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

We study the bifurcation problem

$$ \begin{cases} -\Div(a(x)|Du|^{p-2}Du)+h(x)u^{r-1}= f(\lambda,x,u) & \text{in } \sOm\subset\RR^{N}, \\ a(x)|Du|^{p-2}Du\cdot n+b(x)u^{p-1}=\theta g(x,u) & \text{on }\sGa, \\ u\geq0,\quad u\not\equiv0 & \text{in }\sOm, \end{cases} $$

where $\sOm$ is an unbounded domain with smooth non-compact boundary $\sGa$, $n$ denotes the unit outward normal vector on $\sGa$, and $\lambda>0$, $\theta$ are real parameters. We assume throughout that $p lt r lt p^{*}=pN/ (N-p)$, $1 lt p lt N$, the functions $a$, $b$ and $h$ are positive while $f$, $g$ are subcritical nonlinearities. We show that there exist an open interval $I$ and $\lambda^\star gt 0$ such that the problem has no solution if $\theta\in I$ and $\lambda\in (0,\lambda^\star)$. Furthermore, there exist an open interval $J\subset I$ and $\lambda_0 gt 0$ such that, for any $\theta\in J$, the above problem has at least a solution if $\lambda\geq \lambda_0$, but it has no solution provided that $\lambda\in (0,\lambda_0)$.

AMS 2000 Mathematics subject classification: Primary 35J60; 35P30; 58E05; 58G28

§1. The Lamé Functions of degree n (where n is a positive integer) may be defined as those solutions of the equation

which are polynomials in the elliptic functions sn x, cn x, dn x of real modulus K. Such solutions only exist for certain particular values of the constant a; there are 2n + 1 such values and 2n + 1 corresponding Lamé functions.

We study the existence of extremal periodic solutions for nonlinear evolution inclusions defined on an evolution triple of spaces and with the nonlinear operator establish A being time-dependent and pseudomonotone. Using techniques of multivalued analysis and a surjectivity result for L-generalized pseudomonotone operators, we prove the existence of extremal periodic solutions. Subsequently, by assuming that A(t, ·) is monotone, we prove a strong relaxation theorem for the periodic problem. Two examples of nonlinear distributed parameter systems illustrate the applicability of our results.

Let X be a compact Riemannian manifold. If f:X→ℝ is a nondegenerate Morse function in the sense of Bott [2] then one has Morse inequalities which can be expressed in the form

where Pt(X) is the Poincaré polynomial Σtidim Hi(X;ℚ of X ann {Cβ|β ∈B} are the connected components of the set of critical points for f For any polynomial Q(t)∈ℤ[t] we write Q(t)≧0 if all the coefficients of Q are nonnegative.

1. The following is an account of a theorem whose origin has been traced to Prof. Morley of Johns Hopkins University.

In the course of certain vector analysis, some 14 years ago, Prof. Morley found that if a variable cardioide touch the sides of a triangle the locus of its centre, that is, the centre of the circle on which the equal circle rolls, is a set of 9 lines which are three by three parallel, the directions being those of the sides of an equilateral triangle. The meets of these lines correspond to double tangents; they are also the meets of certain pairs of trisectors of the angles, internal and external, of the first triangle. This result was never published, and it was only the particular case of the internal trisectors that reached the present writers, the existence of the enveloping cardioides and the set of 9 lines being quite unknown to them.

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)

We introduce a definition of a Schunck class of periodic abelian-by-finite soluble groups using major subgroups in place of the maximal subgroups used in Finite groups. This allows us to develop the theory as in the finite case proving the existence and conjugacy of projectors. Saturated formations are examples of Schunck classes and we are also able to obtain an infinite version of Gaschütz Ω-subgroups.

Asymptotic formulae are given for the distribution of Liusternik-Schnirelman eigenvalues of certain pairs of nonlinear functionals generalising the usual Weyl theory for linear pairs of elliptic operators. In particular an application is made to the von Kármán theory of buckled plates.

In this paper, the centraliser of an arbitrary element of a wreath product is determined. One application of this is to find the breadth of a wreath product (Theorems 21 and 22), a problem which was raised in discussion with Dr. I. D. Macdonald. Another application is to groups generated by elements generating their own centralisers (Theorem 20).

Let A and B be two groups. Define

AB = {f : B → A; f(b) = e for all but a finite number of elements of B} to be a group by defining the product pointwise

What is meant will best be understood by the following:-

1. Suppose we have an unlimited straight line XX′ (fig.24) with a finite point O on it. Take two points on the line, C and C1 in the same sense with respect to O such that OC·OC1=a2. Then C and C1 are connected by inversion, and O is the centre inversion. Without loss of generality for what follows we may, for convenience, take a2=12=1, so that OC·OC1=1.

The method employed in this paper is first to ascertain in how many ways a cube can be cut into tetrahedra without making new corners, and then, taking each of these divisions of the cube as the type of a genus of divisions of the general parallelepiped, determining the number of species in each genus.

In a short course of lectures on “Railway Practice,” the author was recently called upon to deal with the mechanical principles involved in the design of retaining walls. What has come to be known as Rankine's method had to be explained, at all events, in its practical application. But the time at the author's disposal did not admit of the general consideration of the theory of stress by which Rankine in characteristic fashion leads up (in his Applied Mechanics) to the particular problem under discussion. The author had therefore to choose between omitting a demonstration or making a short cut, which at the same time should be of a rigorous nature.