We investigate a locally full HNN extension of an inverse semigroup. A normal form theorem is obtained and applied to the word problem. We construct a tree and show that a maximal subgroup of a locally full HNN extension acts on the tree without inversion. Bass-Serre theory is employed to obtain a group presentation of the maximal subgroup as a fundamental group of a certain graph of groups associated with the D-structure of the original semigroup.

We investigate certain norm and continuity conditions that provide us with ‘uniqe Hahn-Banch Theorems’ from (nc0) to (nℓ∞) and from N(nE) to N(nE″). We show that there is a unique norm-preserving extension for norm-attaining 2-homogeneous polynomials on complex c0 to ℓ∈ but there is no unique norm-preserving extension from (3c0) to (3ℓ∈).

Let T be a continuous t-norm (a suitable binary operation on[0, 1]) and Δ + the space of distribution functions which are concertratede on [0,∞. theτT product of any F, G in Δ+ is defined at any real x by , and the pair (Δ+, τT) forms a semigroup. Thus, given a sequence {Fi} in Δ+, the n-fold product τT(F1 … Fn) is well-defined for each n. Moreover, that resulting sequence {τT(F1, …, Fn)} is pointwise non-increasing and hence has a weak limit. This paper establishes a convergence theorem which yields a representation for this weak limit. In addition, we prove the Zero-One law that, for Archimedean t-norms, the weak limit is either identically zero or has supremum 1.

If A and B are two complex matrices and if U is a complex unitary matrix such that UAUCT = B (where UCT denotes the conjugate transpose of U), then A and B are said to be unitarily similar. Necessary and sufficient conditions that two matrices be unitarily similar have been dealt with in [5] (from the point of view of group representation theory) and in [2] (from the point of view of developing a canonical form under unitary similarity).

Nonsingular derivations of modular Lie algebras which have finite multiplicative order play a role in the coclass theory for pro-p groups and Lie algebras. We study the orders of nonsingular derivations of finite-dimensional non-nilpotent Lie algebras of characteristic p > 0. The methods are essentially number-theoretic.

1991 Mathematics subject classification (Amer. Math. Soc): primary 17B50; secondary 12E20.

We say that a regulär semigroup S is a coetension of a (regular) semigroup T by rectangular bands if there is a homomorphism ϕ: S → T from S onto T such that, for each e = e2 ∈ S, e(ϕ ϕ-1) is a rectangular band. Regular semigroups which are coextesions of pseudo-inverse semigroups by rectangular bands may be characterized as those regular semigroups S with the property that, for each e = e2 ∈ S, ω(e) = {f = f2 ∈ S: ef = f} and ωl(e) = {f = f2 ∈ S: fe = f} are bands: this paper is concerned with a study of such semigroups.

The functions where ξ > 0, are essentially Fourier convolutions of Gaussian and Cauchy type distributions, and have thus found application in many fields. In reactor theory, and no doubt elsewhere, the need has arisen to analyse complicated integrals and functional equations involving ψ and φ, the so-called Voigt profiles.

Two graphs, the edge crossing graph E and the triangle graph T are associated with a simple lattice polygon. The maximal independent sets of vertices of E and T are derived including a formula for the size of the fundamental triangles. Properties of E and T are derived including a formula for the size of the maximal independent sets in E and T. It is shown that T is a factor graph of edge-disjoint 4-cycles, which gives corresponding geometric information, and is a partition graph as recently defined by the authors and F. Harary.

We describe a generalization of the classical Julia-Wolff-Carathéodory theorem to a large class of bounded convex domains of finite type, including convex circular domains and convex domains with real analytic boundary. The main tools used in the proofs are several explicit estimates on the boundary behaviour of Kobayashi distance and metric, and a new Lindelöf principle.

This paper is concerned with the numerical range and some related properties of the operator Δ/ S: T → AT – TB(T∈S), where A, B are (bounded linear) operators on the normed linear spaces X and Y. respectively, and S is a linear subspace of the space ℒ (Y, X) of all operators from Y to X. S is assumed to contain all finite operators, to be invariant under Δ, and to be suitably normed (not necessarily with the operator norm). Then the algebra numerical range of Δ/ S is equal to the difference of the algebra numerical ranges of A and B. When X = Y and S = ℒ (X), Δ is Hermitian (resp. normal) in ℒ (ℒ(X)) if and only if A–λ and B–λ are Hermitian (resp. normal) in ℒ(X)for some scalar λ;if X: = H is a Hilbert space and if S is a C *-algebra or a minimal norm ideal in ℒ(H)then any Hermitian (resp. normal) operator in S is of the form Δ/ S for some Hermitian (resp. normal) operators A and B. AT = TB implies A*T = TB* are hyponormal operators on the Hilbert spaces H1 and H2, respectively, and T is a Hilbert-Schmidt operator from H2 to H1.

We prove certain Lp estimates (1<p<∞) for nonisotropic singular integrals along surfaces of revolution. The singular integrals are defined by rough kernels. As an application we obtain Lp boundedness of the singular integrals under a sharp size condition on their kernels. We also prove a certain estimate for a trigonometric integral, which is useful in studying nonisotropic singular integrals.

Recently A. L. Šmel'kin [14] proved that a product variety1 is generated by a finite group if and only if is nilpotent, is abelian, and the exponents of and are coprime. Alternatively, by the theorem of Oates and Powell [13], we may say that a Cross variety is decomposable if and only if it is of the above form.

The following paper is a sequel to the author's earlier paper [2]. In that paper some general results were obtained which described the motion of a fluid with a free surface subsequent to a given initial state and prescribed boundary conditions of a certain type. The analysis was based on a linearized theory but gravity effects were included. Viscosity, compressibility and surface tension effects were neglected. Among the problems treated was that of the normal symmetric entry of a thin wedge into water at rest. This Water entry problem has attracted a considerable amount of attention since the pioneer paper by Wagner [5]. Both linear and non-linear approximations have been used but all papers apart from [2] neglect gravity on the assumption that in the early stages of the penetration this is unimportant. One of the objects of [2] was to determine the solution with the gravity terms retained. A formal solution was obtained but no attempt was made to analyse this quantitatively. In the present paper we examine the extent of this effect in some detail. It will be of help to the reader to have some familiarity with the first three or four sections of [2] but in order to make the present paper self-contained we shall first reintroduce the notation used there and quote the necessary results from that paper without proof.

The study of topological semirings, initiated by Selden [5], arises naturally from the theory of topological semigroups. It is of interest to take a known multiplication and investigate the possible additions. Selden has done this in [5] for several compact semirings.

Burkill (1957) introduced extended Riemann-Stieltjes integrals of the form ∫baƒ(x)dkg(x)/dxk-1 to provide an alternative approach to the theory of distributions. We will denote Burkill's integral by (B)∫baƒ(x)dkg(x)/dxk-1 Burkill's paper (1957) partially motivated the study of the Riemann-Stieltjes integral in Russell (1970). He showed that the two integrals are not equivalent.

An inhomogeneous version of a general form of the Jarník-Besicovitch Theorem is proved.

The work establishes the estimates of linear independence, transcendence, and algebraic independence measures of the values at algebraic points of a set of E-functions, satisfying linear differential equations with coefficients from C(z), rather close to their natural bounds.

A topology on a set X is defined by specifying a family of its subsets which has the properties (i) arbitrary set intersections of members of belong to , (ii) finite set unions of members of belong to and (iii) the empty set □ and the set X each belong to . The members of are called the closed subsets of X. If X is any subset of X then denotes the closure of X, that is, the set intersection of all closed subsets which contain X, however when X = {x} contains one point only we will denote by . The pair (X, ) is called a topological space or, in what follows, a T-space. By a T-lattice we mean a complete distributive lattice of sets in which arbitrary g.l.b. means arbitrary set intersection, finite l.u.b. means finite set union and which contains the empty set □ It is well-known, for example Birkhoff [1], that if (X, ) is a T-space and the members of are partially ordered by set inclusion then is a T-lattice.

The results described herein were obtained in connection with the development of some lattice theoretic machinery needed by Randall and Foulis in their treatment of the so-called “logic of empirical science” (See [3] for an introduction to this subject). The final product, however, turned out to be of sufficient interest to warrant its independent presentation.

We give a detailed exposition of the theory of decompositions of linearised polynomials, using a well-known connection with skew-polynomial rings with zero derivative. It is known that there is a one-to-one correspondence between decompositions of linearised polynomials and sub-linearised polynomials. This correspondence leads to a formula for the number of indecomposable sub-linearised polynomials of given degree over a finite field. We also show how to extend existing factorisation algorithms over skew-polynomial rings to decompose sub-linearised polynomials without asymptotic cost.