If a linear operator A in a Banach space satisfies certain conditions, then the spectrum of any almost periodic solution of the differential equation u′ = Au + f is shown to be identical with the spectrum of f, where f is a Stepanov almost periodic function.

Analogues of the Krein-Milman theorem for order-convexity have been studied by several authors. Franklin [2] has proved a set-theoretic result, while Baker [1] has proved the theorem for posets with the Frink interval topology. We prove two Krein-Milman results on a large class of posets, with the open-interval topology, one for the original order and one for the associated preorder. This class of posets includes all pogroups. Cellular-internity defined in Rn by Miller [3] leads to another notion of convexity, cell-convexity. We generalize the definition of cell-convexity to abelian l-groups and prove a Krein-Milman theorem in terms of it for divisible abelian l-groups.

The aim of the present paper is to introduce a definition of the Perron-Stieltjes integral employing the notion of approximate derivative with respect to a nondecreasing function ω and to study some of the properties of the integral. Various authors have studied the Perron integral and Perron-Stieltjes integral in different ways, most of which can be found in the references appended in the list of the bibliography. Among them Ridder [10] uses the concept of approximate co-derivative but he assumes that the monotone function a associated with co is continuous. Finally we consider a more general type of integral, the co-approximately continuous Denjoy-Stieltjes integral, defined descriptively by the method of Saks [11].

It is a trivial consequence of Magnus' solution to the word problem for one-relator groups [9] and the existence of finitely presented groups with unsolvable word problem [4] that not every finitely presented group can be embedded in a one-relator group. We modify a construction of Aanderaa [1] to show that any finitely presented group can be embedded in a group with twenty-six defining relations. It then follows from the well-known theorem of Higman [7] that there is a fixed group with twenty-six defining relations in which every recursively presented group is embedded.

Let Bd(e) denote the Burnside group with d ≥ 2 generators a1,.a2, …, a, and exponent e > 0, i.e., the free group of rank d of the Burnside variety of exponent e. It is known that Bd(e) is infinite for all sufficiently large odd values of e; cf. Novikov and Adyan [3] or Britton [1]. In particular Bd(pk), where p is an odd prime, is infinite for all sufficiently large k. It is not known whether or not Bd(2k) is infinite for all sufficiently large k infiniteness would imply that Bd(n) is infinite for all sufficiently large n, as has been conjectured by Novikov [2].

In a paper [1] with the same title Barnes has shown that if Q(x, y, z) is an indefinite ternary quadratic form of determinant d ≠ 0 then there exist integers x1, y1, z1, x2,···z3 satisfying for which Furthermore, unless Q is equivalent to a multiple of or two other forms Q2, Q3 then the constant ⅔ in (1.2) can be replaced by 1/2.2. For Q1 equality is needed on at least one side of (1.2) while for Q2, Q3 the constant ⅔ can be reduced to 12/25 but no further.

The usual technique for dealing with an abelian W*-algebra is to consider it, via the Gelfand theory, as the algebra of all continuous complex-valued functions on an extremally disconnected compact Hausdorff space with a separating family of normal linear functionals. An alternative approach, outlined in [2] and [10], is to develop the theory within the framework of Riesz spaces (linear vector lattices) where the order properties of the self-adjoint operators play an important and natural role. It has been known for a long time that the self-adjoint part of an abelian W*-algebra is a Dedekind complete Riesz space under the natural ordering of self-adjoint operators, but it is only relatively recently that a proof of this fact has been given that is independent of the Gelfand theory, and the interested reader may consult [2] or [10] for the details. This approach is essentially foreshadowed in [6] and provides a very satisfying introduction to the theory of commutative rings of operators. From this point of view, the spectral theorem for self-adjoint operators falls naturally into place as an easy consequence of the spectral theorem of H. Freudenthal. In this paper, the line of approach via Riesz spaces is developed further and several well known results are shown to follow as elementary consequences of the order structure of the algebra.

A well known theorem ([1] page 432) in the study of finite groups states that if P is a Sylow p-subgroup of the finite group G, and if P0 is a normal subgroup of P such that whenever two elements, σ and τ, of P are conjugate in G, the cosets σP0 and τP0 are conjugate in P/P0, then there is a normal subgroup K of G such that G = KP and K ∩ P = P0. In this note we will extend this result to allow P to be any Hall subgroup if G is solvable. More precisely, following theorem will be the proved.

In [5] Magill showed that compactness is a composable property of Hausdorff spaces. (That is, if α,β are compact subsets of X × X, then α ○ β is also compact when X is T2.) Also Magill gave an example to show that the composition of connected relations need not be connected. Subsequently, he characterized Hausdorff k-spaces in terms of the semigroup of compact relations in X.

Let S, T be self-mappings on a (non-empty) complete metric space (X, d). Let ai, i = 1, 2, …, 5, be non-negative real numbers such that < 1 and for any x, y in X,

Spitzbart [1] has considered a generalization of Hermite's interpolation formula in one variable and has obtained a polynomial p(x) of degree n + Σnj=0 = rj in x which interpolates to the values of a function and its derivatives up to order rj at xj, j = 0, 1,···n. Ahlin [2] has considered a bivariate generalization of Hermite's interpolation formula. He has developed a bivariate osculatory interpolation polynomial which agrees with f(x, y) and its partial and mixed partial derivatives up to a specified order at each of the nodes of a Cartesian grid. However, the above interpolation problem considered by Ahlin assumes that the values of partial and mixed partial derivatives of the same fixed order k – 1 are available at every point of the rectangular grid. It may also be observed that Ahlin's formula is essentially a Cartesian product of a special case of Spitzbart's formula in one variable.

It is well known (see Latyšev [1] for finite dimensional case) that the universal envelope of Lie Alaebra g over a commutative field ť of characteristic 0 is a PI-alaebra (i.e. possesses a nontrivial identity) if and only if this Lie algebra is abelian. On the other hand the recent results due to Passman [2] describe the conditions under which the group algebra of a group over an arbitrary commutative field is a PI-algebra. A. L. Šmel'kin suggested that I should find necessary and sufficient conditions for a Lie algebra g over a field of nonzero characteristic under which its universal envelope Ug should be a PI-algebra. These conditions are given in the following theorem.

Let R be a ring and M a left R-module. We investigate when the functors HomR(M, —), HomR( —, M), and — ⊗RM are exact for certain restricted subcategories of modules.

A semiring (S, +,·) is a nonempty set S, endowed with associative operations of addition and multiplication, such that the multiplicative semigroup (S, ·) distributes over the addition. That is: x(y +z) = xy + xz and (x + y)z = xz + yz for all x, y and z in S. A topological semiring is a semiring, defined on a Hausdorff space, such that both of the operations are jointly continuous.

Magill in [4] first proved that two Boolean rings are isomorphic if and only if their respective endomorphism semigroups are isomorphic. His proof, however, relied on topological techniques. More recently Maxson has published a proof of the above using purely algebraic techniques [5]. In this paper, structure theorems are given which allow us to extend the above result to the pk-rings of Foster [1]. As a special case, the result is shown to apply also to p-rings. An example is given to show that a further extension to J-rings is impossible.

A system or family (Aγ: γ∈ N) of sets Aγ, indexed by the elements of a set N, is called an (a, b)-system if ¦N¦ = a and ¦Aγ¦ = b for γ ∈ N. Expressions such as “(a, <b)-system” are self-explanatory. The system (Aγ: γ∈N) is called a δ-system [1] if Aμ∩Aγ = Ap ∩ Aσ whenever μ, γ, ρ, σ ∈ N; μ≠ γ; ρ ≠ σ. If we want to indicate the cardinality ¦N¦ of the index set N then we speak of a δ(¦N¦) system. In [1] conditions on cardinals a, b, c were obtained which imply that every (a, b)-system contains a δ(c)-subsystem. In [2], for every choice of cardinals b, c such that the least cardinal a = fδ(b, c) was determined which has the property that every (a, < b)-system contains a δ(c)-subsystem.

The aim of this paper is to generalize Theorem 2.10 (i) of [2]. As stated in [2] this theorem deals with the semigroup of all selfmaps on a discrete space and provides a characterization of H-classes which contain an idempotent. We will generalize this theorem to the case of other semigroups of functions on a discrete space, some semigroups of continuous functions on non-discrete topological spaces, and one semigroup of binary relations. The results in this paper form the main part of chapter 3 of [1]. Some results will be quoted from [1] without proof; the required proofs can easily be supplied by the reader.

The following theorem is the focal point of the present paper. It stipulates an algebraic condition equivalent, in any finitely generated group, to the solubility of the word problem.

THEOREM I. A necessary and sufficient condition that a finitely generated group G have a soluble word problem is that there exist a simple group H, and a finitely presented group K, such that G is a subgroup of H, and H is a subgroup of K.

Suppose that A = [αnk], (n, k = 0, 1, 2, …), is an indinite matrix with complex entries. A transforms a complex sequence a = {an to a complex sequence {bn = b = Aa where assuming that the series in (1) converges. Each sequence a = {an is uniquely associated with a power series In this way the matrix A transforms a power series into a power series. Specifically, the power series (2) is mapped to the power series where the bn's are given by (1). We are only interested in matrices having the property that each power series analytic in Δ = {z: |z| < 1} maps to a power series analytic in Δ.

The aim of this note is to give an analogue, for an inverse semigroup S, of the theorem for a group G which says that if G is the set of normal subgroups of G, then the map N → (N) = {(a, b) ∈ G x G: ab-1 ∈ N}; for N ∈ G is a 1: 1 order preserving map of G onto ∧(G), the lattice of congruences on G. It will be shown that if E is the semilattice of idempotents of S, P = {E: α ∈ J} is a normal partition of E, and K is a certain collection of self conjugate inverse subsemigroups of S, then the map K →(X) = {(a, b)∈ S x S: a-la, b-1b∈ Eα for some α ∈ J and ab-l ∈ K) for K e Jf is a 1:1 map of K onto the set of congruences on S which induce P.