We construct an explicit set of algebraically independent generators for the center of the universal enveloping algebra of the centralizer of a nilpotent matrix in the general linear Lie algebra over a field of characteristic zero. In particular, this gives a new proof of the freeness of the center, a result first proved by Panyushev, Premet and Yakimova.

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.

An R-module M is called coretractable if  HomR(M/K,M)≠0 for any proper submodule K of M. In this paper we study coretractable modules and their endomorphism rings. It turns out that if all right R-modules are coretractable, then R is a right Kasch and (two-sided) perfect ring. However, the converse holds for commutative rings. Also, for a semi-injective coretractable module MR with S=EndR(M), we show that u.dim(SS)=corank(MR).

Let 𝔻 be the open unit disc, let v:𝔻→(0,∞) be a typical weight, and let Hv∞ be the corresponding weighted Banach space consisting of analytic functions f on 𝔻 such that . We call Hv∞ a typical-growth space. For ϕ a holomorphic self-map of 𝔻, let Cφ denote the composition operator induced by ϕ. We say that Cφ is a bellwether for boundedness of composition operators on typical-growth spaces if for each typical weight v, Cφ acts boundedly on Hv∞ only if all composition operators act boundedly on Hv∞. We show that a sufficient condition for Cφ to be a bellwether for boundedness is that ϕ have an angular derivative of modulus less than 1 at a point on ∂𝔻. We raise the question of whether this angular-derivative condition is also necessary for Cφ to be a bellwether for boundedness.

This note contains a proof of the fact that a Jordan curve in ℝ2 with a continuous tangent line at each point admits a regular reparameterization. We extend the result both to more general curves in ℝn and to higher orders of differentiability.

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.

An almost sure limit theorem for the maxima of multivariate stationary Gaussian sequences is proved under some mild conditions.

Let S be an ideal of the free inverse monoid on a set X, and let ℬ denote the Banach algebra l1(S). It is shown that the following statements are equivalent: ℬ is *-primitive; ℬ is prime; X is infinite. A similar result holds if ℬ is replaced by ℂ[S], the complex semigroup algebra of S.

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.

In this paper, strongly Gorenstein flat modules are introduced and investigated. An R-module M is called strongly Gorenstein flat if there is an exact sequence ⋯→P1→P0→P0→P1→⋯ of projective R-modules with M=ker (P0→P1) such that Hom(−,F) leaves the sequence exact whenever F is a flat R-module. Several well-known classes of rings are characterized in terms of strongly Gorenstein flat modules. Some examples are given to show that strongly Gorenstein flat modules over coherent rings lie strictly between projective modules and Gorenstein flat modules. The strongly Gorenstein flat dimension and the existence of strongly Gorenstein flat precovers and pre-envelopes are also studied.

We study the matrix equation C(BXC)†B=X†, where X† denotes the Moore–Penrose inverse. We derive conditions for the consistency of the equation and express all its solutions using singular vectors of B and C. Applications to compliance matrices in molecular dynamics, to mixed reverse-order laws of generalized inverses and to weighted Moore–Penrose inverses are given.

Let ℱ be a family of meromorphic functions defined in D, all of whose zeros have multiplicity at least k+1. Let a and b be distinct finite complex numbers, and let k be a positive integer. If, for each pair of functions f and g in ℱ, f(k) and g(k) share the set S={a,b}, then ℱ is normal in D. The condition that the zeros of functions in ℱ have multiplicity at least k+1 cannot be weakened.

We consider in this paper the question of when the finite sum of products of two Toeplitz operators is a finite-rank perturbation of a single Toeplitz operator on the Hardy space over the unit disk. A necessary condition is found. As a consequence we obtain a necessary and sufficient condition for the product of three Toeplitz operators to be a finite-rank perturbation of a single Toeplitz operator.

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.

We prove quantitative versions of the Balog–Szemerédi–Gowers and Freiman theorems in the model case of a finite field geometry 𝔽2n, improving the previously known bounds in such theorems. For instance, if is such that ∣A+A∣≤K∣A∣ (thus A has small additive doubling), we show that there exists an affine subspace H of 𝔽2n of cardinality such that . Under the assumption that A contains at least ∣A∣3/K quadruples with a1+a2+a3+a4=0, we obtain a similar result, albeit with the slightly weaker condition ∣H∣≫K−O(K)∣A∣.

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 introduce partial actions of weakly left E-ample semigroups, thus extending both the notion of partial actions of inverse semigroups and that of partial actions of monoids. Weakly left E-ample semigroups arise very naturally as subsemigroups of partial transformation semigroups which are closed under the unary operation α↦α+, where α+ is the identity map on the domain of α. We investigate the construction of ‘actions’ from such partial actions, making a connection with the FA-morphisms of Gomes. We observe that if the methods introduced in the monoid case by Megrelishvili and Schröder, and by the second author, are to be extended appropriately to the case of weakly left E-ample semigroups, then we must construct not global actions, but so-called incomplete actions. In particular, we show that a partial action of a weakly left E-ample semigroup is the restriction of an incomplete action. We specialize our approach to obtain corresponding results for inverse semigroups.

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.

Consider a projective limit G of finite groups Gn. Fix a compatible family δn of coactions of the Gn on a C*-algebra A. From this data we obtain a coaction δ of G on A. We show that the coaction crossed product of A by δ is isomorphic to a direct limit of the coaction crossed products of A by the δn. If A=C*(Λ) for some k-graph Λ, and if the coactions δn correspond to skew-products of Λ, then we can say more. We prove that the coaction crossed product of C*(Λ) by δ may be realized as a full corner of the C*-algebra of a (k+1)-graph. We then explore connections with Yeend’s topological higher-rank graphs and their C*-algebras.