An elation of a design is an automorphism γ of fixing some block X pointwise and some point x on X blockwise. Luneburg [4] and I [2] have proved results which state that a design admitting many dations and having additional properties must be the design of points and hyperplanes of a finite desarguesian projective space. In this note, additional results of this type will be proved and applied to yield a generalization of a previous result on Jordan groups [3]. The proofs were suggested by a result of Hering on dations of finite projective planes [1, pp. 122, 190].

For every ( differentiable) bundle E over a manifold M, Jk(E) denotes the set of all k-jets of local (differentiable) sections of the bundle E. Jk(E) is a bundle over M such that if X is a section of E, then

is a (differentiable) section of Jk(E). If E is a vector bundle, Jk(E) is a vector bundle and we have the canonical exact sequence of vector bundles

where Sk(T*) is the symmetric Whitney tensor product of the cotangent vector bundle T* of M. and π is the canonical morphism which associates to each k-jet of section its jet of inferior order.

A map f : M → N is topologically equivalent tog: X → Y if there exist homeomorphisms α: M → X and β: N → Y such that βfα–1 = g. At x ∊ M, f is locally topologically equivalent to g if, for every neighbourhood W ⊂ M of x, there exist neighbourhoods U ⊂ W of x and V of f(x) such that f|U: U → V is topologically equivalent to g.

1.1. Definition. Given a map f: M → N and x ∊ M, let F be the component of f–1(f(x)) containing x. The singular set Af is defined as follows: x ∊ M – Af if and only if there are neighbourhoods U of F and V of f(x) such that f| U: U → F is topologically equivalent to the product projection map of V × F onto V.

In his fundamental paper [3], Hurwitz showed that the order of a group of biholomorphic transformations of a compact Riemann surface S into itself is bounded above by 84(g – 1) when S has genus g ≧ 2. This bound on the group of automorphisms (as we shall call the biholomorphic self-transformations) is attained for Klein's quartic curve of genus 3 [4] and, from this, Macbeath [7] deduced that the Hurwitz bound is attained for infinitely many values of g.

After genus 3, the next smallest genus for which the bound is attained is the case g = 7. The equations of such a curve of genus 7 were determined by Macbeath [8] who also gave the equations of the transformations. The equations of these transformations were found by using the Lefschetz fixed point formula. If the number of fixed points of each element of a group of automorphisms is known, then the Lefschetz fixed point formula may be applied to deduce the character of the representation given by the group acting on the first homology group of the surface.

Let be a family of sets in a linear space X. A hyperplane π is called a k-secant of if π intersects exactly k members of . The existence of k-secants for families of compact sets in linear topological spaces has been discussed in a number of recent papers (cf. [3–7]). For X normed (and a finite family of two or more disjoint non-empty compact sets) it was proved [5] that if the union of all members of is an infinite set which is not contained in any straight line of X, then has a 2-secant. This result and related ones concerning intersections of members of by straight lines have since been extended in [4] to the more general setting of a Hausdorff locally convex space.

In [2] Sandler studied generalizations of projective planes called pseudo planes. These structures gave rise to ternary operations for which addition and multiplication are loop operations. Our aim in this paper is to investigate the pseudo planes for which the operations of addition and multiplication give rise to nearfields. The pseudo planes for which this is true will be called π-systems in this paper, and a list of postulates for such systems is given in § 2. In that section some results on collineations of π-systems are given which are stronger than analogous results for general pseudo planes.

Bruck and Kleinfeld [3] proved that any alternative ring with characteristic prime to 2 must satisfy the identity

where the associator (x,y,z) is defined by (x, y, z) = (xy)z – x(yz) and . Linearization of the identity (x2, y, z) = 2x · (x, y, z) yields for characteristic prime to 2 an equivalent identity

(1)

Using the right alternative law (x, y, z) = –(y, x, z) and the flexible law (x, y, z) = –(z, y, x) which is satisfied in any alternative ring we obtain

(2)

and

(3)

In this paper we prove that the order of the collineation group of the Veblen-Wedderburn plane of order nine is 311,040. This result was stated by Hall [3] in 1943 and proved by Pierce [9] in 1964. Hall assumed that there were 10 · 8 · 6 · 4 · 2 = 3840 collineations which permute points on the ideal line L and 81 collineations which leave L pointwise fixed. In 1955 André [1] verified this assumption. When it was realized that a harmonic homology with axis L had been overlooked, the number of central collineations with axis L doubled and hence the order of the collineation group became 3840 · 162 = 622,080. This latter figure has been assumed to be correct as recently as 1965 ([6]).

Here it is proved that there are 1920 collineations which move points on L and 162 collineations which leave L pointwise fixed, thus giving the figure 311,040. Pierce's proof of this fact is established from a different viewpoint.

One possible first step in considering the structure of a class of semigroups is to study the ideal extensions (here simply called “extensions“) of simple or 0-simple semigroups in by another if the latter are of known structure. Extensions of a semigroup by another were first studied by Clifford (see [1, 4.4 and 4.5]). In his constructions, an extension of a semigroup S by a semigroup T with zero is given by a function (satisfying certain conditions) from T* = T\0 into the translational hull of S.

We use certain results (refining those of Clifford) established in [2] and a description of the translational hull of a Brandt semigroup given in [9] (see also [8]), to construct all extensions V of a Brandt semigroup S having a finite number of idempotents by any Brandt semigroup T (cf. [10]).

The concept of extending to a topological space X a continuous pseudometric defined on a subspace S of X has been shown to be very useful. This problem was first studied by Hausdorff for the metric case in 1930 [9]. Hausdorff showed that a continuous metric on a closed subset of a metric space can be extended to a continuous metric on the whole space. Bing [4] and Arens [3] rediscovered this result independently. Recently, Shapiro [15] and Alo and Shapiro [1] studied various embeddings. It has been shown that extending pseudometrics can be characterized in terms of extending refinements of various types of open covers. In this paper we continue our study of extending pseudometrics. First we show that extending pseudometrics can be characterized in terms of σ-locally finite and σ-discrete covers. We then investigate when can certain types of covers be extended.

This note is an addendum to my earlier paper [8]. The class of adjoint abelian operators discussed there was small because the compatibility relation between the operator and the duality map was too restrictive. (In effect, the relation is appropriate for Hilbert space, but ill-suited for other Banach spaces where the unit ball is not round.) However, the techniques introduced in [8] permit us to readily obtain a spectral theory (of the Dunford type) for a wider class of operators on Banach spaces, as we shall show.

A duality system for the operator T is an ordered sextuple

(i) T is a bounded linear operator mapping the Banach space B into B,

(ii) ϕ is a duality map from B to B*. Thus, for x ∊ B, ϕ(x) = x* ∊ B*, where ‖x‖ = ‖x*‖ and x*(x) = ‖x‖2. The existence of ϕ follows easily from the Hahn-Banach Theorem.

In [5], Zenor stated the definition of weakly normal. In the main, since weak normality does not imply either normality or regularity, various properties related to either normality or regularity will be considered in the context of weak normality.

Throughout this paper the word “space” will mean topological space. The closure of a point set M will be denoted by cl(M). The closure of a point set M with respect to the subspace K will be denoted by cl(M, K).

Definition 1. A space S is weakly normal provided that if is a monotonically decreasing sequence of closed sets in S with no common part and H is a closed set in S not intersecting H1, then there is a positive integer N and an open set D such that HN ⊂ D and cl(D) does not intersect H.

In this paper we prove that the proper Dirichlet subalgebras of the disc algebra discovered by Browder and Wermer [1] are maximal subalgebras of the disc algebra (Theorem 2). We also give an extension to general function algebras of a theorem of Rudin [4] on the existence of maximal subalgebras of C(X). Theorem 1 implies that every function algebra defined on an uncountable metric space has a maximal subalgebra.

A function algebra A on X is a uniformly closed, point-separating subalgebra of C(X), containing the constants, where X is a compact Hausdorff space. If A and B are function algebras on X, A ⊂ B, A ≠ B, we say A is a maximal subalgebra of B if whenever C is a function algebra on X with A ⊂ C ⊂ B, either C = A or C = B.

Let p be a rational prime, G a non-trivial finite p group, and K the field of p elements, regarded as a trivial G-module according to context; then we define:

d(G) = dimKH1(G, K), the minimal number of generators of G,

r(G) = dimKH2(G, K),

r′(G) = the minimal number of relations required to define G,

where, in the last equation, it is sufficient to take the minimum over those presentations of G with d(G) generators. It is well known (see § 2) that the following inequalities hold:

We shall consider only finite p-groups, so that the class of groups with r = d coincides with that consisting of those groups whose Schur multiplicator is trivial.

1. Definitions and notation. A generalized Steiner system (t-design, tactical configuration) with parameters t, λt, k, v is a system (T, B), where T is a set of v elements, B is a set of blocks each of which is a k-subset of T (but note that blocks bi and bj may be the same k-subset of T) and such that every set of t elements of T belongs to exactly λt of the blocks. If we put λt = u we denote by Su(t, k, v) the collection of all systems with these parameters. Thus Q ∈ Su(t, k, v) means Q = (T, B) is a system with the given parameters. If λt = u = 1, we write S(t, k, v) instead of S1(t, k, v) and refer to the system as a Steiner system. If t = 2, the system is called a balanced incomplete block design.

A complex function on the real line is said to be bounded in mean square if it is locally in L2 (i.e. on each finite interval) and satisfies

(1.1)

The set of all such functions clearly forms a linear space over the complex numbers and is a Banach space B under the norm ‖·‖B defined by (1.1). This space, among others, has been discussed by Beurling in [1], where it was shown to be the dual, in the Banach space sense, of a certain Banach (convolution) algebra of functions. We have used Beurling's characterization of B and others of his results throughout this paper, and indeed the essence of one or two of the proofs has been derived from his theorems.

In this paper we investigate indefinite Finsler spaces in which the metric tensor has signature n — 2. These spaces are a generalization of Lorentz manifolds. Locally a partial ordering may be defined such that the reverse triangle inequality holds for this partial ordering. Consequently, the spaces we study may be made into what Busemann [3] terms locally timelike spaces. Furthermore, sufficient conditions are obtained for an indefinite Finsler space to be a doubly timelike surface (see [2; 4]). In particular, all two-dimensional pseudo-Riemannian spaces are shown to be doubly timelike surfaces.

Let F be the free group on n generators x1, …, Xn and let G be an arbitrary group. An element ω ∈ F determines a function x → ω(x) from n-tuples x = (x1, x2, …, xn) ∈ Gn into G. In a recent paper [5] Solomon showed that if ω1, ω2, …, ωm ∈ F with m < n, and K1, …, Km are conjugacy classes of a finite group G, then the number of x ∈ Gn with ωi(x) ∈ Ki for each i, is divisible by |G|. Solomon proved this by constructing a suitable equivalence relation on Gn.

Another recent application of an unusual equivalence relation in group theory is in Brauer's paper [1], where he gives an elementary proof of the Frobenius theorem on solutions of xk = 1 in a group.

Let V be an irreducible algebraic variety of dimension > 1 defined over a field k in an affine n-space over k, and let H be the generic hyperplane defined by u0 + u1X1 + … + unXn = 0, where u0, u1, …, un are indeterminates over k. It is well known that:

(1) if V is normal over k, then V ∩ H is normal over k(u0, …, un) (see [6]), and

(2) if P is in the intersection V ∩ H, then P is absolutely simple on V ∩ H over k(u0, …, un) if and only if P is absolutely simple on V over k (see [2; 5]).

In this paper we prove:

(1′) if V is factorial over k, then V ∩ H is also factorial over k(u0, …, un) (Theorem 3), and

(2′) if P is in V ∩ H, then P is normal on V ∩ H over k(u0, …, un) if and only if P is normal on V over k (Theorem 2).

The theory of gentle perturbations was introduced by Friedrichs [3] as a tool to study the perturbation theory of the absolutely continuous spectrum of a self-adjoint operator H0 and developed in an abstract form by Rejto [7; 8]. Two examples of gentle structures are well knowTn. In the first of these, the gentle operators have Hölder continuous complex or operator-valued kernels, and in the second, the kernels are Fourier transforms of L1 functions [4].

The gentle structure has traditionally been verified in the case when H0 is in its spectral representation, that is, when H0 is the simple differentiation operator. This is not the natural setting for the second example mentioned above where one should consider the simple differentiation operator in a suitable L2-space and perturbations with L1 kernels.