Let

where λ is rational and not an integer. The author investigates lower estimates for example for

where the αi are distinct rational numbers not 0, and where x1, …, xk, are integers and

It is shown that for any almost periodic linear differential system asymptotic stability and uniform stability together imply uniform asymptotic stability.

The (logarithmic) capacity or transfinite diameter is originally defined for compact sets in the complex plane. An extension may be made by defining the capacity of a given arbitrary set in the plane as the supremum of the capacities of all compact sets contained in the given set. Convergence in capacity is defined analogously to convergence in measure. It is shown in this paper that properties of convergence in capacity are also analogous to those of convergence in measure.

It is proved that if G is a permutation group on a set Ω every orbit of which contains more than mn elements, then any pair of subsets of Ω containing m and n elements respectively can be separated by an element of G.

In recent years fixed point theorems have been proved for non-expansive and similar mappings on uniformly convex Banach spaces. The only role the linear structure plays in the statement of these results occurs in the definition of uniform convexity. It is therefore natural to ask whether the results depend essentially on the linear structure, or whether an extension of the notion of uniform convexity to metric spaces would allow the hypothesis of linear structure on the underlying space to be removed.

Let G be a finite group. It is shown that G is 2-closed if and only if

(a) every 2-block of G has full defect, and

(b) every Sylow 2-intersection is centralized by a Sylow 2-subgroup of G.

As a consequence it is shown that G is a TI-group if and only if every 2-block of G has either full defect or defect zero and (b) holds. This result and a theorem of Kwok yield complete characterizations of finite groups with certain relations being satisfied by every nonprincipal irreducible character.

Weak duality and direct duality theorems are proved, under appropriate assumptions, for the following pair of programming problems in complex space:

The objective function may be nondifferentiable and the constraints are of a more general nature than those considered earlier by various authors. Several well-known results are shown to be special cases of the results proved here.

Oscillation criteria are obtained, for the nonlinear delay differential inequality u(u″+f(t, u(t), u(g(t)))) ≤ 0. The main theorems give sufficient conditions (and in some cases sufficient and. necessary conditions) for all solutions u(t) to have arbitrary large zeros. Generalizations to more general cases are discussed.

It is shown that every finitely generated field K of characteristic 0 may be embedded in infinitely many p-adic fields in such a way that the images of any given finite set C of non-zero elements of K are p-adic units. The result is suggested by Lech's proof of his generalization of Mahler's theorem on recurrent sequences. It also provides a simple proof of Selberg's theorem about torsion-free normal subgroups of matrix groups.

The purpose of this paper is to show that a well-known method for constructing “queer” sets in models of ZF set theory is also applicable to certain algebraic structures. An infinite set is called “quasi-minimal” if every subset of it is either finite or cofinite. In Section 1 I set out the two systems of set theory to be used in this paper, and illustrate the technique in its most fundamental form by constructing a model of set theory containing a quasi-minimal set. In Section 2 I show that by choosing the parameters appropriately, one can use this technique to obtain models of set theory containing groups whose carriers are quasi-minimal. In the third section various independence results are deduced from the existence of such models: in particular, it is shown that it is possible in ZF set theory to have an infinite group that satisfies both the ascending and descending chain conditions. The quasi-minimal groups constructed in Section 2 were all elementary abelian; in Section 4 it is shown that this was not just chance, but that in fact all quasi-minimal groups must be of this type. Finally in Section 5 permutations and permutation groups on quasi-minimal sets are examined.

Let Qp and K be the rational p–adic field and an algebraic extension of Qp of finite degree, respectively, and let I and Ik be the subsets of Qp and of K consisting of the p–adic integers of these fields.

It is known that the continuous functions f: I → Qp can be written as

where this series converges uniformly on I, and that such continuous functions need not be differentiable at any point. We here study continuous functions F: IK → K which for all X on IK are the sum of a uniformly convergent series

It is proved that such functions F(X) have at every point of IK derivatives of all orders. In the special case when K is totally ramified, they cannot in general be developed into power series that converge everywhere on IK, but this is possible when K is not totally ramified.

In this paper, we study the oscillation of nth-order differential equations. Recently, Atkinson and the present authors studied (separately) the comparison properties of differential inequalities. Kartsatos treated the nth-order ordinary case and proposed several open problems.

The purpose of this paper is to answer one of them in the affirmative concerning more general functional differential equations. The result is that if under several conditions, the equation

is oscillatory for n even or a solution x(t) of (1) is oscillatory or for n odd, then this is also the case for the equation

In 1919, Adolf Hurwitz formed the quaternion ring R composed of elements whose coordinates were either all integers or halves of odd integers. The objective of this paper is to examine the (two-sided) ideal structure in the hurwitzean polynomial ring R[x], formed by taking all polynomials with coefficients in R. The maximal and prime ideals of R[x] will be characterized with results surprisingly analogous to those in Z[x]. In addition, a canonical basis, of the type developed by G. Szekeres, 1952, for polynomial domains, will be developed for the ideals of R[x].

Let G be a doubly transitive permutation group on a finite set Ω, and for α in Ω suppose that Gα has a set Σ of non-trivial blocks of imprimitivity in Ω - {α}. If Gα is 2-transitive but not faithful on Σ, when is it true that the stabiliser in Gα of a block of Σ does not act faithfully on that block (that is, there is a nontrivial element in Gα which fixes every point of the block)? In a previous paper this question was answered when is the alternating or symmetric group, or a Mathieu group in its usual representation. In this paper we answer the question when , permuting the q + 1 points of the projective line, for some prime power q. We show that the only groups which arise satisfy either

(i) PSL(3, q) ≤ G ≤ PΓL(3, q) in its natural representation, or

(ii) G is a group of collineations of an affine translation plane of order q, and contains the translation group.

A projection morphism ρ: G1 → G2 of finite graphs maps the vertex-set of G1 onto the vertex-set of G2, and preserves adjacency. As an example, if each vertex v of the dodecahedron graph D is identified with its unique antipodal vertex v¯ (which has distance 5 from v) then this induces an identification of antipodal pairs of edges, and gives a (2:1)-projection p: D → P where P is the Petersen graph.

In this paper a category-theoretical approach to graphs is used to define and study such double cover projections. An upper bound is found for the number of distinct double covers ρ: G1 → G2 for a given graph G2. A classification theorem for double cover projections is obtained, and it is shown that the n–dimensional octahedron graph K2,2,…,2 plays the role of universal object.

Evidence is presented which suggests that the following assertions about a variety ⊻ of groups may be equivalent:

(a) ⊻ is locally finite,

(b) all ⊻-groups satisfying the maximal condition for normal subgroups are finite, and

(c) all ⊻-groups satisfying the minimal condition for normal subgroups are finite.

A code not requiring a distinct symbol to separate words is called comma-free. Two codes are isomorphic if one can be obtained from the other by a permutation of the underlying alphabet. Since subcodes of comma-free codes are comma-free, we investigate only maximal comma-free codes. All isomorphism classes of maximal comma-free codes with words of length 2 are determined and a natural representative of each class is given.

Suslev's constraint multipliers are used to derive the equations of motion of dynamical systems (holonomic or nonholonomic) in the form of Poincaré-Četaev equations and in the canonical form. For holonomic systems defined by redundant variables, the constraint multipliers occuring in the canonical equations are determined and a modification of the Hamilton-Jacobi Theorem for integrating the canonical equations is presented.

Some fixed point results are derived for mappings of contractive type in metric and topological spaces.

It is shown that all solutions of ẍ + 2x3 = p(t) are bounded, the notation indicating that p is periodic. It is not necessary to have a small parameter multiplying p.

The essential step is to show by appeal to Moser's theorem that, under the mapping (of the initial-value plane) which corresponds to the equation, there are invariant simple closed curves. This implies also that there is an uncountable infinity of almost-periodic solutions and, for each positive integer m, an infinity of periodic solutions of least period 2mπ (2π being taken as the least period of p ).

It is suggested that for a large class of equations the same attack would show all solutions of ẍ + g(x) = p(t) bounded. However, in order to show the method clearly, no generalisation is attempted here.