In a previous paper (2), the inner product of two S-functions {λ}.{μ} was defined for (λ),(μ) partitions of the same integer n. Briefly, the ordinary product {λ}.{μ} of two S-functions corresponds to the analysis of the direct product of two corresponding representations of the full linear group, while the inner product {λ}.{μ} corresponds to the analysis of the direct product of two representations of the corresponding symmetric group.

Murnaghan (9) has proposed the following method of analyzing the Kronecker product of two symmetric group representations. If (λ) = (λ1, λ2, … , λi) is a partition of p, the representation of the symmetric group on n symbols corresponding to the partition (n — p, λ1 , … , λi) is denoted by [λ] and is said to be of depth p.

If [λ] is of depth p and [μ] of depth q, then the terms in the Kronecker product [λ] X [μ] of depth p + q are terms which correspond to the terms in the product of S-functions {λ} {μ}).

In 1942 Thrall (3) introduced what he called the Lie character to study the structure of the free Lie ring. These characters are defined by associative representations of the full linear group GLq(C). Their importance is seen in the fact that, for each n, the splitting of the submodule of Lie forms of degree n into irreducible invariant subspaces is completely specified by the Lie character ζn.

For A an associative algebra with identity over a field K, [A : K] < ∞, and d an integer, we define gΛ(d) to be the number of inequivalent indecomposable Λ-modules of degree d over K. Following (6), we define Λ to be of finite representation type if . Λis said to be of bounded representation type if there exists d0 such that gΛ(d) = 0 for d ⩾ d0; Λ is of unbounded representation type if not of bounded type.

In this paper we study generalized homomorphisms between two algebras, namely the binary relations whose graphs are subalgebras of the direct product of the given algebras. In 1897 Goursat proved that every subgroup of the direct product of two groups is determined by an isomorphism between factor groups of subgroups of the given groups (10, §§11, 12; 25, pp. 15, 16). A like result is here shown for a general class of algebras, including loops and quasigroups, by a method due to Riguet (22). This result is used to obtain general forms of the Zassenhaus lemma and the Jordan- Hälder-Schreier theorem for normal series (26, §9).

This paper continues a study appearing in (5) of the combinatorial properties of a matrix A of m rows and n columns, all of whose entries are 0's and 1's. Let the sum of row i of A be denoted by ri and let the sum of column i of A be noted by st. We call R = (r1, … , rm) the row sum vector and S = (s1, … , sn) the column sum vector of A. The vectors R and S determine a class consisting of all (0, 1)-matrices of m rows and n columns, with row sum vector R and column sum vector S. Simple arithmetic properties of R and S are necessary and sufficient for the existence of a class (1 ; 5).

Let R = (rij) be an m × n matrix and let S = (sik) be dip × q matrix denned over a field F. The Kronecker product R × S of R and S is denned as follows:

Definition 1.1. The Kronecker product R × S of the matrices R and S is given by where rij S; i = 1, 2, … , m; j = 1, 2, … , n, is itself a p × q matrix (1, 69-70).

A difference set (,D) is defined in (2) as a subset D of k elements in a group of order υ with the following properties :

(1) if x ∊ , x ≠ 1, there are exactly λ distinct ordered pairs (d1d2) of elements of D such that x = 1d2;

(2) if x ∊ , x ≠ 1, there are exactly λ distinct ordered pairs (3d4) of elements of D such that x = 3d4−1.

The theory developed for the study of flows in networks (2; 3; 4; 5; 6; 7) sometimes provides a useful tool for dealing with certain kinds of combinatorial problems, as has been previously indicated in (3; 4; 6; 7). In particular, Hall-type theorems for the existence of systems of distinct representatives which contain a prescribed set of marginal elements (10; 11), or, more generally, whose intersection with each member of a given partition of the fundamental set has a cardinality between prescribed lower and upper bounds (9), can be obtained in this way (7).

We are given two identical symmetrical bodies (e.g., cubes) with independent random orientations; then we can always, in several ways, turn one of these bodies about some axis through its centre of gravity, so as to bring it into the same orientation as the other body. The smallest angle of rotation needed will be called the disorientation, d, of the two bodies, and we shall be concerned with the distribution of d under these conditions.

In a previous paper (6) we have studied those minimal surfaces which admit geodesic mappings without isometries or similarities on another, not necessarily minimal, surface. Here we determine all pairs of minimal surfaces which can be geodesically mapped on each other. We find that two such surfaces are either:

(i) similar Bonnet associates of each other, or

(ii) both Poisson surfaces (that is, isometric to a plane), or

(iii) both Scherk surfaces (2).

In a previous paper (2) we have developed a spectral theory and a unitary equivalence theory for a certain class of non-normal operators. We dealt primarily with operators which were called J2 operators. At present we are interested in studying the uniformly closed rings generated by such operators.

The notation will be the same as in (2).

Recently C. A.Rogers (2, Theorem 4) proved the following theorem which applies to many problems in geometry of numbers:

Let f(X1,X2, … , Xk) be a non-negative B or el-measurable function in the nk-dimensional space of points (X1,X2. Further, let Λo be the fundamental lattice, Ω a linear transformation of determinant 1, F a fundamental region in the space of linear transformations of determinant 1, defined with respect to the subgroup of unimodular transformations and μ(Ω) the invariant measure1 on the space of linear transformations of determinant 1 in Rn.

Let {s(n)} be a real sequence and let x be any number in the interval 0 < x ⩽ 1. Representing x by a non-terminating binary decimal expansion we shall denote by {s(n,x)} the subsequence of {s(n)} obtained by omitting s(k) if and only if there is a 0 in the decimal place in the expansion of x. With this correspondence it is then possible to speak of “a set of subsequences of the first category,” “an everywhere dense set of subsequences,” and so on.

The main problem of the present paper is the existence of the Burkill integral of an interval function ƒ(I) which is not supposed to be continuous. Little is known about this case, though otherwise the theory of the integral can be considered as complete: we may refer to Ringenberg's comprehensive paper (2) in which further references are given.

Let Sumner (4) has discussed for arbitrary real λ and h, whereƒ(Z) is an entire function of exponential type I shall show that in this case an alternative definition of ∇λλ, which leads to Sumner's results more quickly, is equivalent to Sumner's. (However, Sumner's definition is, in principle, applicable to a wider class of functions.)

A mixed problem in the theory of partial differential equations is an auxiliary data problem wherein conditions are assigned on two distinct surfaces having an intersection of lower dimension. Such problems have usually been formulated in connection with hyperbolic differential equations, with initial and boundary conditions prescribed. In this paper a study is made of the conditions appropriate to a system of R linear partial differential equations of first order, in R dependent and N independent variables.

For power series

(1.1) for which

(1.2),

it has been known for four decades (1) that ƒ(z) is regular and univalent or schlicht in |z| < 1. This theorem, due to J. W. Alexander, has more recently been studied by Remak (5) who has shown that w = ƒ(z), under the hypothesis (1.2), maps |z| < 1 onto a star-like region, and if (1.2) is not satisfied=(z) need not be univalent in |z| < 1 for a proper choice of the amplitudes of the coefficients an.

Various sets of necessary and sufficient conditions are known in order that a function ƒ(s), analytic for Re s > 0, be represented as the Laplace transform of a function in Lp(0,∞), 1 < p ⩽ ∞ . Most of these theories are based on the properties of some inversion operator for the transformation— see, for example, (7, chap. 7). However in the case p = 2 a number of representation theorems of a much simpler type are available.

The theory of solutions of partial differential equations

(1.1) with analytic coefficients can be based upon the theory of analytic functions of a complex variable; the basic tool in this approach is integral operators which map the set of solutions of (1.1) onto the algebra of analytic functions. For certain classes of operators this mapping which is first defined in the small, can be continued to the large, cf. Bergman (3).