We generalize the classical example, due to Abraham, of a train algebra that is not special train, to non necessarily commutative right nil algebras of index n.

The problem considered is that of long gravity waves approaching, from an arbitrary direction, a semi-infinite barrier, the whole system being in rotation. It is shown that the rotation gives rise to a wave in the shadow region whose amplitude depends upon the angle of incidence, but whose form is independent of it and which travels along the barrier without attenuation in that direction. The work is an extension and simplification of previous work by Crease, involving the use of methods previously developed by the author.

In this paper, we use the Riemann zeta function ζ(x) and the Bessel zeta function ζμ(x) to study the log behaviour of combinatorial sequences. We prove that ζ(x) is log-convex for x > 1. As a consequence, we deduce that the sequence {|B2n|/(2n)!}n ≥ 1 is log-convex, where Bn is the nth Bernoulli number. We introduce the function θ(x) = (2ζ(x)Γ(x + 1)) 1/x, where Γ(x) is the gamma function, and we show that logθ(x) is strictly increasing for x ≥ 6. This confirms a conjecture of Sun stating that the sequence is strictly increasing. Amdeberhan et al. defined the numbers an(μ) = 2 2n+1 (n + 1)!(μ+ 1)nζμ(2n) and conjectured that the sequence {an(μ)}n≥1 is log-convex for μ = 0 and μ = 1. By proving that ζμ(x) is log-convex for x > 1 and μ > -1, we show that the sequence {an(≥)}n>1 is log-convex for any μ > - 1. We introduce another function θμ,(x) involving ζμ(x) and the gamma function Γ(x) and we show that logθμ(x) is strictly increasing for x > 8e(μ + 2)2. This implies that

Based on Dobinski’s formula, we prove that

where Bn is the nth Bell number. This confirms another conjecture of Sun. We also establish a connection between the increasing property of and Holder’s inequality in probability theory.

In any extension theory for semigroups one must determine the basic building blocks and then discover how they fit together to create more complicated semigroups. For example, in group theory the basic building blocks are simple groups. In semigroup theory however there are several natural choices. One that has received considerable attention, particularly since the seminal work on inverse semigroups by Munn ([14, 15]), is the notion of a fundamental semigroup. A semigroup is called fundamental if it cannot be [shrunk] homomorphically without collapsing some of its idempotents (see below for a precise definition).

It is a fundamental fact in the theory of radicals of associative rings that if S is a radical and I is a two-sided ideal of R then S(I)⊆S(R). In view of this result it seems to be interesting to investigate radicals satisfying such or similar connections for other type of subrings. There are many works devoted to similar problems (2, 8, 8, 10). In this paper we try to get a uniform description of some facts in this area.

This paper is concerned with the existence and uniqueness of solutions for the Picard boundary value problem

in a real Hilbert space. Our theorems improve corresponding results of Mawhin for |k| large.

We consider the problem of maximizing the sum of squares of the leading coefficients of polynomials (where Pj(x) is a polynomial of degree j) under the restriction that the sup-norm of is bounded on the interval [ −b, b] (b>0). A complete solution of the problem is presented using duality theory of convex analysis and the theory of canonical moments. It turns out, that contrary to many other extremal problems the structure of the solution will depend heavily on the size of the interval [ −b, b].

A strip of radius $r$ in the hyperbolic plane is the set of points within distance $r$ of a given geodesic. Wedefine the density of a packing of strips of radius $r$ and prove that this density cannot exceed

$$ \mathcal{S}(r)=\frac{3}{\pi}\sinh r\mathrm{arccosh}\biggl(1+\frac{1}{2\sinh^2r}\biggr). $$

This bound is sharp for every value of $r$ and provides sharp bounds on collaring theorems for simple geodesics onsurfaces.

AMS 2000 Mathematics subject classification: Primary 51M09; 52C15. Secondary 51M04

We deal with a class of $p$-Laplacian Dirichlet boundary-value problems where the combined effects of ‘sublinear’ and‘superlinear’ growths allow us to establish the existence of at least two positive solutions.

AMS 2000 Mathematics subject classification: Primary 35J25; 35J60. Secondary 35J65; 35J70

Methods for solving boundary value problems in linear, second order, partial differential equations in two variables tend to be somewhat rigidly partitioned in some of the standard text-books. Problems for elliptic equations are sometimes solved by finding the fundamental solution which is defined as a solution with a given singularity at a certain point. Another approach is by way of Green's functions which are usually defined as solutions of the original homogeneous equations now made inhomogeneous by the introduction of adelta function on the right hand side. The Green's function coincides with the fundamental solution for elliptic equations but exhibits a totally different type of singularity for parabolic or hyperbolic equations. Boundary value problems for hyperbolic equations can often by solved by Riemann's method which depends on the existence of an auxiliary function called the Riemann or sometimes the Riemann-Green function. The main object of this paper is to show the close relationship between Riemann's method and the method of Green's functions. This not only serves to unify different methods of solution of boundary value problems but also provides an additional method of determining Riemann functions for given hyperbolic equations. Before establishing these relationships we shall survey the general approach to boundary value problems through the use of the Green's function.

It is known that ℚ-derived univariate polynomials (polynomials defined over ℚ, with the property that they and all their derivatives have all their roots in ℚ) can be completely classified subject to two conjectures: that no quartic with four distinct roots is ℚ-derived, and that no quintic with a triple root and two other distinct roots is ℚ-derived. We prove the second of these conjectures.

AMS 2000 Mathematics subject classification: Primary 11G30

Let $R$ be a commutative ring. Let $M$ respectively $A$ denote a Noetherian respectively Artinian $R$-module, and $\mathfrak{a}$ a finitely generated ideal of $R$. The main result of this note is that the sequence of sets $(\mathrm{Att}_R\mathrm{Tor}_1^R((R/\mathfrak{a}^{n}),A))_{n\in\mathbb{N}}$ is ultimately constant. As a consequence, whenever $R$ is Noetherian, we show that $\mathrm{Ass}_R\mathrm{Ext}_R^1((R/\mathfrak{a}^{n}),M)$ is ultimately constant for large $n$, which is an affirmative answer to the question that was posed by Melkersson and Schenzel in the case $i=1$.

AMS 2000 Mathematics subject classification: Primary 13E05; 13E10

In the 1970s, Feldman and Moore classified separably acting von Neumann algebras containing Cartan maximal abelian self-adjoint subalgebras (MASAs) using measured equivalence relations and 2-cocycles on such equivalence relations. In this paper we give a new classification in terms of extensions of inverse semigroups. Our approach is more algebraic in character and less point-based than that of Feldman and Moore. As an application, we give a restatement of the spectral theorem for bimodules in terms of subsets of inverse semigroups. We also show how our viewpoint leads naturally to a description of maximal subdiagonal algebras.

We introduce a general definition for coloured cyclic operads over a symmetric monoidal ground category, which has several appealing features. The forgetful functor from coloured cyclic operads to coloured operads has both adjoints, each of which is relatively simple. Explicit formulae for these adjoints allow us to lift the Cisinski–Moerdijk model structure on the category of coloured operads enriched in simplicial sets to the category of coloured cyclic operads enriched in simplicial sets.

We introduce the notion of functionally compact sets into the theory of nonlinear generalized functions in the sense of Colombeau. The motivation behind our construction is to transfer, as far as possible, properties enjoyed by standard smooth functions on compact sets into the framework of generalized functions. Based on this concept, we introduce spaces of compactly supported generalized smooth functions that are close analogues to the test function spaces of distribution theory. We then develop the topological and functional–analytic foundations of these spaces.

It is proved that if G is any compact connected Hausdorff group with weight w(G)≦c, ℝ is the topological group of all real numbers and n is a positive integer, then the topological group G × ℝn can be topologically generated by n + 1 elements, and no fewer elements will suffice.

In this paper we investigate the weighted convolution algebras lp(ωn), where 1≦p<∞ and {ωn} is a sequence of positive weights satisfying the following conditions. If p = 1 we require ω0 = 1, and ωt+s≦ωtωs.Then

The main result of this paper is the following

Theorem A. Let G be a π-separable finite group with Hall πsubgroup H. Suppose θεIrr(H). Then there exists a unique subgroup M, maximal with the property that it contains H and θ can be extended to a character of M.

Let H1(U2) be the Hardy space of the bidisc as described in (3). Each function f ∈ H1(U2) has a Taylor expansion of the form . For 0<p<∞, a doubly-indexed sequence is said to be a multiplier of H1(U2) into lp if

This paper is concerned with the cases p = 2 and p = 1. Theorem 1 characterises the multipliers of into l2 and is an analogue in two variables of an old result of Hardy and Littlewood. Theorem 2 characterises the sequences (an)n≥0 such that (an+m)n,m≥0 is a multiplier of H1(U) into l1

If letters a, b, c, … are used to denote points of a nondegenerate plane cubic curve, other than the singular point if any, and if the product ab is defined as the third point of the curve collinear with a and b, we obtain an algebraic system having nonassociative multiplication (ab . c ≠ a . bc in general). It is in fact a totally symmetric entropic quasigroup (these terms are defined below). This idea, which was put forward at a meeting of the Edinburgh Mathematical Society a few years ago, will be exploited in a forthcoming paper. Such quasigroups have many properties which can be interpreted geometrically. Or, conversely, known properties of cubic curves suggest theorems about totally symmetric entropic quasigroups, which if established will involve a gain in generality, since not every such quasigroup can be “placed” on a cubic.