New estimates are obtained for the maximum modulus of the generalized logarithmic derivatives f(k)/f(j), where f is analytic and of finite order of growth in the unit disc, and k and j are integers satisfying k>j≥0. These estimates are stated in terms of a fixed (Lindelöf) proximate order of f and are valid outside a possible exceptional set of arbitrarily small upper density. The results obtained are then used to study the growth of solutions of linear differential equations in the unit disc. Examples are given to show that all of the results are sharp.

We formulate and prove two versions of Miyachi’s theorem for connected, simply connected nilpotent Lie groups. This allows us to prove the sharpness of the constant 1/4 in the theorems of Hardy and of Cowling and Price for any nilpotent Lie group. These theorems are proved using a variant of Miyachi’s theorem for the group Fourier transform.

A commutative complex unital Banach algebra can be represented as a space of continuous complex-valued functions on a compact Hausdorff space via the Gelfand transform. However, in general it is not possible to represent a commutative real unital Banach algebra as a space of continuous real-valued functions on some compact Hausdorff space, and for this to happen some additional conditions are needed. In this note we represent a commutative real Banach algebra on a part of its state space and show connections with representations on the maximal ideal space of the algebra (whose existence one has to prove first).

Descloux and Geymonat considered a model problem in two-dimensional magnetohydrodynamics and conjectured that the essential spectrum has an explicitly given band structure. This conjecture was recently proved by Faierman, Mennicken, and Möller by reducing the problem to that for a 2×2 block operator matrix. In a subsequent paper Faierman and Mennicken investigated the essential spectrum for the problem arising from a particular type of perturbation of precisely one of the operator entries in the matrix representation cited above of the original problem considered by Descloux and Geymonat. In this paper we extend the results of that work by investigating the essential spectrum for the problem arising from particular types of perturbations of all but one of the aforementioned operators. It remains an open question whether one can perturb the exceptional operator in such a way as to leave the essential spectrum unchanged.

For an increasing weight w in Bp (or equivalently in Ap), we find the best constants for the inequalities relating the standard norm in the weighted Lorentz space Λp(w) and the dual norm.

Let E/k be a function field over an infinite field of constants. Assume that E/k(x) is a separable extension of degree greater than one such that there exists a place of degree one of k(x) ramified in E. Let K/k be a function field. We prove that there exist infinitely many nonisomorphic separable extensions L/K such that [L:K]=[E:k(x)] and AutkL=AutKL≅Autk(x)E.

Let f be a polynomial of degree n≥2 with f(0)=0 and f′(0)=1. We prove that there is a critical point ζ of f with ∣f(ζ)/ζ∣≤1/2 provided that the critical points of f lie in the sector {reiθ:r>0,∣θ∣≤π/6}, and ∣f(ζ)/ζ∣<2/3 if they lie in the union of the two rays {1+re±iθ:r≥0}, where 0<θ≤π/2.

Assuming a conjecture intermediate in strength between one of Chowla and one of Heath-Brown on the least prime in a residue class, we show that for any coprime integers a and m≥1, there are infinitely many Carmichael numbers in the arithmetic progression a mod m.

Let X be a geometrically irreducible smooth projective curve defined over ℝ, of genus at least 2, that admits a nontrivial automorphism, σ. Assume that X does not have any real points. Let τ be the antiholomorphic involution of the complexification xℂ of X. We show that if the action of σ on the set 𝒮(X) of all real theta characteristics of X is trivial, then the order of σ is even, say 2k, and the automorphism of Xℂ has a fixed point, where is the automorphism of X×ℝℂ defined by σ. We then show that there exists X with a real point and admitting a nontrivial automorphism σ, such that the action of σ on 𝒮(X) is trivial, while X/〈σ〉≠ℙ1ℝ. We also give an example of X with no real points and admitting a nontrivial automorphism σ, such that the automorphism has a fixed point, the action of σ on 𝒮(X) is trivial, and X/〈σ〉≠ℙ1ℝ.

Finite groups with abelian Sylow p-subgroups for certain primes p are characterized in terms of arithmetical properties of commutators.

We prove that all g-natural contact metric structures on a two-point homogeneous space are homogeneous contact. The converse is also proved for metrics of Kaluza–Klein type. We also show that if (M,g) is an Einstein manifold and is a Riemannian g-natural metric on T1M of Kaluza–Klein type, then is H-contact if and only if (M,g) is 2-stein, so proving that the main result of Chun et al. [‘H-contact unit tangent sphere bundles of Einstein manifolds’, Q. J. Math., to appear. DOI: 10.1093/qmath/hap025] is invariant under a two-parameter deformation of the standard contact metric structure on T1M. Moreover, we completely characterize Riemannian manifolds admitting two distinct H-contact g-natural contact metric structures, with associated metric of Kaluza–Klein type.

A new notion of L(n)-hyponormality is introduced in order to provide a bridge between subnormality and paranormality, two concepts which have received considerable attention from operator theorists since the 1950s. Criteria for L(n)-hyponormality are given. Relationships to other notions of hyponormality are discussed in the context of weighted shift and composition operators.

According to the Skitovich–Darmois theorem, the independence of two linear forms of n independent random variables implies that the random variables are Gaussian. We consider the case where independent random variables take values in a second countable locally compact abelian group X, and coefficients of the forms are topological automorphisms of X. We describe a wide class of groups X for which a group-theoretic analogue of the Skitovich–Darmois theorem holds true when n=2.

Let K be any compact set. The C*-algebra C(K) is nuclear and any bounded homomorphism from C(K) into B(H), the algebra of all bounded operators on some Hilbert space H, is automatically completely bounded. We prove extensions of these results to the Banach space setting, using the key concept ofR-boundedness. Then we apply these results to operators with a uniformly bounded H∞-calculus, as well as to unconditionality on Lp. We show that any unconditional basis on Lp ‘is’ an unconditional basis on L2 after an appropriate change of density.

Given a metrizable locally convex-solid Riesz space of measurable functions we provide a procedure to construct a minimal Fréchet (function) lattice containing it, called its Fatou completion. As an application, we obtain that the Fatou completion of the space L1(ν) of integrable functions with respect to a Fréchet-space-valued measure ν is the space L1w(ν) of scalarly ν-integrable functions. Further consequences are also given.

A shift automorphism algebra is one satisfying the conditions of the shift automorphism theorem, and a shift automorphism variety is a variety generated by a shift automorphism algebra. In this paper, we show that every shift automorphism variety contains a countably infinite subdirectly irreducible algebra.

We study some classes of planar harmonic mappings produced with the shear construction devised by Clunie and Sheil-Small in 1984. The first section reviews the basic concepts and describes the shear construction. The main body of the paper deals with the geometry of the classes constructed.

According to the classical Borel lemma, any positive nondecreasing continuous function T satisfiesT(r+1/T(r))≤2T(r) outside a possible exceptional set of finite linear measure. This lemma plays an important role in the theory of entire and meromorphic functions, where the increasing function T is either the logarithm of the maximum modulus function, or the Nevanlinna characteristic. As a result, exceptional sets appear throughout Nevanlinna theory, in particular in Nevanlinna’s second main theorem. In this paper, we consider generalizations of Borel’s lemma. Conversely, we consider ways in which certain inequalities can be modified so as to remove exceptional sets. All results discussed are presented from the point of view of real analysis.

Let A be a unital C*-algebra. Let (B,E) be a pair consisting of a unital C*-algebra B containing A as a C*-subalgebra with a unit that is also the unit of B, and a conditional expectation E from B onto A that is of index-finite type and of depth 2. Let B1 be the C*-basic construction induced by (B,E). In this paper, we shall show that any such pair (B,E) satisfying the conditions that A′∩B=ℂ1 and that A′∩B1 is commutative is constructed by a saturated C*-algebraic bundle over a finite group. Furthermore, we shall give a necessary and sufficient condition for B to be described as a twisted crossed product of A by its twisted action of a finite group under the condition that A′∩B1 is commutative.

We discuss n4 configurations of n points and n planes in three-dimensional projective space. These have four points on each plane, and four planes through each point. When the last of the 4n incidences between points and planes happens as a consequence of the preceding 4n−1 the configuration is called a ‘theorem’. Using a graph-theoretic search algorithm we find that there are two 84 and one 94 ‘theorems’. One of these 84 ‘theorems’ was already found by Möbius in 1828, while the 94 ‘theorem’ is related to Desargues’ ten-point configuration. We prove these ‘theorems’ by various methods, and connect them with other questions, such as forbidden minors in graph theory, and sets of electrons that are energy minimal.