The tensor product A ⊗ B of the distributive lattices A and B was first investigated by Fraser in [4] and [5]. In this paper, we present some results relevant to the structure and construction of this tensor product. In particular, we establish a sufficient condition for join-irreducibility in the tensor product and show that this condition characterizes join-irreducibility in the case that A and B satisfy the descending chain condition. We also show that if A and B satisfy the descending chain condition then so does A ⊗ B; this insures the compact generation of A ⊗ B by its join-irreducibles. We conclude with some examples and applications of our results to the tensor product of finite distributive lattices.

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.

In [6] Sands proved that the semisimple classes of associative rings are exactly the coinductive and closed under ideals and extensions classes. This characterization was transferred to the alternative case by Van Leeuwen, Roos and Wiegandt in [3]. Answering a question of [9], Sands [7] has recently proved that in the associative case the condition of being closed under ideals can be replaced by the regularity of the class. The same result for alternative rings has been proved by Anderson and Wiegandt in [2]. Thus the following result holds.

In what follows, character means irreducible complex character.

Let G be a finite group and let % be a character of a normal subgroup N. If χ extends to a character of G then χ is stabilised by G, but the converse is false. The aim of this paper is to prove the following theorem which gives a sufficient condition for χ to be extended to a character of G.

In this paper the study of radicals of finite near-rings is initiated. The main result (Theorem 4.3) gives a description of hereditary radicals having hereditary semisimple classes too. Also it is shown that there exist non-hereditary radicals having hereditary semisimple classes.

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].

The problem of deciding which graded polynomial algebras over the field of p elements can occur as the -cohomology of a space has played a central rôle in the development of algebraic topology beginning as early as 1950. In the case where the polynomial generators do not occur in dimensions divisible by p, Adams and Wilkerson [1] have given a complete solution by showing that the spaces constructed by Clark and Ewing [3] suffice to realize all such algebras as -cohomology rings. The main result of Adams and Wilkerson for odd primes can be stated as follows.

The general heat equation is defined as

where v is a fixed positive number and α is a fixed number. If v = α = 0, then (1.1) reduces to the ordinary heat equation

where u(x,t) is regarded as the temperature at a point x at time t, in an infinite insulated rod extended along the x-axis in the xt-plane. If we set , then (1.1) becomes

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.

A (k – l)-field on Sn-1 may be given as a section ϕ of the fibre bundle

with fibre Vn-1, k-1 or, equivalently, as a semi-orthogonal map, i.e., a map

which is isometric in the second variable and such that for the basis vector e1∈Rk and every x∈Rn

By analogy with the concept of “inverse semi-group” in semi-group theory, in this paper we introduce the concept of “generalized near-field” in near-rings. A near-ring N is called a generalized near-field (GNF) if for each a ε N there exists a unique b ε N such that a = aba and b = bab, that is (N, ·) is an inverse semi-group. Surprisingly, this concept in rings coincides with that of “strong regularity”. But this is not true in the case of near-rings. Every GNF is strongly regular, but in general the converse is not true.

Let Tm, Vmn be Hermitean linear operators on complex Hilbert spaces Hm, m=1…k. A nonzero column vector satisfying

will be called an eigenvalue. This type of problem has been studied extensively by Atkinson [2] from the viewpoint of determinantal operators on the tensor product We shall connect his work with more recent investigations [5,7] of eigenvalue indices based on minimax principles for , which can be viewed as an operator on .

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

In [4] Elmer Rees proves that the symplectic group Sp(n) can be smoothly embedded in Euclidean space with codimension 3n, and the unitary group U(n) with codimension n. These are special cases of a result he obtains for a compact connected Lie group G. The general technique is first to embed G/T, where T is a maximal torus, as a maximal orbit of the adjoint representation of G, and then to extendto an embedding of G by using a maximal orbit of a faithful representation of G. In thisnote, we observe that in the cases G = Sp(n) or SU(n) an improved result is obtained byusing the “symplectic torus” S3 x … x S3 in place of T = S1 x … x S1. As in Rees's construction, the normal bundle of the embedding of G is trivial.

The strong maximum principle for harmonic functions is usually arrived at by appealing to the mean value theorem (c.f. [2], p. 53). It is also of course possible simply to appeal to the Hopf maximum principle [2], but using sledge hammers to kill flies is generally viewed as aesthetically unpleasing. In contrast to the case of harmonic functions, the only proof of the strong maximum principle for the heat equation that is known to me is to invoke Nirenberg's strong maximum principle for parabolic equations [2]. As in the case of harmonic functions, it seems desirable to provide a direct proof of this result without having to go through the subtle comparison arguments that are employed in the more general case. The purpose of this note is to provide a proof of the strong maximum principle for the heat equation based on a mean value theorem for solutions of the heat equation which we derive below. Such an approach provides a straightforward and simple proof of the strong maximum principle which avoids most of the detailed estimates of the proof of the maximum principle for more general parabolic equations. Unfortunately the proof of the maximum principle for the heat equation using the mean value theorem is not as short as the proof in the corresponding case of harmonic functions. It nevertheless seems worthwhile to show that such an alternate proof is possible, and it is to this purpose that we address this paper.

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].

Let (S, Σ, μ) and (T, Θ, v) be two measure spaces of finite measure where we assume S, T are compact Hausdorff spaces and μ, v are regular Borel measures. We construct the product measure space (T x S, >, Φ σ) in the usual way. Let G = [gl, g2, …, gp] and H = [hl, h1, …, hm be finite dimensional subspaces of C(S) and C(T) respectivelywhere G and H are also Chebyshev with respect to the L1-norm. Note that a subspace Y of a normed linear space X is Chebyshev if each x ∈X possesses exactly one best approximation y ∈Y. For example, in C(S) with the L1-norm, the subspace of polynomials of degree at most n is a Chebyshev subspace. This is an old theorem of Jackson. Now set

If G is a finite group and A is a group of automorphisms of G, the “centralizer” nearring C(A, G) consists of the identity-preserving maps from G to itself which commute with the action of A. The main concern of this paper will be with the additive structur of C(A, G) in the case that this near-ring is semisimple.

Let A and B be C*-algebras and A ⊗ B denote the minimal C*-tensor product of A and B. T. Huruya [1] gave examples of C*-tensor products A ⊗ B with C*-subalgebras A1⊗B2 and A2⊗B2 such that strictly contains , answering a question of S. Wassermann [3, Remark 23]. In this short note, we show that the same situation can occur even if A1 = A2