If the functions be written as in the above figure, then

1. Of any three functions written consecutively on a circular arc or on a diameter, the mean is the product of the extremes. Thus sinA = tanAcosA, 1 = sinAcosecA, and so on.

2. In each triangle the square of the function written at the vertex turned downwards is equal to the sum of the squares of the functions written at the other two vertices. Thus I2 = sin2A + cos2A, and so on.

A pair of coupled nonlinear differential equations is studied and asymptotic properties of its non-oscillatory solutions are obtained. In particular, we provide classification schemes for these solutions which are justified by existence criteria.

Introduction. Nous nous proposons ici de pousser un peu plus loin les conséquences que Broderick a tirées d'un théorème dont il nous a fait connaître l'existence au dernier Colloque de St Andrews.

L'un des buts de Broderick était de prouver que certaines formules de King restent valables quand les événements considéréd ne sont plus indépendants, démonstration dont il n'avait pas trouvé trace dans la littérature, sauf relativement à des cas particuliers.

Let $\rho:G\hookrightarrow\GL(n,\F)$ be a representation of a finite group $G$ over a finite field $\F$ and $f_1,\dots,f_n\in\F[V]^G$ such that the ring of invariants is a polynomial algebra $\F[f_1,\dots,f_n]$. It is known that in this case the algebra of coinvariants $\F[V]_G$ is a Poincaré duality algebra, and if, moreover, the order of $G$ is invertible in $\F$, that a fundamental class is represented by the Jacobian determinant $\mathchoice{\det\biggl[\frac{\partial f_i}{\partial z_j}\biggr]}{\det[\partial f_i/\partial z_j]}{\det[\partial f_i/\partial z_j]} {\det[\partial f_i/\partial z_j]}$, and is therefore a $\det^{-1}$-relative invariant. In this note we deduce what happens in the modular case. As a bonus we obtain a new criterion for an unstable algebra over the Steenrod algebra to be a ring of invariants.

AMS 2000 Mathematics subject classification: Primary 13A50; 55S10

In the first part of the paper we prove several results on the existence of invariant closed ideals for semigroups of bounded operators on a normed Riesz space (of dimension greater than 1) possessing an atom. For instance, if S is a multiplicative semigroup of positive operators on such space that are locally quasinilpotent at the same atom, then S has a non-trivial invariant closed ideal. Furthermore, if T is a non-zero positive operator that is quasinilpotent at an atom and if S is a multiplicative semigroup of positive operators such that TS ≤ ST for all S ∈ S, then S and T have a common non-trivial invariant closed ideal. We also give a simple example of a quasinilpotent compact positive operator on the Banach lattice l∞ with no non-trivial invariant band.

The second part is devoted to the triangularizability of collections of operators on an atomic normed Riesz space L. For a semigroup S of quasinilpotent, order continuous, positive, bounded operators on L we determine a chain of invariant closed bands. If, in addition, L has order continuous norm, then this chain is maximal in the lattice of all closed subspaces of L.

The problem of statistical “selection” is concerned with the alteration induced in a frequency distribution in several variables by an alteration of the parameters in a subsection of the distribution. It may be illustrated by a simple trivariate case, as follows:

From a population characterised by variables x, y, z, correlated and normally distributed, with means 0, 0, 0, variances and product variances r12σ1σ2, r13σ1σ3, r23σ2σ3, a sub-population is extracted by selection in x alone, in such a way that after selection x is still normally distributed, but with mean h and variance s2 . It is required to determine the new values, in the selected population, of the means and variances of y and z, and of the product variances.

A Banach space which is not reflexive may or may not be equivalent (in Banach's sense) to an adjoint space. For example, it is an elementary fact that the space (l), though not reflexive, is equivalent to (co)*, where (co) is the space of all sequences that converge to zero, normea in the usual way. On the other hand, (co) itself is not equivalent to any adjoint space : this can be proved by means of the Krein-Milman theorem, but here we obtain the result by an elementary argument which is scarcely more complicated than the standard proof that (co) is not reflexive.

Casey, in an extensive and complete mémoire, has discussed very thoroughly the bicircular quartic. Among the unfinished notes of Dr Clifford the properties of the curve are established by means of a tetracyclic system of coordinates. Such a system of coordinates has been extensively used by Coolidge. The object this paper is to present a valid discussion of the curve in terms a tricyclic system of “power” coordinates.

Let $p$ be an odd prime. The primary purpose of this paper is to determine the excess of the conjugates of the Steenrod operations $\mrm{P}[k;f]$, which are defined as $\mrm{P}[k;f]:=\mrm{P}(p^{k-1}f)\cdot\mrm{P}(p^{k-2}f)\cdot\cdots\cdot\mrm{P}(pf)\cdot\mrm{P}(f)$. The result is then used to obtain sufficient conditions for an element in the polynomial algebra $\mathbb{F}_p[x_1,\dots,x_s]$ to be in the image under the standard action of the Steenrod algebra. Results and methods are generalizations of previous work by Judith Silverman and by myself with Judith Silverman.

AMS 2000 Mathematics subject classification: Primary 55S10. Secondary 55S05

The present paper is a continuation of the work initiated in [l]-[5]. In [5] I gave an expansion of the form

for the second order C.F. associated with

where U8, V8, W8 satisfy a fourth-order recurrence relation, there being a similar expansion for third order C.F.'s. I shall now give simple expressions for U8, V8, W8 (or related forms) in terms of χ2s(Z1), χ2s (Z2), ω2s(Z1), ω2s(Z2), where

and show that there is a remarkable relation between the recurrence formula for the first order C.F. and that satisfied by U3, V3, W3. The generalised form of these results will be stated and proved.