The classical Dvoretzky-Rogers theorem states that if E is a normed space for which l1(E)=l1{E} (or equivalently , then E is finite dimensional (see [12] p. 67). This property still holds for any lp (l<p<∞) in place of l1 (see [7]p. 104 and [2] Corollary 5.5). Recently it has been shown that this result remains true when one replaces l1 by any non nuclear perfect sequence space having the normal topology (see [14]). In this context, De Grande-De Kimpe [4] gives an extension of the Devoretzky-Rogers theorem for perfect Banach sequence spaces.

A study of nonselfadjoint algebras of Hilbert space operators was begun by considering special types of such algebras, namely those determined by a commuting family of rank one operators. A first step in this direction was made by Erdos in [1] and is continued more extensively in [2].

In many biological diffusion-reaction studies, it was found that one should include the effect of density dependent rates, drift terms and spatially varying growth rates, in order to obtain more accurate results. (See e.g. [7],[10], [8] , [3]). On the other hand, many recent mathematical results on reaction-diffusion systems do not include such general setting. This article investigates the behaviour of competing-species reaction-diffusion model under this more general situation. Efforts are made to obtain results concerning coexistence, survival and extinction, by methods similar to that in [5], [6].

Let p be an odd prime and let GL(2, p) denote the general linear group of invertible 2x2 matrices with entries in the field of p elements. The groupPGL(2, p) is the factor of GL(2,p) by its centre and has derived group PSL(2,p) with derived factor C2, the cyclic group of order 2.

In this note it is shown that Theorem 2.8 of [2] is erroneous. Throughout this note nea-ring means a right near-ring.

There is now a huge volume of literature dealing with exact and approximate solutions of the equations governing the motion of an inviscid, incompressible fluid. Major difficulties in finding exact solutions arise when the fluid is bounded by a socalled "free" surface on which the pressure is constant because the equation of such a surface is not known in advance. A boundary condition of this type arises when a mathematical analysis of water wave theory is attempted. Different approximations are, of course, appropriate to different circumstances and here we deal with a well known approximation for waves in shallow water which will be described in detail in the next section.The main ideas underlying this particular approximation have been known for some time but it is probably true to say that they were first brought into real prominence by Stoker [3] in a long article in the Communications of Pure and Applied Mathematics. The main ideas in this article were later developed in considerable detail in Stoker's book “Water Waves” [4]. More recently this approximation has been used by Sewell and Porter [2] as one of several examples in fluid and solid mechanics which are given a geometrical interpretation in terms of “constitutive surfaces” which has close links with catastrophe theory. An excellent account of this interesting approach, which lists many associated references, is given by Sewell [1].

The problem of finding a suitable representation for a fractional power of an operator defined in a Banach space X has, in recent years, attracted much attention. In particular, Balakrishnan [1], Hovel and Westphal [3] and Komatsu [4] have examined the problem of defining the fractionalpower (–A)α for closed densely-defined operators A such that

The number of self-complementary (s.c.) graphs and digraphs with a given number of vertices was found by R. C. Read in [1]. That paper used a special case of De Bruijn's generalisation of Polya's theorem that involved the cycle-index of Gn, the group of permutations of pairs of vertices induced by permutations of the vertices. We obtain Read's formulae by using only well-known elementary facts about s.c. graphs and their complementing permutations.

The transpositions that generate a symmetric group can be represented as real reflections: symmetry operations of a regular simplex. Analogous unitary reflections serve to generate other factor groups of the braid group; they are symmetry operations of regular complex polytopes. Certain relationships among these groups have, as geometric counterparts, unexpected plane sections of the polytopes, beginning with the square sections of the regular tetrahedron. In Section 6, 5-dimensional coordinates will be used to exhibit pentagonal sections of the 4-dimensional regular simplex. The most spectacular instance of such “equatorial” sections occurs in the case of the Witting polytope in complex 4-space, so exquisitely drawn by Peter McMullen for the frontispiece of Regular Complex Polytopes [6]. This has a plane section 3{<5}3 which appears thare as Fig. 4.8B on page 48. Shephard [9, p. 92] called it 3(360)3. Its 120 vertices will be seen to be situated “inside” 120 of the 2160 faces 3{3}3 of the Witting polytope. These “faces” are self-inscribed octagons [7, p. 290].

The purpose of this paper is to expand upon the results obtained in [4]. We consider the set H of differential equations

where p and r are continuous real-valued functions of period ω (ω being fixed throughout). The equation (1.1) is denoted by P or (p,r), and we regard H as the set of pairs of continuous functions of period ω.On H we define a norm:

In recent years the geometry of generic submanifolds of Euclidean space has been theobject of much study. Thorn hinted in [7] that the focal set of such a submanifold couldprofitably be studied by using the family of distance squared functions on thesubmanifold from points of the ambient space. For a generic submanifold the focal set isthe catastrophe or bifurcation set of this family. The key to obtaining results on thelocal structure of this focal set is a transversality theorem of Looijenga [5]; for analternative exposition see [8].

Recall that a norm and monotone if and monotone if If the norm is both absolute and monotone, itis called a Riesz norm. It is easy to show that a norm is Riesz if and only if whenever A Banach lattice is a vector lattice equipped with a complete Riesznorm.

In this paper, we study the boundary-initial value problem for a linear elastic body ina bounded domain, when the body force depends on the displacement vector u in asublinear way.

Recently, much attention has been given to nonlinear body forces not only to studythe fundamental solutions of the equations governing linear elastodynamics, see e.g.Kecs [3], but also to derive global non existence results in abstract problems with directapplications to nonlinear heat diffusion or to the nonlinear wave equation, see e.g. Ball[1], Levine and Payne [10].

In connection with algebras of unbounded operators, Lassner showed in [4] that, if T is a densely defined, closed linear operator in a Hilbert space such that its domain is contained in the domain of its adjoint T* and is globally invariant under T and T*,then T is bounded. In the case of a Banach space (in particular, a C*-algebra) weshowed in [6] that a densely defined closed derivation in a C*-algebra with domaincontaining its range is automatically bounded (see the references in [6] and [7] for thetheory of derivations in C*-algebras).