The quintuple product identity has appeared many times in the literature. Indeed, no fewer than 12 proofs have been given. We establish a more general identity from which the quintuple product identity follows in two ways.

We prove that every Banach space X with characteristic of uniform convexity less than 2 has the fixed point property whenever X satisfies a certain orthogonality condition.

On analogy with functions if Lebesuge class Lα on the real line the author considers those multiplicative arthmetic functions which are bounded in mean α>1. Necessary and sufficient conditions are obtained in order that they should have a mean-value, zero or non-zero. An application is made to Ramanujan's τ-function.

This paper contains the complete classification of the finite p-groups G where p is an odd prime, G is generated by two elements, and the commutator subgroup of G is cyclic. These groups are a special kind of two-generator metabelian group, a class that has been studied by Szekeres (1965). He determined the defining relations of such groups but, as he noted, a “residual isomorphism problem” remains. The cyclic commutator groups are simple when considered from this first point of view; they have a short, easily derived set of defining relations. However, the isomorphism problem is a bit complicated for the defining relations contain nine parameters and each of these parameters might and can be an invariant of the group.

We establish a duality between distributive bisemilattices and certain compact left normal bands. The main technique in the proof utilizes the idea of Plonka sums.

Let M be a reductive algebraic monoid with zero and unit group G. We obtain a description of the submonoid generated by the idempotents of M. In particular, we find necessary and sufficient conditions for M\G to be idempotent generated.

We consider algebraic independence properties of series such as We show that the functions fr(z) are algebraically independent over the rational functions Further, if αrs (r = 2, 3, 4, hellip; s = 1, 2, 3, hellip) are algebraic numbers with 0 < |αrs|, we obtain an explicit necessary and sufficient condition for the algebraic independence of the numbers fr(αrs) over the rationals.

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.

The authors establish the boundedness on the Herz spaces and the weak Herz spaces for a large class of rough singular integral operators and their corresponding fractional versions. Applications are given to Fefferman's rough singular integral operators, their fractional versions, their commutators with BMO() functions and Ricci-Stein oscillatory singular integral operators. Some new results are obtained.

Let N be a nilpotent simply connected Lie group, and A a commutative connected d-dimensional Lie group of automorphisms of N which correspond to semisimple endomorphisms of the Lie algebra of N with positive eigenvalues. Form the split extension S = N × A ≅ N × a, a being the Lie algebra of A. We consider a family of “rectangles” Br in S, parameterized by r > 0, such that the measure of Br behaves asymptotically as a fixed power of r. One can construct the Hardy-Littlewood maximal function operator f → Mf relative to left translates of the family {Br}. We prove that M is of weak type (1, 1). This complements a result of J.-O. Strömberg concerning maximal functions defined relative to hyperbolic balls in a symmetric space.

Examples of hyperbolic knots in S3 are given such that their complements contain quasi-Fuchsian non-Fuchsian surfaces. In particular, this proves that there are hyperbolic knots that are not toroidally alternating.

M. H. Stone raised the problem ([1] Problem 70) of characterising the class of distributive pseudo-complemented lattices ℒ = 〈L; ∨, ∧, 0, 1〉 in which a* ∨ a** = 1 holds identically. Several solutions to this problem have now been offered — the first being by G. Grätzer and E. T. Schmidt [6], who gave this class of lattices the name Stone lattices. Later solutions were given by J. Varlet [11], O. Frink [4] and G. Grätzer [5]; see also G. Bruns [2].

Our first main result shows that a graph product of right cancellative monoids is itself right cancellative. If each of the component monoids satisfies the condition that the intersection of two principal left ideals is either principal or empty, then so does the graph product. Our second main result gives a presentation for the inverse hull of such a graph product. We then specialize to the case of the inverse hulls of graph monoids, obtaining what we call ‘polygraph monoids’. Among other properties, we observe that polygraph monoids are F*-inverse. This follows from a general characterization of those right cancellative monoids with inverse hulls that are F*-inverse.

New methods are introduced here to show that if n is a quasiperfect number and ω(n) the number of its distinct prime factors, then ω(n) ≥ 7 and n > 1035, and if further 3 ∤ n then ω(n) ≥ 9 and n > 1040.

We consider the existence of multiple positive solutions of a nonlinear two-point boundary value problem by modifying a “time map” technique introduced by J. Smoller and A. Wasserman. We count the number of positive solutions and find their Conley indices and thus determine their stabilities.

In a paper of nearly thirty years ago (Mahler 1937) I first studied approximation properties of algebraic number fields relative to their full system of inequivalent valuations. I now return to these questions with a slightly improved method and establish a number of existence theorems for such fields.

In a Riemannian manifold M, elastica are solutions of the Euler-Lagrange equation of the following second order constrained variational problem: find a unit-speed curve in M, interpolating two given points with given initial and final (unit) velocities, of minimal average squared geodesic curvature. We study elastica in Lie groups G equipped with bi-invariant Riemannian metrics, focusing, with a view to applications in engineering and computer graphics, on the group SO(3) of rotations of Euclidean 3-space. For compact G, we show that elastica extend to the whole real line. For G = SO(3), we solve the Euler-Lagrange equation by quadratures.

We study the bifurcation of steady-state solutions of a scalar reaction-diffusion equation in one space variable by modifying a “time map” technique introduced by J. Smoller and A. Wasserman. We count the exact number of steady-state solutions which are totally ordered in an order interval. We are then able to find their Conley indices and thus determine their stabilities.

A similarity method is used to develop a solution of the wave equation within a sector with mixed boundary conditions. In this manner the field which results from the diffraction of an incident pulse of step function time dependence is found.

We establish a general algebraic independence theorem for the solutions of a certain kind of functional equations. As a particular application, we prove that for any real irrational ζ, the numbers are algebraically independent, for multiplicatively independent algebraic αi with 0<|<| <1.