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Showing papers in "Annals of Mathematics in 1938"



Journal ArticleDOI
TL;DR: In this article, it was shown that the relativistic equations of gravitation for empty space are sufficient to determine the motion of matter represented as point singularities of the field.
Abstract: Introduction. In this paper we investigate the fundamentally simple question of the extent to which the relativistic equations of gravitation determine the motion of ponderable bodies. Previous attacks on this problem' have been based upon gravitational equations in which some specific energy-momentum tensor for matter has been assumed. Such energy-momentum tensors, however, must be regarded as purely temporary and more or less phenomenological devices for representing the structure of matter, and their entry into the equations makes it impossible to determine how far the results obtained are independent of the particular assumption made concerning the constitution of matter. Actually, the only equations of gravitation which follow without ambiguity from the fundamental assumptions of the general theory of relativity are the equations for empty space, and it is important to know whether they alone are capable of determining the motion of bodies. The answer to this question is not at all obvious. It is possible to find examples in classical physics leading to either answer, yes or no. For instance, in the ordinary Maxwell equations for empty space, in which electrical particles are regarded as point singularities of the field, the motion of these singularities is not determined by the linear field equations. On the other hand, the well-known theory of Helmholtz on the motion of vortices in a non-viscous fluid gives an instance where the motion of line singularities is actually determined by partial differential equations alone, which are there non-linear. We shall show in this paper that the gravitational equations for empty space are in fact sufficient to determine the motion of matter represented as point singularities of the field. The gravitational equations are non-linear, and, because of the necessary freedom of choice of the coordinate system, are such that four differential relations exist between them so that they form an overdetermined system of equations. The overdetermination is responsible for the existence of equations of motion, and the non linear character for the existence of terms expressing the interaction of moving bodies. Two essential steps lead to the determination of the motion.

799 citations


Journal ArticleDOI
TL;DR: In this paper, it was shown that two invariants are inactive in the formation of field equations and thus may be omitted from the integrand of the action principle, i.e., I, = Ra, 6Ra# and 12 = R2.
Abstract: Introduction. If the geometry of nature is Riemannian and the field equations of this geometry are controlled by a scale-invariant action principle, there are four a priori possible and algebraically independent invariants which may enter in the integrand of the action principle. This abundance of invariants hampers the mathematical development and the logical appeal of the theory. The present paper shows that two of these invariants are inactive in the formation of field equations and thus may be omitted. Only the two invariants I, = Ra,6Ra# and 12 =R2

641 citations


Journal ArticleDOI

295 citations




Journal ArticleDOI

188 citations


Journal ArticleDOI
TL;DR: In this paper, the authors introduce the class of functions of finite p-variation, corresponding to the classical criterion for a function's being the indefinite integral of a function of class LV.
Abstract: This paper is concerned with the study of certain classes of abstractly-valued functions which arise naturally in connection with the theory of integration of functions with values in a Banach space. These classes correspond to the usual classes C, LP, M, etc. which are well known in analysis. The subject matter of the paper is divided into seven sections as follows: ?2 contains most of the technical apparatus, which consists of the Lebesgue integral of abstract functions as defined by Bochner,2 and several types of integrals based on the notion of the Riemann-Stieltjes integral. Since, in the case of abstractly-valued functions, there may be a distinction between an indefinite integral and an absolutely continuous function,3 we have found it important to introduce the class of functions of finite p-variation, corresponding to the classical criterion for a function's being the indefinite integral of a function of class LV. For certain Banach spaces it is known that the distinction mentioned above does not arise.4 Such a space is said to satisfy condition (D) [for the precise statement see ?21. We have obtained some information about the implications of this condition and its relation to other properties of the space [see especially ??4, 5 and 7]. ?3 contains the determination of the most general linear functionals on the various function classes under discussion. ?4 is devoted to a discussion of weak convergence and its relation to condition (D). In ?5 the interrelations of condition (D), weak completeness, and reflexiveness of Banach spaces is taken up. In ?6 we have results on the Parseval relation and properties of the Fourier coefficients of abstract functions. Counter examples to show the divergence from classical theory are given. In ?7 we return to the question of reflexivity and condition (D), obtaining a criterion for spaces which satisfy this condition. Finally, in ?8 it is shown how by the methods of Fourier series it is possible to characterize

111 citations


Journal ArticleDOI

101 citations




Journal ArticleDOI
TL;DR: In this article, the authors give a complete answer to the question of best possible projections in the case where the dimensionality of the subspace on which we project is by 1 less than the dimension of the entire space.
Abstract: By a well known theorem of Hahn-Banach every linear functional of norm M defined over a closed linear subspace of a Banach space' can be extended linearly to the entire space without increasing its norm. For operations the corresponding problem takes the following form: Given a linear operation u(x) whose domain of definition is a closed linear subspace of a Banach space B, and whose range lies in a Banach space B2, does there exist an operation U(x) defined over B1, with range in B2 and which coincide with u(x) over the subspace? How small can the norm of the extended operation be made? In the particular case where u(x) = x, B1 = domain of definition of u; the existence of an extension U(x) is equivalent to the existence of a complementary subspace or, as F. J. Murray2 has shown, to the existence of a projection of B, on the subspace. Conversely, if A(X) is such a projection and u(x) any linear operation, the linear operation U(X) = u(A(X)) is an extension of u(x) and its norm is < 11 u 11.11 A 11. The question of extension is thus equivalent to the discussion of best projections, i.e. of projections of least norm. It is relatively easy to exhibit examples of Banach spaces for which certain closed subspaces have no projections; even more, F. J. Murray3 has shown that this occurs within the function spaces L, and the sequence spaces lp! For such subspaces an extension of an operation is not always possible. If the Banach space is finite dimensional, i.e. if we are dealing with a Minkowski space, of n dimensions say, the existence of projections is trivial but the discussion of the best possible projections (i.e. those with a minimal norm) is interesting and may help to explain why, in certain infinitely dimensional spaces, projections fail to exist. We give in this paper a complete answer to the question of best possible projections in the case where the dimensionality of the subspace on which we project is by 1 less than the dimensionality of the entire space. The results obtained lead to some interesting theorems on convex regions; they are considered in section 6.

Journal ArticleDOI
TL;DR: Hilbert and Rohn as mentioned in this paper showed that an algebraic curve of order 6 cannot have all its ovals lying outside each other, unless one of these ovals must lie within another oval.
Abstract: As early as 1876 Harnack' showed that the maximal number of components (maximal connected subsets) of a real algebraic curve of order n in the projective plane is precisely I(n 1) (n 2) + 1. At the same time Harnack proposed a process for the construction of curves with this maximal number of components. Such curves we shall call in the sequel, M-curves. Harnack showed that these M-curves have no singular points. Take a sphere in the three-dimensional space in which the projective plane containing our algebraic curve is situated, and join the centre of this sphere to every point of the projective plane by a straight line. We thus project the plane on the sphere. A component of an algebraic curve is called an "oval" (or an "even" component) if its projection on the sphere consists of two ordinary closed curves. If this projection on the sphere S consists of a single closed curve the corresponding component is called "odd." Algebraic curves having no real2 singular points possess at most one odd component. Hence every algebraic curve (having no real singular points) of even order consists of ovals only while a curve of odd order has (besides ovals) exactly one odd component. In 1891 D. Hilbert3 proposed a new method of constructing M-curves. In the same work Hilbert announced without proof that an M-curve of order 6 cannot have all its ovals lying outside each other. At least one of these ovals must lie within another oval. Here the words "an oval lies within another oval" mean that the cone projecting the first oval on S lies within the projecting cone of the second oval. Hilbert considers this a remarkable fact, since it proves that M-curves cannot have a too simple topological structure. In his report to the International Mathematical Congress in 1900 on modern problems of mathematics Hilbert considers the investigation of the topology of M-curves and of the corresponding algebraic surfaces as most timely.4 After a series of attempts the above mentioned theorem announced by Hilbert was at last proved in 1911 by K. Rohn.5 In the same work Rohn proved that an

Journal ArticleDOI
TL;DR: In this paper, the equations of motion for the case of two comparable masses were integrated to the same approximation, emphasizing the effects on the orbit of a double star of possible astronomical interest.
Abstract: In the preceding paper Einstein, Infeld and Hoffmann' have developed a most ingenious and useful method for obtaining, by successive approximations, the gravitational field and equations of motion of n bodies in the general theory of relativity. They have carried out the derivation to that order which leads. in the well known solution of the one body problem, to the perihelion advance of an infinitesimal planet, and have given explicitly the equations of motion for the case of two bodies of comparable masses.2 It is the purpose of this note to integrate these latter equations, to the same approximation, emphasizing the effects on the orbit of a double star of possible astronomical interest. Their equations of motion (17.2) differ from the classical equations of the two body problem by the appearance on the right of the specifically relativistic terms, of order m/r compared with those on the left. It is therefore expedient to introduce in place of the coordinates tn , An the six variables

Journal ArticleDOI
TL;DR: In this paper, it was shown that if a function f(z) = f (z1, *, Zk) is analytic and bounded within a convex tube T, then it also exists and is bounded within T. In fact, there are functions of this kind which have the period 27ri in each variable z5.
Abstract: 0 xX = Xx ; _x < ye < ??, K = 1, **k where (x?, ... , x) is an arbitrary point of S. A tube is an (open) domain if and only if its basis is a domain. We shall say that a tube T' lies within a tube T if the closure of T' is part of the interior of T. The convex closure (in the usual sense) of the tube T will be denoted by T. Obviously T is again a tube and its base S is the convex closure of S. In the paper loc. cit. it was shown that if a function f(z) = f(z1, * , Zk) is analytic and bounded within T it also exists and is bounded within T. In the present paper we shall establish the theorem omitting the property of boundedness. THEOREM. Any function which is analytic in a tube T is analytic in its convex closure T. On the other hand, any convex tube T is the natural domain of analyticity for some function. In fact, there exists functions of this kind which have the period 27ri in each variable z5 . Combining this fact with our theorem we are led to the following statement: the envelope of regularity (Regularitatshiille)2 of T is T. From the viewpoint of the theory of analytic functions of real variables our theorem may be formulated as follows. A function f(x,, * * , Xk) which is analytic over a domain S has, by definition, an extension

Journal ArticleDOI
TL;DR: In this paper, it was shown that the only approximable Lie groups are the compact Abelian groups, which are the groups whose degree depends only on the group approximated.
Abstract: A certain sense in which a finite group may be said to approximate the structure of a metrical group will be discussed. On account of Jordan's theorem on finite groups of linear transformations' it is clear that we cannot hope to approximate a general Lie group with finite subgroups. I shall show that we cannot approximate even with groups which are 'approximately subgroups': in fact the only approximable Lie groups are the compact Abelian groups. The key to the situation is again afforded by Jordan's theorem, but it is not immediately applicable. It is necessary to find representations of the approximating groups whose degree depends only on the group approximated.

Journal ArticleDOI
TL;DR: In this article, the authors consider the case M = a compact 2-dimensional manifold without boundary, and give a characterization of the class of all images G of M under monotone transformations.
Abstract: If M and G are topological spaces, a continuous transformation T(M) = G of M onto G is said to be monotone if the inverse image of each point of G is connected. In this paper we consider the case M = a compact 2-dimensional manifold without boundary, and we give a characterization of the class of all images G of M under monotone transformations. This problem has already been solved by R. L. Moore' in the case M = a 2-sphere. His result is that G is a cactoid,2 and every cactoid is a monotone image of a 2-sphere. Our results are highly analogous (see Theorem 5). If T(M) = G is monotone, the set of inverse images of points of G forms an upper semi-continuous collection of continua filling M. Conversely if G is an upper semi-continuous collection of continua filling M, the set G can be so topologized that the transformation sending P e M into the continuum of G containing P is a continuous monotone transformation of M onto G. Thus the study of monotone transformations is equivalent to the study of upper semicontinuous collections of continua. In addition to the characterization of G in the general case we find the effect on G of certain restrictions on the elements of G, considered as continua on M. Theorems 1, 2, 3, respectively give conditions both necessary and sufficient that G be (1) homeomorphic to M, (2) a 2-manifold, and (3) a generalized cactoid.



Journal ArticleDOI
TL;DR: In this article, it was shown that if a normal subgroup N contains a certain matrix C, then N must contain also the canonical matrix A =I+ me2l for some positive integer m. The problem of describing normal subgroups falls naturally into two parts.
Abstract: This paper is a continuation of two other papers with the same title [1, 2]. Let GLn[R] be the group of n x n invertible matrices with elements from a ring R. In [1], R was the ring of integers modulo a prime power pr = v, and the normal subgroups of GLn[R] were described. For each ideal W in the ring R, two normal subgroups N*,n, Nvn were defined by means of congruences: Nvn {M det M = 1, M-I, identity, mod W}; N*,n = { M M mod W is scalar}. It was shown that every normal subgroup of GLn[R] (n>2) contains Nw n and is contained in N*,n for suitable W. This result was generalized in [8] to the case where R is any valuation ring. The important case where R is the ring of rational integers, and GLn[R] is the unimodular group, is not covered by either of these references. A study of the latter case is the primary purpose of this paper. The methods used here work reasonably well when R is any euclidean ring (even a non-commutative one); certain results hold in a still wider class of rings. The general problem of describing normal subgroups falls naturally into two parts. In the first part (?? 2, 3) the discussion is restricted to the case n > 2. There we show that if a normal subgroup N contains a certain matrix C, then N must contain also the canonical matrix A =I+ me2l for some positive integer m. The second part (?? 5, 6) includes the case n = 2. The problem here is to find the least normal subgroup Qmn of GLn[R], or of the subgroup SLn[R] of the latter, which contains A. For R = J, we conjecture that Qm,n is Nm,n = N(m),n, and establish the validity of this conjecture for m 2) into a form parallel to and generalizing the solution given in [1]; v need not be a prime power (? 4). The results of [8]

Journal ArticleDOI
TL;DR: In this article, a semigroup S of a domain of integrity D is defined, and a set S in which a multiplication ab is defined is called an arithmetic if this multiplication is associative and commutative, if an identity element is present in S, and if the cancellation law holds.
Abstract: A set S in which a multiplication ab is defined is called a semigroup if this multiplication is associative and commutative, if an identity element is present in S, and if the cancellation law holds: ab = ac implies b = c. For example, the non-zero elements of a domain of integrity constitute a semigroup under multiplication. The notions of divisibility and irreducible elements are defined in the usual way. A semigroup S will be called an arithmetic if every element of S is uniquely decomposable into irreducible elements. The problem is to give necessary and sufficient conditions that a semigroup must satisfy in order that it can be embedded in an arithmetic. Any arithmetic z containing a semigroup S will be called an ideal arithmetic of S. This is an exact formulation of the problem of "restoring unique decomposition by the adjunction of ideal elements." The statement of the problem is in no way altered if we are considering a domain of integrity D rather than a semigroup. In that case we endeavor to embed the multiplicative semigroup S of D in an arithmetic A, or rather the semigroup S of principal ideals in D, which may be regarded as arising from S by identifying elements which divide each other. In Dedekind's theory of algebraic numbers, 2 is the set of all integral ideals in D, and it should be observed that z is not again a ring but only a semigroup-we can multiply two ideals but we cannot add them! This observation should make it clear that the problem is actually one of multiplication alone.' The general solution of the problem has been effected by means of what Krull calls v-ideals.2 A subset a of a domain of integrity D is a v-ideal if it contains every element of D which is divisible by all common divisors of a in the quotient-field of D. (Every v-ideal is also a Dedekind ideal, but not in general conversely.) Since this definition involves only the notion of divisibility, it can be applied to a semigroup S; of course we replace "quotient-field of D" by "quotient-group of S," the group of all formal quotients of elements



Journal ArticleDOI
TL;DR: In this paper, it was shown that Ug is necessarily continuous in the strong topology of the ring of bounded operators on Hilbert space, i.e., if a group Ut(X < t < cX) of unitary operators on H is measurable in the sense that (Utx, y) is measurable for each x, y in H, then Ug is always continuous.
Abstract: In connection with the theorem of M. H. Stone2 on the representation of one parameter groups of unitary transformations on Hilbert space J. von Neumann3 has shown (without using any representation theorem) that if a group Ut(X < t < cX) of unitary operators on Hilbert space H is measurable in the sense that (Utx, y) is measurable for each x, y in H, then Ug is necessarily continuous in the strong topology of the ring of bounded operators on Hilbert space, i.e.,


Journal ArticleDOI
TL;DR: In this article, the authors consider certain simple topological properties of periodic homeomorphisms of a space S into itself and show that these properties are related to the homology groups of the so-called modular space of S, the space obtained by identifying points which are images of each other under powers of T.
Abstract: In this paper we shall consider certain simple topological properties of periodic homeomorphisms of a space S into itself. Two homeomorphisms T and To of S into itself are said to belong to the same topological type if there is a third homeomorphism r such that To = rTT 1. For arbitrary T's, many characteristic properties of types can be described in terms of the structure of certain invariant sets, the recurrence properties under repeated iteration and so on, (see for example [1]); but the problem of complete classification of types even for the case where S is, say, a sphere is too general to be of great interest. If, however, one considers only homeomorphisms of finite period, various algebraic type invariants can be defined. For orientable surfaces, for example, such invariants are actually sufficient to enable one to determine whether or not two sense-preserving periodic transformations belong to the same type (Nielsen [6]). We shall here consider such invariants as can be readily defined in terms of homology theory. For the most part, we shall maintain a purely combinatorial point of view throughout and shall take S to be a finite simplicial complex and T to be a homeomorphism of S into itself which carries simplexes into simplexes, T being then necessarily of finite period. In the definition of "type," r is also understood to be a simplicial homeomorphism. It will be seen that some of our considerations resemble those which have recently led Reidemeister and de Rham to the (strictly combinatorial) classification of lensand cyclic-spaces. While our invariants do not have this sharpness, they do have the merit of being more topological in character; some of our theorems have meaning and hold true equally well for very general spaces. This implies in particular that when S is a complex, T and r do not really need to be simplicial. Although the modifications necessary to place our results in a really topological setting will not be carried out here, it will readily be seen from our papers [7] and [8] what the required procedure would be. Our invariants are closely related, as we shall show, to the homology groups of the so-called modular space of S the space obtained by identifying points which are images of each other under powers of T. In the particular case in