Let A and B be function algebras. The well-known Nagasawa theorem [5] states that A and B are isometric if and only if they are isomorphic in the category of Banach algebras. In [2] it was shown that this theorem is stable in the sense that if the Banach–Mazur distance between the underlying Banach spaces of A and B is close to one then these algebras are almost isomorphic, that is there exists a linear map T from A onto B such that . On the other hand one can get from Theorems 1 and 3 of [3] that the Nagasawa theorem can be extended to some operator algebras as follows:

If G is a group and N a ring, the elements of the group ring NG can be thought of either as formal sums or as functions Φ:G→Nwith finite support. If N is a nearring, problems arise in trying to construct a group near-ring either way. In the first case, Meldrum [7] was abl to exploit properties of distributively generated near-rings (N, S) to build free (N,S)-products and hence a near-ring analogue of a group ring. For the latter case, Heatherly and Ligh [3] observed that the set of functions could be made into a near-ring under multiplication given by provided N satisfies

for all ai,bin∈N and k∈Z+. Such near-rings are called pseudo-distributive. In fact these are precisely the conditions under which the set Nk of k x k matrices over N is also a near-ring and then both NG and Nk are pseudo-distributive.

Consider the ordinary linear matrix differential system

ψ(x) is a scalar mapping, X and A(x) are n by n matrices. Both belong to C1([a,∞)) for some integer l. The stability and asymptotic behaviour of its solutions have been subject to much investigation. See Bellman [2], Levinson [24], Hartman and Wintner [20], Devinatz [9], Fedoryuk [11], Harris and Lutz [16,17,18] and Cassell [30]. The special interest in eigenvalue problems and in the deficiency index problem stimulated a continued interest in asymptotic integration. See e.g. Naimark [36], Eastham and Grundniewicz [10] and [8,9]. Harris and Lutz [16,17,18] succeeded in explaining how to derive many known theorems in asymptotic integration by repeatedly using certain “(1 + Q)” linear transformations.

Let G be a group with identity 0 and let be a group of automorphisms of G. The centralizer near-ring determined by G and is the set for all α∈ and f(0)=0}, forming a near-ring under function addition and function composition. This class of nea-rings has been extensively studied (for example see [1], [2], [5] and [6]) and it is known that every finite simple near-ring with identity which is not a ring is isomorphic to C(;G) for a suitable pair (,G) see [6] page 131, Corollary 4.59 and Theorem 4.60.

The classical Hamburger moment problem can be formulated as follows: Given a sequence {cn:n=0,1,2,…} of real numbers, find necessary and sufficient conditions for the existence of a distribution function ψ (i.e. a bounded, real-valued, non-decreasing function) on (– ∞,∞) with infinitely many points of increase, such that , n = 0,1,2, … This problem was posed and solved by Hamburger [5] in 1921. The corresponding problem for functions ψ on the interval [0,∞) had already been treated by Stieltjes [15] in 1894. The characterizations were in terms of positivity of Hankel determinants associated with the sequence {cn}, and the original proofs rested on the theory of continued fractions. Much work has since been done on questions connected with these problems, using orthogonal functions and extension of positive definite functionals associated with the sequence. Accounts of the classical moment problems with later developments can be found in [1,4,14]. Good modern accounts of the theory of orthogonal polynomials can be found in [2,3].

Let X be an infinite set, be the group of all bijections of X and S be a semigroup of total transformations of X with the composition of transformations f and g in S defined by the formula

We say that S is a -normal semigroup if

This article is intended to enlarge the study of spaces satisfying the countable neighbourhood property and to clarify the incidence of this property in the stability of some locally convex properties of tensor products.

We shall use the standard notations of locally convex spaces as in [17] and [18]. The word space will always mean separated locally convex space. If (£, t) is a space, the set of all continuous seminorms on it will be denoted by cs(E). The linear hull and the absolutely convex hull of a subset C of a space will be written lin(C) and г(C) respectively.

This note is an exposition of the simple and elegant approach to Grothendieck's inequality given in [2] and lsqb;4], with one further simplification. The process of factorizing through L2 ([2], p. 21) introduces a factor of into the final constant. We show that this step can be avoided.

For bounded operators, the theory of the joint numerical range has been developed by various authors [1,2,3,4,5]. Especially, the properties of commuting normal n-tuples are discussed in detail. Our purpose here is to show that many results in the above references still hold in the case of unbounded normal operators (see Theorem 2.3, Corrollary 3.5, Theorem 4.1, Theorem 4.2). Besides, the operator algebras are closely related to the theory of joint spectrum and joint numerical ranges in the boundedcase (cf. [1,3]). How about unbounded operators? It seems that one must consider unbounded operator algebras. Some work has been done in this direction for the joint spectrum of unbounded normal operators [9]. In the last section of this paper, we provide some intimate relations between the joint numerical range and the unbounded operator algebras for unbounded normal operators.

The theory of joint spectra for commuting operators in a Hilbert space has recently been studied by several authors (Vasilescu [11,12], Curto [4,5], and Cho-Takaguchi[2,3]). In this paper we willuse the definition by Taylor [10] of the joint spectrum to show that thejoint spectrum is determined by the action of certain "Laplacians"(cf. Curto [4,5]) of a chain-complex of Hilbert spaces.

In this paper we analyse the electrical behaviour within systems of long and short coupled nerve axons by using a geometric approach to obtain a priori bounds on solutions. In [4[ we developed a general model for a bundle of n-uniform unmylinated nerve fibres. If FitzHugh-Nagumo dynamics, [3[ are used to describe the ionic membrane currents, then the model takes the form

Here W=(w1,…wn)T denotes the membrane action potentials for each fibre in the bundle and Z=(Z1,…Zn)T represents the recovery variables for each fibre, which control the return to the resting equilibrium after any transmission of signals.

Recently, there has been renewed interest in the homology of connective covers of the classifying spaces BU and BO, and their associated Thom spectra-see e.g. [4,6,9,10,15]. There are now numerous families of generators as well as structural results on the action of the Steenrod algebra. However, these two areas have not been well related since the methods used have tended to emphasise one goal rather than the other. In this paper we show that there are in fact canonical Hopf algebra decompositions for the sub-Hopf algebras of the homology of BU, and BO constructed by S. Kochman in [9], generalising those of [8]. Furthermore, these are clearly and consistently related to the Steenrod algebra action, and provide canonical sets of algebra generators. They should thus allow calculations of the type exemplified in [6] to be carried out in all cases, although of course the complexity of the answer increases rapidly! A by-product of our approach is that we can easily obtain results on these homologies as Hopf algebras, such as selfduality and a computation of endomorphism groups over the Steenrod algebra. We feel that the methods will also give interesting information in the case of some other familiar spaces even if their homology is not self dual (or bipolynomial); we intend to return to this in a sequel.