Now assume |G : H| is infinite and that H is definable in G (but H need no longer be infinite or G-normal). Let Z = Z◦(G), and note that (a) implies that Z is infinite. If |Z : H ∩Z| is infinite, then |HZ : H| is infinite as well. In this case we are done since H ≤ HZ ≤ NG(H). Otherwise, |Z : H ∩Z| is finite, so the connectedness of Z implies that Z ≤ H. We can now proceed by induction on the r...

In the present paper we prove the integrability (in the sense of existence of formal symmetry of infinite rank) for a class of (1+1)-dimensional evolution systems generalizing the triangular systems considered by Bakirov, Beukers, Sanders and Wang, and by Sanders and van der Kamp. The important consequence of this result is the existence of formal symmetry of infinite rank for “almost integrabl...

We show that if G is a stable group of finite weight with no infinite, infinite-index, chains of definable subgroups, then G is superstable of finite U rank. Combined with recent work of Palaćın and Sklinos, we conclude that (Z,+, 0) has no proper stable expansions of finite weight. A corollary of this result is that if P ⊆ Zn is definable in a finite dp-rank expansion of (Z,+, 0), and (Z,+, 0,...

Given an n-tuple {a1, ..., an} of self-adjoint operators on an infinite dimensional Hilbert space H, we say that a projection p in B(H) is locally minimal for {a1, ..., an} if each pajp (for j = 1, ..., n) is a scalar multiple of p. In Theorem 1.8 we show that for any such {a1, ..., an} and any positive integer k there exists a projection p of rank k that is locally minimal for {a1, ..., an}. I...