We give a necessary and sufficient condition for a class of row-decreasing infinite matrices to be partially well-ordered with regard to the component-wise ordering. Then, using these matrices, we determine all the classes of abelian groups, closed under taking subgroups, direct limits, and isomorphic groups.

From recent results of Lausch, it is easy to establish necessary and sufficient conditions for a Fitting class to be maximal in the class of all finite soluble groups. We use Lausch's methods to show that there are normal Fitting classes not contained in any Fitting class maximal in the class of all finite soluble groups. We also find conditions on Fitting classes and for to be maximal in .

Let X be a compact, connected, oriented topological G-manifold, where G is a compact connected Lie group. Assume that the fixed point set is finite but nonempty, the action is otherwise free, and the orbit space is a manifold. It follows that either G = U(1) = S1 and dimX =4 or G = Sp(1) = S3 and dimX = 8, and the number of fixed points is even. The authors prove that these ∪(1)-manifolds (respectively, Sp

(1)-manifolds) are classified up to orientation-preserving equivariant homeomorphism by (1) the orientation-preserving homeomorphism type of their orbit 3-manifolds (respectively, 5-manifolds), and

(2) the (even) number of fixed points.

Both the homeomorphism type in (1) and the even number in (2) are arbitrary, and all the examples are constructed. The smooth analog for U(1) is also proved.

In a previous paper (Acta Arith. 21 (1972), 89–97), I had proved that the sum of the absolute values of the coefficients of the mth transformation polynomial Fm (u, v) of the Weber modular function j(ω) of level 1 is not greater than 2(36n+57)2n when m = 2n is a power of 2. The aim of the present paper is to give an analogous bound for the case of general m. This upper bound is much less good and of the form where c > 0 is an absolute constant which can be determined effectively. It seems probable that also in the general case an upper bound of the form eO(m10gm) should hold, but I have not so far succeeded in proving such a result.

Results are obtained which enable one to assess the extent to which the waiting times of queuing customers depend on the number of servers which provide them with service.

In the domain of real symmetric matrices ordered by the positive definiteness criterion, the symmetric matrix Riccati differential equation has the unusual property of preserving the ordering of its solutions as the independent variable changes, Here is is shown that, subject to a continuity restriction, the Riccati equation is unique among comparable equations in possessing this property.

An algorithm is presented for computing the unique stationary distribution of an infinite regular stochastic matrix of a structural form subsuming both upper-Hessenberg and generalized renewal matrices of this kind. Convergence is elementwise, monotone from above, from information within finite truncations, of increasing order.

By using a variational method, we study the structure of the Padé table for a formal power series. For series of Stieltjes, this method is employed to study the relations of the Padé approximants with orthogonal polynomials and gaussian quadrature formulas. Hence, we can study convergence, precise locations of poles and zeros, monotonicity, and so on, of these approximants. Our methods have nothing to do with determinant theory and the theory of continued fractions which were used extensively in the past.

Sufficient conditions, involving the existence of a Lyapunov function which is a submartingale of special type, are given for the instability of stochastic discrete time systems.

An example of Wehrfritz is pointed out to show that GL(4, Q) is not coherent. This answers a question of Serre. It is shown that finitely generated soluble coherent subgroups of GL(2, Q) need not be polycyclic, in sharp contrast to the fact that all soluble subgroups of GL(n, Z) are polycyclic and so automatically coherent.

The concept of a tetradiffric function is introduced. This new scheme for defining discrete analytic functions is shown to retain the algebraic simplicity of monodiffric functions, while introducing to the theory a symmetry similar to the Schwarz Reflection Principle.

In this paper we construct the Wiener-Hopf equation for diffraction of a planetary wave by an infinite strip and obtain its solution in the form of an integral representation. We also discuss the asymptotic character of the diffracted wave using the saddle point method.

The sum of an infinite series involving the 4F3 function is found by considering an infinite series whose terms involve the product of two Bessel functions of the first kind. Furthermore the infinite series involving the 4F3 function can be utilized to find an approximation to the 4F3 function of unit argument, for particular values of the parameters.