The examples are unconnected.

According to some published footnotes, examples are known of Serre fibrations which are not Hurewicz fibrations. However, as far as I know no such example has been published. The following example, which was stimulated by a more complicated example of G. Horrocks, seems worth mentioning, since it is extremely simple and does suggest some further questions.

In an earlier paper (3), a non-abelian cohomology theory (in the dimensions 0 and 1) for associative algebras was developed. One of the objectives was to obtain equivalence classes of crossed homomorphisms by considering inner automorphisms of the coefficient-algebra. This paper is an adaptation of the methods employed there to the case of Lie algebras. Throughout, all our Lie algebras will be over the field of real numbers, and finite-dimensional.

In this paper we discuss the structure of a partially ordered set X by studying in detail certain subsets of X.

Improved sufficient conditions are developed for the functional completeness of a set F of functions with variables and values ranging over N = {1, 2, …, n}, where n ≥ 3. In particular, F is complete if it generates a triply transitive group of permutations of N and contains either (i) only a single function, or (ii) at least one function satisfying the Slupecki conditions, the latter apart from certain exceptional cases. These are shown by a detailed investigation to occur only when n = 3 or when n is a power of 2 and all functions of F are linear in each variable, relative to some representation of N as a vector space over Z2.

In (3) W. Gaschütz introduced the concept of a formation of finite soluble groups, generalizing the results of R. W. Carter on the existence and conjugacy of nilpotent self-normalizing subgroups in finite soluble groups (2). Carter's results have already been extended to two classes of infinite groups in (9), (10), and the object of the present work is to extend the results of Gaschütz to a class of infinite groups, thereby generalizing (9).

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.

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

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.

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.

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.

A gap in the proof of the main theorem of (1) has been pointed out by C. D. Papakyriakopoulos and R. H. Fox. That proof was incomplete in that the table of homeomorphisms on page 776 of (1) did not cover all the possible cases (of which there are an infinite number). The corrected proof, given below, is intended to be a re-written version of the first half of section 3 of (1), the notation and references being those of (1).

We refer to a beautiful and important result of Tutte(1), in the theory of graphs; that a linear graph G is prime if and only if it contains a set ∑ of vertices, such that u(G∑) > n(∑); where n(∑) is the number of vertices in ∑, G∑ is the graph obtained from G by deleting the star of ∑ (all the vertices of G in ∑, together with all the edges of G meeting vertices of ∑), and u(G∑) is the number of connected components of G∑ having an odd number of vertices.

I asserted in (1), p. 35, that the topology of the hyperspace H(μX)of a uniform space μX determines the uniformity μ, and sketched a proof. D. Hammond Smith has shown (2) that the proof indicated is wrong, even for metric spaces. However, Smith proved the assertion for metric spaces and some others, and left open the question whether it was true in general. This note gives a counter example. I should note that I began with a transfinite construction based on Smith's example; whether that would have worked or not, the following construction is simpler.

Rohlin (7) has shown that Kolmogorov automorphisms are mixings of all degrees. Later Sucheston(9) introduced regular automorphisms and after showing that they are mixings of all degrees asks if the converse of either of the above theorems holds. Since the equivalence of Kolmogorov and regular automorphisms follows from (1) and (8) we have only one problem. We show that the converse of either of the above theorems is false by constructing a stationary Gaussian process which is a mixing of all degrees but which is not a Kolmogorov automorphism.

An important and much-investigated class of measures is the class of Hausdorff measures, first defined by Hausdorff (1). These measures form a subclass of the class of translation invariant measures, but just how wide a class they form is not known.

Suppose that a sequence of discs

arranged in decreasing order of diameters, forms a packing within the unit plane square I2. It has been shown, by Florian(1), that the area of

is at least O(a), where a is the radius of θn. However, Gilbert (2) has produced some empirical results for the Apollonius packing 71 of discs which seem to suggest that for such a packing, the area of the set

is at least O(as) for some positive real number s, less than one. As Gilbert remarks, it is difficult to imagine that the Apollonius packing is not the extremal case, and so, that it would seem likely that there exists a positive real number s, less than one, such that for a general packing, the area of

is at least O(as). The purpose of this paper is to establish this result by showing that 0·97 is an allowable value for s.

Corresponding to a fixed sequence {μn}, the Hausdorff method of summability (H, μn) is defined by the sequence-to-sequence transformation†

where we write

The quasi-Hausdorff method (H*, μn) is defined by the transformation

thus the matrix of the (H*, μn) transformation is the transpose of that of the (H*, μn) transformation. A method introduced by Ramanujan (9), which we will call‡ (S,μn) is given by the transformation

Thus the elements of row n of the matrix of the (S, μn) transformation are those of the corresponding row of the (H*, μn) transformation moved n places to the left.

Throughout this paper we shall denote the space Lp(− ∞, ∞) by Lp, and the numbers p and p′ will be related by 1/p + 1/p′ = 1.

The Poisson operator Pa is defined by

Let f = f(x1, …, xn) be a positive definite quadratic form in n ( ≤ 8) variables; then we consider the old problem of estimating the minimum of f (its least value for integers xi not all 0) in terms of the determinant Δ(f). Normalizing by supposing the minimum to be 1, the known results may be stated as

each inequality being best possible. Further, for each n, all forms for which equality holds in (1) have integral coefficients and are equivalent to each other.

A well-known result in the interpolation theory of integral functions (see Whittaker (16, 17), Pólya (14), Iyer (2), Pfluger (10)) states that an integral function of at most type K < ½π of order 2 bounded at the lattice points m + in (m, n = 0, ± 1, ± 2, … ) is necessarily constant. That the value ½π cannot be increased is shown by the Weierstrass σ-function. The result has, however, been generalized in several ways; the lattice points being replaced by more general sets and the bounded-ness condition by one of restricted rate of growth.