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24 Jan 2023: Further Questions. (12) Is every abelian group the additive group of some ring? ... a) Show that Z$ is a subring of the field Q of rational numbers.(b) Show that any subring R of Q is of the form Z$ for some set $

23 Jul 2023: In this talk, we will discuss some prospecting on how to userotation invariance to obtain new information on the critical random cluster models on Z2. ... Howdoes the answer depend on A,B,C? What if there are more than two species, with some

28 Nov 2023: converges to some operator Φ(f) B(H). This does not depend on the choiceof (sn): if (tn) is another sequence of simple functions converging to f, thenagain by (2), we ... Let U B(H) be a unitary operator.Then U = eiQ for some hermitian operator Q.

12 Sep 2023: If r = 3 then ker H = 〈e12,e13,ae23 be14〉 for some a,b k. ... If r = 3 then ker H = 〈e12,e23 ae13,e14 be13〉 for some a,b k.

29 Nov 2023: Thennecessarily ρ(0) = idV , and ρ(1) is some invertible linear map α on V. ... representation then ϕf,V = λ idV for some λ Cby Schur’s Lemma.

14 Sep 2023: By part (a),every such ν is the image of nΣX for some n. ... Y ||U|. as required. 2.2. Some stabilisers in G0 and their linear characters.

1 Feb 2023: The same forαβ. Proposition 1.7. Let α F be algebraic. Then for some integer b 1, bα is analgebraic integer. ... 5. Remark. Some people take (ii) as the definition of norm and trace.

7 Jan 2023: 50. Exercise 4.3.19. (i) We have au bv = 1 for some integers a and b so. ... If a, b S , then a = 10n 1, b = 10m 1 for some integers n, m 0.We have.

28 Nov 2023: It follows that every Banach algebra is isometricallyisomorphic to a closed subalgebra of B(X) for some Banach space X. ... B. Then. σB(x) σA(x) and σB(x) σA(x). It follows that σB(x) is the union of σA(x) together with some of the

18 Jan 2023: 1,n1C ),. having c1(L) H2(CPn CPn; Z). By the Künneth theorem, we must have c1(L) =Ax 1 B1 x for some A,B Z. ... ai xi bi yi) c xn. for some scalars ai, bi and c.