The geometries considered here are the Möbius plane M() (W. Benz [1]), the Laguerre plane L() (W. Benz and H. Mäurer [7]) and the Minkowski plane A() (W. Benz [5], G. Kaerlein [18]) over a field . All of them are geometries of an algebra with identity over a field.

The characterization of the projective plane over a field by the proposition of Pappus first gave a close relation between algebraic and geometric structures. B. L. v. d. Waedern and L. J. Smid [28] presented a further example by characterizing the Möbius and Laguerre plane with incidence axioms and the "complete" proposition of Miquel.

Let V denote a ring of subsets of an abstract space X, let R denote the real numbers, and let N denote the positive integers. Denote by a(V, R) (respectively ca(V, R)) the space of real valued, finitely additive (respectively countably additive) functions on the ring V and denote by ab(V, R) the subspace consisting of those members of the space a(V, R) with finite variation on each set in the ring V. Members of the space a(V, R) are referred to as charges and members of the space ab(V, R) are referred to as locally bounded charges. We denote by cab(V, R) the intersection of the spaces ab(V, R) and ca(V, R).

Let V denote a ring of subsets of an abstract space X, let R + denote the nonnegative reals, and let N denote the set of positive integers. We denote by C(V) the space of all subadditive and increasing functions, from the ring V into R +, which are zero at the empty set. The space C(V) is called the space of contents on the ring V and elements are referred to as contents.

A sequence of sets An ∊ V, n ∊ N is said to be dominated if there exists a set B ∊ V such that An ⊆ B, for n = 1, 2, A content p ∊ C(V) is said to be Rickart on the ring V if limn p(An ) = 0 for each dominated, disjoint sequence An ∊ V, n ∊ N.

We consider effective homogeneous spaces M = G/H where G is a compact connected Lie group, H is a closed subgroup and G acts effectively on M (i.e., H contains no non-trivial subgroup normal in G). It is known that dim G ≦ m 2/2 + m/2 where m = dim M and that if dim G = m 2/2 + m/2, then M is diffeomorphic to the standard sphere Sm or the standard real projective space RPm [1]. In addition it has been shown that for fixed m there are gaps in the possible dimensions for G below the maximal bound [4; 5].

The classical comparison and interlacing theorems of Sturm were originally proved for the equations

under the assumption that all coefficients are real-valued, continuous, and p > 0, P > 0. Atkinson [1, Chapter 8] has carried out the standard theory for eigenvalue problems involving (1), under the more general hypothesis

Let S 1 . . . ,Sr be r line segments, each of non-zero length, in n-dimensional euclidean space Rn . If a polytope Z is defined as the vector (Minkowski) sum

(1) Z = S 1 + . . . + Sr ,

then the segments Si will be called the components of Z. Since we do not wish to exclude the possibility that some of the components may be parallel, the polytope Z may be written in the form (1) in many different ways. For this reason it is convenient to define a zonotope to be the polytope Z together with some specified set of components {S1 , . . . , Sr }. Figures 1, 2 and 3 show some zonotopes of 1, 2 and 3 dimensions with 4, 5 and 6 components.

This paper is divided into two sections. In the first, we characterize the subgroups N of a reduced abelian primary group for which all pure N-high subgroups are bounded. This condition on pure N-high subgroups occurs in several instances, for instance, all pure N-high subgroups of a primary group G are bounded if G is the smallest pure subgroup of G containing N; all N-high subgroups are bounded if N ≠ 0 and all N-high subgroups are closed in the p-adic topology.

Certain phases are associated with the Kronecker squares and cubes of representations of the finite and of the compact semi-simple groups. These phases are important in giving the symmetry properties of the 1 — jm and 3 — jm symbols of the groups [4; 9]. It is our primary purpose to evaluate these phases.

The Frobenius-Schur invariant [12, p. 142] for an irreducible representation of group G

(1.1)

We consider here singular differential operators, and for convenience the finite singularity is taken to be zero. One operator discussed is the operator L defined by

where q 0 > 0 and the coefficients q t are real, locally Lebesgue integrable functions defined on an interval (a, b). For a given positive, continuous weight function h, conditions are given on the functions qi for which the number of linearly independent solutions y of L(y) = λhy (Re λ = 0) satisfying.

The permanent of an n-square complex matrix P = (pij ) is defined by

where the summation extends over Sn , the symmetric group of degree n. This matrix function has considerable significance in certain combinatorial problems [6; 7]. The properties and many related problems about the permanent are presented in [3] along with an extensive bibliography.

Let-R be a commutative ring with 1, and let

be the symmetric algebra of an R-module M. We regard the isomorphisms S 0(M) ≅ R and S 1(M) ≅ M a s identifications. There is a unique R-algebra homomorphism Δ : S(M) → S(M) ⊗R S(M) (called the comultiplication) satisfying Δ(m) = m ⊗ 1 + 1 ⊗ m for all m ∊ M; any element x ∊ S(M) for which Δ(x) = x ⊗ 1 + 1 ⊗ x is said to be primitive. The set of all primitive elements in S(M) is denoted P(M).

Every completely regular space has at least one Hausdorff compactification and much research in Topology has been devoted to methods of constructing the compactifications of completely regular spaces. These methods fall into two general categories: internal methods and external methods. External methods are characterized by their reliance on structures which are not topological or outside the immediate topological structure of the space in question; examples of the former are A. Weil [20] - uniform structures and Yu. Smirnov [15] who uses the proximity structures of Efremovich; using the continuous real-valued functions to produce an embedding of the space serves as an example of the latter.

The notion of almost convergence introduced by Lorentz [15] has been generalized in several directions (see, for example [1; 8; 11 ; 14; 17]). I t is the purpose of this paper to give a generalization based on the original definition in terms of invariant means. This is effected by replacing the shift transformation by an "ergodic" semigroup of positive regular matrices in the definition of invariant mean. The resulting "- invariant means" give rise to a summability method which we dub -almost convergence.

Let (X, ) be a topological space. By we denote the family of all Lodato proximities on X which induce . We show that is a complete distributive lattice under set inclusion as ordering. Greatest lower bound and least upper bound are characterized. A number of techniques for constructing elements of are developed. By means of one of these constructions, all covers of any member of can be obtained. Several examples are given which relate to the lattice of all compatible proximities of Čech and the family of all compatible proximities of Efremovič. The paper concludes with a chart which summarizes many of the structural properties of , and .

Let X be a compact Hausdorff space and C(X) the set of all continuous complex-valued functions on X. A function algebra A on X is a uniformly closed, point separating subalgebra of C(X) which contains the constants. Equipped with the sup-norm, A becomes a Banach algebra. We let MA denote the maximal ideal space and SA the Shilov boundary.

The word ring is used to mean commutative ring. Just as valuations on fields are used to study domains, so valuations on rings can be used to study rings; these rings need not have units [12]. We introduce slightly weaker conditions than having identity in order to get a more general theory. A Prufer ring A is one in which every finitely generated regular ideal is invertible. If we replace invertibility in the total quotient ring K, by invertibility in a ring R where A ⊆ R ⊆ K we get an R-Prufer ring. These rings do occur, for example the Witt ring of a non-Pythagorean field or a ring of bounded continuous functions.

In the following paper ring will always mean commutative ring which may or may not have an identity. We use the letter N exclusively for nilpotents of the ring A.

A ring such that, given any two ideals L and M with L ⊆ M there exists an ideal Q such that L = QM is called a multiplication ring. For references to early papers on multiplication rings by Krull and Mori the reader is referred to [2]. A ring in which every regular ideal is invertible is called a Dedekind ring.

We let G denote an infinite compact group, and Ĝ its dual. We use the notation of our book [3, Chapters 7 and 8]. Recall that A(G) denotes the Fourier algebra of G (an algebra of continuous functions on G), and denotes its dual space under the pairing 〈f, ϕ〉 , (f ∊ A (G), ϕ ∊ ). Further, note is identified with the C*-algebra of bounded operators on L 2(G) commuting with left translation. The module action of A (G) on is defined by the following: for f ∊ A (G), ϕ ∊ , f · ϕ ∊ by

Let us associate to each open set U ⊂ R n+1 the space HU of real functions f which are twice continuously differentiable in x 1 . . . xn and once continuously differentiable in x n+1 and which satisfy the heat equation: Δf = ∂f/∂x n+1 where Then we have what in the axiomatic of Bauer is called a strong harmonic space [2, p. 61]. We will call functions of HU harmonic in U.

The desire to study constructive properties of given mathematical structures goes back many years; we can perhaps mention L. Kronecker and B. L. van der Waerden, two pioneers in this field. With the development of recursion theory it was possible to make precise the notion of "effectively carrying out" the operations in a given algebraic structure. Thus, A. Frölich and J. C. Shepherdson [7] and M. O . Rabin [13] studied computable algebraic structures, i.e. structures whose operations can be viewed as recursive number theoretic relations. A. Robinson [18] and E. W. Madison [11] used the concepts of computable and arithmetically definable structures in order to establish the existence of what can be called non-standard analogues (in a sense that will be specified later) of certain subfields of R and C, the standard models for the theories of real closed and algebraically closed fields respectively.