Extension theory has been intensively studied for completely regular spaces and is fairly well developed for T0-topological spaces. (See, for example, [1] and [5]). However, except for definitions of some of the basic concepts in [4] and results on embedding of closure spaces in cubes in [2] and [7], ours is the first study of the general theory of extensions of G0-closure spaces. (Definitions will be given following these introductory paragraphs).

One of the attractions of finite combinatorics is its explicit constructions. This paper is part of a program to enlarge the domain of finite combinatorics to certain infinite structures while preserving the explicit constructions of the smaller domain. The larger domain to be considered consists of the recursive structures. While recursive structures may be infinite they are still amenable to explicit constructions. In this paper we shall concentrate on recursive colorings of highly recursive graphs.

A function f: Nk → N, where N is the set of natural numbers, is recursive if and only if there exists an algorithm (i.e., a finite computer program) which upon input of a sequence of natural numbers , after a finite number of steps, outputs . A subset of Nk is recursive provided that its characteristic function is recursive. For a more thorough definition of recursive functions and recursive relations see [10].

We compute the category of perverse sheaves on Hermitian symmetric spaces in types $\text{A}$ and $\text{D}$, constructible with respect to the Schubert stratification. The calculation is microlocal, and uses the action of the Borel group to study the geometry of the conormal variety $\Lambda$.

Consider a round robin tournament in which each of n players is required to play precisely one game with each other player, and assume that each game ends in a win or a loss. The results of such a tournament can be conveniently recorded in a square (0, 1)-matrix T = (tij) of order n by setting tij = 1 if player i defeats player j , tij = 0 if player i loses to player j , and tii = 0. Thus T has 0's along the main diagonal, and in the off-diagonal positions T satisfies the "skew-symmetry" condition that tij = 1 if and only if tji = 0. We call such a (0, 1)-matrix T a tournament matrix.

A balanced incomplete block design is defined as an arrangement of v objects in b blocks, each block containing k objects all different, so that there are r blocks containing a given object and λ blocks containing any two given objects. Such designs have been studied for their combinatorial interest, as in (3), and also for their application to statistics, where the objects are usually varieties.

In (1) Bruck introduced the notion of a difference set in a finite group. Let G be a finite group of v elements and let D = {di}, i = 1, . . . , k be a k-subset of G such that in the set of differences {di-1dj} each element ≠ 1 in G appears exactly λ times, where 0 < λ < k <v— 1. When this occurs we say that (G, D) is a v, k, λ group difference set. Bruck showed that this situation is equivalent to the one where the differences {didj-1} are considered instead, and that a v, k, λ group difference set is equivalent to a transitive v, k, λ configuration, i.e., a v, k, λ configuration which has a collineation group which is transitive and regular on the elements (points) and on the blocks (lines) of the configuration. Among the parameters v, k and λ, then, we have the relation shown by Ryser (5)

An orientable map is often presented as a realization of a finite connected graph G in an orientable surface so that the complementary domains of G, the “faces” of the map are topological open discs. This is not the definition to be used in the paper. But let us contemplate it for a while.

On each edge of G we can recognize two opposite directed edges, or “darts”. Let θ be the permutation of the dart-set S that interchanges each dart with its opposite. The darts radiating from a vertex v occur in a definite cyclic order, fixed by a chosen positive sense of rotation on the surface. The cyclic orders at the various vertices are the cycles of a permutation P of S. The choice of P rather than P–l, which corresponds to the other sense of rotation, makes the map “oriented”.

In this paper a ring will always mean a commutative ring with identity element. Furthermore, a ring R is called a multiplication ring if, whenever A and B are ideals of R and A is contained in B, there is an ideal C such that A = BC. Noetherian multiplication rings have been studied by Asano (1), Krull (4, 5), and Mori (6, 7). Krull also studied non-Noetherian multiplication rings (3). In (8, 9), Mori studied non-Noetherian multiplication rings which did not necessarily contain an identity element.

Our aim in this paper is to prove inequalities of the form

1

or

2

for all real values of the parameters α, β and all non-negative (in some cases all positive) xi. Obviously, an is finite in all cases, and we shall show that An is finite if α and α + β are both non-negative. In all cases, we obtain sharp values of the constants an, An (when finite), as well as bounds for these constants, and their behaviour as n → ∞. In case a < 0, we naturally consider only positive xi, otherwise the xi may be non-negative. Although we always write xi ≧ 0 in the following, this should be read as xi > 0 in case α < 0; similar remarks apply to the parameter t introduced below.

In (3), J. Hartmanis raised the question whether the lattice of all topologies in a given set is complemented and gave the affirmative answer for the case of a finite set. H. Gaifman (2), has extended this result to denumerable sets. Using Gaifman's paper, Anne K. Steiner (4) has proved that the lattice is always complemented. Our aim in this article is to give an alternative proof, independent of Gaifman's results. So far, Steiner's proof has not been available to the author.

Norm closed (or weakly closed) Jordan algebras of self-adjoint operators on a Hilbert space were initially studied by Topping, Effros, and Stormer [15], [4], [12], [13]. These works are very “spatial”, in that the algebras are considered in one given representation. The introduction of their abstract counterparts, the JB- and JBW-algebras, has led to an increased interest in this subject. The author hopes this paper will support the view that a more “space-free” approach is fruitful, even if only the “concrete” algebras are under study. In accordance with this view, a “JC-algebra” in this paper will mean a normed Jordan algebra over the reals, which is isometrically isomorphic to a norm closed Jordan algebra of self-adjoint operators.

Some of the results in this paper are closely related to, or rewordings of, results in the above-mentioned papers. However, I feel that the present approach is sufficiently different to be of interest in itself. In particular, many of the technical difficulties associated with earlier approaches are avoided.

Here we define and prove some properties of the semi-classical wavefront set. We also define and study semi-classical Fourier integral operators and prove a generalization of Egorov’s theorem to manifolds of different dimensions.

Several studies have been made to obtain larger classes of non-associative algebras from classes of algebras with a known structure. Thus, we have right alternative algebras (2)* and non-commutative Jordan algebras (6), (7), (8), and (9). These algebras are defined by a subset of the set of identities of the algebras from which they derive their names. Also, Albert (1), among others has studied Jordan admissible algebras. This paper is concerned with algebras which are related to Lie algebras in that they satisfy some of the identities of a Lie algebra and are Lie admissible. Theorem 2 answers a question raised by Albert in (1).

A well-known result of Dembowski and Wagner (4) characterizes the designs of points and hyperplanes of finite projective spaces among all symmetric designs. By passing to a dual situation and approaching this idea from a different direction, we shall obtain common characterizations of finite projective and affine spaces. Our principal result is the following.

Theorem 1. A finite incidence structure is isomorphic to the design of points and hyperplanes of a finite projective or affine space of dimension greater than or equal to4 if and only if there are positive integers v, k, and y, with μ > 1 and (μ – l)(v — k) ≠ (k — μ)2such that the following assumptions hold.

• (I) Every block is on k points, and every two intersecting blocks are on μ common points.

• (II) Given a point and two distinct blocks, there is a block containing both the point and the intersection of the blocks.

• (III) Given two distinct points p and q, there is a block on p but not on q.

• (IV) There are v points, and v– 2 ≧ k > μ.

Let {sn(z)} be a given sequence of linear fractional transformations (or simply l.f.t.'s) of the form

1.1

and let

1.2

The sequence of l.f.t.'s {Sn(z)} is called a continued fraction generating sequence (or simply a c.f.g. sequence).

We study $K$-orbits in $G/P$ where $G$ is a complex connected reductive group, $P\,\subseteq \,G$ is a parabolic subgroup, and $K\,\subseteq \,G$ is the fixed point subgroup of an involutive automorphism $\theta$. Generalizing work of Springer, we parametrize the (finite) orbit set $K\,\backslash \,G/P$ and we determine the isotropy groups. As a consequence, we describe the closed (resp. affine) orbits in terms of $\theta$-stable (resp. $\theta$-split) parabolic subgroups. We also describe the decomposition of any $(K,\,P)$-double coset in $G$ into $(K,\,B)$-double cosets, where $B\,\subseteq \,P$ is a Borel subgroup. Finally, for certain $K$-orbit closures $X\,\subseteq \,G/B$, and for any homogeneous line bundle $\mathcal{L}$ on $G/B$ having nonzero global sections, we show that the restriction map $\text{re}{{\text{s}}_{X}}\,:\,{{H}^{0}}\,\left( G\,/\,B,\,\mathcal{L} \right)\,\to \,{{H}^{0}}\,\left( X,\,\mathcal{L} \right)$ is surjective and that ${{H}^{i}}\,\left( X,\mathcal{L} \right)\,=\,0$ for $i\,\ge \,1$. Moreover, we describe the $K$-module ${{H}^{0}}\left( X,L \right)$. This gives information on the restriction to $K$ of the simple $G$-module ${{H}^{0}}\,\left( G\,/\,B,\mathcal{L} \right)$. Our construction is a geometric analogue of Vogan and Sepanski’s approach to extremal $K$-types.

The stacking method (see [1] and [5, Section 6]) has been used with great success in ergodic theory to construct a wide variety of examples of ergodic transformations (see, for example, [1 ; 3 ; 4; 5; 7]). However very little is known in general about the class of transformations which can be constructed by the stacking method using single stacks. In particular there is no simple characterization of the class .

We take the definition of a Witt ring to be that given in [13], i.e., it is what is called a strongly representational Witt ring in [8]. The classical example is obtained by considering quadratic forms over a field of characteristic ≠ 2 [17], but Witt rings also arise in studying quadratic forms or symmetric bilinear forms over more general types of rings [5,7, 8, 9]. An interesting problem in the theory is that of classifying Witt rings in case the associated group G is finite. The reduced case, i.e., the case where the nilradical is trivial, is better understood. In particular, the above classification problem is completely solved in this case [4, 12, or 13, Corollary 6.25]. Thus, the emphasis here is on the non-reduced case. Although some of the results given here do not require |G| < ∞, they do require some finiteness assumption. Certainly, the main goal here is to understand the finite case, and in this sense this paper is a continuation of work started by the second author in [13, Chapter 5].

In an earlier paper [14] I have adumbrated a method for establishing that the zeta-function of a Shimura variety associated to a quaternion algebra over a totally real field can be expressed as a product of L-functions associated to automorphic forms. Now I want to add some body to that sketch. The representation-theoretic and combinatorial aspects of the proof will be given in detail, but it will simply be assumed that the set of geometric points has the structure suggested in [13]. This is so at least when the algebra is totally indefinite, but it is proved by algebraic-geometric methods that are somewhat provisional in the context of Shimura varieties. However, contrary to the suggestion in [13] the general moduli problem has yet to be treated fully. There are unresolved difficulties, but they do not arise for the problem attached to a totally indefinite quaternion algebra, which is discussed in detail in [17].

Let X be an arbitrary compact subset of the real line R which has at least two points. For each finite complex valued function f on X we denote by V(f; X) (and call it the weak variation of f on X) the least upper bound of the numbers ∑i|f(bi) – f(ai)| where {[ai, bi]} is any sequence of non-overlapping intervals whose end points belong to X. A function f is said to be of bounded variation (BV) on X if V(f; X) < ∞. A function f is said to be absolutely continuous (AC) on X, if given any ∈ > 0 there exists an n > 0 such that for every sequence of non-overlapping intervals {[au bi]} whose end points belong to X, the inequality

implies that

([7], p. 221, 223).