In this paper we study some properties of the semigroups associated with the linear retarded functional differential equations (FDE) in the setting of Banach spaces Rn X Lp( — h, 0, Rn), 1 < p < ∞. Earlier investigations of these equations via semigroups defined on the customary space C([ — h, 0], Rn) played an important role in problems of stability, oscillations, bifurcation, asymptotic behavior etc. [15].

Let G be a compact abelian group and form the spaces LP(G) with respect to the normalized Haar measure on G.

One often encounters problems that are difficult as they are, but become manageable when translated to a different category. Thus very often, problems on Boolean algebras are answered by first transferring them to problems on Boolean spaces. (See, for example, [7]). It is with this spirit that we approach in this paper two problems on Boolean algebras. These problems are two decades old, and are considered to be outstanding problems in the field.

Paul Erdôs has suggested an investigation of infinite groups from the point of view of the partition relations of set theory. In particular, he suggested that given a group G, one considers the graph T with vertex set G whose edges are the pairs ﹛g, h﹜ which do not commute.

All structures considered in this paper will be finite.

The product στ of two permutations σ and τ of a set V is defined by στ(x) = στ(X)) for every x ∈ V. The set Sv of all permutations of F is a group under this operation. A permutation group on F is a subgroup of Sv.

The theory of minimal models, as developed by Sullivan [6; 8; 16] gives a method of computing the rational homotopy groups of a space X (that is, the homotopy groups of X tensored with the additive group of rationals Q). One associates to X a free, differential, graded-commutative lgebra , over Q, called the minimal model of X, from which one can read off the rational homotopy groups of X.

Let be an infinite matrix. We call a sequence A-limitable (denoted by s ∈ (A)) if the sequence anksk exists and converges.

Let Tn denote a tree with n nodes that is rooted at node r. (For definitions not given here see [4] or [10].) The altitude of a node u in Tn is the distance α = α (u, Tn) between r and u in Tn. The width of Tn at altitude is the number Wk = Wk(Tn) of nodes at altitude in Tn, where = 0, 1, …

Generalized weighing (GW) matrices are orthogonal matrices whose nonzero entries are roots of unity. Several families are constructed with the aid of finite geometries which include as special cases interesting examples of conference matrices and weighing matrices.

This paper studies the construction of Frobenius algebras. We begin with a description of when a graded -algebra has a Frobenius algebra as a homomorphic image. We then turn to the question of actual constructions of Frobenius algebras. We give a, method for constructing Frobenius algebras as factor rings of special tensor algebras. Since the representation theory of special tensor algebras has been studied intensively ([6], see also [2; 3; 4]), our results permit the construction of Frobenius algebras which have representations with prescribed properties. Such constructions were successfully used in [9].

Let be a complex Banach space and the algebra of bounded linear operators on . In this paper we study functions from the complex numbers to of the form

A Baer ring is a ring in which every right (and left) annihilator ideal is generated by an idempotent. Generalizing quite naturally from the fact that the endomorphism ring of a vector space is a Baer ring, Wolfson [5; 6] investigated questions such as when the endomorphism ring of a free module is a Baer ring, and when the ring of continuous linear transformations on a pair of dual vector spaces is a Baer ring. A further generalization was made in [7], where the question of when the endomorphism ring of a torsion-free module over a semiprime left Goldie ring is a Baer ring was treated.

The following problem was the starting point for this investigation: Can every desarguesian affine Hjelmslev plane be embedded into a desarguesian projective Hjelmslev plane [8]? An affine Hjelmslev plane is called desarguesian if it can be coordinatized by a right chain ring R with a maximal ideal J(R) consisting of two-sided zero divisors. A projective Hjemslev plane is called desarguesian if the coordinate ring is in addition a left chain ring, i.e. a chain ring. This leads to the algebraic version of the above problem, namely the embedding of right chain rings into suitable chain rings. We can prove the following result.

The following is our main result.

(A) THEOREM. Let (M, g) be a closed connected Einstein space, n = dim M ≧ 2 (with constant scalar curvature R). Let K0 be the lower bound of the sectional curvature. Then either (M, g) is isometrically diffeomorphic to a sphere and the first nonzero eigenvalue ƛ1of the Laplacian fulfils

Let G be a finite simple group and let F be an algebraically closed field. A faithful projective F-representation of G of smallest possible degree often cannot be lifted to an ordinary representation of G, though it can of course be lifted to an ordinary representation of some central extension of G. It is a natural question to ask whether by considering non-central extensions, it is possible in some cases to decrease the smallest degree of a faithful projective representation.

Let E be an n-dimensional real affine space, V its vector space of translations and A(E) the affine group of E. Suppose that (. , .) is a nondegenerate symmetric bilinear form on F of signature (n — 1, 1), O(V) its orthogonal group and S(V) its group of similarities.