Showing papers in "Annals of Mathematics in 1943"

Transformation Groups of Spheres

Abstract: 1. The compact Lie group G is said to be a transformation group of the space TW or to act on the space W if the following conditions are satisfied: a) to every element g of G there is associated a homeomorphism g(x) [x in TV] of W onto itself. b) if gl and g2 are elements of G then g9g2(X)] = (glg2)(X). c) the point g(x) depends continuously on the pair (g, x). Conditions a) and b) imply that to the identity element of G is associated the identity homeomorphism. The group G is said to act transitively if in addition to a), b), and c) the following fourth condition is satisfied: d) for any two points x and y of TW there is an element g in G such that g(x) = y. When d) is satisfied we say that TV is a homogeneous space under G. In this paper we take for W the n-dimensional sphere S' and study the question of what compact connected Lie groups can act transitively and effectively, (see 2 a) below), on S'. In I we prove a theorem on the structure of such a group which shows us that our main concern in the study of this problem is with simple groups. In II we study the question for simple groups using the Killing-Cartan classification, and we find that in general only those simple groups can be transitive and effective on SL which are well known to be so. In III we use our methods to draw some conclusions about the structure of certain subgroups of the rotation group of the n-dimensional sphere which we denote by Rn . Otherwise expressed Rn is the group of orthogonal transformations of determinant 1 on n + 1 real variables.

On the Expansion of the Partition Function in a Series

Abstract: 1. A geometric property of the Farey series, discovered by L. R. Ford (1) is used in this note for the construction of a new path of integration to replace the circle carrying the Farey dissection, first introduced by Hardy and Ramanujan in their classical paper (2). This new path of integration will bring about an essential simplification in the treatment of the partition function and, in general, in the determination of the coefficients of modular functions of nonnegative dimension. It seems to me that the new path exhibits more clearly than the Farey arcs do the different contributions of the approximation functions near the roots of unity. Moreover, only two estimations have to be performed, and they are direct consequences of the obvious statements (3.2) and (4.1) concerning the circle over the diameter 0 to 1. Ford's theorem referred to above can be enunciated as follows: If in a complex r-plane we mark the points corresponding to the reduced fractions h/k and draw about the points

On Families of Mutually Exclusive Sets

Abstract: In this paper we shall be concerned with a certain particular problem from the general theory of sets, namely with the problem of the existence of families of mutually exclusive sets with a maximal power . It will turn out-in a rather unexpected way that the solution of these problems essentially involves the notion of the so-called \"inaccessible numbers .\" In this connection we shall make some general remarks regarding inaccessible numbers in the last section of our paper .

Homology With Local Coefficients

Abstract: In a recent paper [16] the author has had occasion to introduce and use what he believed to be a new type of homology theory, and he named it homology with local coefficients. It proved to be the natural and full generalization of the Whitney notion of locally isomorphic complexes [18]. Whitney, in turn, credits the source of his idea to de Rham's homology groups of the second kind in a nonorientable manifold [13]. It has since come to the author's attention that homology with local coefficients is equivalent in a complex to Reidemeister's Uberdeckung [10]. Since this new homology theory (which includes the old) seems to have such wide applicability, a complete review of the older theory is needed to determine to what extent and in what form its theorems generalize. The object of this paper is to make such a survey. The general conclusion is that all major parts of the older theory do extend to the new. In addition the newer theory fills in several gaps in the old. The most noteworthy of these is a full duality and intersection theory in a non-orientable manifold (?14). For the sake of completeness, some of the results of Reidemeister have been included. The new approach and new definitions make for easier and more intuitive proofs. They lead also to results not obtained by Reidemeister. The most important is a proof of the topological invariance of all the homology groups obtained.' In addition developments are given of the subjects of multiplications of cycles and cocycles, chain mappings, continuous cycles, and Cech cycles. Part I contains an abstract development of systems of local groups in a space entirely apart from their applications to homology. Any fibre bundle over a base space R [18] determines many such systems in R (one for each homology group, homotopy group, etc., of the fibre). These are invariants of the bundle. They should prove to be of some help in classifying fibre bundles. Part II, which contains the extended homology theory, presupposes on the part of the reader a knowledge of the classical theory such as can be found in the books of Lefschetz [7] and Alexandroff-Hopf [1].

Automorphism Rings of Primary Abelian Operator Groups

Abstract: The representation of a ring as the ring of all the automorphisms of a primary abelian operator group expresses significant inner properties of this ring, since there exists essentially at most one such representation of a given ring. Thus it is the object of this investigation to expose invariant qualities of these automorphism rings. They are found to be pecularities of their ideal theory (Chapter III); and from these we select a complete set of postulates for the class of all the automorphism rings of primary abelian operator groups (Sections 12 and 13); thus proving incidentally that the structure of these rings is completely determined by their ideal theory (Section 14). The properties of the ideals in these rings reflect so completely the structure of the underlying operator group that we are able to prove-provided the "rank" of this group is at least three-the essential identity of the group of a-utomorphisms of the ring and of the group of projectivities (= biunivoque and monotone increasing transformations) of the system of admissible subgroups of the underlying operator group (Chapter II). Two [extreme] special cases may serve as an-i illustration for these theorems. Every projective geometry of finite dimension not less than three may be represented as the set of admissible subgroups of a suitable primary abelian operator group; and our characterization of the automorphisms of the automophism ring specializes in this case to the (well known) theorem that each automorphism of the automorphism ring (= ring of square matrices with coefficients from a suitable (not necessarily commutative) field) is induced by a so-called seini-linear transformation. If secondly G is an ordinary primary abelian group with the property that the least common multiple of the orders of the elements in G is a prime power p"' and that G contains at least three independent elements of maximum order, then our theorem states that every automorphism of the automorphism ring is an inner auomorphism.

On Some Convergence Properties of the Interpolation Polynomials

Abstract: It is well known that there exist continuous functions whose Lagrange interpolation polynomials taken at the roots of the Tchebycheff polynomials T„ (x) diverge everywhere in (-1, + 1) .' On the other hand a few years ago S . Bernstein proved the following result' : Let f(x) be any continuous function ; then to every c > 0 there exists a sequence of polynomials ~p„(x) where ~0 ,(x) is of degree n 1 and it coincides with f(x) at, at least n cn roots of T, (x) and gyp, (x) -p f( .c) uniformly in (-1, + 1) . Fejer proved the following theorem' : Let the fundamental points of the interpolation be a normal 4 point group

On Permutation Groups of Prime Degree and Related Classes of Groups

Abstract: The transitive permutation groups of prime degree p appear as the Galois groups of the irreducible algebraic equations f(x) = 0 of degree p. This is the reason that these groups have been the subject of a large number of investigations.' However, only few results of a general nature have been obtained. In the present paper, the theory of group representations2 will be applied in order to derive some new theorems concerning the structure of these groups. Actually, the method can be used for the study of a wider class of groups, viz. the groups 5 of finite order g which have the following property: (*) The group (M contains elements P of prime order p which commute only with their own powers Pi. It is clear that transitive permutation groups of degree p have the property (*). Secondly, the doubly transitive permutation groups of degree p 1 are of this type.3 A third example is furnished by the irreducible linear groups in a p-dimensional vector space whose center consists of the unit element only, in particular by the simple linear irreducible groups in p dimensions (cf. section 7). It is easily seen (section 1) that the order g of a group (M with the property (*) is of the form

Corrections to Two of My Papers

Abstract: In my paper " On th.e dit~ergence properties of th.e Lagran.ge interpolation poly-n.umiuZs, " (Annals of R4at. cos i7r (p and q odd), and the fundamental points of the interpolation are the roots of the Tchebicheff polynomial TR(z), then t'here exists a continuous fun&ion f(x) such that lim L,v(qJ) = W. Dr. Schijnberg has point,ed out, that the proof there given is not correct. There is a trivial error in lemma 1; namely, it is possible that ~1~) = zot). Nevertheless it is possible to save almost everything, practically without mod&-ing the proof. We prove t.he following slightly weaker. THEOREM. There exists a continuous function f($) such tka.t if x0 = cos % ?T, zvhere p and q are odd, then lim 1 L,(j&)) 1 = CCI. Proof. We need LEMMA 1. If xjrn) # xjtl) then 1 xam)-xjn) 1 >-$ for m 2 n. Proof. As in the paper. Everything is now unchanged until the bottom of page 311. 'We have t'here where E,, = fl and will be determined later; t.he definition of f%(z) is the same as in the paper. Ln(~2(~~)) = 0 still holds (p. 313 top). It suffices t,o show that, for r > YL, fr(x: ") = 0. And this is true, for otherwise either W Xl (n) =xk , which is impossible since (21-1, r) = 1, or we have which does not hold by lemma 1. Define now en = signum L,,(ql(nt~)); then clearly and the rest of the proof is unchanged. At present I cannot, decide whet,her a continuous function f(x) exists such that lim Lo) = =, or whether a continuous f(x) exists with lim L,cf(zO)) = a, where a # f(x& 647

Some Remarks on Algebras Over an Algebraically Closed Field

Abstract: The theory of rings with radicals is an interesting and far reaching problem of modern algebra.' In this paper we have examined some aspects of algebras which may have radicals and whose coefficient fields are algebraically closed. Some of the methods employed clearly could be used for less restricted algebras, but a full extension of the results requires the solution of a number of problems still under investigation.2 The authors feel that the theory of algebras over an algebraically closed field has some interest and value in itself, particularly in view of its immediate application to the representation theory of finite groups. Moreover, this restricted case is a valuable testing ground for theorems for more general rings. In the first part of the paper we have studied the concept of basic algebra. The basic algebras are semi-primitive subalgebras (i.e. modulo their radicals are direct sums of division algebras, in our case, direct sums of fields) which for the algebras under discussion play a role in some respects analogous to that of division algebras for simple algebras. Related to the basic algebras are the Cartan basis systems,3 and systems of elementary modules.4 The commutator algebras of matrix representations of an algebra, or what is equivalent, the algebras of homomorphisms of the related representation spaces, can be analyzed in a rather simple manner. We shall say that a linear function sp of an algebra a is symmetric, if for every a, E c a, (p(af) = sp(Oa). In the case where a is over an algebraically closed field, and is also semisimple, the characters of the irreducible representations of a form a complete set of symmetric functions of a. When a has a radical, this is no longer true. In Part 2 of the paper, we discuss symmetric functions of algebras with radical. In Part 3 of the paper the regular representations are written in terms of elementary modules.

Contributions to the Analytic Theory of Continued Fractions and Infinite Matrices

Abstract: in which the a, are real and positive, and the b, are complex numbers with nonnegative imaginary parts. In addition, we are concerned with certain infinite matrices connected with this continued fraction. In the earlier investigations of continued fractions of this form, beginning with the classical work of Stieltjes [15]', the b, have been supposed real. For this case, and with some additional restrictions, Stieltjes was able to connect the continued fraction with one or more integrals of the form

On Some Algebraical Properties of Operator Rings

Abstract: [1] J. VON NEUMANN: "Zur Algebra der Funktionaloperatoren . ," Math. Annalen, vol. 102, pp. 370-427 (1929). [2] J. VON NEUMANN: "Uber adjungierte Funktionaloperatoren," Annals of Math., vol. 33, pp. 294-310 (1932). [31 F. J. MURRAY: "Linear transformations between Hilbert spaces," Trans. Amer. Math. Soc., vol. 37, 301-338 (1935). [4] F. J. MURRAY AND J. VON NEUMANN: "On rings of operators," Annals of Math., vol. 37, pp. 116-229 (1936). [5] F. J. MURRAY AND J. VON NEUMANN: "On rings of operators IV," Annals of Math., vol. 44, (following immediately).

On the Sum of Two Sets of Integers

Abstract: In his beautiful paper: "A proof of the fundamental theorem on the density of sums of sets of positive integers"' Mr. Mann succeeded in proving the (a, A)hypothesis and a generalization of it that had been conjectured for more than ten years. We found that his method can be simplified considerably and even yields some stronger results. Let A, B respectively be sets of nonnegative integers a, b. Let C = A + B be the set of all integers of the form a + b. Let A(x), B(x), C(x) denote the number of positive integers of the sets

On Hodge's Theory of Harmonic Integrals

Abstract: The attempt which WV. V. D. Hodge made in Chapter III of his beautiful book' to establish the existence of harmonic integrals with preassigned periods has not been entirely successful because the proof is partly based on a false statement (p. 136) concerning the behavior of the solution of a non-homogeneous integral equation when the spectrum parameter approaches an eigen value. In a Princeton seminar on the subject, H. F. Bohneriblust pointed out that counter examples are readily available even for linear equations with a finite number of unknowns. For instance the equation Xx + Ax = c with

On Homotopy Type and Deformation Retracts

Abstract: It has recently been shown by J. H. C. Whitehead' that two complexes X and Y belong to the same homotopy type2 if and only if there is a third complex W of which both X and Y are deformation retracts.3 I shall show that this theorem holds not merely for complexes but for the most general spaces for which continuity has a meaning. The proof which I give is direct and constructive and avoids the extraneous notions of relative homology and relative homotopy groups which complicate Whitehead's proof. The concept of homotopy type splits naturally into two concepts which I shall call rightand left-homotopy inversion. In theorems 3.3 and 3.4 I show that right-and-left inversion correspond respectively to deformation and retraction, thus replacing Whitehead's theorem by two "component" theorems. The necessary preliminary study of deformation, retraction, and inversion is carried out in ??1 and 2, and the mapping cylinder, the fundamental tool of our theory, is defined in ?3. It should be noted that Whitehead's definition4 of mapping cylinder is not really satisfactory for the general spaces considered here. The fundamental theorems of this paper are theorems 3.1 and 3.2. They are generalizations of the theorems (3.3 and 3.4) discussed above. In ?4 these fundamental theorems are applied (in another direction) to the Hopf-Pannwitz deformation and also to yield a new characterization of the closure of a homogenous n-dimensional polyhedron. These theorems (3.1 and 3.2) are of considerable interest in themselves. They exhibit a duality which is quite striking and seem to indicate a relatively unexplored region which I might designate as "algebra of mapping classes". In this connection they should be compared with the fundamental theorem of fibre spaces5 to which they bear an evident analogy. In ??5 and 6 certain specializations are considered. They are to be regarded as trends in the following two directions (a) bridging the gap between homotopy type and nucleus' (b) bridging the gap between homotopy type and topological type. In ?7 I develop an n-dimensional analogue of ?3. This is in line with

Some Remarks on Set Theory

Abstract: Let there be given n ordinals a,, a2, , * , a,n. It is well known that every ordinal can be written uniquely as the sum of indecomposable ordinals. (An ordinal is said to be indecomposable if it is not the sum of two smaller ordinals.) Denote by +(a) the largest of these indecomposable ordinals belonging to a. ( (aj may have a coefficient c in the decomposition of a.) Put y = mini6, (4ai), and assume that there are k a's with o(ac) =,y. Denote these a's by a1, a2, * * *, ak. If in the sum ail+ai,+ +ain, ij, i2, * *, i,, a permutation of 1, 2, *, n, none of the ai, i < k appear at the end, they get absorbed in the following summands, and we get exactlyf(n-k) different sums. Assume next that exactly r of the ai's, r

Foundations of the Theory of Lie Groups With Real Parameters

Abstract: The material presented below amounts roughly to the so-called fundamental theorems of Lie and their implications concerning Lie algebras, Lie subgroups and subalgebras. By straightforward and fairly elementary steps, we shall extend the concept of Lie group to include groups admitting coordinate systems in which the functions defining ab, the group product of elements a, b, possess continuous first order derivatives in the coordinates of a and satisfy a Lipschitz condition in those of b. That such groups are equivalent to classical (i.e. analytical) Lie groups was announced in 1936 by van Kampen, although apparently van Kampen published nothing by way of proof beyond a certain decisive uniqueness theorem concerning systems of ordinary differential equations. (This is contained in our theorem (12.1); the proof given below is essentially that of van Kampen [3]). It is conceivable that the condition relative to coordinates of b could be weakened or dispensed with entirely. For, the only part the condition in question plays in our development is to make it certain that the inverse of a certain mapping-the mapping into "canonical coordinates"-is single-valued. In this connection mention should be made of the paper [1J of Garrett Birkhoff in which it is shown that the existence of continuous first order derivatives of ab with respect to the coordinates of both the a's and the b's is sufficient for defining a Lie group. We shall consider only groups with a finite number of real parameters (whereas the groups considered by Birkhoff are not necessarily finite dimensional). This makes it possible to use the standard existence theorems for systems of ordinary real differential equations. We make no use whatever of the theory of partial differential equations. We maintain throughout a purely local point of view in the sense that we consider only what happens in the neighborhood of the identity. Historical notes. The proof of the uniqueness of square roots (8.5) is due to Claude Chevalley (unpublished). The proof given below of Lie's theorem that a linear system of vector fields which is closed under commutation defines a Lie group is, we believe, due to van der Waerden [4]. We have supplied a number of preliminary lemmas which make the proof rigorously applicable to the case in which the vector fields are only assumed to possess continuous first order derivatives. The theorem in ?17 that every local subgroup of a Lie group is a Lie group is due essentially to E. Cartan [2]. 481

A Note on Cesaro Summability of Fourier Series

Abstract: 1. It is known that the (C, a) continuity of a function is not sufficient for the (C, a) sumnmability of its Fourier series at a point. Many sufficient conditions for (C, a) summability of a Fourier series were found by many authors.' The object of this note is also to find a sufficient condition which arises from a convergence criterion for Fourier series recently proved by the author.2 Let +(t) be an even integrable periodic function with period 2ir and its Fourier series be