We consider the half linear Sturm-Liouville problem

on the interval [0,1] subject to separated boundary conditions (which may be eigenparameter dependent at x = 1) and use Prüfer techniques to produce an oscillation theory for this problem. Both right definite (r > 0) and left definite (r of both signs) cases are discussed.

The study of radicals in general categories has followed several lines of development. The problem of defining radical properties in general categories has been considered by Kurosh and Shul'geifer, see (7). Under mild conditions on their categories they obtain sufficient conditions for the existence of radical functors which are closely related to radical properties. Another approach is by Maranda (5) and Dickson (3) who studied idempotent radical functors and torsion theories in abelian categories. Our aim has been to study radical functors in as general a category as possible. To this end we introduce the concept of an R-category. The categories of rings, modules, near-rings, groups and Jordan algebras are all examples of R-categories.

We prove two inequalities which relate the Lp modulus of continuity of n-th order, ωn(f,·)p, of an Hp function f with the p-th mean values of the n-th derivative f(n). Using these inequalities we extend classical results of Hardy and Littlewood [5], Gwiliam [4], Zygmund [13] and Taibleson [12] as well as a recent result of Oswald [6].

For a large class of C*-algebras including all von Neumann algebras, the central Haagerup tensor product of the multiplier algebra with itself has an isometric representation as completely bounded operators.

In his notable memoir on the Bessel Functions, Sonine proves the elegant result that

m > –½, provided a plane triangle can be drawn with sides a, b, c; otherwise the value of the integral is zero.

We establish various martingale inequalities in a rearrangement-invariant (RI) Banach function space. If $X$ is an RI space that is not too small, we associate with it RI spaces $\mathcal{H}_p(X)$ $(1\leq p\lt\infty)$ and $K(X)$, and discuss martingale inequalities in these spaces. One of our results is as follows. Let $1\leq p\lt\infty$, let $f=(f_n)$ be an $L_p$-bounded martingale, and let $|f|^p=g+h$ be the Doob decomposition of the submartingale $|f|^p=(|f_n|^p)$ into a martingale $g=(g_n)$ and a predictable non-decreasing process $h=(h_n)$ with $h_0=0$. Then, in the case where $1\ltp\lt\infty$, we obtain the inequalities

$$ \|h_{\infty}^{1/p}\|_X\leq2\|f_{\infty}\|_{\mathcal{H}_p(X)}\quad\text{and}\quad \Big\|\sup_n|g_n|^{1/p}\Big\|_X\leq4\|f_{\infty}\|_{\mathcal{H}_p(X)}, $$

and, in the case where $p=1$, we obtain the inequalities

$$ \|h_{\infty}\|_X\leq\sup_{n\in\mathbb{Z}_{+}}\|f_n\|_{K(X)}\quad\text{and}\quad \sup_{n\in\mathbb{Z}_{+}}\|g_n\|_X\leq2\sup_{n\in\mathbb{Z}_{+}}\|f_n\|_{K(X)}. $$

For some specific choices of $X$, we can give explicit expressions for $\mathcal{H}_p(X)$ and $K(X)$. For example, $\mathcal{H}_1(L_1)=L\log L$, $\mathcal{H}_p(L_{p,\infty})=L_{p,1}$, and so on. Furthermore, if the Boyd indices of $X$ satisfy $0\lt\alpha_{X}\leq\beta_{X}\lt1/p$ (respectively, $0\lt\alpha_{X}$), then $\mathcal{H}_p(X)=X$ (respectively, $K(X)=X$). In any case, $\mathcal{H}_p(X)$ is embedded in $K(X)$, and $K(X)$ is embedded in $X$.

AMS 2000 Mathematics subject classification: Primary 60G42; 60G46. Secondary 46E30

We consider associative algebras filtered by the additive monoid ℕp. We prove that, under quite general conditions, the study of Gelfand-Kirillov dimension of modules over a multi-filtered algebra R can be reduced to the associated ℕp-graded algebra G(R). As a consequence, we show the exactness of the Gelfand-Kirillov dimension when the multi-filtration is finite-dimensional and G(R) is a finitely generated noetherian algebra. Our methods apply to examples like iterated Ore extensions with arbitrary derivations and “homothetic” automorphisms (e.g. quantum matrices, quantum Weyl algebras) and the quantum enveloping algebra of sl(v + 1)

The study of the cohomological dimensions of algebraic varieties has produced some interesting results and problems in local algebra: the general local problem is that posed by Hartshorne and Speiser in (4, p. 57). We consider a (commutative, Noetherian) local ring A (with identity), a proper ideal a of A, and ask the following question.

In (5) the author showed how to construct all inverse semigroups from their trace and semilattice of idempotents: the construction is by means of a family of mappings between ℛ-classes of the semigroup which we refer to as the structure mappings of the semigroup. In (7) (see also (8) and (9)) K. S. S. Nambooripad has adopted a similar approach to the structure of regular semigroups: he shows how to construct regular semigroups from their trace and biordered set of idempotents by means of a family of mappings between ℛ-classes and between ℒ-classes of the semigroup which we again refer to as the structure mappings of the semigroup. In the present paper we aim to provide a simpler set of axioms characterising the structure mappings on a regular semigroup than the axioms (R1)-(R7) of Nambooripad (9). Two major differences occur between Nambooripad's approach (9) and the approach adopted here: first, we consider the set of idempotents of our semigroups to be equipped with a partial regular band structure (in the sense of Clifford (3)) rather than a biorder structure, and second, we shall enlarge the set of structure mappings used by Nambooripad.

R. A. Rankin [2] and J. Lehner [1] considered the non-vanishing of Poincaré series for the classical modular matrix group and for an arbitrary fuchsian group, respectively.

In this paper we consider the non-vanishing of Poincaré series for the congruence group

We are interested in entire solutions for the semilinear biharmonic equation Δ2 u = f(u) in ℝN , where f(u) = eu or –u –p (p > 0). For the exponential case, we prove that for the polyharmonic problem Δ2m u = eu with positive integer m, any classical entire solution verifies Δ2m–1 u < 0; this completes the results of Dupaigne et al. (Arch. Ration. Mech. Analysis208 (2013), 725–752) and Wei and Xu (Math. Annalen313 (1999), 207–228). We also obtain a refined asymptotic expansion of the radial separatrix solution to Δ2 u = eu in ℝ3, which answers a question posed by Berchio et al. (J. Diff. Eqns252 (2012), 2569–2616). For the negative power case, we show the non-existence of the classical entire solution for any 0 < p ⩽ 1.

It is shown that over an arbitrary countable field there exists a finitely generated algebra that is nil, infinite dimensional and has Gelfand–Kirillov dimension at most 3.

In this note, we examine some of the properties of Hermitian operators on complex unital Banach Jordan algebras, that is, those operators with real numerical range. Recall that a unital Banach Jordan algebra J, is a (real or complex) Jordan algebra with product a ˚ b, having a unit 1, and a norm ∥·∥, such that J, with norm ∥·∥, is a Banach space, ∥1∥ = 1, and, for all a and b in j,

We prove if B is a cluster-tilted algebra, then B is τB-tilting finite if and only if B is representation-finite.

By the rank r(S) of a finite semigroup S we shall mean the minimum cardinality of a set of generators ofS. For a group G, as remarked in [3], one has r(G)≦log2|G|, the bound being attained when G is an elementary abelian 2-group. By contrast, we shall see that there exist finite semigroups S for which r(S)≧|S| – 1. In the hope that it will not be considered too whimsical, we shall refer to a finite semigroup S of maximal rank (i.e. for which r(S) = |S|) as royal; a semigroup of next-to-maximal rank (i.e. for which r(S) = |S|–1) will be called noble.

This paper concerns criteria for assuring that every solution of a real fourth order nonselfadjoint differential equation

is oscillatory at x = ∞. Our technique is a generalisation of that used by Whyburn (1) for the study of the selfadjoint equation,

combined with the theory of H-oscillation of vector equations as introduced by Domšlak (2) and studied by Noussair and Swanson (3). Whyburn's technique consists of representing (1.2) as a dynamical system of the form

and then studying (1.3) in terms of polar coordinates in the y, z-plane. In Section 2 below we show how to represent (1.1) as a dynamical system of the form

We show that the flow generated by the totally competitive planar Lotka–Volterra equations deforms the line connecting the two axial equilibria into convex or concave curves, and that these curves remain convex or concave for all subsequent time. We apply the observation to provide an alternative proof to that given by Tineo in 2001 that the carrying simplex, the globally attracting invariant manifold that joins the axial equilibria, is either convex, concave or a straight-line segment.

An unexpected relationship between indecomposable involutive set-theoretic solutions to the Yang–Baxter equation and one-generator braces has recently been discovered by Agata and Alicja Smoktunowicz. We extend these results and answer three open questions which arose in this context.

In 1950, Wintner (11) showed that if the function f(x) is continuous on the half-line [0, ∞) and, in a certain sense, is “ small when x is large ” then the differential equation

does not have L2 solutions, where the function y(x) satisfying (1) is called an L2 solution if

When the plane wave equation is expressed in terms of parabolic co-ordinates x, y, the variables are separable, and the elementary solutions have the form

where x, y, μ are real. In this context, therefore, the functions Dν (z) which are directly significant are those where amp z = ± π/4 and ν + ½ is purely imaginary, rather than those where z is real and ν is a positive integer. The expansion of an arbitrary function in terms of the latter sort of D-function (substantially, in terms of Hermite polynomials) is well known. This paper is concerned with the expansion in terms of the former sort of D-function.