The gradual disengagement of mathematics from logic. Beginning with a historical review of the development of mathematical thought, we have to consider successively (cf. 10, pp. 139-140):

(1) The observational period. For some familiar regularities of (outer or inner) experience of time and space, which, to any attainable degree of approximation, seemed invariable, absolute and sure invariability was postulated. These regularities were called axioms and were put into language. Thereupon extensive systems of properties were developed from the linguistic substratum of the axioms by means of reasoning, guided by experience but linguistically following and using the principles of classical logic. This logic was considered autonomous, and mathematics was considered more or less dependent on logic.

The gamma function Γ(z + 1) = П(z) has been defined in different ways:

(1) (Weierstrass)

(2) (Kuler)

(3) (Gauss)

(4) (Euler)

(5) (Lerch)

Let p be a fixed prime > 3. The first factor of the field R(ζ), where R is the rational field and , is determined by means of

1.1 ,

where

r is a primitive root (mod p), and ri is the least positive residue of ri (mod p).

We shall consider methods of summation A, B, … defined by matrices of real elements (amn), (bmn), (m, n = 1, 2, …) which are regular, that is, have the three well-known properties of Toeplitz (4, p. 43). A method A is said to be core-consistent with the methodBfor bounded sequences if the A-core (3, p. 137; and 4, p. 55) of each real bounded sequence is contained in its B-core. B is totally included in A, B ≪A, if each real sequence which is B-summable to a definite limit (this limit may be finite or infinite of a definite sign) is also A-summable to the same limit.

From a symmetric balanced incomplete block design we may construct a derived design by deleting a block and its varieties. But a design with the parameters of a derived design may not be embeddable in a symmetric design. Bhattacharya (1) has such an example with λ = 3 . When λ = 1, the derived design is a finite Euclidean plane and this can always be embedded in a corresponding symmetric design which will be a finite projective plane.

1. Introduction. Under the non-parametric assumption that a set of observations is a sample from an absolutely continuous distribution, the order statistics are known to form a complete sufficient statistic. It is proved in this note that it suffices to have the class of uniform distributions over finite numbers of intervals or the class of uniform distributions over sets of a ring which is a basis for the σ-algebra of Borel sets. This result is derived as a particular caseof that of several samples from more general distributions.

In §2 a result in measure theory is obtained. The remainder of this paper, §3 to §11, contains results in the branch of statistics called non-parametric theory; these results in part are based on the measure result of §2.

The measure result concerns a class of probability distributions—those distributions having a probability density function on the real line and for which a fraction p of the probability is on the negative axis and a fraction q = 1 - p is on the positive axis. Corresponding to a sample of n the functional form is obtained for a statistic having expectation zero for all distributions in the class; such a statistic is referred to as an unbiased estimate of zero.

We consider the problem of finding a unique canonical form for complex matrices under unitary transformation, the analogue of the Jordan form (1, p. 305, §3), and of determining the transforming unitary matrix (1, p. 298, 1. 2). The term “canonical form” appears in the literature with different meanings. It might mean merely a general pattern as a triangular form (the Jacobi canonical form (8, p. 64)). Again it might mean a certain matrix which can be obtained from a given matrix only by following a specific set of instructions (1). More generally, and this is the sense in which we take it, it might mean a form that can actually be described, which is independent of the method used to obtain it, and with the property that any two matrices in this form which are unitarily equivalent are identical.

The coefficients of the second derivatives in an elliptic or hyperbolic differential equation of the second order determine a Riemannian metric on the space of the independent variables. A Riemannian space has been called harmonic if the Laplace equation corresponding to it has a solution which is a function only of geodesic distance in that space. Harmonic spaces have been studied in some detail (1; 3). In this note are examined the circumstances under which a non-parabolic second order linear equation may have a “geodesic” solution of the type described. It will be shown that the equation must be self-adjoint, that the Riemann space corresponding to the equation must be harmonic and that the coefficient of the dependent variable must be a constant. Conversely, if these conditions are satisfied, the equation has two geodesic solutions, one of which is an elementary solution. In the case of elliptic equations, the second solution is connected with a certain mean value property which is valid in harmonic spaces.

Two polynomials θ(G, n) and ϕ(G, n) connected with the colourings of a graph G or of associated maps are discussed. A result believed to be new is proved for the lesser-known polynomial ϕ(G, n). Attention is called to some unsolved problems concerning ϕ(G, n) which are natural generalizations of the Four Colour Problem from planar graphs to general graphs. A polynomial χ(G, x, y) in two variables x and y, which can be regarded as generalizing both θ(G, n) and ϕ(G, n) is studied. For a connected graph χ(G, x, y) is defined in terms of the “spanning” trees of G (which include every vertex) and in terms of a fixed enumeration of the edges.

In earlier papers (4; 5; 6), certain space curves, invariant under cyclic involutions of periods three, five, and seven, have been mapped. Lucien Godeaux (2; 3) in 1916 mapped plane cubic curves, invariant under an involution of period three, onto a cubic surface in ordinary three-space. Mlle. J. Dessart (1) in 1931 mapped plane quintic curves, invariant under an involution of order five, onto a fifth order surface in a space of four dimensions.

In the Euclidean plane a curve C has a one-parameter family of parallel involutes and a unique evolute C* which coincides with the locus of the centres of the osculating circles of C. If is parallel to C, C* is also the evolute of . We will study parallel curves in n-dimensional Euclidean space and obtain generalizations of the properties given above.

We will study parallel curves in n-dimensional Euclidean space and obtain generalizations of the properties given above.

α 1α 2, …, αp are n-dimensional vectors,

, ;

they are arranged to form a closed polygon

.

Denote by R(α 1, α 2, …, α p) the radius of the smallest circumscribed hypersphere with centre at 0 ; by R(α 1, α 2, …, α p) the minimum of

(α 1, α 2, …, α p)

for all possible reorderings

of α 2, …, α p−1 and by cn the least possible constant such that

for all possible choices of p and α 1α 2, … , α p.

If we define the weight b of a Young diagram containing n nodes to be the number of removable p-hooks where n = a + bp, then three fundamental theorems stand out in the modular representation theory of the symmetric group Sn .

This paper contains an account of a simple method of deriving the coordinates of the vertices of any uniform polytope or honeycomb (degenerate polytope) whose symmetry group is generated by reflections.

Let be a finite group of transformations of three-dimensional Euclidean space, such that the distance between any two points is preserved by all transformations of the group. Such a group is a group of orthogonal linear transformations of three variables, or, geometrically speaking, a group of rotations and rotatory inversions. Thirty-two groups of this type are important in crystallography and are known as the crystallographic classes.

Le théoréme qui suit résoud une question qui m'avait été posée indépendamment par M. H. Mirkil et Professeur E. Farah, dans le cas particulier de p = 2 :

Théorème 1. Soit M un espace localement compact muni d'une mesure bornée μ, et soit 1 ≤ p < + ∞. Soit H un sous-espace vectoriel de L∞(μ), fermé dans LP(μ). Alors H est de dimension finie.

The literature already contains several theories of limits which have great generality (10; 12; 1; 5; 6; 13; 3; 14; 4, p. 34). Nevertheless, the intrinsic importance and frequent use of the concept may justify the publication of another variant, provided that it has advantages in ease of application withoutsacrifice of generality. The theory of convergence studied in this note includes the other theories and their applications in a smooth way, without artifice.

Let L denote the formal ordinary differential operator

,

where we assume the p k are complex-valued functions with n-k continuous derivativeson an open real interval a < x < b (a = — ∞, b = + ∞, or both may occur), p 0(x) ≠ 0 on a < x < b, and L coincides with its Lagrange adjoint L + given by

.

In an earlier paper (1), published in this journal, a necessary condition was given which the reciprocal of a polynomial without multiple roots must satisfy in order to be a characteristic function. This condition is, however, valid for a wider class of functions since it can be shown (2, theorem 2 and corollary to theorem 3) that it holds for all analytic characteristic functions. The proof given in (1) is elementary and has some methodological interest since it avoids the use of theorems on singularities of Laplace transforms. Moreover the method used in (1) yields some additional necessary conditions which were not given in (1) and which do not seem to follow easily from the properties of analytic characteristic functions.