Our main object is to prove the following result.

THEOREM C. Let A be an affine translation plane of order qr ≧ q2 suchthatl∞, the line at infinity, coincides with the translation axis of A. Suppose G is a solvable autotopism group of A that leaves invariant a set Δ of q + 1 slopes and acts transitively on l∞ \ Δ.

Then the order of A is q2.

An autotopism group of any affine plane A is a collineation group G that fixes at least two of the affine lines of A; if in fact the fixed elements of G form a subplane of A we call G a planar group. When A in the theorem is a Hall plane [4, p. 187], or a generalized Hall plane ([13]), G can be chosen to be a planar group.

Regularity and complete regularity are important topological properties and several generalizations of them occur in the literature on separation axioms. The properties of certain of these variants of (complete) regularity are similar to those of (complete) regularity and their theories run, either in part or in the whole, parallel to the theory of (complete) regularity. All the more, analogies inherent in their definitions as well as the nature of results obtained in the process of their study suggest the need of formulating a coherent unified theory encompassing the theory of (complete) regularity and its generalizations. An attempt leading towards the fulfillment of this need constitutes the theme of the present paper.

Section 2 is devoted to preliminaries and basic definitions. In Section 3 we devise a framework which leads to the formulation of a unified theorv of (complete) regularity, almost (complete) regularity, ([26,], [27], [28]), (complete) s-regularity [13], (functionally) Hausdorff spaces, R1-spaces [3], and others.

We correct some errors in [1] and extend some of the results there. Generally, we shall follow the terminology and notation of [1]. There is an error in the proof of Lemma 3.13 there, and consequently the subsequent results which depend on it are incorrect as stated. However, they are correct if we replace the condition “a-admissible” by “strongly a-admissible” (see [3] where this notion was introduced). We also show that the results in [1] are correct if the semilattices are assumed to be modular.

We shall change the terminology in [3] slightly.

Definition 1 (see [3]). Let A be a Boolean algebra and let D be a meet semilattice with 1. An admissible map f.A X D → D is called strongly admissible if

where a’ is the complement of a in A.

Let A = (Aij)l≦ij≦l be a Cartan matrix, i.e., Aii = 2 for all i and Aij is an integer ≦ 0 for i ≠ j, with Aij = 0 if Aji = 0. The size l of A is called its rank, for Lie-theoretic reasons, and may be larger than its matrix rank. We associate to A its Dynkin diagram, with vertices 1, 2, … , l, with AijAji lines joining i to j, and with an arrow pointing from i to j if Aij/Aji < 1, i.e., pointing toward the shorter root (see below). The Cartan matrix A is indecomposable if its diagram is connected, and symmetrizable if there exist positive rational numbers q1 … , ql with

qiAij = qjAji for all i and j.

Symmetrizability is automatic if the diagram contains no cycle.

By a projectivityof vector spaces Xand Yover fields F and G is meant an isomorphism Ψ:(X) → (Y) of their lattices of subspaces. A basic theorem of projective geomtry [2, p. 44] asserts that, for spaces of dimension at least 3, any such projectivity is of the form Ψ(M) = SM for a bijection S:X → Y which is semi-linear in the sense that S is an additive mapping for which there exists an isomorphism σ:F→ G such that

S(λx) = σ(λ)Sx for all λ ∈ Fand all x∈ X.

In [12] Mackey obtained a continuous version of this result: for real normed linear spaces Xand Y, the lattices and of closed subspaces are isomorphic if and only if X and Yare isomorphic (i.e., via a bicontinuous linear bijection).

Soit M un facteur de type IIIλ 0 ≦ λ< 1, agissant sur un espace hilbertien à base dénombrable. Le propos de ce travail est l'étude des états normaux d'un tel facteur à équivalence unitaire près. L'espace quotient n'est pas séparé; l'espace séparé associé est obtenu comme quotient de l'espace des états normaux par la relation d'équivalence R dont les classes sont les fermetures des orbites précédentes.

Pour étudier cet espace, on définit un calcul fonctionnel sur M+* (espace des formes linéaires positives normales sur M) à valeurs dans M. Ce calcul fonctionnel est déterminé par la donnée d'une C* algèbre d'applications de M+ * dans M “continues” en un certain sens, notée C*(M).

J. M. Borwein has given in [1] a practical necessary and sufficient condition for a convex operator to be continuous at some point. Indeed J. M. Borwein has proved in his paper that a convex operator with values in an order topological vector space F (with normal positive cone F+) is continuous at some point if and only if it is bounded from above by a mapping which is continuous at this point. This result extends a previous one by M. Valadier in [16] asserting that a convex operator is continuous at a point whenever it is bounded from above by an element in F on a neighbourhood of the concerned point. Note that Valadier's result is necessary if and only if the topological interior of F+ is nonempty. Obviously both results above are generalizations of the classical one about real-valued convex functions formulated in this context exactly as Valadier's result (see for example [5]).

Let k be a commutative field, and A a finite-dimensional k-algebra. By a module will always be meant a finitely generated right module. Following [8], we shall call a module TA a tilting module if (1) pdTA ≦ 1, (2) Ext1A(T, T) = 0 and (3) there is a short exact sequence

with T’ and T” direct sums of direct summands of T. Given a tilting module TA, the full subcategories

and

of the category modA of A -modules are respectively the torsion-free class and the torsion class of a torsion theory on modA[8]. The aim of the present paper is to find conditions on a torsion theory in order that it be induced by a tilting module.

Similarity invariants of certain families of Volterra operators acting in Lp[0, 1] were first determined by Kalisch in [4] and [5]. Subsequently, Kantorovitz made an extensive study of perturbations of the form Tα = M + αN, where M is the operation of multiplication by x, and N is the Volterra operator

both acting in Lp[0, 1], and determined that Re α is a similarity invariant for the class of Tα's (cf. [7], and earlier work cited there). This result was eventually generalized to unbounded M (cf. [6]), and one direction was proved in [2] for both M and N unbounded: if Re α = Re β, then Tα is similar to Tβ. Properties of the Riemann-Liouville semigroup of fractional integration play a key role, for in all cases the ensuing similarity turns out to be implemented by N(i(lm(α — β))), where {N(iγ)} is the boundary group of the semigroup of fractional integration.

A large number of important integral transforms, such as Laplace, Fourier sine and cosine, Hankel, Stieltjes, and Riemann- Liouville fractional integral transforms, can be put in the form

1.1

where f(t) and the kernel, h(t), are locally integrable functions on (0,∞), and x is a positive parameter. Recently, two important techniques have been developed to give asymptotic expansions of I(x) as x → + ∞ or x → 0+. One method relies heavily on the theory of Mellin transforms [8] and the other is based on the use of distributions [24]. Here, of course, the integral I(x) is assumed to exist in some ordinary sense.

If the above integral does not exist in any ordinary sense, then it may be regarded as an integral transform of a distribution (generalized function). There are mainly two approaches to extend the classical integral transforms to distributions.