Let k be a field and X a k-form of the projective line. We classify all the isomorphism classes of vector bundles over X.

Given an integer n≥2, let λ(n):=(log n)/(log γ(n)), where γ(n)=∏ p∣np, denote the index of composition of n, with λ(1)=1. Letting ϕ and σ stand for the Euler function and the sum of divisors function, we show that both λ(ϕ(n)) and λ(σ(n)) have normal order 1 and mean value 1. Given an arbitrary integer k≥2, we then study the size of min {λ(ϕ(n)),λ(ϕ(n+1)),…,λ(ϕ(n+k−1))} and of min {λ(σ(n)),λ(σ(n+1)),…,λ(σ(n+k−1))} as n becomes large.

Let (N,ℝ,θ) be a centrally ergodic W* dynamical system. When N is not a factor, we show that for each nonzero real number t, the crossed product induced by the time t automorphism θt is not a factor if and only if there exist a rational number r and an eigenvalue s of the restriction of θ to the center of N, such that rst=2π. In the C* setting, minimality seems to be the notion corresponding to central ergodicity. We show that if (A,ℝ,α) is a minimal unital C* dynamical system and A is not simple, then, for each nonzero real number t, the crossed product induced by the time t automorphism αt is not simple if there exist a rational number r and an eigenvalue s of the restriction of α to the center of A, such that rst=2π. The converse is true if, in addition, A is commutative or prime.

In any regular semigroup with an orthodox transversal, we define two sets R and L using Green’s relations and give necessary and sufficient conditions for them to be subsemigroups. By using R and L, some equivalent conditions for an orthodox transversal to be a quasi-ideal are obtained. Finally, we give a structure theorem for regular semigroups with quasi-ideal orthodox transversals by two orthodox semigroups R and L.

For two given projections p and q in a C*-algebra, we investigate how to express the Drazin inverses of the product pq and the difference p−q, and give applications. As a special case, we obtain the results of [C. Y. Deng, ‘The Drazin inverses of products and differences of orthogonal projections’, J. Math. Anal. Appl.335 (2007) 64–71], with considerably simpler proofs.

The norm of a function f in the Hardy space Hp(𝔻) is by definition the limit of as r→1. We show that grows at most like o(1/log r) as r→1.

We establish two-sided bounds for the complexity of two infinite series of closed orientable three-dimensional hyperbolic manifolds, the Löbell manifolds and the Fibonacci manifolds. The manifolds of the two series are indexed by an integer n and the corresponding complexity estimates are both linear in n.

We show that under a suitable transversality condition, the intersection of two rational subtori in an algebraic torus (ℂ*)n is a finite group which can be determined using the torsion part of some associated lattice. We also give applications to the study of characteristic varieties of smooth complex algebraic varieties. As an example we discuss A. Suciu’s line arrangement, the so-called deleted B3-arrangement.

For ε>0, let Σε={z∈ℂ:∣arg z∣<ε}. It has been proved (D. E. Marshall and W. Smith, Rev. Mat. Iberoamericana15 (1999), 93–116) that ∫ f−1(Σε)∣f(z)∣ dA(z)≃∫ 𝔻∣f(z)∣ dA(z) for every ε>0, uniformly for every univalent function f in the classical Bergman space A1 that fixes the origin. In this paper, we extend this result to those conformal maps on 𝔻 belonging to weighted Bergman–Orlicz classes such that f(0)=∣f′(0)∣−1=0.

In this paper we prove a realizability theorem for Quinn’s mapping cylinder obstructions for stratified spaces. We prove a continuously controlled version of the s-cobordism theorem which we further use to prove the relation between the torsion of an h-cobordism and the mapping cylinder obstructions. This states that the image of the torsion of an h-cobordism is the mapping cylinder obstruction of the lower stratum of one end of the h-cobordism in the top filtration. These results are further used to prove a theorem about the realizability of end obstructions.

It is shown that the dnth Chebyshev polynomials on the Julia set JP, and on the equipotential curve ΓP(R), of the polynomial P(z)=zd−c, are identical and exactly equal to the nth iteration of P(z) itself. As an application, the capacity of the Julia set JP is deduced, a result that was first obtained by Brolin.