Spectral and dynamical properties of some one-dimensional continuous Schrödinger and Dirac operators with a class of sparse potentials (which take non-zero values only at some sparse and suitably randomly distributed positions) are studied. By adapting and extending to the continuous setting some of the techniques developed for the corresponding discrete operator cases, the Hausdorff dimension of their spectral measures and lower dynamical bounds for transport exponents are determined. Furthermore, it is found that the condition for the spectral Hausdorff dimension to be positive is the same for the existence of a singular continuous spectrum.

This research deals with properties of polynomial regular functions, which were introduced in a recent study concerning Wiman-Valiron theory in the unit disc. The relation of polynomial regular functions to a number of function classes is investigated. Of particular interest is the connection to the growth class Gα, which is closely associated with the theory of linear differential equations with analytic coefficients in the unit disc. If the coefficients are polynomial regular functions, then it turns out that a finite set of real numbers containing all possible maximum modulus orders of solutions can be found. This is in contrast to what is known about the case when the coefficients belong to Gα.

The theory of Inversion presents one of the simplest examples of those Birational Transformations of plane figures, whose general theory is due to Cremona. It has a distinguishing feature to which it owes its name. If the point P “inverts” into Q, then Q inverts into P. It is therefore a simple case of these involutive point transformations much of the general theory of which was developed by the late Admiral de Jonquières in a paper printed as late as 1864 in the Nouvelles Annales, but which had originally been addressed to the Institute of France in 1859. This memoir is not only highly interesting, but is eminently readable and very ingenious.

Let ν0 be a valuation of a field K0 with residue field k0 and value group Z, the group of rational integers. Let K0(x) be a simple transcendental extension of K0. In 1936, Maclane [3] gave a method to determine all real valuations V of K0(x) which are extensions of ν0. But his method does not seem to give an explicit construction of these valuations. In the present paper, assuming K0 to be a complete field with respect to ν0, we explicitly determine all extensions of ν0 to K0(x) which have Z as the value group and a simple transcendental extension of k0 as the residue field. If V is any extension of ν0 to K0(x) having Z as the value group and a transcendental extension of k0 as the residue field, then using the Ruled Residue theorem [4, 2, 5], we give a method which explicitly determines V on a subfield of K0(x) properly containing K0.

Dieudonné (4) has constructed an example of a Banach space X and a complete Boolean algebra of projections on X such that has uniform multiplicity two, but for no choice of x1, x2 in X and non-zero E in is EX the direct sum of the cyclic subspaces clm {Ex1:E∈} and clm {Ex2:E∈}. Tzafriri observed that it could be deduced from Corollary 4 (9, p. 221) that the commutant ′ of is equal to A(), the algebra of operators generated by in the uniform operator topology. A study of (3) suggested the direct proof of the second property given in this note. From this there follows a simple proof that has the first property.

Most of the solutions of Laplace's equation in common use in mathematical physics have been expressed in the integral form given by Whittaker, viz

We give an abstract description of the kernel of a proper primitive inverse congruence on a categorical semigroup. More specifically, we show that it is a *-reflexive, *-unitary, *-dense subsemigroup, and that on a given categorical semigroup there is a one–one correspondence between such subsemigroups and the proper primitive inverse congruences. Our results allow us to give a description of the minimum proper primitive inverse semigroup congruence on a strongly E*-dense categorical semigroup.

We prove that spaces with an uncountable $\omega$-independent family fail the Kunen–Shelah property. Actually, if $\{x_i\}_{i\in I}$ is an uncountable $\omega$-independent family, there exists an uncountable subset $J\subset I$ such that $x_j\notin\overline{\co}(\{x_i\}_{i\in J\setminus\{j\}})$ for every $j\in J$. This improves a previous result due to Sersouri, namely that every uncountable $\omega$-independent family contains a convex right-separated subfamily.

AMS 2000 Mathematics subject classification: Primary 46B20; 46B26

Throughout we consider operators on a reflexive Banach space X. We consider certain algebraic properties of F(X), K(X) and B(X) with the general aim of examining their dependence on the possession by X of the approximation property. B(X) (resp. K(X)) denotes the algebra of all bounded (resp. compact) operators on X and F(X) denotes the closure in B(X) of its finite rank operators. The two questions we consider are:

(1) Is K(X) equal to the set of all operators in B(X) whose right and left multiplication operators on F(X) (or on B(X)) are weakly compact?

(2) Is F(X) a dual algebra?

In this paper, we are concerned with the non-existence of positive solutions of a Hartree–Poisson system:

\begin{equation*}\left\{\begin{aligned}&-\Delta u=\left(\frac{1}{|x|^{n-2}}\ast v^p\right)v^{p-1},\quad u \gt 0\ \text{in} \ \mathbb{R}^{n},\\&-\Delta v=\left(\frac{1}{|x|^{n-2}}\ast u^q\right)u^{q-1},\quad v \gt 0\ \text{in} \ \mathbb{R}^{n},\end{aligned}\right.\end{equation*}

$n \geq3$

$\min\{p,q\} \gt 1$

The closed wedges in C(X) (the space of real continuous functions on a compact Hausdorff space X) which are also inf-lattices have been characterized by Choquet and Deny (2); see also (5). The present note extends their result to certain wedges of affine continuous functions on a Choquet simplex, the generalization being in the same spirit as the generalization of the Kakutani- Stone theorem obtained by Edwards in (4).

I should like to thank my supervisor, Dr D. A. Edwards, for suggesting this problem and for his subsequent help. I am also grateful to the referee for correcting several slips.

We investigate in this paper a certain special family of quadric varieties, that is of in [R]. Now among the more important properties of a quadric in [R] is that it possesses a system or systems of “ generators,” i.e., the quadric may be taken as the locus of certain families of subspaces, the behaviour of these depending on the parity of R. If R is even, a quadric V2n–1 in [2n] contains a single family of [n—l]'s, so it seems likely that in discussing special families of quadrics in [2n] an important type will be obtained by constraining the quadric to pass through a number of [n — l]'s.

Let R be an associative, commutative ring with identity, and let A be a (unitary) R-module. It is well known that if A is a Noetherian R-module then every submodule of A has a primary decomposition in A. The object of the present paper is to dualise this result; that is, to show that if A is an Artinian R-module then every submodule of A can be expressed as the sum of a finite number of coprimary submodules of A.

In a previous paper (q.v.) by Mr W. L. Marr and the present writer, it was shown that, in accordance with Morley's Theorem, the angles A+2pπ, B+2pπ, C+2rπ of the triangles ABC be trisected, the three groups of six lines at the vertices give rise to 27 triangles DEF, the biangular coordinates of D with respect to BC being (B/3+2qπ/3, C/3+2rπ/3) or (βq, γr), and similarly for and F with respect to CA and AB.

A simple approximate formula is obtained for the capacity of the condenser formed by a “small” conductor placed inside a much larger one. The formula involves a constant whose choice is, to a certain extent, arbitrary and it is shown that, for problems involving spheroids inside cylinders and between parallel plates, the constant may be found in a simple fashion so as to give very accurate results. A similar formula is obtained for the loss in potential energy due to a crack or cavity in a circular beam or a thick plate. For the particular cases of the boundary value problems considered which have been treated by other means very close numerical agreement is obtained between those results and ones deduced in the present paper.

The Poisson kernel is defined for z in the open unit disc D and ζ in the unit circle ∂D. As usually employed, it is integrated with respect to the second variable and a measure on ∂D to yield a harmonic function on D. Here, we fix a σ-finite positive Borel measure m on D and integrate the Poisson kernel with respect to the first variable against a function φ in L1(m) to obtain a function Tmφ on ∂D. We ask for what measures m the range of Tm is L1(∂D), for what m the kernel of Tm is non-zero, and for what m every positive continuous function on ∂D is of the form Tmφ with φ non-negative. When m is the counting measure of a countably infinite subset {ak:k∈ℕ} of D, the function (Tmφ)(ζ) is of the form with . The main results generalize results previously obtained for sums of this form. A related mapping from Lp(m) into Lp(∂D) with 1 <p<∞ is briefly considered.

This paper is an appendix to a joint paper with Professor Macbeath. In (3), it was proved that the invariant measure, m(k), of the set of real n × nmatrices τ, with determinant 1 and norm satisfying ∥τ∥≦ k, had the property that

This paper concerns a Jacobson-type radical for the near-ring N. This radical, denoted by Js(N) has an external representation on a type-0N-group of a very special kind. Such N-groups are said to be of type-s. The main objective of this paper is to decompose Js(N) as a sum Js(N) = J1/2(N) + A + B for N satisfying the descending chain condition for N-subgroups. In this decomposition J1/2(N) is nilpotent and A is the unique minimal ideal modulo which Js(N) is nilpotent.

Throughout this paper, we suppose that T and R are continuous linear operators on the Banach spaces X and Y, respectively. One of the basic problems in the theory of automatic continuity is the determination of conditions under which a linear transformation S: X → Y which satisfies RS = ST is continuous or is discontinuous. Johnson and Sinclair [4], [6], [11; pp. 24–30] have given a variety of conditions on R and T which guarantee that all such S are automatically continuous. In this paper we consider the converse problem and find conditions on the range S(X) which guarantee that S is automatically discontinuous. The construction of such automatically discontinuous S is then accomplished by a simple modification of a technique of Sinclair's [10; pp. 260–261], [11; pp. 21–23].

It is shown that every spherically symmetric metric can be transformed into the isotropic form. As illustration an example is given.