The purpose of this paper is twofold. In order to state our first aim, let U denote the ‘infinite’ unitary group, and let BU be a classifying space for U. Then Bott (2),(3) has shown that we propose to investigate the ‘Postnikov system’ of BU.

In (10), (11) Wall determined the structure of the cobordism ring introduced by Thom in (9). Among Wall's results is a certain exact sequence relating the oriented and unoriented cobordism groups. There is also another exact sequence, due to Rohlin(5), (6) and Dold(3) which is closely connected with that of Wall. These exact sequences are established by ad hoc methods. The purpose of this paper is to show that both these sequences are ‘cohomology-type’ exact sequences arising in the well-known way from mappings into a universal space. The appropriate ‘cohomology’ theory is constructed by taking as universal space the Thom complex MSO(n), for n large. This gives rise to (oriented) cobordism groups MSO*(X) of a space X.

Ω will denote the space of all plane paths ω, so that ω is a short way of denoting the curve . we assume that there is a probability measure μ defined on a Borel field of (measurable) subsets of Ω, so that the system (Ω, , μ) forms a mathematical model for Brownian paths in the plane. [For details of the definition of μ, see for example (9).] let L(a, b; μ) be the plane set of points z(t, ω) for a≤t≤b. Then with probability 1, L(a, b; μ) is a continuous curve in the plane. The object of the present note is to consider the measure of this point set L(a, b; ω).

The theorems of Bott (4), (5) on the stable homotopy of the classical groups imply that the sphere Sn is not parallelizable for n ≠ 1, 3, 7. This was shown independently by Kervaire(8) and Milnor(7), (9). Another proof can be found in (3), §26·11. The work of J. F. Adams (on the non-existence of elements of Hopf invariant one) implies more strongly that Sn with any (perhaps extraordinary) differentiable structure is not parallelizable if n ≠ 1, 3, 7. Thus there exist already four proofs for the non-parallelizability of the spheres, the first three mentioned relying on the Bott theory, as given in (4), (5). The purpose of this note is to show how the refined form of Bott's results given in (6) leads to a very simple proof of the non-parallelizability (only for the usual differentiable structures of the spheres). We shall prove in fact the following theorem due to Milnor (9) which implies the non-parallelizability.

A group G will be called measurable if there exists a real-valued, non-negative set function μ(A) defined on all subsets A ⊂ G which satisfies .

Fuchs((1), Problem 22) has asked the following question: Given positive integers ri (1 ≤ i ≤ 4) such thatr1 + r2 = r3 + r4, r1 ≠ r3, r1 ≠ r4, do there exist indecomposable torsion-free Abelian groups Gi (1 ≤ i ≤ 4) such that, where Gi is of rank ri (1 ≤ i ≤ 4)?* We shall show in this note that there do always exist groups Gi with the desired properties; in fact, we shall prove the following stronger result

The purpose of this paper is to investigate the structure of certain types of semigroups. Rees(6),(7) has determined the structure of a completely simple semigroup, and has shown that such a system may be realized as a type of matrix semigroup. Clifford (2) and Schwarz (8) have found conditions, namely, the existence of minimal left and minimal right ideals, under which a simple semigroup is completely simple, and have made a more detailed study of such semigroups. Preston (4), (5) has studied inverse semigroups, in which each non-zero element has a unique relative inverse, and has also considered inverse semigroups which contain minimal right or left ideals. In the present paper we obtain a set of conditions on a simple semigroup, each of which is equivalent to the semigroup being both completely simple and inverse. Section 2 defines the terms used and gives a brief resume of the main results which have already been proved. Section 3 is devoted to our present considerations.

It is worth remarking that if e, f are two primitive idempotents in a semigroup S which commute, then: (i) if S has a zero 0, either e = f or ef = 0; (ii) if S has no zero, e = f. Clearly the product ef is under each of the primitive idempotents e and f.

Littlewood has shown(1) that any rotation in 4 dimensions can be expressed as the resultant of a right-handed and a left-handed rotation, and that every right-handed rotation commutes with every left-handed rotation. It follows that the rotation group in 4 dimensions is expressible in some way as the product of two groups of right-handed and left-handed rotations, respectively.

Drazin (2) has recently introduced the concept of a pseudo-invertible element of an associative ring or semigroup. In this note we first show that such an element of a semigroup S may be characterized by the fact that some power of it lies in a subgroup of S. The main result (Theorem 2) provides a new criterion for complete simplicity: a simple semigroup is completely simple if and only if all its elements are pseudo-invertible. This is applied to the principal factors of a general semigroup and two previously known results are obtained as corollaries.

Let G be an extension of the normal subgroup K by its quotient group H. Suppose we are given free resolutions for H, K (see below for definition). We shall show how to construct from them, by a ‘twisted tensor product’, a free resolution for G, which may also be used to give the spectral sequence of the extension.

Let G be a monoid; that is to say, G is a set such that with each pair σ, τ of elements of G there is associated a further element of G called the ‘product’ of σ and τ and written as στ. In addition it is required that multiplication be associative and that G shall have a unit element. The so-called ‘Homology Theory’† associates with each left G-module A and each integer n (n ≥ 0) an additive Abelian group Hn (G, A), called the nth homology group of G with coefficients in A. It is natural to ask what can be said about G if all the homology groups of G after the pth vanish identically in A. In this paper we give a complete answer to this question in the case when G is an Abelian group. Before describing the main result, however, it will be convenient to define what we shall call the homology type of G. We write Hn(G, A) ≡ 0 if Hn(G, A) = 0 for all left G-modules A.

Let p be a prime and let Pi denote the Steenrod p-power if p ≠ 2 and the Steenrod square Sqi if p = 2. If p ≠ 2 let β denote the Bockstein co-boundary (defined so as to anticommute with suspension).

Notation. We consider an additive, right exact functor T(B, C) where B is a variable Λ-module, C a variable Γ-module and the values of T are in the category of Δ-modules. Here Λ, Γ, Δ are rings (which need not be commutative) each possessing an identity element.

We shall define the norm h(A) of a regular summability matrix A = (amn) by Two matrices are said to be b-equivalent if every bounded sequence summable by on matrix is summable by the other. If A sums all bounded sequences that are summable by B, A is said to be b-stronger than B. The norm of a method is defined as , where the inf is taken over all the matrices equivalent to for bounded sequences. These norms have been investigated by Brudno(1). One of his main results is that, if sums all bounded sequences that are summable, then In paper we shall prove the following.

Much is known about power series with integral exponents, and, as in the case of the Theta-functions, about series whose exponents are the squares of the natural numbers. About other types of power series little is known. I prove here that a certain power series whose exponents are the square roots of the natural numbers is a symmetrical Fourier kernel.

Recently Ragab ((3), p. 538) evaluated an integral involving the product of an associated Legendre function and an E-function. In this note we derive an integral which is more general than the one given by Ragab.

Formulae for Bessel derivatives with respect to order have generally been obtained by differentiating integral forms of the Bessel functions. In this note an asymptotic expansion will be derived from the differential equation satisfied by the functions, and it will be obtained in a form which terminates when the order is half an odd integer. Previous discussion of these functions has been restricted to the order one half. (See, for example, Ansell and Fisher (1) and Oberhettinger(2).)

Several writers have dealt with the use of Chebyshev polynomials as aids to interpolation. The present paper considers the theory with associated Legendre polynomials replacing those of Chebyshev. Interpolation formulae of both Bessel and Everett type are derived, and formulae and expansions are given for evaluating the coefficients required in them either directly or in terms of derivatives or central differences. The introduction summarizes a number of formulae for polynomial interpolation and indicates the relationship between the new formulae and the old.

In genetics it is of considerable interest to be able to calculate the probabilities that a mutant gene will either be lost to the population or that its allele will be lost, since, in general, if no further mutation occurs, one of these two events must ultimately take place in any finite population. A good deal of research has been done on this problem by Fisher (l) and Kimura (3) and the purpose of this paper is to extend their results to more general situations and to discuss the validity of their approximations.