The theory of knots undertakes the task of giving a complete survey of all existing knots. A solid mathematical foundation was not laid to this theory until our century. A mathematician of the rank of Felix Klein thought it to be nearly hopeless to treat knot problems with the same exactness as we are accustomed to from classical mathematics. We want to give here a short summary of the modern topological methods enabling us to approach the knot problem in a mathematical way.

In order to exclude pathological knots, as for instance knots being entangled an infinite number of times, we will define a knot as a polygon lying in the space. In other words: a knot is a closed sequence of segments without double points. In Figure 1 some examples of knots are given in plane projection.

In this paper we shall consider integral ideals in finite algebraic extensions(, …) of the field of rational numbers.

Two ideals in the same field are said to be equal if and only if they contain the same numbers.

1. Historical remarks. In this paper we extend some well-known extremum properties of the regular polygons to the regular polyhedra. We start by mentioning some known results in this direction.

First, let us briefly consider the problem which has received the greatest attention among all the extremum problems for polyhedra. It is the determination of the polyhedron of greatest volume F of a class of polyhedra of equal surface areas F, i.e., the isepiphan problem.

This paper deals with a particular class of distributive properties which appear to be important for Analysis and which I call frequencies. They can be defined for any kind of sets but it essential for proper application that a condition L (the statement of Lindelöf's lemma) is satisfied. From this condition follows Theorem 1, which is characteristic for frequencies but does not hold for other distributive properties. For every frequency F of a space Σ, one can build up an Analysis mod F of Σ the classical case is the Analysis mod F0. It is convenient to introduce the words “nearly every” with such a meaning that “every” and “almost every” are the special cases which, when we use the notation of this paper, correspond to F = F0 and F = FC.

1. Introduction. A graph G consists of a set V(G) of objects called nodes and a set M (G) of objects called links, V(G) and M (G) having no members in common. With each link A there is associated just two nodes said to be the ends of A, or to be incident with A, or to be joined by A. The sets V(G) and M(G) may be finite or infinite. There may be nodes with which no link is incident. Such nodes are said to be isolated.

Certains théorèmes démontrent pour l'anneau des polynomes A [x1, …, xn] une propriété supposée valable dans l'anneau A, par exemple le théorème de la base finie de Hilbert pour tout idéal d'un anneau commutatif qui possède un élément unité. (Voir P. Dubreil [2] p. 210, ou B. L. Van der Waerden [7] §80, ou le mémoire original de Hilbert [3], p. 473.) E. Lasker [5] a montré que tout idéal de l'lanneau K[x1, x2 … , xn] des polynomes à n variables à coefficients dans un corps K commutatif est l'intersection d'un nombre fini d'idéaux primaires.

In this paper we shall give an elementary proof of the theorem

(1.1)

where φ(k) denotes Euler's function, and

(1.2)

where p denotes the prime, and and are integers with (,) = 1, positive.

Introduction. The irreducible representations of the symmetric group Sn, were shown by A. Young to be in one-to-one correspondence with certain arrays of n nodes. E.g. for n = 12 and the partition λ = [4, 4, 3, 1] we have the array which we call a “Young diagram.” The question arises as to the manner in which various properties of the representations are reflected in their corresponding Young diagrams.

Let it be required to arrange v elements into v sets such that every set contains exactly k distinct elements and such that every pair of sets has exactly elements in common . This combinatorial problem is studied in conjunction with several similar problems, and these problems are proved impossible for an infinitude of v and k. An incidence matrix is associated with each of the combinatorial problems, and the problems are then studied almost entirely in terms of their incidence matrices. The techniques used are similar to those developed by Bruck and Ryser for finite projective planes [3]. The results obtained are of significance in the study of Hadamard matrices [6;8], finite projective planes [9], symmetrical balanced incomplete block designs [2; 5], and difference sets [7].

1. Introduction. The notion spherical geometry is suggested by the familiar geometry of the Euclidean 2-sphere in which the role of path is played by “arc of great circle”. The first postulational treatment of the subject seems to be that of Halsted [10] for the two-dimensional case. Kline [11] under the name double elliptic geometry, gave a greatly simplified foundation for the three-dimensional case based on the primitive notions point and order. Halsted and Kline study not merely descriptive (that is positional, non-metrical) properties of figures but also introduce metrical notions by postulating or defining congruence. Kline includes a continuity postulate designed to yield real spherical geometry.

1. General remarks. The heuristic strength of the general principle of relativity lies in the fact that it considerably reduces the number of imaginable sets of field equations; the field equations must be covariant with respect to all continuous transformations of the four coordinates. But the problem becomes mathematically well-defined only if we have postulated the dependent variables which are to occur in the equations, and their transformation properties (field-structure). But even if we have chosen the field-structure (in such a way that there exist sufficiently strong relativistic field-equations), the principle of relativity does not determine the field-equations uniquely. The principle of “logical simplicity” must be added (which, however, cannot be formulated in a non-arbitrary way). Only then do we have a definite theory whose physical validity can be tested a posteriori.

1. Introduction. The equations of dynamics were put into a general form by Lagrange, who expressed them in terms of a set of generalized coordinates and velocities. An alternative general form was later given by Hamilton, in terms of coordinates and momenta. Let us consider the relative merits of the two forms.

Introduction. In a paper called “A Theorem in Finite Projective Geometry and some Applications to Number Theory” [Trans. Amer. Math. Soc, vol. 43 (1938), 377-385], J. Singer proved that the finite projective geometry PG(s — 1,pn), that is the projective geometry of dimension s — 1 whose coordinate field is the Galois field GF(pn), admits a collineation L of period q = (psn — 1)/ (pn — 1). Since this q is the number of points of PG(s — 1, pn), Singer's result states that the points of PG(s — 1, pn) are cyclically arranged. Singer's construction of L uses the notion of a “primitive irreducible polynomial of degree 5 belonging to a field GF(pn) which defines a PG(s — 1, pn).”

In the present paper, the authors extend the Cousin theorems and the continuity theorem, using some previous results on analytic functions connected with open Riemann surfaces.

The Cousin theorems, concerning the existence of analytic functions of several complex variables with prescribed poles and zeros in a given domain, have been generalized in various manners, but only in the case where the domain is schlicht. The authors proceed to the case where the given domain is the direct product of n open Riemann surfaces. They prove the following two theorems.

If there is a nonsingular symmetric bilinear form defined on a distributive algebra A with basis elements C1, C2, … , Cr over a field F such that and , then the so called Casimir operator [1,2]

Let P(n) denote the number of unrestricted partitions of the positive integer n. Ramanujan conjectured that

(1.1)

if . He also indicated that such congruences could be deduced from identities of the type

1. Introduction. A dissection of a rectangle R into a finite number n of non-overlapping squares is called a squaring of R of order n. The n squares are called the elements of the dissection. If there is more than one element and the elements are all unequal the squaring is called perfect and R is a perfect rectangle. (We use R to denote both a rectangle and a particular squaring of it). If a squared (perfect) rectangle is a square we call it a squared (perfect) square.

1. Introduction. It is the object of this paper to describe in more detail than has hitherto been done the general methods by which a square may be dissected into smaller unequal non-overlapping squares. Examples of such dissections are given

This is the third in a series of problems in the study of surface waves which have been disturbed by the presence of a plane barrier and to which a solution may be provided. We assume as in part I, that the fluid is incompressible and non-viscous, and that motion is irrotational.

In some work of Ramanujan certain results are given which are equivalent to the following.

If

(1)