A theory T is said to κ-stable if, given a pair of models U ⊂ B of T with U of power κ, there are only κ types of elements of B over U (types are defined below). This notion was introduced by Morley (1965) who gave a powerful analysis of ω-stable theories. Shelah (1971) showed that there are only four possibilities for the set of κ in which a countable theory is stable. This partition of all theories into four classes (ω-stable, superstable, stable, and unstable theories) has proved to be of great value. However, most familiar examples of theories are unstable.

This paper investigates the logical stability of various groups. Theorem 1: If a group G is stable and locally nilpotent then it is solvable. Theorem 2: Every non-Abelian variety of groups is unstable.

The purpose of this note is to apply the work of Kasteleyn (1967) on the enumeration of I-factors of a graph to derive a quick proof of a theorem of Szekeres (1973). In the following, G is understood to be a finite, directed graph. If u, v are adjacent vertices of G, we denote by (u, v) an edge of G directed from u to v. Let{f1, f2,…, fm} be a set of 1-factors of G, and for all i write where n is half the number of vertices of G. (Here we regard a 1-factor as a set of edges). Then Kasteleyn associates with fi a plus sign if ui1vi1ui2vi2…uinvin is an even permutation of u11v11u12v12…u1nv1n, and a minus sign otherwise. The symmetric difference of two 1-factors is a collection of circuits, called alternating circuits. An alternating circuit of G is said to be clockwise even if the number of its edges that are directed in agreement with the clockwise sense is even; otherwise it is clockwise odd. Since the length of any alternating circuit is even, these definitions are not dependent on the sense designated as clockwise. It follows from the work of Kasteleyn that two given 1-factors in a directed graph agree in sign if and only if the number of clockwise even alternating circuits in their symmetric difference is even.

Since it is no longer fashionable to publish per se results which depend only on algebraic manipulation, many useful and complicated results of quite general interest languish unseen hidden as lemmata in specialist papers. In particular, the theory of transcendental numbers is rich in ingenious techniques for evaluating determinants. These techniques are apparently not well known even to workers in the field of transcendental numbers, let alone to researchers on other areas where the results might find application. This note accordingly discusses a variety of interesting results on determinants and is to be viewed as an appendix to the encyclopaedic volumes of Muir (1911, 1933) which the researcher might approach in order to obtain information in this area. For reasons of motivation a brief mention is made of the context in which the determinants arise.

In this paper we examine the linear differential equations x′ = Ax + P(φ)x, φ′ = ω where x ∈ Rn, φ ∈ Rm, A and ω are constant and P(φ) is real analytic and periodic in φ. We use the method of accelerated convergence to overcome the small divisors problem and reduce this system to the system y′ = By, φ′ = ω with constant coefficients.

This problem has already been examined by Mitropolśki and Samolenko but the calculations and details in their work are formidable and difficult to follow. Besides being simpler our method provides more precise estimates at all stages and can be extended to the differentiable case to provide a significant improvement over previous results.

Let C be a nontrivial full subcategory of the category F of finite discrete groups and continuous homomorphisms, closed under subobjects, quotient and finite products. We consider the category PC of pro-C-groups and continuous homomorphisms (i.e. inverse limits of C-groups) which forms a variety in category PF of profinite groups and continuous homomorphisms. The study of pro-Cgroups is motivated by their occurrence as Galois groups of filed extensions in algebraic number thory (see Serre (1965)). The purpose of this paper is to study the tripleableness of the forgetful functors from PC to various underlying categories. It is also shown that PC is equivalent to the category of algebras of the theory of the forgetful functor from C to S (the category of sets and mappings).

The well known theorem of E. H. Brown on the representation of Homotopy Functors is extended to a larger domain category. This is accomplished by requiring the functor to satisfy a further condition, that of being locally trivial: the condition is both necessary and sufficient for the extension and is satisfied by functors arising from locally trivial fibre bundles such as K-theory.

The pointwise order makes the group A(Ω) of order-preserving permutations of a totally-ordered set Ω a lattice-ordered group. We give some criteria for determining the compatible tight Riesz orders on A(Ω) in the case of Ω being a totally-ordered field, and then obtain various adjunctionshellip one between tight Riesz orders on A(Ω) and certain ideals of the fixed point lattice Φ(Ω), and a second between maximal tangents and certain filters of Φ(Ω). We also establish a correspondence between tight Riesz orders and first-order properties. Finally, we make use of our results to say what we can in the case of the automorphisms of the real field, and to pose several open problems.

New characterizations of maximal topologies are given for a class of topologies including the compact, Lindelöf and m-compact topologies.

Let M be a vector subspace of the Banach space C(Ω) of all real-valued continuous functions on a compact space Ω, and suppose that M contains a subset L and the constant functions. Then in order that L be dense in M it is necessary and sufficient that L, M satisfy a filtering property and that each m in M can be approximated on every two points of Ω by functions in L.

With respect to a given family of functions F, a function is said to be F-convex, if it is supported, at each point, by some member of F. For particular choices of F one obtains the convex functions and the generalized convex functions in the sense of Beckenbach. F-convex functions are characterized and studied, retaining some essential results of classical convexity.

Let X be a real Banach space and let K be a bounded closed convex subset of X. We prove that the set of strongly exposing functions K^ of K is a (norm) dense G8 in X* if and only if for any bounded closed convex subset C such that K⊄C, there exists a point x in K which is a strongly exposed point of conv (C ∪ K). As an application, we show that if X* is weakly compact generated, then for any weakly compact subset K in X, the set K^ is a dense G8 in X*.

It is proved that there exists no compatible tight Riesz order on a complemented modular lattice. An example is provided of a complemented lattice with a compatible tight Riesz order.

Let Γ be a graph with finite vertex set V. Γ is homogeneous if whenever U1, U2 ⊆ V are such that the vertex subgraphs (U1), (U2) are isomorphic, then every isomorphism from (U1) to (U2) extends to an automorphism of Γ; homogeneous graphs were studied by Sheehan (1974) and were classified by the author. Γ is locally homogeneous if whenever U ⊆ V, then every automorphism of (U) extends to an automorphism of Γ. We prove that every locally homogeneous graph is homogeneous.

Throughout, R will denote an associative ring with center Z. For elements x, y of R and k a positive integer, we define inductively [x, y]0 = x, [x, y] = [x, y]1 = xy − yx, [x, y, y, hellip, y]k = [[x, y, y, hellip, y]k−1, y]. A ring R is said to satisfy the k-th Engel condition if [x, y, y, hellip, y]k = 0. By an integral domain we mean a nonzero ring without nontrivial zero divisors.

In this paper we investigate the following problem proposed by Lausch and Nöbauer: Let be a variety of universal algebras, B an algebra of and A a subalgebra of B. If a system of algebraic equations over A is solvable in , is it then solvable over B? We show that the answer is affirmative in certain varieties but negative in the general case.