towards proving Theorem 6.7.

12 Jan 2023: Example 2.20 (continuing 2.16). (i) Let G = PSL2(Z) y H2, and let. ... 20. are constants K,C, such that for all a,b MCG(S),. dMCG(S)(a,b) K,CYS.

1 Feb 2023: What is. it?We have dK = 20 so can take c =. 80π< 9. π< 3. So every ideal class contains. an ideal of norm 2. ... x1x2 = R. (Picture). 20. So apply Minkowski’s theorem, we need to choose a convex symmetric setcontained in this region.

25 Oct 2023: If |an1| n/2,then. P(t) = an1tn1 Q(t). 20. where Q(t) =n. ... Proof of Example 11.20. Let n = ′n ′′n and Kn = K′n K′′n where.

28 Nov 2023: 20. 3. If X is a Banach space and A = B(X), then for T A the spectrum of T inA has the usual meaning: λ σA(T) if and only