Showing papers in "Annals of Mathematics in 1936"

The Logic of Quantum Mechanics

Abstract: One of the aspects of quantum theory which has attracted the most general attention, is the novelty of the logical notions which it presupposes It asserts that even a complete mathematical description of a physical system S does not in general enable one to predict with certainty the result of an experiment on S, and that in particular one can never predict with certainty both the position and the momentum of S, (Heisenberg’s Uncertainty Principle) It further asserts that most pairs of observations are incompatible, and cannot be made on S, simultaneously (Principle of Non-commutativity of Observations)

On Equivalent Sets of Elements in a Free Group

Abstract: together with the 'simple automorphism' which replaces a, by its inverse. Relative to this kind of equivalence we have very little to add to a paper by J. Nielsen,2 in which he gives a mechanical process for deciding whether or no two sets are equivalent and also a process for reducing (a) to one of a finite number of normal forms. When reduced in this way we shall describe a set of elements as reduced (N), and we recall that (a) is reduced (N) if it contains no two words of the form (AB)" and (AC)" respectively,3 where 1(A) > 1(B) or > I(C), and if the last half of every word with an even number of letters is an 'isolated ending.' That is to say, if a AB and 1(A) = 1(B) no other word in (a) ends with B or begins with B'. In ?2 it is assumed that any empty words which appear' during the process of reduction are discarded, as in J. N., while in ?4 they are retained. Theorem 1 below is essentially a restatement of various arguments used by Nielsen, while Theorem 2 adds a detail to J. N. The second kind of equivalence refers to the effect on (a) of automorphisms5 of G, two ordered sets of elements (a) and (I), both of which contain the same number of words, being equivalent if ax, corresponds to #X(X = 1, 2, *. * ) in some automorphism of G. That is to say they are equivalent if there is an automorphism

New Foundations of Projective and Affine Geometry

Abstract: Projective geometry is often called geometry of projection and section (Geometrie des Verbindens und Schneidens). In this paper foundations of projective geometry are given in terms of these two operations. We start from a single class of undefined entities, corresponding to the linear parts of a space, and two undefined operations denoted by + and ., corresponding to the join and the intersection, respectively, of these linear parts. Thus if A, B are two undefined entities, A + B corresponds to the least dimensional part of which both A and B are parts, while A • B corresponds to the highest dimensional part which is part both of A and of B. In this way we obtain a far-reaching analogy with abstract algebra where, in defining a field, one also starts with a Single class of undefined elements and two undefined operations, addition and multiplication. Moreover, we obtain an analogy with the algebra of logics, in particular with the calculus of classes.1 In fact, this paper presents what might be called an algebra of elementary geometry.

Almost Periodic Functions and Compact Groups

Abstract: The theory of any given almost periodic function on a group G can be reduced to the study of a continuous function on a uniquely determined compact group F. Then there is a one-to-one correspondence between continuous functions on F and a certain class of a.p. functions on G, such that convolutions of corresponding functions are again corresponding functions. This class of functions on G is the smallest modul containing the given function. In II we repeat the definition of modul and give, with some indications for the proofs, certain of their properties that are not affected by the knowledge of the compact group F. In III we give the definition of the compactified group F and in IV we prove our statement concerning the class of functions on G related to the continuous functions on F. The approximation theorem for moduls of a.p. functions proved by Bochner and von Neumann' is put in an essentially different light if considered on the compact group. We give a proof of this theorem in V, together with a generalization of the theorem concerning Bohr a.p. functions with positive Fourier coefficients. In VI we prove the product theorem for expansions of a.p. functions. This theorem might just as well have been proved on the basis of the approximation theorem given in (1) p. 475. In VII we give some definitions and results on the relation between different moduls on one group, applying this in VIII to moduls on compact groups. As a consequence we find in IX a theorem (th. 6) implicitly contained in Pontrjagin's paper (3), but giving a somewhat better insight into the character of a compact group. The proof of theorem 7 is nothing but an adaption of Pontrjagin's proof of the same theorem (cf. (3)), using theorem 6 as a starting point. Theorem 7 and its proof are a refinement of von Neumann's work in (6). Theorem 6 can be used as the foundation for an analysis of the structure of a compact connected group (7). In X and XI we give two convergence theorems for a.p. functions and an analysis of infinite direct products of compact groups. Theorem 8 will be found in (2) as theorem 34, while theorem 10 is related to theorem 35 of (2). Theorems 4 and 10 are generalizations of results of H. Bohr: Math. Zeitschr., 23 (1925), pp. 38-44. Though abelian compact groups have not been specially mentioned, a com-

Multiplications on a Complex

Abstract: In their communications at the First International Topological Conference (Moscow, September 1935), J. W. Alexander and A. Kolmogoroff introduced the notion of a dual cycle1 and defined a product of a dual p-eycle and a dual q-eycle, this product being a dual (p + q)-eyele. A different multiplication of the same sort is considered in this paper. It may be shown that the Alexander-Kolmogoroff product, augmented by the dual boundary of a suitable (p + q - 1)- chain, is equal to the \( \left( {_{p}^{{p + q}}} \right)th \) multiple of the product here introduced.2 Moreover, I consider also a product of an ordinary n-cycle and a dual p-eycle (n ≥ p), this product being an ordinary (n — p)-cycle. There is a simple algebraic relationship between the two kinds of multiplication, which I shall explain elsewhere. As an application of the general theory, I give a new approach to the duality and intersection theory of a combinatorial manifold, given in a simplicial subdivision. The theory works exclusively in the given subdivision.

On a Certain Topology for Rings of Operators

Abstract: 1. The notation to be used in this paper agrees with the one which is described in ?1.1 of the following paper On rings of operators. The quotations refer to the bibliography of that paper, quotations from the paper itself will be marked RO. We will make free use of the results of Chapters I and II in RO, excepting however Lemma 2.3.7 in ?2.3. This Lemma is based on the results of the present paper, but it will not be used in the other parts of those sections. Our subject is the introduction of a new topology in the ring of operators B (cf. (18), p. 370) of a space ! (cf. RO, ?1.1, point (b)), which is more appropriate for use under certain conditions than the three topologies of B defined in (18), pp. 378-388. Apart from its independent interest, we need it for application in RO, Lemma 2.3.7 in ?2.3, and in Chapter III. This topology is technically superior to the three above mentioned ones in this: For the "uniform" topology in B the results of our ?4 are true, but those of our ?3 are not. For the "strong" as well as for the "weak" topology in B, ?3 is true, but ?4 is not. (We will give an example for this in ?5.) The "strongest" topology however, as we are going to define it, will have all these properties.

On the Connectivity Ring of an Abstract Space

Abstract: (integrands of multiple integrals) that are independent, in the large, modulo the derived forms. The geometrical method of approach has been extended to compact metric spaces by Vietoris' and to still more general spaces by tech.2 Moreover, this branch of the theory has been very greatly perfected by the introduction of Pontrjagin's cycles with real coefficients reduced modulo 1. Now, if we use Pontrjagin's cycles, the kth connectivity group of a compact, metric space becomes a compact, metric group. Moreover, by a theorem of Pontrjagin,3 every such group may be identified with the character group of a countable, discrete group. This immediately suggests the advisability of regarding the discrete group, rather than its equivalent (though more complicated) metric character group, as the kth invariant of the space, and of looking for a revised theoretical treatment leading simply and directly to this group. We give such a treatment below, based on a suitable combinatory adaptation of the second, or analytic, method of approach. One decided advantage of taking the discrete groups rather than their metric character groups as the fundamental connectivity groups of the space is that we can then define the product4 (as distinguished from the sum) of two elements of the same or of different groups. The combined groups of all dimensionalities (or, more precisely, their direct sum) will thus become a connectitvity ring, as distinguished from a set of isolated connectivity groups.

Manifolds with Abelian Fundamental Groups

Abstract: 1. In a recent paper we showed' that the rank' of the (abelian) fundamental group of an n-dimensional group manifold can not exceed n. At the time of writing that paper it seemed to the author that this theorem might hold for any manifold with an abelian fundamental group. This is of course true-at least for compact manifolds-when n = 2. We shall show below that it is also true for n = 3 but is not true in general when n > 3. In order to construct counter examples for n > 3 we shall first determine the fundamental group of the symmetric product k,, of a finite complex Kn by itself.

Linear Differential Equations with Almost Periodic Coefficients

Abstract: d ~ ( t ) a , l ( t ) ~ ( t ) + . . . + a~,~(t)~,(t) + fl~(t); d t in which the funct ions a, , (t) and fir are real or complex a.p.~ functions of the real variable t, and the fit(t) may or may not be identically zero. We shall seek to determine conditions under which the solutions of (I. II) are of a ra ther general type involving a.p. functions. Before characterizing this type of solution more explicitly, we shall introduce a shorter vector terminology. I . z . Troughout this paper we shall use the letters x, y, z, and b to denote n-dimensional vectors (or matrices of n rows and one column) having the compod t ., nents ~1 . . . . , ~,~; V~, . . . , Vn; ~,, . . . , ~ ; and i l l , . . . , fl~. The vector ~-t~L(), . . d d--t~(t) will be denoted by D Ix]; the n-by-n matr ix whose elements are a , , will