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Suppose that f : S 1 → R 2 f:{S^1} \to {R^2} is an immersion, i.e., a C 1 {C^1} map such that f ′ f’ is never zero. We call f normal if there are only finitely many self-intersections and these are transverse double points. A normal immersion f can be topologically determined by a finite number of combinatorial invariants. Using these invariants it is possible to give considerable information about extensions of f to D 2 {D^2} . In this paper we give a new set of invariants, inspired by the work of S. Blank, to solve several problems concerning the existence of certain kinds of extensions. The problems solved are as follows: (1) When does f have a light open extension F : D 2 → R 2 F:{D^2} \to {R^2} ? (Recall that light means F − 1 ( y ) {F^{ - 1}}(y) is totally disconnected for all y and open means F maps open sets of the interior of D 2 {D^2} to open sets of R 2 {R^2} .) Because of the work of Stoïlow, the question is equivalent to the following: when does there exist a homeomorphism h : S 1 → S 1 h:{S^1} \to {S^1} , such that fh has an analytic extension to D 2 {D^2} ? (2) Suppose that F : D 2 → R 2 F:{D^2} \to {R^2} is light, open, sense preserving, and, at each point of S 1 {S^1} , F is a local homeomorphism. At each point of the interior of D 2 {D^2} , F is locally topologically equivalent to the power mapping z m {z^m} on D 2 , m ≥ 1 {D^2},m \geq 1 . Points where m > 1 m > 1 are called branch points and m − 1 m - 1 is the multiplicity of the point. There are only a finite number of branch points. The problem is to discover the minimum number of branch points of any properly interior extension of f. Also we can ask what multiplicities can arise for extensions of a given f. (3) Given a normal f, find the maximum number of properly interior extensions of f that are pairwise inequivalent. Since each immersion of the disk is equivalent to a local homeomorphism, the problem of immersion extensions is a special case of this. It is Blank’s solution of the immersion problem that prompted this paper.

Quasi-primal algebras (which include finite simple polyadic and cylindric algebras) were introduced by A. S. Pixley. In this paper equational classes generated hy quasi-primal algebras are investigated with respect to the following concepts: the congruence extension property, the amalgamation property and the amalgamation class, weak injectives and weak injective hulls, the standard semigroup of operators. A brief discussion of monadic algebras is included to illustrate the results of the paper.

In his Introduction to a Form of General Analysist E. H. Moore has called attention to the great importance of developments A in analysis, and has used them in a general theory which includes the theories of continuous functions and convergent series. The authors of the present paper have made further studies of the properties of developments in relation to the theory of Moore.: It is the purpose of the present paper to develop the theory of developments along lines inaugurated by Frechet and developed by him and other investigators. The methods of analysis suggested by this theory have led the authors to results in the Calcul Fonctionnel, some of which have been published previously.? The theory of developments A is placed, in the present paper, into close relation with the theories of systems (V) and (Q3) of Frechet|| and the topological space of Hausdorff. ? We develop the general theory in terms of five completely independent properties of a development A which together suffice to make the developed class $ a compact metric space (cf. Hausdorff, loc. cit.). The theory is applied to determine necessary and sufficient conditions that a topological space be a compact metric space. A further application is made to spaces s satisfying axiom systems :1 or :2 introduced by R. L.

In his paper, G. Shimura quotes a proposition [2, Proposition 2.5] which is derived directly from a transformation formula of theta functions due to KrazerPrym and suggests proving this proposition by means of a generalized Poisson summation formula obtained by A. Weil [4, Theoreme 4]. The purpose of this paper is to execute his idea. Let us explain briefly the result. Let V be a 2n dimensional real vector space and let D be a discrete subgroup of rank 2n in V. Suppose that we have a nondegenerate alternate bilinear form A on V x V which assumes integral values on D x D. The form A represents an integral cohomology class on the torus T= VID. A complex vector space structure J on V induces a complex structure on the torus, which may be denoted by the same notation. Such a complex structure J is said to be admissible if there is a positive divisor on the complex torus (T, J) whose cohomology class is A. Making use of the theory of theta functions on the complex vector space (V, J), we assign to each admissible complex structure J on T a holomorphic map 0(J) of the complex torus (T, J) into a complex projective space P of a certain dimension depending only on the form A, in such a way that the cohomology class A is a rational multiple of the image of the cohomology class of hyperplane sections on P under the cohomology homomorphism 0(J)*. Consider the group S(D) of real linear transformations of V which leave the bilinear form A and the subgroup D invariant. If a E S(D), then a induces an isomorphism a of T onto itself. If J is an admissible complex structure on T, there is a unique admissible complex structure Ja on T such that a: (T, Ja) -(T, J) is a holomorphic isomorphism of these two complex tori. Now, the result asserts that if we make a suitable choice of the assignment 0(J) to each admissible complex structure J and if a belongs to a congruence subgroup in S(D) of sufficiently high level with respect to an appropriate base of D, then the following diagram is commutative

The present paper may be considered as a sequel to our previous paper in the Proceedings of the Cambridge Philosophical Society, Theorie generale de noyaux reproduisants-Premiere partie (vol. 39 (1944)) which was written in 1942-1943. In the introduction to this paper we outlined the plan of papers which were to follow. In the meantime, however, the general theory has been developed in many directions, and our original plans have had to be changed. Due to wartime conditions we were not able, at the time of writing the first paper, to take into account all the earlier investigations which, although sometimes of quite a different character, were, nevertheless, related to our subject. Our investigation is concerned with kernels of a special type which have been used under different names and in different ways in many domains of mathematical research. We shall therefore begin our present paper with a short historical introduction in which we shall attempt to indicate the different manners in which these kernels have been used by various investigators, and to clarify the terminology. We shall also discuss the more important trends of the application of these kernels without attempting, however, a complete bibliography of the subject matter. (KAR) P. 2

Introduction. In the theory of differentiable fibre bundles, with a Lie group as structure group, the notion of a connection plays an important role. In this paper we shall consider complex analytic connections in complex analytic fibre bundles. The situation is then radically different from that in the differentiable case. In the differentiable case connections always exist, but may not be integrable; in the complex analytic case connections may not exist at all. In both cases we are led therefore to certain obstructions, an obstruction to the integrability of a connection in the differentiable case, an obstruction to the existence of a connection in the complex analytic case. It is a basic theorem that, if the structure group is compact, the obstruction in the differentiable case (the curvature) generates the characteristic cohomology ring of the bundle (with real coefficients). What we shall show is that, in a large class of important cases, the obstruction in the complex analytic case also generates the characteristic cohomology ring. Using this fact we can then give a purely cohomological definition of the characteristic ring. This has a number of advantages over the differentiable approach: in the first place the definition is a canonical one, not depending on an arbitrary choice of connection; secondly we remain throughout in the complex analytic domain, our characteristic classes being expressed as elements of cohomology groups with coefficients in certain analytic sheaves; finally the procedure can be carried through without change for algebraic fibre bundles. The ideas outlined above are developed in considerable detail, and they are applied in particular to a problem first studied by Weil [17], namely the problem of characterizing those fibre bundles which arise from a representation of the fundamental group. We show how Weil's main result fits into the general picture, and we discuss various aspects of the problem. As no complete exposition of the theory of complex analytic fibre bundles has as yet been published, this paper should start with a basic exposition of this nature. However this would be a major undertaking in itself, and instead we shall simply summarize in ?1 the terminology and results on vector bundles which we require, and for the rest we refer to Grothendieck [8], Serre [12], and Hirzebruch [9].

0. Introduction and notation. In this paper we study the condition that the maximal right quotient (MRQ) ring Q [10, p. 106] of a right nonsingular ring R with 1 is flat as a left R-module. It is known [11, p. 134] that if Q is the classical right quotient ring of R, then Q is flat as a left R-module. This is not always the case with the MRQ ring of R: in ?2 we obtain an ideal theoretic characterization (Theorem 2.1) and a module theoretic characterization (Theorem 2.2) of a right nonsingular ring R, all of whose regular right quotient rings are flat as left R-modules; we also indicate the existence of a class of commutative rings R, whose singular ideal is zero and for which the maximal quotient ring is not R-flat. Throughout this paper R denotes an associative ring with identity. A right R-module M is denoted MR; all R-modules are unitary. Let NR and MR be modules such that NR. (MR. We say that NR is large in MR (MR is an essential extension of NR) if NR intersects nontrivially every nonzero submodule of MR. A right ideal I or R is large in R if IR is large in RR. For any module MR, L(MR) denotes the lattice of large submodules of MR. Let MR be a module. We denote by Z(MR) the singular submodule of MR. If for any x E M we set (0: x) ={r E R I xr =0}, then

In an earlier paper(') [3] we noted certain identities which connect addition, scalar multiplication, commutation ([ab] = ab ba), and pth powers in an arbitrary associative algebra of characteristic p ( 0). These lead naturally to the definition of a class of abstract algebras called restricted Lie algebras which in many respects bear a closer relation to Lie algebras of characteristic O than ordinary Lie algebras of characteristic p. As is shown in the present paper any restricted Lie algebra S may be obtained from an associative algebra by using the operations mentioned above. In fact 2 determines a certain associative algebra U, called its u-algebra, such that 2 is isomorphic to a subalgebra of U1, the restricted Lie algebra defined by 2; and if Q8 is any associative algebra such that Y13 contains a subalgebra homomorphic to 2 and Q8 is the enveloping algebra of this subset then U is homomorphic to Q3. The algebra U has an anti-automorphism relative to which the elements corresponding to those in 2 are skew. For ordinary Lie algebras an algebra having these properties has been defined by G. Birkhoff [2] and by Witt [5]. In their case however, the associative algebra has an infinite basis even when the Lie algebra has a finite basis whereas here U has a finite basis if and only if 2 has. Consequently every restricted Lie algebra 2 with a finite basis has a (1-1) representation by finite matrices. The theory of representations of 2 can be reduced to that of the associative algebra U. Thus, for example, there are only a finite number of inequivalent irreducible representations. The most natural way to obtain a restricted Lie algebra is as a derivation algebra of an arbitrary algebra 2, i.e., as the set of transformations D: aaD in 2I such that

Introduction. The purpose of this paper is to clarify the structure of finite groups satisfying the following condition: (CN): the centralizer of any nonidentity element is nilpotent. Throughout this investigation we consider only groups of finite order. A group is called a (P)-group if it satisfies a group theoretical property (P). In this paper we shall clarify the structure of nonsolvable (CN)-groups and classify them as far as possible. This goal has been attained in a sense which we shall explain later. If we replace in (CN) the assumption of nilpotency by being abelian we get a stronger condition (CA). The structure of (CA)-groups has been known. In fact after an initial attempt by K. A. Fowler in his thesis [8], Wall and the author have shown that a nonsolvable (CA)-group of even order is isomorphic with LF(2, q) for some q=2n>2. A few years later the author [12] has succeeded in proving a particular case of Burnside's conjecture for (CA)groups, namely a nonsolvable (CA)-group has an even order. Quite recently Feit, M. Hall and Thompson [7] have proved the Burnside's conjecture for (CN)-groups. We can therefore consider groups of even order and focus our attention to the centralizers of involutions. We consider the condition (CIT): (CIT): a group is of even order and the centralizer of any involution is a 2-group. There is no apparent connection between the class of (CN)-groups and the class of (CIT)-groups. But a nonsolvable (CN)-group is a (CIT)-group (Theorem 4 in Part I). This theorem reduces the study of nonsolvable (CN)groups to that of (CIT)-groups. Both properties (CN) and (CIT) are obviously hereditary to subgroups (provided that we consider only subgroups of even order in the case of (CIT)). Although it is true that a homomorphic image of a (CN)-group is also a (CN)-group (this statement is false for infinite groups), it is not an obvious statement. On the other hand it is not difficult to show that a factor group of a (CIT)-group is a (CIT)-group, provided that the order is even. This is due to the following characterization of (CIT)-groups: namely a (CIT)-group is a group of even order containing no element of order 2p with p>2 and vice versa. This makes the study of (CIT)-groups somewhat easier. The large part of this paper concerns the structure of (CIT)groups.

This paper is an attempt at developing a theory of algebraic systems that would correspond in a natural fashion to the No-valued propositional calculus(2). For want of a better name, we shall call these algebraic systems MV-algebras where MV is supposed to suggest many-valued logics. It is known that the classical two-valued logic gives rise to the study of Boolean algebras and, as can be expected, every Boolean algebra will be an MValgebra whereas the converse does not hold. However, many results for Boolean algebras can be appropriately carried over to MV-algebras, although in some cases the proofs become more subtle and delicate. The motivation behind the present study is to find a proof of the completeness of the Novalued logic by using some algebraic results concerning MV-algebras; more specifically, it is known that the completeness of the two-valued logic is a consequence of the Boolean prime ideal theorem and we wish to exploit just some such corresponding result for MV-algebras(3). It will be seen that our effort in duplicating this result is only partially successful. In the first four sections of this paper we present various theorems concerning both the arithmetic in MV-algebras and the structure of these algebras. In the last section we give some applications of our results to the study of completeness of NO-valued logic and some related topics. We point out here that the treatment of MV-algebras as given here is not meant to be complete and exhaustive. 1. Axioms of MV-algebras and some elementary consequences. An MV

Introduction. In his papers [5; 6] Eichler demonstrated the significance for the study of automorphic forms of Bol 's discovery [3 ] of some remarkably simple differential operators taking automorphic forms into automorphic forms. In [6] in particular Eichler discussed a relation between the automorphic forms associated to a transformation group 9 on a Riemann surface D and some purely algebraic constructions involving the group 9, the first cohomology groups of 9 with certain modules of polynomials as coefficients; the cocycles appeared as the periods of the automorphic forms under iterated indefinite integration, generalizing the classical interpretation of the periods of the abelian integrals on D/g (which can of course be considered as automorphic forms on D) as cocycles of the group 9 or alternatively of the space ID/9. The object of studying such a relation is the development of tools for calculating the dimensions of spaces of automorphic forms and the traces of the Hecke operators on automorphic forms. The aim of the present paper is the study of a more general form of this relation in somewhat greater detail for one complex variable, but in such a manner that the results can be extended to several complex variables; the actual extension to several complex variables, as well as the application to the study of the Hecke operators, will be discussed elsewhere. As for the contents of this paper, ? 1 is devoted to an exposition of Bol's differential operators in a form more useful in the present context than that of [3]. In ?2 these differential operators are applied to give an exact cohomology sequence containing, in a rather more transparent form, the relation of Eichler discussed above. The interpretations of the terms appearing in this exact sequence are discussed in ??3 through 5; the only point of difficulty arises in ?4, Theorem 3 of that section really being a form of the Serre duality theorem [12] appropriate to the occasion. These results are combined in ?6 to give a formal statement of the fundamental result of the paper. 1. Differential operators preserving automorphic forms. Let SC be the group of 2 X 2 real matrices

In a paper published in 1906t, Professor G. D. Birkhoff treated the meanvalue and remainder theorems belonging to polynomial interpolation, in which the linear differelntial operator lt(n) played a particular role. It is natural to expect that a generalization of many of the ideas of that paper may apply to the general linear differential operator of order n, and the author is attempting such a program. This generalization throws fundamentally new light on the theory of trigonometric interpolation. A very elegalnt paper by G. Polyae has just appeared treating mean-valute theorems for the general operator in a restricted interval. It is the special aim of the present paper to develop a general mean-value theorem, and to show how it can be specialized to obtain Polya's results. We consider a linear differential expression of order n

This paper is a contribution to the structure and representation theory of finitedimensional cylindric and polyadic equality algebras. The main result is that every singulary algebra is representable, where an algebra is singular y if it is generated by elements supported by singletons. In particular, all prime algebras are representable (a prime algebra has no proper subalgebra, and every cylindric or polyadic equality algebra has a prime subalgebra). Since every polyadic algebra can be embedded in a polyadic equality algebra the results also apply to them. Every infinite-dimensional singulary algebra is locally finite and hence is known to be representable, so we shall be concerned just with the finite-dimensional case, where previously very little was known about representation. It may be mentioned that for any dimension greater than one there are nonrepresentable cylindric and polyadic equality algebras(2). For some special kinds of singulary algebras we obtain a direct construction of the representation, which brings out the structure of the algebra clearly. The methods used essentially constitute an algebraic version of Behmann's solution of the decision problem for the singulary predicate calculus with equality(3). Since only finitely many variables are available, the algebraization has some novel features. The essential ideas can be seen in the case of prime algebras, for which no advanced results about cylindric or polyadic algebras are needed. In the general case, however, use is made of the fact that a cylindric algebra is representable if every finitely generated subalgebra of it is. In the last section of the paper we discuss the logical counterpart of the methods and results of the algebraic part of the paper.

1. The object of this paper is to develop some of the simpler properties of division algebras, that is to say, linear associative algebras in which division is possible by any element except zero. The determination of all such algebras in a given field is one of the most interestiiig problems in the theory of linear algebras. Early in the development of the subject, Frobenius showed that quaternions and its subalgebras form the only division algebras in the field of real numbers and, with the exception of the single theorem that there is no non-commutative division algebra in a filnite field, no further definite result of importance was known till Dickson discovered the algebra referred to in ? 4. It is shown in the present paper that the Dickson algebra is the only noncommutative algebra of order 9 so that the only division algebras of order not greater than 9 are (i) the Dickson algebras of order 4 and 9, (ii) the ordinarv commutative fields, (iii) algebras of order 8 which reduce to a Dickson algebra of order 4 when the field is extended to include those elements of the algebra which are commutative with every other element. ? 2. LEMMA 1. If B is a subalgebra of order b in a division algebra A of order a, there exists a complex C of order c such that

1. Motivation. Many variants of the notion of automaton have appeared in the literature. We find it convenient here to adopt the notion of E. F. Moore [7]. Inasmuch as Rabin-Scott [9] adopt this notion, too, it is convenient to refer to [9] for various results presumed here. In particular, Kleene's theorem [5, Theorems 3, 5] is used in the form in which it appears in [9]. It is often perspicacious to view regular expressions, and this notion is used in the sense of [3]. In general, we are concerned with the problems of automatically designing an automaton from a specification of a relation which is to hold between the automaton's input sequences and determined output sequences. These "design requirements" are given via a formula of some kind. The problems with which we are concerned have been described in [1]. With respect to particular formalisms for expressing "design requirements" as well as the notion of automaton itself, the problems are briefly and informally these: (1) to produce an algorithm which when it operates on an automaton and a design requirement produces the correct answer to the question "Does this automaton satisfy this design requirement?", or else show no such algorithm exists; (2) to produce an algorithm which operates on a design requirement and produces the correct answer to the question "Does there exist an automaton which satisfies this design requirement?", or else show no such algorithm exists; (3) to produce an algorithm which operates on a design requirement and terminates with an automaton which satisfies the requirement when one exists and otherwise fails to terminate, or else show no such algorithm exists. Interrelationships among problems (1), (2), (3) will appear in the paper [1]. This paper will also indicate the close connection between problem (1) and decision problems for truth of sentences of certain arithmetics. The paper [1 ] will also make use of certain results concerning weak arithmetics already obtained in the literature to obtain answers to problems (1) and (3). Thus

An inversive difference field K \mathcal {K} is a field K together with a finite number of automorphisms of K. This paper studies inversive extensions of inversive difference fields whose underlying field extensions are separable algebraic. The principal tool in our investigations is a Galois theory, first developed by A. E. Babbitt, Jr. for finite dimensional extensions of ordinary difference fields and extended in this work to partial difference field extensions whose underlying field extensions are infinite dimensional Galois. It is shown that if L \mathcal {L} is a finitely generated separable algebraic inversive extension of an inversive partial difference field K \mathcal {K} and the automorphisms of K \mathcal {K} commute on the underlying field of K \mathcal {K} then every inversive subextension of L / K \mathcal {L}/\mathcal {K} is finitely generated. For ordinary difference fields the paper makes a study of the structure of benign extensions, the group of difference automorphisms of a difference field extension, and two types of extensions which play a significant role in the study of difference algebra: monadic extensions (difference field extensions L / K \mathcal {L}/\mathcal {K} having at most one difference isomorphism into any extension of K \mathcal {K} ) and incompatible extensions (extensions L / K , M / K \mathcal {L}/\mathcal {K},\mathcal {M}/\mathcal {K} having no difference field compositum).

If a subspace W of ?* is closed with respect to this new multiplication, then W is a Jordan algebra relative to the new multiplication. A special Jordan algebra(2) is a Jordan algebra isomorphic to one obtained from an associative algebra in the above manner. The title of this paper is the same as that of a paper by A. A. Albert [3]. However, in that paper Albert considers only special Jordan algebras, while we prove the principal theorem here for the "general" Jordan algebras defined by commutativity and the identity (1.1). In each case the base field is assumed to be of characteristic 0. Considering the Jordan algebra Q3, we define(3)

This paper is concerned with properties of additive set functions defined on lattices of sets. Extensions of results of Brunk and Johansen, Darst, Johansen, and Uhl are obtained. Two fundamental approximation properties for lattices of sets (established in another paper) permit us to translate the setting and consider countably additive set functions defined on sigma lattices of sets. Thereby results for countably additive set functions defined on sigma lattices of sets are used to obtain alternate derivations and extensions of Darst’s results for additive set functions defined on lattices of sets, i.e., we consider the Radon-Nikodym derivative, conditional expectation, and martingale convergence for lattices of sets.

It is the purpose of this paper to develop the theory of the similarity transformation for matrices whose elements belong to a division algebra. In order to get a basis for generalization, the theory of the similarity transformation for matrices whose elements belong to a field is sketched in what seems to the authors a more suggestive method than those used heretofore.: L. A. Wolf's paper entitled Similarity of matrices in which the elements are real quaternions? treats the case of the quaternion algebra over any subfield of the real field, by passing to the equivalent square matrices of order 2n with elements in the subfield. Some of the results of the present paper are given in an abstract and a subsequent paper by N. Jacobson.II Jacobson's results are to a certain extent more general. In the present paper a usable rational process is given for determining the equivalence or non-equivalence of matrices whose elements belong to a division algebra, and a theorem is developed concerning the rank of a polynomial in a matrix, which is not indicated by Jacobson. Some of the results contained herein were given by Ingraham at the summer meeting of the Society in 1935.?

possesses an almost complex structure. The properties of these manifolds have been investigated by Calabi [6], Gray [11], and Yano and Sumitomo [22]. The almost complex structure, which is a generalization of the one on S6, is defined in terms of a vector cross product on R7, which is a generalization of the ordinary Gibbs vector cross product on R3. In this paper we give a general definition of vector cross product (?2), and then study vector cross products on manifolds. Vector cross products are interesting for three reasons: first, they are themselves natural generalizations of the notion of almost complex structure; secondly, a vector cross product on a manifold M gives rise to unusual almost complex structures on certain submanifolds of M; thirdly, vector cross products provide one approach to the study of Riemannian manifolds with holonomy group G2 or Spin (7). Vector cross ptoducts on vector spaces have been studied from an algebraic standpoint in Brown and Gray [5] and from a topological standpoint in Eckmann [9] and Whitehead [20]. We consider the topological existence of vector cross products on manifolds in ?2. Then in ?3 we give conditions for the existence of vector cross products on vector bundles over CW-complexes in terms of characteristic classes. As an extra feature of our investigations we determine the behavior of the triality automorphism of Spin (8) on Spin characteristic classes. In ?4 we discuss vector cross products from a differential geometric point of view and in ?5 we develop some important relations between vector cross products and curvatures. The rest of the paper (?6 and ?7) is devoted to almost complex structures on manifolds. We show in ?6 that every orientable 6-dimensional submanifold of R8 possesses an almost complex structure. Such an almost complex structure is defined by means of a 3-fold vector cross product on R8. We also show that all of Calabi's manifolds can be obtained by this method. However, the almost complex structures constructed here are more general in several ways. In the first place, we consider 6-dimensional orientable submanifolds M of arbitrary 8-dimensional pseudo-Riemannian manifolds M possessing 3-fold vector cross products, e.g., parallelizable 8-dimensional manifolds. Secondly, the metric of M we use may be indefinite provided it has signature (4, 4). (Here we

Of the many theorems contained in STURM'S famous memoir in the first volume of Liouville's Journal (1836), p. 106, two, which I have called the Theorems of Comparison, may be regarded as most fundamental. I have recently shownt how the methods which STURM used for establishing these theorems can be thrown into rigorous form. In the present paper I propose to prove these theorems by a simplert and more direct method. This method was suggested to me by a passage, to which Professor H. BURKHARDT kindly called my attention, in one of D'ALEMBERT' S papers on the vibration of strings.? D'ALEMBERT'S fundamental idea, and indeed all that I here preserve of his method, consists in replacing the linear

This paper has arisen from attempts to sharpen Specker's result, which is sharpened in 7.1 and 7.2 (first cf. the definition of H(m; a) in ?7 prior to 7.1). The writer's efforts along these lines led to developments in the theory of numerations (defined in the first paragraph of ?2) of independent interest, and most of this paper is concerned with these developments. The content of this paper may be developed in an axiomatic set theory of the von Neumann-Bernays-Godel kind (cf., e.g., [1]) modified as follows to allow (but not to imply the existence of) elements which are not sets. Each object is either an element or a class. A set is any element which is a class. An atom is an element which is not a class. The usual axioms may be modified in the obvious way to accommodate atoms. We shall assume all the usual axioms so modified except the restrictive axiom and the axiom of choice (cf. ?7). We indicate briefly our use of some terminology and notation. Elements

In this paper we construct, for an arbitrary principal bundle (with group a real Lie group) over a differentiable manifold, a sheaf of germs of linear differential operators of first order operating on the sheaf of germs of sections of any differentiable vector bundle associated with the given principal bundle. The commutators of these operators define a structure of Lie algebra (over the real numbers) on the operator sheaf. By considering the sheaves of germs of differential forms on the manifold with values in appropriate vector bundles, we obtain a graded sheaf of operators carrying a structure of graded Lie algebra (over the real numbers). In many respects, the present work represents a generalization of the paper of Frolicher and Nijenhuis [2] which characterized the derivations (of all degrees) of the exterior algebra of real-valued differential forms on a differentiable manifold. The case considered by these authors is obtained by choosing the associated vector bundle to be the product bundle with fibre the real numbers. In addition, the usual theory of covariant differentiation, corresponding to a connection in the given principal bundle, appears as a special case. It is hoped that the theory developed here will also have applications in the systematic study of deformation of the structures defined on a differentiable manifold by continuous pseudogroups of transformations. 1. Fundamental sequence of vector bundles. Let M be a differentiable (i.e., C*) manifold and let P-4M be a differentiable principal bundle with group G, where G is a real Lie group. Let T(M), T(P), and T(G) denote the bundle spaces of the tangent bundles to M, P, and G respectively. Then T(G) is also a Lie group, and T(P)--T(M) is a principal bundle with group T(G) [3 ]. Since G is a subgroup of T(G), we may form the quotient bundle T(P)/G --T(M), and we have

This paper is a continuation of [14], developing the representation theory of finite semigroups further. The main result, Theorem 1.24, states that if the group of units U of a mapping semigroup (X, S) is multiply transitive with a sufficiently high degree of transitivity, then for certain irreducible characters χ \chi of U, χ \chi can be “extended” formally to an irreducible character of S. This yields a partial generalization of a well-known theorem of Frobenius on the characters of multiply-transitive groups and provides the first nontrivial explicit formula for an irreducible character of a finite semigroup. The paper also contains preliminary results on the “spectrum” (i.e., the set of ranks of the various elements) of a mapping semigroup.

1. Introduction and summary of results. It is well known (Bers [1], Nakai [5]) that an open Riemann surface is determined up to mirror image by its ring of holomorphic functions. Thus, it is quite natural to expect that one can find an algebraic characterization of the ring of holomorphic functions on an open Riemann surface. Also, one expects that all the information about the conformal structure of an open Riemann surface can be recovered from the ring of holomorphic or the field of meromorphic functions on it. The purpose of this paper is to investigate the relationships between the conformal structure of an open Riemann surface and the algebraic structure of the ring of holomorphic functions on it. Our main result (Theorem III) characterizes conformal structure in terms of simple algebraictopological concepts. In ?5 we obtain a purely algebraic characterization of certain rings of holomorphic functions. It is not as simple or as elegant as one would like. Alternate algebraic characterization of rings of holomorphic and fields of meromorphic functions are discussed in ?6. One of the characterizations is due to the referee. The hypotheses used in the referee's characterization are simpler and more natural than those appearing in any of the other algebraic characterizations discussed in this paper. The author thanks the referee for pointing out this alternate approach and for the various suggestions which have simplified the presentation of this paper. Any nontrivial holomorphic mapping F: X-? Y between two open Riemann surfaces induces a C-monomorphism F': H(Y) -? H(X) between their rings of

Introduction. The aim of this paper is to develop further the algebraic theory of minimal sets begun in [6] and [7] and to apply it to obtain a structure theorem for the collection of group-like extensions of a given minimal set. When this minimal set is taken to be a single point, the resulting theorem reduces to the one obtained by Furstenberg [8]. The paper is divided into six sections. In ??1 and 2 the general algebraic theory is developed. Since I feel that this theory can be fruitfully applied to other problems in topological dynamics besides the one considered here, I have tried to make this exposition self-contained. Thus much of the discussion contained in these two sections is to be found in [6] and [7]. However, some of the notation has been changed for purposes of simplification. The new material, herein, is concerned with the construction and study of various topologies on a certain subgroup of the P-compactification of an arbitrary abstract group. These are the so called topologies of which brief mention was made in [7]. ?3 is a collection of results from various papers inserted in order to make the overall exposition self-contained. In ?4 the algebraic theory is applied to develop a structure theory of group-like extensions. The relation of this structure theory to that of Furstenberg for distal minimal sets [8] is exhibited. This is done by means of the notions of a principal extension and a principal bi-transformation group (analogous to a principal fiber bundle) introduced in this section. The main proposition of this paper is 4.14 which is a general statement about principal group extensions. When applied to the situation at hand it yields a generalization of the Furstenberg results. In ?5 the main proposition is applied to some special cases. Here the Bohr compactification of the abstract group T is exhibited in terms of the algebraic theory previously described. During the course of this paper and also in [6] and [7] various seemingly arbitrary choices are made. ?6 is devoted to showing that these choices are indeed natural.

0. Introduction. Let V be a nonelementary finitely generated Kleinian group, and A1 a simply connected component of the region of discontinuity Q of P. M. Eichler [4], L V. Ahlfors [2], L. Bers [31 and I. Kra [5], [6] have represented periods of Eichler integrals as polynomials of degree at most 2q 2, q> 2 being an integer. By this method, however, period relations for Eichler integrals are very complicated even when 1 is a Fuchsian group of the first kind (Eichler 141). On the other hand, G. Shimura [71 has regarded the periods as column number vectors of length 2q 1. In his paper he gave a certain period relation for Fuchsian groups. By using Shimura's idea, we shall give period relations and inequalities for Eichler integrals for Kleinian groups. These are a generalization of those for abelian integrals. The main results in this paper are Theorems 1 and 2. We shall state some notations in ?I and some lemmas in ?2. In ?3 we shall prove Theorem 1 and in ?4 we shall state the period relations and inequalities, and give an alternate proof for the Kra result [S]. The author wishes to express his deep appreciation to Professor I. Kra, K. Mathumoto and K. Oikawa for encouragement and advice.

of analytic devices all play a role. It will be seen, however, that the functionals making up the algebras under study have obvious representations as integrals with respect to countably additive, totally finite measures that vanish except on countable sets. Thus the refinements of measure and integration theory are avoided, nothing more recondite than infinite series being needed for all of the integral computations employed. The great majority of our results deal only with the commutative case: in view of the genuine difficulties connected with harmonic analysis even on noncommutative groups and the obvious fact that semigroups are less tractable than groups, no apology is perhaps required for this. The present paper may be described as an introduction to harmonic analysis on discrete commutative semigroups. It will no doubt be noted that we state no theorems concerning Tauberian theorems, analogues of the Pontryagin duality theorem, or the Silov boundary. We hope to deal with these topics in a subsequent communication. 1.2. Throughout this paper, we use the terminology, notions, and results of [3 ]. A few points, however, require re-statement, and we make a few new definitions. The symbol "-" is used between semigroups, algebras, etc., to denote the existence of a 1-to-1 correspondence 7r preserving all operations of both systems; and if a product, say, is defined only for some pairs, then the equality 7r(xy) =7r(x)7r(y) is to hold whenever either side is defined. A homomorphism of a semigroup G onto a semigroup H is a single-valued mapping ,u of G onto H such that ,u(xy) =,(x),u(y) for all x, yCG. 1.3. A multiplicative function on a semigroup G is any complex function f on G satisfying the functional equation f(xy) =f(x)f(y) for all x, yCG. A

Let K K be a proper rectilinear triangulation of a 2 2 -simplex S S in the plane and L ( K ) L(K) be the space of all homeomorphisms of S S which are linear on each simplex of K K and are fixed on Bd ( S ) \text {Bd}(S) . The author shows in this paper that L ( K ) L(K) with the compact open topology is simply-connected. This is a generalization of a result of S. S. Cairns in 1944 that the space L ( K ) L(K) is pathwise connected. Both results will be used in Part II of this paper to show that π 0 ( L 2 ) = π 1 ( L 2 ) = 0 {\pi _0}({L_2}) = {\pi _1}({L_2}) = 0 where L n {L_n} is a space of p.l. homeomorphisms of an n n -simplex, a space introduced by R {\mathbf {R}} . Thom in his study of the smoothings of combinatorial manifolds.

Introduction. The present paper is a continuation and amplification of R. M. Thrall's paper On symmetrized Kronecker powers and the structure of the free Lie ring [7 ]J(1), which we shall denote as FR. The notation used in FR shall be adopted in this paper. The author wishes to express her appreciation to Professor R. M. Thrall of the University of Michigan, who suggested the probl'em of the present paper and gave valuable guidance during the preparation of it. We propose to study the structure of the free Lie ring, its characteristic ideals, and certain related right ideals in the group ring of the symmetric group. We operate over a field K of characteristic zero and thus insure that the named group ring be semi-simple (which is not the case for fields of characteristic p). In FR a recursion formula [7, p. 386] was developed from which the irreducible constituents of the mth Lie representation for m ? 10 were obtained. The main result of the present investigation is Theorem III (?2) which gives a direct formula for the character of the mth Lie representation. 1. The free Lie ring and Lie representations of the full linear group. The free non-associative K ring Zn, the free Lie ring L =Ln, and the mth Lie representation are defined in FR [7, pp. 372-373]. It is convenient to make the definition of ideal more explicit than in FR. We begin with the concept of a homomorphism of a ring Zn onto a ring Z *, and then call a subset J of Zn an ideal if it is the kernel of some homomorphism. A homogeneous ideal is defined in FR [7, p. 372 ]. The following result was stated without proof in FR [7, p. 372].

This method is based on the Cartan-Leray spectral sequence, [3; 1 ], and can be generalized to other algebraic situations, as will be shown in a forthcoming paper of Cartan-Eilenberg [2]. Since the details of the Cartan-Leray technique have not been published (other than in seminar notes of limited circulation), we develop them in Chapter I. The auxiliary theorems we need for this purpose are useful also in other connections. In Chapter II, which is independent of Chapter I, we obtain a spectral sequence quite directly by filtering the group of cochains for G. This filtration leads to the same group E2=H(G/K, H(K)) (although we do not know whether or not the succeeding terms are isomorphic to those of the first spectral sequence) and lends itself more readily to applications, because one can identify the maps which arise from it. This is not always the case with the first filtration, and it is for this reason that we have developed the direct method in spite of the somewhat lengthy computations which are needed for its proofs. Chapter III gives some applications of the spectral sequence of Chapter II. Most of the results could be obtained in the same manner with the spectral sequence of Chapter I. A notable exception is the connection with the theory of simple algebras which we discuss in ?5. Finally, let us remark that the methods and results of this paper can be transferred to Lie Algebras. We intend to take up this subject in a later paper.

The main theorems of this paper show that certain conditions (called λ c , λ b , β c {\lambda _c},{\lambda _b},{\beta _c} , and β b {\beta _b} ) are invariant, in the presence of T 0 {T_0} -regularity, under the application of closed continuous peripherally compact mappings. Interest in these conditions lies in the fact that they may be used to characterize certain regular T 0 {T_0} open continuous images of some classes of M M -spaces in the sense of K. Morita, and in the fact that they are preserved by open continuous mappings with certain appropriate additional conditions. For example, the authors have shown that a regular T 0 {T_0} -space is an open continuous image of a paracompact Čech complete space if and only if the space satisfies condition λ b {\lambda _b} [Pacific J. Math. 37 (1971), 265-275]. Moreover, in the same paper it is shown that if a completely regular T 0 {T_0} -space satisfies condition λ b {\lambda _b} then any T 0 {T_0} completely regular open continuous image of it also satisfies λ b {\lambda _b} . These results together with the results of the present paper and certain known results lead to the following theorem: The smallest subclass of the class of regular T 0 {T_0} -spaces which contains all paracompact Čech complete spaces and which is closed with respect both to the application of perfect mappings and to the application of open continuous mappings preserving T 0 {T_0} -regularity is the subclass satisfying condition λ b {\lambda _b} . Similar results are obtained for the regular T 0 {T_0} -spaces satisfying λ c , β b {\lambda _c},{\beta _b} , and β c {\beta _c} . The other classes of M M -spaces involved are the regular T 0 {T_0} complete M M -spaces (i.e., spaces which are quasi-perfect preimages of complete metric spaces), T 2 {T_2} paracompact M M -spaces, and regular T 0 M {T_0}M -spaces. In the last two cases besides the inferiority of the mappings the notion of uniform λ \lambda -completeness, which generalizes compactness of a mapping, enters. (For details see General Topology and Appl. 1 (1971), 85-100.) The proofs are accomplished through the use of two basic lemmas on closed continuous mappings satisfying certain additional conditions.

1. Introduction and summary. The theory of hierarchies deals with the classification of objects according to some measure of their complexity. Such classifications have been fruitful in several areas of mathematics: analysis (descriptive set theory), recursion theory, and the theory of models. Although much of the hierarchy theory of each of these areas was developed independently of the others, Addison, in the series of papers [Ad 1-6], has shown not only that there are deep-seated analogies among these theories, but that indeed many of their results can be derived from those of a general theory of hierarchies. Toward a further consolidation of these theories, this paper will study the relationships and analogies between certain classical hierarchies of descriptive set theory and their counterparts in recursion theory. The roots of modern hierarchy theory lie in the investigations of Baire, Borel, Lebesgue, and others around the turn of the century. As analysts with a concern for the foundations of their subject, they felt that constructions effected by means of the axiom of choice or the set of all countable ordinals were less secure than those carried out by more elementary means. They sought to discover what role these suspect constructions played in analysis and whether or not they could be avoided altogether. Thus descriptive set theory arose with the goal of identifying, classifying, and studying those sets (of real numbers) which were of interest for analysis and for which an "explicit" construction could be given. Needless to say, there was vigorous disagreement as to just what constituted an explicit construction. The first large class of sets studied were the Borel sets. Since each Borel set can be constructed by iteration of the elementary operations of countable union and

On account of its simplicity, its immediate applicability to many functions already in existence, and its suggestiveness with regard to generalizations of known functions or the creation of new, the algebra in question merits the somewhat detailed exposition in Part I. As its use is perhaps best seen from an example, we have sketched briefly in Part II the outlines of a new theory of the relations between the Bernoullian and Eulerian functions of any arguments and any ranks, showing that by means of the algebra all necessary computations are reduced to a minimum. These functions are of two variables, of which one is complex and the other, the rank, a positive integer. The algebra establishes a simple isomorphism between the theory of relations between the functions and the like for the ordinary sine and cosine. (It is necessary here to distinguish between ordinary and umbral circular functions; the nature of the distinction appears presently.) In a paper which I expect to publish later there is a more detailed application of this algebra to certain new functions, suggested by the algebra, of three variables, of which two are complex and the third a positive integer. For unit values of one of the complex variables these functions degenerate to the Bernoullian and Eulerian functions of Part II; for unit values of the other, they become certain polynomials, of considerable importance in the arithmetical theory of quadratic forms, discussed on several occasions by Hermite, Weierstrass and, more recently, by Bulygain and Gruder. In this application there are simple isomorphisms with the circular, the hyperbolic and the elliptic functions. In a third paper, to appear shortly, I have shown how the algebra gives at once the complete theory of the relations between the functions of Spitzer, which include as special cases the Bessel coefficients. Here there is simple isomorphism with the exponential function. By a generalization of the exponential function, Emxm+;/r(m+v), in which x, v are complex, I have shown in a paper not yet published that the algebra is readily applicable to the Bessel functions. The algebra therefore is of considerable utility. Its chief use, how

In this work we study flat modules over commutative noetherian rings under two kinds of restriction: that the modules are either submodules of free modules or that they have finite rank. The first ones have nicely behaved annihilators: they are generated by idempotents. Among the various questions related to flat modules of finite rank, emphasis is placed on discussing conditions implying its finite generation, as for instance, (i) over a local ring, a flat module of constant rank is free, and (ii) a flat submodule of finite rank of a free module is finitely generated. The rank one flat modules already present special problems regarding its endo- morphism ring; in a few cases it is proved that they are flat over the base ring. Finally, a special class of flat modules—unmixed—is discussed, which have, so to speak, its source of divisibility somewhat concentrated in the center of its endo- morphism ring and thus resemble projective modules over flat epimorphic images of the base ring. Introduction. This paper is an attempt to lay a somewhat limited framework for the study of flatness on modules over noetherian rings. As already for abelian groups the general description of the torsion-free modules is practically hopeless, we shall not be too demanding here and shall concentrate in those aspects of the theory which best reflect the nature of the ring and the universal algebra involved. In this vein, after some preliminaries on the rank of flat modules and a generaliza- tion of a theorem of (4) on how well the finite generation of a high nonzero exterior power of a module may disclose the finite generation of the module itself, we discuss : annihilators of flat submodules of free modules, which, in general, have a good behavior in comparison to annihilators of arbitrary flat modules; various con- ditions disguising the finite generation of a flat module; flat modules of rank one and its endomorphism rings—which are, in many cases, flat epimorphic images of the base ring; a discussion of a class of flat modules—unmixed modules—where the source of divisibility lies, in general, in the complement of an affine subschema of the spectrum of the ring. The paper concludes with a list of unsolved problems whose solutions might yield new methods or, more hopefully, uncover other

I present this paper for publication to an American journal and in English for the following reason: The theory of linear algebras has been greatly extended through the work of American mathematicians. Of late, German mathematicians have become active in this theory. In particular, they have succeeded in obtaining some apparently remarkable results by using the theory of algebraic numbers, ideals, and abstract algebra, highly developed in Germany in recent decades. These results do not seem to be as well known in America as they should be on account of their importance. This fact is due, perhaps, to the language difference or to the unavailability of the widely scattered sources. This paper develops a new application of the above mentioned theories to the theory of linear algebras. Of particular importance is the fact that purely algebraic results are obtained from deep-lying arithmetical theorems. In the middle part, an account is given of the fundamental algebraic basis for these arithmetical methods. This account is more extended than is necessary for this paper, and should obviate an extended study of several German papers. I am very grateful to Professor H. T. Engstrom (New Haven) for going through the manuscript and proof with me and anglicising my many literal translations from the German.

Introduction. In this paper a series of propositions are given concerning commutative algebras of operators on a Banach space and more especially commutative algebras of scalar operators. A number of the results are known and due to W. G. Bade [1; 2] but different proofs are given here. The inspiration for this paper, both in the choice of the subject matter and of method, has largely been derived from [2; 7] and [10]. The material presented here is divided into four paragraphs. The first, which is introductory, contains various results on spectral families of measures. The principal propositions are contained in paragraphs 2 and 3. Theorem 1, proved in paragraph 2, is a generalisation of a theorem due to W. G. Bade [2], and almost all the other results of this same paragraph are more or less consequences of it. In paragraph 3 it is shown that, under certain conditions, an algebra of scalar operators can be identified, in a sense made precise below, with a von Neumann algebra. This fact makes it possible to reduce many results concerning algebras of scalar operators or a-complete boolean algebras of projections, in Banach spaces, to corresponding results in Hilbert spaces. Various remarks on spectral families of measures are made in paragraph 4.

This paper deals with the problem of telling whether a given elementary function, in the sense of analysis, has an elementary indefinite integral. In ?1 of this work, we give a precise definition of the elementary functions and develop the theory of integration of functions of a single varia' Z. By using functions of a complex, rather than a real variable, we can limit ourselves to exponentiation, taking logs, and algebraic operations in defining the elementary functions, since sin, tan- 1, etc., can be expressed in terms of these three. Following Ostrowski [9], we use the concept of a differential field. We strengthen the classical Liouville theorem and derive a number of consequences. ?2 uses the terminology of mathematical logic to discuss formulations of the problem of integration in finite terms. ?3 (the major part of this paper) uses the previously developed theory to give an algorithm for determining the elementary integrability of those elementary functions which can be built up (roughly speaking) using only the rational operations, exponentiation and taking logarithms; however, if these exponentiations and logarithms can be replaced by adjoining constants and performing algebraic operations, the algorithm, as it is presented here, cannot be applied. The man who established integration in finite terms as a mathematical discipline was Joseph Liouville (1809-1882), whose work on this subject appeared in the years 1833-1841. The Russian mathematician D. D. Mordoukhay-Boltovskoy (1876-1952) wrote much on this and related matters. The present writer received his introduction to this subject through the book [10] by the American J. F. Ritt

Introduction. 1. In a paper presented to this Society at a previous meeting, t the general structure of linear associative algebra was discussed and certain fundamental theorems of great generality were proved. The present paper is an application of these theorems to the study of what may be called group algebras. By a group algebra is meant that linear algebra whose units are defined to be such that each unit et corresponds to an operator Oi of some given abstract finite group, f and conversely, and such that for each equation of the group 0 0. = 0, corresponds an equation eze. = ek of the algebra. From the syinbolic point of view the algebra differs from the group only in that expressions-for brevity, let us say, numbers-of the form Ex. e. are possible, wherein the coefficients x, are any scalars. ? That this algebra is linear and associative, is obvious from the definition. When no confusion is feared, the notations and terminology of the group and of the algebra will be used interchangeably. 2. In the paper cited as Theory is developed the theorem that the numbers of a linear associative algebra are subject to the laws of matrices, and certain conclusions are drawn from this fact. This method of development enables us to make immediate use of any theorem needed which is true of matrices, and so saves a redevelopment of many such theorems. In the present paper I shall consider the numbers as multiple algebraic entities, referring to the theory of matrices only when some needed theorem is to be translated into an algebraic theorem. 3. In Part 1 of the paper the general form of any grotup algebra is to be dis. cussed and certain general theorems established. In Part 2 a few special cases

In the present paper we present a new method for studying conjugate nets, which possesses many advantages over those employed hitherto. We first refer the sustaining surface to its asymptotic net. We then find, by referring the surface to any one of its conjugate systems as a parametric net, that all of the projective properties of this net are expressible in terms of those quantities which determine the sustaining surface and one other function, which may be chosen arbitrarily, and which then determines the most general conjugate net on the surface. Thus one can tell at a glance which properties of a conjugate net are really peculiar to the net, and which others are due to the character of the sustaining surface. This paper contains, as applications of the method, besides some other things, the demonstrations of a number of new theorems recently discovered by Wilczynski, and discussed by him orally at the meeting of the Society at Chicago in December, 1920. He has withdrawn his own proofs in favor of those here presented on account of the great simplification accomplished thereby.t The method which is developed in this paper was suggested by one of G. M. Green's memoirs, entitled a Memoir on the general theory of surfaces and rectilinear congruences.1 We have in fact preserved the notation which Green used in section 16 of his paper, concerning General theorems on conjugate nets.