Let A = (am, n) be a (regular summability) matrix. Then will denote the set of bounded sequences which are summed by A. If {Ai} (i = 1, 2, …, N) is a finite set of such matrices, and if consists of every bounded sequence then we shall say that the matrices span the bounded sequences. Ifx = {xn} belongs to then we denote the value to which A sums x by A-lim x. If y = {yn} is any sequence, then the A-transform of y (if it exists) is the sequence {Aμ(y)}, where

Theorem 1. Let M3be a compact 3-manifold and let M3 = E3 ∪ R be a standard decomposition where R is a subcomplex in a triangulation of M3((2)). If M3is a homology 3-sphere while R contains no arc that is wild in M3then M3 = S3

In two recent papers of Conner and Floyd ((2)) and ((3)), the additive structure of the SU-cobordism (or bordism) ring was completely determined. The object of this note is to point out that their results can also be used to determine the somewhat complicated multiplicative structure.

For a real number λ, ‖λ‖ is the absolute value of the difference between λ and the nearest integer. Let X represent the m-tuple (x1, x2, … xm) and let

be any n linear forms in m variables, where the Θij are real numbers. The following is a classical result of Khintchine (1):

For all pairs of positive integers m, n there is a positive constant Г(m, n) with the property that for any forms Lj(X) there exist real numbers α1, α2, …, αn such that

for all integers x1, x2, …, xm not all zero.

Introduction. It is a classical problem to construct a complex projective space, using as ‘imaginary points’ objects of some kind from the geometry of real projective space. (The various solutions to this problem have been reviewed in a book by Coolidge ((1)).) The object of the present paper is to solve the corresponding problem for n-dimensional projective spaces over two skew fields, one being a second-rank extension of the other. The method used seems not to have been published before, even in connexion with the classical problem.

If X is a uniformizable space any uniform structure compatible with its topology will give a uniform structure on the collection H(X) of all non-empty closed sets of X ((1), Section 2, Exercises 7, 8, definition 1·6): for each entourage U ∈ we define an entourage U on H(X) by putting U = {(A, B); A ⊂ U(B) and B ⊂ U(A)}. It has been conjectured ((6), p. 35, Exercise 17) that if 1 and 2 are distinct uniform structures on X, 1 and 2 induce indifferent topologies on H(X). Unfortunately the method proposed in (6) fails, even for metrizable structures, as we shall show by an example. We shall, however, establish a number of partial results tending to support the conjecture; in particular we shall establish the result when both the uniform structures are metrizable, or when one is precompact.

Introduction. Let X be a submanifold of Y, in either the topological, smooth, or piecewise linear ( = PL) categories. A normal cell bundle on X in Y is a bundle ξ = (p, E, X) in the category whose fibre is a closed cell, and such that E is a neighbourhood of X in Y and p: E → X is a retraction. The triple (Y, X, ξ) is a tubular neighbourhood, or briefly, a tube. For convenience we may refer to a tube by its cell bundle.

If A = [aij] is an n × n matrix, the permanent of A is the scalar valued function of A defined by

where the summation extends over all permutations (i1, i2, …, in) of the integers 1, 2, …, n. If we assume that A is a non-negative matrix (that is, that A has non-negative entries) then we come upon an extremely interesting situation. Much work has been done in finding significant upper bounds for the permanent and permanental minors of A (see, e.g. (2, 4–7) in (5) an excellent bibliography is given). If B = [bij] is another n × n non-negative matrix, then we may form the product AB and consider the permanent of this matrix. Unlike the determinant, the permanent is not a multiplicative function. In our circumstances here, however, it is easy to verify and indeed a proof was written down in (1) that

Introduction. This paper is a continuation of (1), where definitions may be found. All manifolds, maps, bundles, etc., are to be piecewise linear. Recall that cells are always closed.

The Stone-Weierstrass theorem gives very simple necessary and sufficient conditions for a subset A of the algebra of all real-valued continuous functions on the compact Hausdorff space X to generate a subalgebra dense in namely, this is so if and only if the functions of A strongly separate the points of X, in other words given any two distinct points of X there exists a function in A taking different values at these points, and given any point of X there exists a function in A non-zero there. In the case of the algebra of all complex-valued continuous functions on X, the same result holds provided that we consider the subalgebra generated by A together with Ā, the set of complex conjugates of the functions in A.

Introduction. A famous construction of Wiener and Wintner ((13)), later refined by Salem ((11)) and extended by Schaeffer ((12)) and Ivašev-Musatov ((8)), produces a non-negative, singular, continuous measure μ on [ − π,π[ such that

for every ∈ > 0. It is plain that the convolution μ * μ is absolutely continuous and in fact has Lebesgue–Radon–Nikodým derivative f such that For general locally compact Abelian groups, no exact analogue of (1 · 1) seems possible, as the character group may admit no natural order. However, it makes good sense to ask if μ* μ is absolutely continuous and has pth power integrable derivative. We will construct continuous singular measures μ on all non-discrete locally compact Abelian groups G such that μ * μ is a absolutely continuous and for which the Lebesgue–Radon–Nikodým derivative of μ * μ is in, for all real p > 1.

An error in Dieudonné's paper, Sur les espaces de Köthe, published in Journal d' Analyse Mathématique, 1 (1951), 81–115, is pointed out and a correction put forward.

Statement of results. Let A and B be two matrix summability methods, and S a given set of sequences. We shall say that B is S-stronger than A if B sums every sequence which belongs to S and is summed by A. If each of A, B is S-stronger than the other, A and B will be called S-equivalent. If B is S-stronger than A, but A is not S-stronger than B, we say that B is strictly S-stronger than A.

An inverse AI for an arbitrary matrix A was first given by Moore (4). Since the application to solution of linear equations only depended on the property A AI A = A, Bjerhammar (2) used this equation to define the set of generalized inverses AI. If A is regular, then only the regular inverse A−1 satisfies this definition. If A is a generalized inverse of AI, so that AI = AI AAI, then AI is a reciprocal inverse.

In this paper, we give an extension (Theorem 1) of the following well-known result of Banach ((l)): If X, Y are sets and Θ: X → Y, ψ: Y → X are injective mappings, then there exist partitions X = X1 ∪ X2, Y = Y1 ∪ Y2 such that Θ(X1) = Y1 and ψ(Y2) = X2. Here, as is usual, we say that X = X1 ∪ X2 is a partition of the set X = X1 ∩ X2 = π. Our theorem is applied in sections 2 and 3 to problems concerned with the existence of common representatives for two families of sets.

If α is a positive real number then, for each set E in R3, we define

where U(ρ, E) is any countable collection of convex sets, each of diameter at most ρ, whose union covers E, and Aα is a positive real number. Then the convex α-dimensional measure Λα(E), of E, is . In this paper we shall only be considering the cases α = 1,2 where, as is usual, we take A1 = 1 and A2 = ¼π. The symbols ,Λs(E) will denote the spherical (circular) measure of E, i.e. when the coverings are restricted to being collections of spheres (circles), and E(z) is the intersection of a set E (in R3) with the plane (x, y) at z.

Small letters denote non-negative integers and capital letters denote sets. The set theoretic difference of A and B is A − B. The empty set is ø and |A| denotes the cardinal of A. We say that A is an immediate extension of B if B ⊂ A and |A − B| = 1. If C is a set of sets we put C(n) = {X: X ∈ C, |X| = n}. A set of sets C, has the property P(k, n) if (i) ø ∈ C and (ii) if X ∈ C(i) and i < n, then C(i + 1) contains at least k immediate extensions of X. We assume throughout that k ≥ 1.

The notion of the inverse of a matrix with entries from the real or complex fields was generalized by Moore (6, 7) in 1920 to include all rectangular (finite dimensional) matrices. In 1951, Bjerhammar (2, 3) rediscovered the generalized inverse for rectangular matrices of maximal rank. In 1955, Penrose (8, 9) independently rediscovered the generalized inverse for arbitrary real or complex rectangular matrices. Recently, Arghiriade (1) has given a set of necessary and sufficient conditions that a matrix commute with its generalized inverse. These conditions involve the existence of certain submatrices and can be expressed using the notion of EPr matrices introduced in 1950 by Schwerdtfeger (10). The main purpose of this paper is to prove the following theorem:

Theorem 2. A necessary and sufficient condition that the generalized inverse of the matrix A (denoted by A+) commute with A is that A+ can be expressed as a polynomial in A with scalar coefficients.

Let p be a prime and let f(x) be a quartic polynomial with integral coefficients. I consider the problem of estimating the least non-negative non-residue k of f(x) (modp) (I omit the mod p hereafter), for large primes p, so f(x) ≡ r has a solution for

but not for r = k. The same problem for cubics has been considered by Mordell ((1)), who showed that

as p → ∞, where the constant implied in the O-symbol is independent of the coefficients of the cubic. In fact a more detailed examination of Mordell's proof gives the better estimate

It is the purpose of this paper to show that this same estimate also holds for quartic polynomials.

Introduction. Let sn denote the infimum of the set of real numbers X, where x belongs to X if, and only if, x is the Besicovitch dimension of the residual set of a packing of n-spheres into the unit n-cube In, of Rn. In recent work ((1)) I have shown that s2 is greater than one, and, quite naturally, I have since been asked whether or not the proof can be generalized to prove the analogous result, sn greater than n − 1, in Rn. Whilst it is clear, in theory, that this could be done, in practice the details might become rather complicated. However, such a generalization is unnecessary, for the result sn > n − 1 is a trivial consequence of combining the result s2 > 1 with the following theorem.