John von Neumann

A Model of General Economic Equilibrium

The subject of this paper is the solution of a typical economic equation system. The system has the following properties: (1) Goods are produced not only from “ natural factors of production,” but in the first place from each other. These processes of production may be circular, i.e. good G1 is produced with the aid of good G2, and G2 with the aid of G1. (2) There may be more technically possible processes of production than goods and for this reason “ counting of equations ” is of no avail. The problem is rather to establish which processes will actually be used and which not (being “ unprofitable”).

Numerical inverting of matrices of high order

PREFACE 188 CHAPTER VIII. Probabilistic estimates for bounds of matrices 8.1 A result of Bargmann, Montgomery and von Neumann 188 8.2 An estimate for the length of a vector 191 8.3 The fundamental lemma 192 8.4 Some discrete distributions 194 8.5 Continuation 196 8.6 Two applications of (8.16) 198 CHAPTER IX. The error estimates 9.1 Reconsideration of the estimates (6.42)-(6.44) and their consequences.. 199 9.2 The general Ai 200 9.3 Concluding evaluation 200

Preliminary discussion of the logical design of an electronic computing instrument (1946)

Inasmuch as the completed device will be a general-purpose computing machine it should contain certain main organs relating to arithmetic, memory- storage, control and connection with the human operator. It is intended that the machine be fully automatic in character, i.e. independent of the human operator after the computation starts. A fuller discussion of the implications of this remark will be given in Chapter 3 below.

Almost periodic functions in a group. I

1. The object of the present paper is to extend H. Bohrs famous theory of almost periodic functions [4, I]f to arbitrary groups, and to show that it gives just the maximum range over which the fundamental results of Frobenius-Schur representation theory [21; 22; 30] and its extensions by Peter and Weyl [32 ] hold. We shall see in particular that all bounded linear representations of a group are equivalent to unitary representations and belong to this class. Another point of importance is that we free ourselves completely from all topological assumptions (such as continuity, etc.) by the use of a definition of almost periodicity due to Bochner [2 ]. Thus we find that the general theory, which applies to every group

Über das Verhalten von Eigenwerten bei adiabatischen Prozessen

In vielen Fragen der Quantenmechanik ist es wichtig, die Veranderung der Eigenwerte und Eigenfunktionen bei stetiger Anderung eines oder mehrerer Parameter zu untersuchen. Namentlich interessiert oft der Fall, in dem man fur zwei spezielle Werte der Parameter Eigenwerte und Eigenfunktionen kennt und sich fur das Zwischengebiet interessiert. Man fragt gewohnlich, ob im Zwischengebiet Uberschneidungen der Eigenwerte vorkommen, in welchen Eigenwert ein bestimmter Eigenwert ubergeht, wenn man von dem einen Wertsystem der Parameter kontinuierlich in das andere Wertsystem ubergeht usw. Fragen ahnlicher Art hat F. Hund aufgeworfen1) und insbesondere die letzte Frage fur den Fall eines Parameters — auf Grund von Beispielen — dahin beantwortet, das Uberschneidungen im allgemeinen — wenn dafur kein spezieller Grund vorhanden ist — nicht vorkommen2). Wir wollen hier dies allgemein begrunden, unsere Methode erlaubt dabei gleichzeitig die Untersuchung von Systemen mit mehreren veranderlichen Parametern.

On complete topological spaces

DEFINITION I. If M is a space in which there is defined a metric dist (f, g) satisfying the usual postulates for distance ([1], p. 94), then a sequence F: fl, f2, * * * is fundamental if, for every 3>0, there exists an n = nli (3) such that m, n > ni imply dist (fn, f,) 0, there exists an n2 = n2(8) such that n > n2 implies dist (f, f,) < 8. M is complete if every fundamental sequence is convergent.

The Principles of Large-Scale Computing Machines

The following paper was one of many seminal works on computing prepared by John von Neumann. This specific paper formed the basis for an article that appears in von Neumanns Collected Works (Vol. V), edited by A. H. Taub and published in 1963 by Pergamon Press. Coauthored by Herman H. Goldstine and von Neumann, the article in the volume is a much expanded version of the following paper.

Unsolved Problems in Mathematics

The invitation of the Organizing Committee for me to speak about “Unsolved problems in mathematics” fills me as it should with considerable trepidation and a prevailing feeling of personal inadequacy. Hilbert gave a talk on this subject at the similar congress about 50 years ago and this is a very formidable precedent. He stated about a dozen unsolved problems in another widely separated areas of mathematics, and they proved to be prototypical for much of the development that followed in the next decades. It would be absolutely foolish, if I tried to emulate this quite singular feat. In addition I do not know the future and the future at any rate can only be predicted ex post with any degree of reliability. I will, therefore, define what I am trying to do in a much more narrow way, hoping that in this manner I have a better chance of not failing. I will limit myself to a particular area of mathematics which I think I know and I will talk about it and about what its open ends appear to be, particularly in some directions which are not the ones that the evolution so far has mainly emphasized and which are, I think, quite important.

A Theorem on Unitary Representations of Semisimple Lie Groups

We show that a connected semisimple Lie group G none of whose simple constituents is compact (in particular, any connected complex semisimple group) has no nontrivial measurable unitary representations into a finite factor,-i. e. a factor of type In(n oo ) or II , in the terminology of [3]. This has been known for the case of representations of complex groups into factors of type In X but the existing proofs are not applicable either to real groups or to factors of type II,, and the present proof is therefore necessarily of a different character from the proof for the complex, finite-dimensional case. Our theorem has the relevant consequence that in the reduction of the regular representation of G into factors (see [9] and [5]), those of type In or II, cannot occur. This is in marked contrast with the situations for compact and discrete groups, only I.s occurring in the compact case (as is well-known) and only Ins and IIs in the discrete case (loc. cit.). In order to clarify the statement of our theorem, we make the following definitions. A representation U of a locally compact group G by unitary operators on a Hilbert space X3C (of arbitrary dimension) is called measurable if the inner product (U(a)x, y) is a measurable function of a e G, relative to Haar measure, for all x and y in X. (When X is separable, such a representation is necessarily continuous in the strong operator topology, as follows from a modification of the proof by the second-named author of a special case of this result; details, as well as a more precise result, are given below.) U is said to be into a factor iFif Tiis a factor of which U(a) is an element, for all a E G. Now we state our central result. THEOREM 1. Let G be a connected semisimple Lie group none of whose simple constituents is compact. Then the only measurable unitary representation of G into a finite factor is the identity representation. The following proof applies also to the case of any weakly continuous representation of G into an algebra of operators on a Hilbert space, on which a weakly continuous trace is defined. To outline briefly the general plan of the proof, the non-compactness of the simple constituents of G is used to show that G must contain one of a certain class of 2and 3-dimensional solvable Lie groups. A special study of the unitary representations into finite factors of the members of this class of groups shows that such representations must be trivial on certain subgroups. The proof is concluded by showing that G is generated by such subgroups. An analog of the following lemma is valid for arbitrary topological groups in the large, and can be proved in the same way. A trivial modification of the proof shows also that the same conclusion is valid if G is a cross product of groups A

Zur Erklärung einiger Eigenschaften der Spektren aus der Quantenmechanik des Drehelektrons

ZusammenfassungMit Hilfe der im 1. Teil dieser Arbeit erhaltenen Ergebnisse wird das Aufbauprinzip der Serienspektren, die Auswahlregeln für die innere und magnetische Quantenzahl und der quadratische Starkeffekt behandelt.