In classical dynamics one has usually supposed that when one has solved the equations of motion one has done everything worth doing. However, with the further insight into general dynamical theory which has been provided by the discovery of quantum mechanics, one is lead to believe that this is not the case. It seems that there is some further work to be done, namely to group the solutions into families (each family corresponding to one principal function satisfying the Hamilton-Jacobi equation). The family does not have any importance from the point of view of Newtonian mechanics; but it is a family which corresponds to one state of motion in the quantum theory, so presumably the family has some deep significance in nature, not yet properly understood.

Cette note a pour principal objet de répondre à deux questions posées par G. Birkhoff.

Théorème 1. Les lattices distributives avec O et I sont exactement les systèmes algébriques avec une opération ternaire (a, b, c) et des éléments O et I tels que:

(1.1) (O, a, I) = a,

(1.2) (a, b, a) = a,

(1.3) (d, (a, b, c), e) = (b, (c, d, e), (a, d, e)).

Many sets of postulates have been given for distributive lattices and for Boolean algebra. For a description of some of the most interesting and for references to others the reader is referred to Birkhoff's “Lattice Theory”[1]. In this paper we give sets of postulates which have some intrinsic interest because of their simplicity. In the first two sections binary operations are used to describe a distributive lattice by 2 identities in 3 variables and a Boolean algebra by 3 identities in 3 variables. In the third section a ternary operation is used to describe distributive lattices with 0 and J by 2 identities in 5 variables.

Special interest has recently attached to formulations of Boolean algebra in terms of ring operations [7], [1]. These axiomatizations have not been as brief as those reached through other modes of approach.

The present note will show that the number of axioms when ring operations are used may be as small as in any present version that is not metamathematical, that is, the number of axioms finally employed will be two.

A semigroup is a set of elements which is closed under an associative operation, usually called multiplication. When can a semigroup be embedded in a group, i.e., under what condition is it isomorphic to a subset of a group? A necessary condition for immersibility is clearly the so-called cancellation law:

Let K be an algebraic field of degree n over the rational field, and let b be the ground ideal (différente) of the field. Let

The Farey sequence of order h — 1 consists of the reduced rational fractions from 0 to 1 inclusive, with denominators less than h, and arranged in order of magnitude. Thus, if h = 6, the sequence is

(1) 0/1,1/5, 1/4,1/3, 2/5, 1/2, 3/5, 2/3, 3/4, 4/5, 1/1.

In some parts of the Geometry of Numbers it is convenient to know that certain affine invariants associated with convex regions attain their upper and lower bounds. A classical example is the quotient of the critical determinant by the content (if the region is symmetrical) for which Minkowski determined the exact lower bound 2–n. The object of this paper is to prove that for continuous functions of bounded regions the bounds are attained. The result is, of course, deduced from the selection theorem of Blaschke, and itself is a compactness theorem about the space of affine equivalence-classes.

1. Krall and Frink [2] have recently considered in connection with certain solutions of the wave equation a system of polynomials yn(x), {n = 0, 1, 2, …), where yn is defined as that polynomial solution of the differential equation

Recent studies of turbulent fluid motions have drawn attention to the transfer of vorticity. There have been some attempts to study turbulence by plane models, but these have been criticized justly for failing to reveal the true nature of a phenomenon which depends essentially on threedimensional convection. I have thought it worthwhile to eschew current conjectures regarding turbulence, turning in preference to the method of a master of the theory of vortices and applying it to the discovery of certain average properties of the continuous rotational motion of any medium whatever. This memoir therefore makes no attempt to deal with the problem of turbulence, but it is possible that the theorems presented here, being exact consequences of the kinematical equations, may nevertheless enjoy a certain relevance.

There have been several brief studies made [3, 4, 7, 8, 9, 11] of systems in which a “distance function” is defined on the set of pairs of elements of some abstract set to another abstract set. Frequently both of the sets involved are given algebraic structures. One of the more novel of these systems is the naturally metrized group [3, 7] originated by Karl Menger in 1931. This system is analogous to the Euclidean line in that it assigns to each pair, a, b of elements of an additively written Abelian group the “absolute value”, (a-b, b-a) = (b-a, a-b), of the "difference" of the elements as ”distance“.

A finite net N of degree k, order n, is a geometrical object of which the precise definition will be given in §1. The geometrical language of the paper proves convenient, but other terminologies are perhaps more familiar. A finite affine (or Euclidean) plane with n points on each line is simply a net of degree n+ 1, order n (Marshall Hall [1]). A loop of order n is essentially a net of degree 3, order n (Baer [1], Bates [1]). More generally, for , a set of k —2 mutually orthogonal n ⨯ n latin squares may be used to define a net of degree k, order n (and conversely) by paralleling Bose's correspondence (Bose [1]) between affine planes and complete sets of orthogonal latin squares.

Let be a compact complex analytic variety of the complex dimension n with a positive definite Kâhlerian metric [4] ; the local analytic coordinates on will be denoted by z = (z1z2, … , zn). Now, suppose a meromorphic function f(z) defined on as given. Then the poles and zero-points of f(z) constitute an analytic surface in consisting of a finite number of irreducible closed analytic surfaces Γ1, Γ2, … , Γk, each of which is a polar or a zero-point variety of f(z).

Soient E un espace compact (par exemple l' intervalle et μ une mesure de Radon sur E. Une fonction numérique f intégrable pour μ définit une mesure sur l'ensemble des parties mesurables (pour μ) de E. On peut aussi considérer la forme linéaire qu'elle définit sur l'espace C des fonctions numériques continues dans E, et cette forme linéaire est continue pour la topologie définie par la norme . Si on appelle encore “mesure” une forme linéaire continue sur C, le théoréme de Lebesgue-Nikodym classique caractérise celles de ces “mesures” forme qui sont de la . par une condition de “continuité absolue” par rapport à μ.

Fourier transforms of distribution functions are frequently studied in the theory of probability. In this connection they are called characteristic functions of probability distributions. It is often of interest to decide whether a given function φ(t) can be the characteristic function of a probability distribution, that is, whether it admits the representation

The writer [1] has previously examined the fundamental concepts of distance geometry in a Boolean algebra, B, with distance defined by . Any technical terms from distance geometry which are not defined in this paper may be found in [1]. A Boolean algebra bearing the given distance function is called an autometrized Boolean algebra. It is clear that the set of motions B (congruences of B with itself) form a group under substitution. This group we denote by M(B).

Enriques, in his posthumous magnum opus (1), devotes a chapter to the canonical or (where the genus is small) bi- or tricanonical models of regular surfaces, for various values of the genus and of the linear genus p(1). If, however, the cases he deals with are tabulated as follows (* marking the surfaces described by Enriques):

In two earlier papers * the writer discussed involutions of periods five and seven on certain cubic surfaces in S3. In this paper, a quartic surface containing a cyclic involution of period eleven is considered.

The surface

Let Cn be a curve in real projective n-space which is a continuous 1— 1 image of either the projective line or one of its closed segments. Consequently its points depend continuously on a real variable s for which , with the understanding that s = 0 and s = 1 represent the same curve point in the case that Cn is the image of the complete projective line. The points of Cn will be described by their corresponding real numbers s.