Hadamard defines the “elementary solution” of the general linear partial differential equation of the second order, namely

(Aik, BiC being functions of the n variables x1, x2, .., xn, which may be regarded as coordinates in a space of n dimensions), to be one of those solutions which are infinite to as low an order as possible at a given point and on every bicharacteristic through that point.

A module of a ring of differential operators $\mathcal{D}$ over a smooth surface has order $1$ if it is isomorphic to a factor module of $\mathcal{D}$ by a cyclic ideal generated by an operator of order $1$. Let $k$ be a positive integer. We give conditions under which an indecomposable $\mathcal{D}$-module of order $1$ is GK-critical of length $k$. We also give examples of indecomposable, non-critical, $\mathcal{D}$-modules whose subfactors have order $1$.

AMS 2000 Mathematics subject classification: Primary 16S32. Secondary 37F75

Homological properties of several Banach left L1(G)-modules have been studied by Dales and Polyakov and recently by Ramsden. In this paper, we characterize some homological properties of and as Banach left L1(G)-modules, such as flatness, injectivity and projectivity.

A liquid crystal is a transversely isotropic liquid consisting of large, relatively rigid, elongated molecules which align more or less parallel to their neighbours. Three distinct types of liquid crystal occur, namely nematic, cholesteric and smectic. In the absence of any external influences, nematics tend to orientate with their anisotropic axis uniformly aligned, whereas cholesterics prefer a characteristic helical configuration and smectics are more highly organised in layered structures. However, it is possible to influence the orientation of the anisotropic axis by a variety of external means. In particular, solid surfaces affect the alignment through the action of surface torques, while electromagnetic fields exert body torques which tend to align the anisotropic axis either parallel or perpendicular to the applied field. Detailed descriptions of the physical properties of liquid crystals may be found in the books by de Gennes [1] and Chandrasekhar [2] and the review by Stephen and Straley [3].

We consider a delay predator–prey system without instantaneous negative feedback and establish some conditions for global attractivity of the positive equilibrium of the system which generalize and improve some of the existing ones. When the system is decoupled, one of the main results reduces to the well-known Wright 3/2 stability condition for the delayed logistic equation.

We prove Hardy-type inequalities for a fractional Dunkl–Hermite operator, which incidentally gives Hardy inequalities for the fractional harmonic oscillator as well. The idea is to use h-harmonic expansions to reduce the problem in the Dunkl–Hermite context to the Laguerre setting. Then, we push forward a technique based on a non-local ground representation, initially developed by Frank et al. [‘Hardy–Lieb–Thirring inequalities for fractional Schrödinger operators, J. Amer. Math. Soc.21 (2008), 925–950’] in the Euclidean setting, to obtain a Hardy inequality for the fractional-type Laguerre operator. The above-mentioned method is shown to be adaptable to an abstract setting, whenever there is a ‘good’ spectral theorem and an integral representation for the fractional operators involved.

We consider weighted sums over points of lattice polytopes, where the weight of a point v is the monomial qλ(v) for some linear form λ. We propose a q-analogue of the classical theory of Ehrhart series and Ehrhart polynomials, including Ehrhart reciprocity and involving evaluation at the q-integers. The main novelty is the proposal to consider q-Ehrhart polynomials. This general theory is then applied to the special case of order polytopes associated with partially ordered sets. Some more specific properties are described in the case of empty polytopes.

Let Φ be a reduced irreducible root system of rank greater than or equal to 2, let R be a commutative ring and let I, J be two ideals of R. In the present paper we describe generators of the commutator groups of relative elementary subgroups [E(Φ,R,I),E(Φ,R,J)] both as normal subgroups of the elementary Chevalley group E(Φ,R), and as groups. Namely, let xα(ξ), α ∈ Φ ξ ∈ R, be an elementary generator of E(Φ,R). As a normal subgroup of the absolute elementary group E(Φ,R), the relative elementary subgroup is generated by xα(ξ), α ∈ Φ, ξ ∈ I. Classical results due to Stein, Tits and Vaserstein assert that as a group E(Φ,R,I) is generated by zα(ξ,η), where α ∈ Φ, ξ ∈ I, η ∈ R. In the present paper, we prove the following birelative analogues of these results. As a normal subgroup of E(Φ,R) the relative commutator subgroup [E(Φ,R,I),E(Φ,R,J)] is generated by the following three types of generators: (i) [xα(ξ),zα(ζ,η)], (ii) [xα(ξ),x_α(ζ)] and (iii) xα(ξζ), where α ∈ Φ, ξ ∈ I, ζ ∈ J, η ∈ R. As a group, the generators are essentially the same, only that type (iii) should be enlarged to (iv) zα(ξζ,η). For classical groups, these results, with many more computational proofs, were established in previous papers by the authors. There is already an amazing application of these results in the recent work of Stepanov on relative commutator width.

The lattice of varieties of bands was constructed in [1] by providing a simple system of invariants yielding a solution of the world problem for varieties of bands including a new system of inequivalent identities for these varieties. References [3] and [5] contain characterizations of varieties of bands determined by identities with up to three variables in terms of Green's relations and the functions figuring in a construction of a general band. In this construction, the band is expressed as a semilattice of rectangular bands and the multiplication is written in terms of functions among these rectangular band components and transformation semigroups on the corresponding left zero and right zero direct factors.

In this note we present some rather loosely connected results on Banach algebras together with some illustrative examples. We consider various conditions on a Banach algebra which imply that it is finite dimensional. We also consider conditions which imply the existence of non-zero nilpotents, and hence the existence of finite dimensional subalgebras. In the setting of Banach algebras quasinilpotents figure more prominently than nilpotents. We give an example of a non-commutative Banach algebra in which 0 is the only quasinilpotent; this resolves a problem of Hirschfeld and Zelazko (4).

It has been established by G. Lallement (3) that the set of idempotent-separating congruences on a regular semigroup S coincides with the set ∑() of congruences on S which are contained in Green's equivalence on S. In view of this and Lemma 10.3 of A. H. Clifford and G. B. Preston (1) it is obvious that the maximum idempotent-separating congruence on a regular semigroup S is given by

We study the boundedness and compactness of Toeplitz operators Ta on Bergman spaces , 1 < p < ∞. The novelty is that we allow distributional symbols. It turns out that the belonging of the symbol to a weighted Sobolev space of negative order is sufficient for the boundedness of Ta. We show the natural relation of the hyperbolic geometry of the disc and the order of the distribution. A corresponding sufficient condition for the compactness is also derived.

Let R be a ring, a linear sequential machine over R is a quintuple ℳ = (Q, A, B, F, G) where

Q, A, B are R-modules

F:Q × A →Q and G:Q × A→B are R-homomorphisms.

The asymptotics behaviour of solutions of the Becker-Döring cluster equations is determined for cases in which coagulation dominates fragmentation. We show that all non-zero solutions tend weak* to zero.

Precise bounds are given for the quantity

$$ L(\alpha)=\frac{\limsup_{m\rightarrow\infty}(1/m)\ln q_m}{\liminf_{m\rightarrow\infty}(1/m)\ln q_m}, $$

where $(q_m)$ is the classical sequence of denominators of convergents to the continued fraction $\alpha=[0,u_1,u_2,\dots]$ and $(u_m)$ is assumed bounded, with a distribution.

If the infinite word $\bm{u}=u_1u_2\dots$ has arbitrarily large instances of segment repetition at or near the beginning of the word, then we quantify this property by means of a number $\gamma$, called the segment-repetition factor.

If $\alpha$ is not a quadratic irrational, then we produce a specific sequence of quadratic irrational approximations to $\alpha$, the rate of convergence given in terms of $L$ and $\gamma$. As an application, we demonstrate the transcendence of some continued fractions, a typical one being of the form $[0,u_1,u_2,\dots]$ with $u_m=1+\lfloor m\theta\rfloor\Mod n$, $n\geq2$, and $\theta$ an irrational number which satisfies any of a given set of conditions.

AMS 2000 Mathematics subject classification: Primary 11A55. Secondary 11B37

Using orbifold metrics of the appropriately signed Ricci curvature on orbifolds with a negative or numerically trivial canonical bundle and the two-dimensional log minimal model programme, we prove that the fundamental group of special compact Kähler 3-folds is almost abelian. This property was conjectured in all dimensions by Campana in 2004, and also for orbifolds in 2007, where the notion of specialness was introduced. We briefly recall the definition, basic properties and the role of special manifolds in classification theory.

In this paper the “hypercore” of a semigroup S is defined to be the subsemigroup generated by the union of all the subsemigroups of S without non-universal cancellative congruences, provided that at least one such subsemigroup exists: otherwise it is taken to be the empty set. It is shown first that if the hypercore of S is nonempty (which holds, for example, when S contains an idempotent) then it is the largest subsemigroup of S with no non-universal cancellative congruence, is full and unitary in S, and is contained in the identity class of every group congruence on S (Theorem 1).

The following work is a sequel to three previous communications, and more particularly to the first. The present object is to shew the effect of repeated operation with the matrix differential operator , when it acts upon a scalar matrix formed from an n rowed determinant |xij|, or sums of principal minors, the n2 elements xij being treated as independent variables. Thus when z is a scalar quantity ω z means the matrix [∂z/∂xij], whose ijth element is the derivative.

We obtain heat-kernel estimates for higher-order operators with measurable coefficients that can be singular or degenerate. Precise constants are given, which are sharp for small times.

AMS 2000 Mathematics subject classification: Primary 35K30; 47D06. Secondary 35K25; 35K35

We discuss the problem of finding those integers which may be represented by (x + y + z)3/xyz, and also those which may be represented by x/y + y/z + z/x, where x, y, z are integers. For example,

satisfy (x + y + z)3/(xyz) = -47, and

satisfy x/y + y/z + z/x = -86.