A doodle is a collection of immersed circles without triple intersections in the 2-sphere. It was shown by second author and Tayler that doodles induce commutator identities (identities amongst commutators) free group. In this paper, we observe idea more closely concentrating on with proper noose systems elementary identities. particular, show there bijection between cobordism classes colored w...

Let R be a prime ring of char R = 2, d a nonzero derivation of R, ρ a nonzero right ideal of R and 0 = b ∈ R such that b[[d2[x, y], [x, y]]n = 0 for all x, y ∈ ρ, n ≥ 1 fixed integer. If [ρ, ρ]ρ = 0 then either bρ = 0 or d(ρ)ρ = 0. Mathematics Subject Classification: 16W25, 16R50, 16N60

Let Mb denote the operation of multiplication by b. Let Hj denote the Hilbert transform performed in the jth coordinate, for j = 1, 2, . . .. Consider the jth order iterated commutator Cb,j = [H1, · · · , [Hj , Mb] · · · ], j = 1, 2, . . . We show that the operator norm of Cb,3 on L (R) is comparable to the norm of b in Chang–Fefferman product BMO(⊗31R 2 +). This fact has some well known equiva...

We prove that the stable commutator lengths of Dehn twists are positive, implying that the natural map from the second bounded cohomology to the ordinary cohomology is not injective, that mapping class groups are not uniformly perfect and that the growth rate of Dehn twists are linear. We then prove that the dimensions of the second bounded cohomology of hyperelliptic mapping class groups and o...

For α > −1, let Aα denote the corresponding weighted Bergman space of the unit ball. For any self-adjoint subset G ⊂ L∞, let T(G) denote the C∗−subalgebra of B(Aα) generated by {Tf : f ∈ G}. Let CT(G) denote the commutator ideal of T(G). It was showed by D. Suárez (in 2004 for n = 1) and by the author (in 2006 for all n ≥ 1) that CT(L∞) = T(L∞) in the case α = 0. In this paper we show that in t...

Abstract. Let A and B be almost commuting (i.e., the commutator AB−BA belongs to trace class) self-adjoint operators. We construct a functional calculus φ 7→ φ(A,B) for functions φ in the Besov class B ∞,1(R ). This functional calculus is linear, the operators φ(A,B) and ψ(A,B) almost commute for φ, ψ ∈ B ∞,1(R ), and φ(A,B) = u(A)v(B) whenever φ(s, t) = u(s)v(t). We extend the Helton–Howe trac...