Introduction. Let Mn+1 denote a closed, orientable, combinatorial (n + 1)-manifold. Let Kn denote an n-dimensional subcomplex of Mn+1. If Mn+1 – Kn is connected, then by ‘cutting’ Mn+1 along Kn we define in a natural way a fundaments domain of a certain regular covering of Mn+1. In case Kn is a spine ((5)) of Mn+1 the any covering space of Mn+1 may be constructed. This construction is quite precis and is combinatorial in spirit.

It is well known that a path-connected Hausdorff space is arc-connected. Indeed, given a path P, that is, a continuous map. f of the unit interval I0(0 ≤ t ≤ 1) into the Hausdorff space X, there is an arc joining the points f(0), f(1) (supposed distinct) which is obtained by ‘cutting out loops in P’. More precisely, there exist a continuous increasing map α of I0 onto itself and a homeomorphism φ of I0 onto φ(I0) ⊂ X, such that if [t0, t1] is any maximal interval of constancy of α (including the case t0 = t1) then f(t0) = f(t1) = φοα(t0). The function φ, α, can be defined by a constructive process.

1. In a previous paper, ((1)), we have discussed problems which arise in finding a matrix which is in some sense stronger than each of a set of regular summability matrices. We intend in this paper to clear up other problems in this subject. We shall retain the notation and definitions of (1) throughout, but shall later modify one set of definitions in the light of some of our results. In this paper, the term matrix will be reserved for regular summability matrices. Also, u will be used to denote the unit sequence, u = {un} = {1} and h(A) will be used to denote the norm of the matrix A = (amn), i.e.

Let G be a group, and let φ (x1, …, xn) be any word in the variables x1, …, xn. We denote by φ(G) and φ*(G) respectively the corresponding verbal and marginal subgroups of G, which are defined as follows.

(i) φ(G)is the subgroup generated by all elements of the form φ(a1; …, an) with ai in G(for i = 1, 2, …, n);

(ii) an element b of G lies in φ*(G) if and only if

for all choices of a1, …, an in G and i= 1, 2, …,n.

It has been known for some time that the product of a non-metrizable Hausdorff space and any (non-trivial) Hausdorff space cannot be the continuous image of an ordered continuum. (For a survey of this and related problems, see Mardešić and Papić ((1)).) Further, it has been shown by Treybig ((2)) (and independently by the present author) that the product of two Hausdorff spaces cannot even be the continuous image of an ordered compactum unless both the spaces are metrizable or one is finite. It is therefore of some interest to give a simple example of a space X which is the continuous image of an ordered compactum K and contains the product of a non-metrizable space and an infinite discrete space, imbedded in such a way as to form a sequence of homeomorphic subsets with a connected (non-trivial) topological limit.

Let M denote a connected, compact, unbounded, piecewise linear (= P L) manifold of dimension 4. Denote by the normal Stiefel-Whitney class in dimension 3, with integer coefficients (twisted if M is not orientable).

In the development of homological algebra, one has to prove at some point that, in defining the derived functors of ⊗ and of Hom, it makes no difference whether we resolve both variables or only one of them. Taking ⊗ ( = ⊗R) as a typical example, what has to be proved is

(A) If the complex F is projective, or even flat, as a right R-module, and if f: P → A is a projective resolution of the left R-module A, then

is an isomorphism.

For the group of real numbers R, an exponential monomial is defined as a function of the form xr(−ixz), for some non-negative integer r and some complex number z. Similarly, an exponential polynomial is a function P(x) exp (−ixz), for a polynomial P. In a now famous paper ((15)), Schwartz proved that every closed translation invariant subspace (variety) of the space of continuous functions on R is determined by the exponential monomials it contains. His techniques do not generalize to groups other than R as they use the theory of functions of one complex variable. A shorter proof of this result, using the Carleman transform of a function, was given by Kahane in his thesis ((9)). Ehrenpreis ((5)) proved results similar to those of Schwartz for certain varieties in the space of analytic functions of n complex variables, and Malgrange ((13)) proved the related result that any solution in ℰ(Rn) (for the notation see (16)) of the homogeneous convolution equation μ*f = 0, for some μ∈ℰ′, belongs to the closure of the exponential polynomial solutions of the equation.

In this note it is shown that any positive root of the transcendental equation

is definable as a continuous increasing function of the real variable ν, provided ν is positive. Here Jν and Yν denote respectively the Bessel functions of the first and second kind of order ν, and k is a positive constant. Watson ((4)) has established the corresponding result for the simpler equations Jν(z) = 0 and Jν(z) cosα − Yν(z) sinα = 0, where α is a constant. The extension of the result to the positive roots of equation (1) is important because this equation occurs quite frequently in physical problems.

We shall present this paper in the framework and terminology of differential topology though all our arguments are valid in the piecewise linear ease also, under local un-knottedness hypotheses. In particular we use Rp for Euclidean space of dimension p, Sp−1 for the standard unit sphere in it, and Dp for the disc which it bounds.

In (8), we have shown how, in general, the problem of identifying the differentials in the Adams spectral sequence (see(1),(4)) is equivalent to that of calculating certain higher-order cohomology operations (in the sense of (6)). However, we propose to investigate here a slightly different method, based on the naturality of the spectral sequence, which can be used to show that certain elements are never boundaries, for any differential.

This paper deals with the existence of a quadratic form as a Liapunov function for a linear homogeneous vector differential equation with constant coefficient matrix such that the total derivative of the Liapunov function is strictly negative semi-definite and not identically equal to 0 for every non-trivial solution of the given equation. A theorem is proved which guarantees the existence of a strictly semi-definite quadratic form which is not identically equal to 0 for every non-trivial solution if and only if the coefficient matrix of the equation is not proportional to the unit matrix. Furthermore, it is proved that if the equilibrium of the given linear equation is asymptotically stable and if the coefficient matrix is not proportional to the unit matrix, then there exists a positive definite quadratic form as a Liapunov function whose total derivative is strictly negative semi-definite and not identically equal to 0 for every non-trivial solution of the given equation. It is also shown how this theorem fits into the system of already known theorems concerned with quadratic forms as Liapunov functions for a linear equation with constant coefficient matrix.

Introduction and statement of results. We shall describe how, for successive integers N, the points {nξ}, with n = 0, 1, …,N – 1, are distributed in the closed unit interval U = [0, 1]; by showing how successive points {Nξ,} modify the partition of U produced by the previous points. The simple generalization to the k-dimensional sequence {nξ} = ({nξ(1)},{nξ(2)}, …,{nξ(k)}), in the unit hypercube Uk, is also made.

The purpose of this note is to answer a question asked in an earlier paper ((2), page 315). Let X be a closed, connected, non-orientable 2-manifold and f: X → X a homeomorphism. It was shown in (2) that f is isotopic to a product of homeomorphisms of two basic types, the c-homeomorphism, which will now be called a twist, and the Y-homeomorphism (see (2) for definitions). It was also proved that f cannot, in general, be expressed as a product of twists alone. The following result is proved here:

For an n-vector x = (xi) and n × n matrix A = (aij) with complex elements, let |x|2 = Σi|xi|2,|A|2 = ΣiΣj|aij|2. Also, M(A), m(A) denoteℜλ1,ℜλn, respectively, where λ1,…,λA are the eigenvalues of A arranged so that ℜλ1 ≥ … ≥ ℜλn. Throughout this paper A(t) denotes a matrix whose elements aij(t) are complex valued Lebesgue integrable functions of t in (0, T) for all T > 0. Then M(A(t)), m(A(t)) are also Lebesgue integrable in (0, T) for all T > 0. The characteristic exponent μ of a non-zero solution x(t) of the n × n system of differential equations

can be defined, following Perron ((12)), as

where ℒ denotes lim sup as t → + ∞. When |A(t)| is bounded in (0,∞), μ is finite; in other cases it could be ± ∞.

In ((4)) Slater gave expansions of the generalized Whittaker functons pFp(x). (She gave this name to the generalized hypergeometric function pFp(x), since it is a generalization of the well-known Whittaker function 1F1(x).) In another paper ((2)), Ragab gave a series of products of generalized Whittaker functions in terms of such functions or in terms of generalized hypergeometric functions pFq(x). In the present paper, integrals involving products of generalized Whittaker functions will be evaluated in terms of such functions or of other generalized hypergeometric functions.

If n(p) is the least quadratic non-residue of a given prime p then it is known ((1)) that n(p) = O(pα) for any α > 1/(4 √e). LeVeque ((2), page 122) gives the following bound with an explicit constant: n( p) < √ p for p ≠ 2, 3, 7, 23. In the present paper an elementary and self-contained proof is given of a result slightly stronger than LeVeque's. Some numerical results, which indicate the extent to which the result proved falls short of what actually obtains, are appended.

In this paper, a method is given for solving certain types of integral equations. It involves a technique of building up the solution of the given integral equation from that of a much simpler auxiliary equation and a consideration of this auxiliary equation forms a large section of the work. The method of solution is applied to a particular problem in semi-circular electron focusing in metals.

Notations. If G denotes an Abelian group and L a locally compact group topology on G, G(L) will denote the resulting topological group, and will denote the character group with the usual Pontrjagin topology (which is locally compact). Rn will denote the real n-dimensional vector group. Finally, for any set X, |X| will denote its cardinal number.

In the usual notation let

where (a)n = a(a + 1)(a + 2)…(a + n − 1), (a)0 = 1.

Dr L. J. Slater ((3), page 628) gave an expansion of the form

for all values of n and t, real or complex.