It is well known that if f(t) is (a) integrable in the Lebesgue sense, or more generally (b) integrable in the Perron sense, over every interval (α, β) interior to (a, b), and if

exists, then f(t) is integrable in the Perron sense over (a, b) to the value (1·1).

The question of whether an injective module E over a noncommutative noetherian ring R remains injective after localization with respect to a denominator set X⊆R is addressed. (For a commutative noetherian ring, the answer is well-known to be positive.) Injectivity of the localization E[X-1] is obtained provided either R is fully bounded (a result of K. A. Brown) or X consists of regular normalizing elements. In general, E [X-1] need not be injective, and examples are constructed. For each positive integer n, there exists a simple noetherian domain R with Krull and global dimension n+1, a left and right denominator set X in R, and an injective right R-module E such that E[X-1 has injective dimension n; moreover, E is the injective hull of a simple module.

By a divisibility semigroup we mean an algebra (S,., ∧) satisfying (Al) (S,.) is a semigroup; (A2) (S, ∧) is a semilattice; (A3) .

A divisibility semigroup is called representable if it admits a subdirect decomposition into totally ordered factors.

In this paper various types of representable divisibility semigroups are investigated and characterized, admitting a representation in general or even a special decomposition, like subdirect sums of archimedean factors, for instance.

We define an infinite class of fractals, called horizontally and vertically blocked labyrinth fractals, which are dendrites and special Sierpiński carpets. Between any two points in the fractal there is a unique arc α; the length of α is infinite and the set of points where no tangent to α exists is dense in α.

It is well known that a module M has finite length if and only if it is semi-artinian and Noetherian or, equivalently, semi-noetherian and artinian. Our main result shows that finite length is often achieved by just assuming that M is semi-artinian, semi-noetherian and has finitely generated socle.

Let N(λ) be a square matrix polynomial, and suppose det N is a polynomial of degree d. Subject to a certain non-singularity condition we construct a d by d Hermitian matrix whose signature determines the numbers of zeros of N inside and outside the unit circle. The result generalises a well known theorem of Schur and Cohn for scalar polynomials. The Hermitian “test matrix” is obtained as the inverse of the Gram matrix of a natural basis in a certain Krein space of rational vector functions associated with N. More complete results in a somewhat different formulation have been obtained by Lerer and Tismenetsky by other methods.

An inverse transversal of a regular semigroup S is an inverse subsemigroup that contains precisely one inverse of each element of S. In the literature there are three known types of inverse transversal, namely those that are multiplicative, those that are weakly multiplicative, and those that form quasi-ideals. Here, by considering natural ways in which certain words can be simplified, we reveal four new types of inverse transversal. All of these can be illustrated nicely in examples that are based on 2 × 2 matrices.

Let n be a positive integer and D a division algebra of finite dimension m over its centre. We describe in detail the structure of a soluble subgroup G of GL(n,D). (More generally we consider subgroups of GL{n,D) with no free subgroup of rank 2.) Of course G is isomorphic to a linear group of degree mn and hence linear theory describes G, but the object here is to reduce as far as possible the dependence of the description on m. The results are particularly sharp if n=l. They will be used in later papers to study matrix groups over certain types of infinite-dimensional division algebra. This present paper was very much inspired by A. I. Lichtman's work: Free subgroups in linear groups over some skew fields, J. Algebra105 (1987), 1–28.

We are concerned with the following classical version of the Borsuk–Ulam theorem: Let f:Sn→Rk be a map and let Af = {x∈Sn|fx= f(−x)}. Then, if k≦n, Af≠φ. In fact, theorems due to Yang [17] give an estimation of the size of Af in terms of the cohomology index. This classical theorem concerns the antipodal action of the group G=ℤ2 on Sn. It has been generalized and extended in many ways (see a comprehensive expository article by Steinlein [16]). This author ([9, 10)] and Nakaoka [14] proved “continuous” or “parameterized” versions of the theorem. Analogous theorems for actions of the groups G=S1 or S3 have been proved in [11], and [12]; compare also [4, 5, 6].

Throughout this paper D denotes a division ring with centre F and n a positive integer. A subgroup G of GL(n,D) is absolutely irreducible if the F-subalgebra F[G] enerated by G is the full matrix ring Dn ×n. It is completely reducible (resp. irreducible) if row n-space Dn over D is completely reducible (resp. irreducible), as D–G bimodule in the obvious way. Absolutely irreducible skew linear groups have a more restricted structure than irreducible skew linear groups, see for example [7],[8], [8] and [10]. Here we make a start on elucidating the structure of locally nilpotent suchgroups.

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.

Dual integral equations of the form

where f(x) and g(x) are given functions, ψ(x) is unknown, k≧0, μ, v and α are real constants, have applications to diffraction theory and also to dynamical problems in elasticity. The special cases v = −μ, α = 0 and v = μ = 0, 0<α2<1 were treated by Ahiezer (1). More recently, equations equivalent to the above were solved by Peters (2) who adapted a method used earlier by Gordon (3) for treating the (extensively studied) case μ = v, k = 0.

The definitions of finite dimensional baric, train, and special train algebras, and of genetic algebras in the senses of Schafer and Gonshor (which coincide when the ground field is algebraically closed, and which I call special triangular) are given in Worz-Busekros's monograph [8]. In [6] I introduced applications requiring infinite dimensional generalisations. The elements of these algebras were infinite linear forms in basis elements a0, a1,… and complex coefficients such that In this paper I consider only algebras whose elements are forms which only a finite number of the xi are non zero.

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.

In this paper we prove five structure theorems for groups with dihedral 3-normalisers. The interest in these theorems lies not so much in the results themselves as in what can be proved from them. The original versions of the results are contained in our doctoral thesis (1) where they are used to prove the following theorem, of which this paper, together with (2), (3) and other papers in preparation, will constitute a published proof:

A conjecture of Huneke and Wiegand claims that, over one-dimensional commutative Noetherian local domains, the tensor product of a finitely generated, non-free, torsion-free module with its algebraic dual always has torsion. Building on a beautiful result of Corso, Huneke, Katz and Vasconcelos, we prove that the conjecture is affirmative for a large class of ideals over arbitrary one-dimensional local domains. Furthermore, we study a higher-dimensional analogue of the conjecture for integrally closed ideals over Noetherian rings that are not necessarily local. We also consider a related question on the conjecture and give an affirmative answer for first syzygies of maximal Cohen–Macaulay modules.

In 1964 Green and Rivlin (1) introduced a theory of simple force and stress multipoles founded on conventional kinematics. Using a work formula, the force and stress multipoles were defined with the help of the velocity field and its spatial derivatives. More recently, within the framework of this general study, Bleustein and Green (2) examined the theory of the simplest multipolar fluid, the dipolar fluid, and formulated constitutive equations for a homogeneous incompressible linear dipolar fluid.

In this paper we present an existence result for a class of nonlinear evolutions inclusions. A result on the compactness of the solution set for a differential inclusion is also established.

Analogous to the Banach-Mazur distance between Banach spaces, we study the completely bounded Banach-Mazur distance between operator spaces.

In many cases of Banach spaces and Hilbert spaces we show that the infimum is attained when T is the identity map, and X, Y have the same base space. This provides a machinery to compute and estimate dcb(X, Y). Later, using symmetric norming functions we construct counterexamples to show that distinct infinite dimensional homogeneous operator spaces may have finite cb-distance, and that two homogeneous Hilbertian operator spaces may not coincide even if they coincide over all 2-dimensional subspaces.

A form of integration of tensors will be introduced here, which will preserve the character of a tensor when so integrated.