Showing papers in "Transactions of the American Mathematical Society in 1961"

Decision problems of finite automata design and related arithmetics

Abstract: 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

The $G$ and $H$ functions as symmetrical Fourier kernels

Abstract: The kernels are said to be symmetrical if k(x)=h(x) and unsymmetrical if k(x)9£h(x). The symmetrical case only will concern us here. Various sets of conditions have been discovered which ensure the validity of (1), (2), the set we use here consists of convergence conditions on/(x), k(x) and h(x) together with a functional equation satisfied by the Mellin transforms of k(x) and h(x). K(s) is said to be the Mellin transform of k(x) if

Semi-inner-product spaces

Abstract: function as a particular Banach space (whose norm satisfies the parallelogram law), but rather as an inner-product space. It is in terms of the innerproduct space structure that most of the terminology and techniques are developed. On the other hand, this type of Hilbert space considerations find no real parallel in the general Banach space setting. Some time ago, while trying to carry over a Hilbert space argument to a general Banach space situation, we were led to use a suitable mapping from a Banach space into its dual in order to make up for the lack of an innerproduct. Our procedure suggested the existence of a general theory which it seemed should be useful in the study of operator (normed) algebras by providing better insight on known facts, a more adequate language to "classify" special types of operators, as well as new techniques. These ideas evolved into a theory of semi-inner-product spaces which is presented in this paper (together with certain applications)(1). We shall consider vector spaces on which instead of a bilinear form there is defined a form [x, y] which is linear in one component only, strictly positive, and satisfies a Schwarz inequality. Such a form induces a norm, by setting |x| = ([x, x])112; and for every normed space one can construct at least one such form (and, in general, infinitely many) consistent with the

The core of a cooperative game without side payments

Abstract: The core of an M-person game, though used already by von Neumann and Morgenstern [15], was first explicitly defined by Gillies [5]. Gillies's definition is restricted to cooperative games with side payments and unrestrictedly transferable utilities(2), but the basic idea is very simple and natural, and appears in many approaches to game theory. We consider a certain set of \"outcomes\" to a game, and define a relation of \"dominance\" (usually not transitive) on this set. The core is then defined to be the subset of outcomes maximal with respect to the dominance relation; in other words, the subset of outcomes from which there is no tendency to move away—the equilibrium states. To turn this intuitive description of the core notion into a mathematical definition, we need precise characterizations of (a) the kind of game-theoretic situation to which we are referring (cooperative game, noncooperative game, etc.); (b) what we mean by \"outcome\"; and (c) what we mean by \"dominance.\" Different ways of interpreting these three elements yield different applications of the generalized \"core\" notion, many of them well-known in game theory. Gillies's core, Luce's ^-stability [lO], Nash's equilibrium points [12], Nash's solution to the bargaining problem [l3](3), and the idea of Pareto optimality—to mention only some of the applications—can all be obtained in this way. Here we shall be concerned exclusively with cooperative games without side payments(4). Our procedure will be to generalize von Neumann's fundamental notion of characteristic function to this case, and on the basis of this generalization to define the core in a way that generalizes and parallels the core in the classical theory—i.e., Gillies's core. The generalization of the characteristic function is of interest for its own sake also; for example, a theory of \"solutions\" has been developed that generalizes and parallels the classical theory of solutions and is based on the characteristic function [3; 16].

On the distribution of first hits for the symmetric stable processes.

Abstract: Here t>0, x and t are points in RN, dS is N-dimensional Lebesgue measure, (x, t) is the usual inner product in RN, and j | 2=Q(, t). Throughout this paper integrals will be over all of RN unless explicitly stated otherwise. Of course, to determine our process we must also specify the distribution of X(O). We will always assume that our process starts from some fixed point x in RN; that is, X(O) = x with probability one. We will write P. and E. for probabilities and expectations under the condition X(O) = x. We will assume that the sample functions are normalized to be right continuous and to have left limits everywhere. See [2, ?2] for a complete description of this setup. Define

First passage times for symmetric stable processes in space

Abstract: Here £ and x are points in RN, dx is A-dimensional Lebesgue measure, (x, £) is the usual inner product in RN, and |a:|2 = (*, x). Throughout this paper integrals will be over all of RN unless explicitly indicated otherwise. If f>0 define TYr) = inf {í : \\X(t)\\ >r). Thus T(r) is the first passage time of the process X(t) to exterior of the ball {x: \\x\\

The cohomology ring of a finite group

Abstract: Using the classifying space of the group, Venkov [9] has provided a short and elegant proof of the finite generation of H*(G, Z) for G in a class of groups including the class of finite groups. (Actually Venkov proves a theorem for coefficients in Z/pZ. However, if emphasis is placed on modules with ascending chain condition rather than on finitely generated rings, it is easy to modify his proof so as to avoid restriction of the coefficient domain.) In this paper I present a proof of the finite generation of H*(G, Z) for G a finite group; this proof is independent of the above mentioned efforts and its main interest is that it is completely within the cohomology theory of groups proper-i.e., it is purely algebraic. In such a context it is natural to generalize the theorem as to coefficient domain so as to prove the "natural" theorem for the theory. I also derive certain consequences of this theorem which are of interest in themselves. The proof is as follows. The problem is reduced from an arbitrary finite group to a p-group by the Sylow subgroup argument in cohomology suitably modified. For the case of a p-group it is possible to proceed inductively by use of the spectral sequence of a group extension. One may prove successively that each E, r > 2, is a finitely generated ring (or, equivalently, a noetherian ring), but whether or not this remains true for the E. term is not determined. The problem is reduced thereby to showing that the spectral sequence stops; i.e., Er = E for some r < X . In the special case that the normal subgroup is cyclic and in the center of the group this amounts to showing that restriction to that subgroup is surjective in some positive even dimension. This basic lemma is proved by a group theoretical argument. The group is imbedded in a certain wreath product so that the central cyclic subgroup gets carried into the center. This reduces the problem to a certain computation for the cohomology of wreath products. Some remarks about notation are necessary. The symbol E will always refer to direct sums unless otherwise stipulated. 0 will refer to tensor product over the ring of integers. An asterisk (*) in place of an index will indicate

Symmetrization of rings in space

Abstract: holds. We then estimate mod R' either by means of the space analogues of the Grötzsch and Teichmüller rings or by means of spherical annuli. The two bounds we obtain are given in Theorem 3 of §17 and in Theorem 4 of §22. In a later paper we will show how these upper bounds can be used to derive a number of important distortion theorems for quasiconformal mappings in space. For a summary of these results see [4].

On the definition and properties of certain variational integrals

Abstract: Here R denotes an open set in the number space En and u = u(x) is a realvalued function defined on R. The integrand f(x, it, p) is assumed to be nonnegative and continuous. Moreover, we suppose throughout the paper that f is convex in p so that I[u] is the integral of a regular variational problem. For the purposes of the calculus of variations, and also for aesthetic reasons, it is natural to want the class of admissible functions u to be as large as possible. Now I[u], as it stands, is certainly well-defined for continuously differentiable functions, but once we go beyond this class there is some question as to the meaning of the integral. If measurable partial derivatives can be associated with u, then one can define I[u] simply as the Lebesgue integral of f(x, u, ut;). This procedure cannot be used indiscriminately, however, for it assigns the absurd value ff(x, u, O)dx to any nonconstant function u whose partial derivatives are zero almost everywhere. As an alternate definition of the integral, we have introduced in [13] a certain lower semicontinuous functional which in general agrees with I[u] whenever u is continuously differentiable, but which at the same time is defined for a much larger class of functions. For convenience in discussing these two integrals the former will be denoted simply by I[u] and the latter by g[u], (a formal definition of these quantities will be given in ?1). Both functionals I[i] and 4 [u] are of interest in the calculus of variations; it is the purpose of this paper to clarify the relation between them. An important illustration of the present situation may be found in the theory of area of a nonparametric surface. Indeed, let us denote by a [u] the Lebesgue area of a surface z=u(x, y) over a region R in the ordinary (x, y) plane, and set

Locally compact transformation groups(

Abstract: In ?1 of this paper it is shown that a variety of conditions implying nice behavior for topological transformation groups are, in the presence of separability, equivalent. In ?2 the continuity properties of the stability subgroups are studied. The conditions of ?1 exclude the line acting on the torus in such a way that each orbit is dense. They exclude the integers acting on the circle by rotation through multiples of an irrational angle and they exclude the group of those sequences of zeros and ones which have all but a finite number of their terms equal to zero when this group acts on the space of all sequences of zeros and ones by coordinatewise addition (mod 2). As we shall see in the proof of Theorem 1, the latter transformation group is a prototype for all excluded transformation groups. This is analogous to the following fact in the theory of Rings of Operators: Every factor of type II, contains a hyperfinite factor of type II,. The conditions were suggested by [3, Theorem 1] and the proof of their equivalence is somewhat analogous to the proof of [3, Theorem 1]. However, the proof does not depend upon [3] nor upon the theory of C*-algebras.

Finite groups with nilpotent centralizers

Abstract: 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.

On quotient varieties and the affine embedding of certain homogeneous spaces

Abstract: We recall that if an algebraic group G operates regularly on a variety V, by a quotient variety is meant a pair (V/G, r), where V/G is a variety and r: V-* V/G is a rational map, everywhere defined and surjective, such that two points of V have the same image under r if and only if they have the same orbit on V, and such that, for any xE V, any rational function on V that is G-invariant (i.e., constant on orbits) and defined at x is actually (under the natural injection of function fields Q(V/G) -*Q(V), Q denoting the universal domain) a rational function on VIG that is defined at rx (cf. [1, expose 8]). Q(V/G) must therefore consist precisely of all G-invariant elements of Q(V), so r is separable. A quotient variety need not exist (obvious necessary condition: all orbits on V must be closed), but when it exists it is clearly unique to within an isomorphism; in this case, for any open subset UC V/G, T-1 U/G exists and equals U. PROPOSITION 1. Let the algebraic group G operate regularly on the variety V, all defined over the field k. Suppose there exists a quotient variety r: V-* V/G. Suppose also that for each point p of V that is algebraic over k there exists an open affine subset of V/G containing the image under r of each of the conjugates of p over k (a vacuous condition if V/G. can. be embedded in a projective space or if V/G and r are known to be defined over a regular extension of k, in particular if k is algebraically closed). Then V/G and r could have been taken so as to be defined over k. The G-invariant elements of Q(V) are generated by those in k(V), in other words there exists a variety W and a generically surjective rational map V-)W, both defined over k, such that for any field K between k and Q, K(W) is the field of G-invariant elements of K(V) [3, Theorem 2]. We have here a field descent problem, and supposing V/G and r to be defined over the extension field K of k, there are two cases to consider: K a regular extension of k, and K algebraic over k. In view of the unicity to within isomorphism of the quotient variety, the criteria of Weil [6] take care of the first case. [Of course this can also be done directly; e.g., supposing k algebraically closed, if

A convergence equivalence related to polynomials orthogonal on the unit circle

Abstract: Presented to the Society, December 27, 1960 under the title On the asymptotic behavior of Toeplitz determinants, and January 9, 1961 under the title On the asymptotic behavior of Toeplitz determinants. II; received by the editors January 23, 1961. (') This work was supported by the United States Air Force through the Air Force Office of Scientific Research of the Air Research and Development Command, under contract No. AF AFOSR-61-4. Reproduction in whole or in part is permitted for any purpose of the United States Government. 471

Filtered algebras and representations of Lie algebras

Abstract: Introduction. There is a general question as to how much can be said about a filtered object through the knowledge of its associated graded object. We consider here a particular case of this general problem. We take the symmetric algebra S(L) of a free K-module L and look for filtered K-algebras whose associated graded algebras are isomorphic to S(L). Some such algebras are already known. In fact if g denotes an arbitrary Lie algebra on L, the "Poincar6-Witt Theorem" asserts that the universal enveloping algebra of g is one such. It turns out that this gives "almost" a general solution of our problem. Indeed, the algebras we seek are suitable generalizations of the usual enveloping algebras and can in fact be defined as universal objects for certain "generalized representations" of Lie algebras on L. The rest of our results are on the cohomology of these algebras. For a Lie algebra g over a commutative ring K and a 2-cocycle f on its standard complex with values in K, we define in ?1 the notion of an frepresentation. The usual representations of g correspond to the case f= 0. We introduce in ?2 the filtered K-algebra gf which is a universal model for f-representations. We deduce the "Poincare-Witt Theorem" (Theorem 2.6) for gf as an easy consequence of the usual Poincare-Witt Theorem, proved in [1, p. 271]. It is then clear that if g is K-free, there is a graded K-algebra isomorphism 4t'f: S(g) -*E0(gf), where S(g) denotes the symmetric algebra of the K-module g and EO(gf) the graded algebra associated with gf (Theorem 2.5). In ?3, we define, for a fixed graded K-algebra S, the category whose objects are pairs (A, 41A), where A is a filtered K-algebra, IPA: SEO(A) an isomorphism of graded algebras and whose maps are defined in an obvious manner. If S is the symmetric algebra of a free K-module L, then there is a 1-1 correspondence between isomorphism classes of such objects and pairs (g, f), where g is a Lie algebra on L and f H2(g, K). For a cocyclef in the class f, the pair (gf, ifr) is an object in the class corresponding to (g, f) (Theorem 3.1). The fourth section is devoted to the computations of certain of the usual homology and cohomology groups of a finite dimensional Lie algebra g. This amounts to a study of g& for f= 0. These computations are used in the next

On the Kronecker products of irreducible representations of the 2×2 real unimodular group. I

Abstract: Introduction. The purpose of the present paper is to determine the decomposition of the Kronecker product of two irreducible representations of the real 2X2 unimodular group into a continuous direct sum of irreducible representations. The irreducible unitary representations of this group have been determined first by V. A. Bargmann [l](1), and those of the 2X2 complex unimodular group by I. M. Gel'fand and M. A. Nalmark [3]. In both cases the list of these representations contains two continuous series; first, the principal continuous series, the members of which can be described by a pair (m, p) of two variables, m with a discrete, p with a continuous range; and secondly, the representations of the exceptional interval, characterized by a single parameter, varying over a finite interval. In the real case in addition to these there exists a discrete series of representations characterized by integers. Concerning the representations of the exceptional interval it has been proved that they do not occur in the decomposition of the left regular representations of these groups into a continuous direct sum of irreducible representations. The problem of finding the irreducible parts for the Kronecker product of two of these representations by the Reduction Theory of von Neumann [9] was taken up first by G. W. Mackey, in the complex case, for two factors taken from the principal series [4; 5]. W. F. Stinespring applied the same method to the discussion of the analogous case for the real group(2). Recently, M. A. Nalmark attacked the same problem in the complex case, and gives a complete discussion of all possibilities [10](3). In Parts I, II, and III of the present work we give the decomposition of the product of any two irreducible unitary representations of the real 2X2 unimodular group. To sketch our method, we restrict ourselves, for the sake

Extreme eigenvalues of Toeplitz forms and applications to elliptic difference equations

Abstract: and obtained some partial results. These enabled us to obtain estimates on the rates of convergence of the "two-line" iterative methods of the Laplace and biharmonic difference equations in rectangular domains(2). In the case of Laplace's equation we obtained an exact asymptotic result. However, in the case of the biharmonic equation we obtained only a "one-sided" estimate. The purpose of this report is two-fold. In ??2, 3, and 4 we extend the results of Kac, Murdoch and Szego, and Widom. We will make very strong use of Widom's results and technique. In ?5 we discuss the application of the preceding results to the general problem of the extreme eigenvalues of "block" Toeplitz matrices. These include the matrices of elliptic difference equations Presented to the Society, January 26, 1961; received by the editors October 18, 1960. (') Some of these results were obtained while the author was at the Brookhaven National Laboratory, summer 1959. (2) The "two-line" iterative methods for the Laplace and biharmonic difference equations were studied by R. S. Varga [18] at the same time. His approach is totally different from the one we investigated in [14]. His approach to the solution of the iteration equations is more general and probably preferable. Varga also estimated the rate of convergence in the Laplace case using the theory of non-negative matrices. That theory does not apply to the biharmonic case.

Uniform distribution of sequences of integers

Abstract: For example any arithmetic progression { an + b; n =1, 2, 3, } is uniformly distributed modulo m if and only if g.c.d. (a, m) = 1. Further we say that the sequence A is uniformly distributed in case A is uniformly distributed modulo m for every integer m _ 2. The case m = 1 is omitted because A (n, j, 1) = n, and so (1.1) holds for every sequence A in case m = 1. Furthermore the language "uniformly distributed modulo 1" has a well-established meaning for a sequence of real numbers I an{o; cf. [1] or [2, p. 72]. These definitions apply to any sequence of integers. Our interest will be primarily in sequences of positive integers satisfying ai

The homology of twisted cartesian products

Abstract: Introduction. In a recent paper [2], E. H. Brown introduced the notion of a twisted tensor product. Briefly, the definition is as follows. Let K be a D.G.A. (differential, graded, augmented) coalgebra, A a D.G.A. algebra, and M a D.G.A. A-module. The twisted tensor product K 0 M of K with M is, except for the differential, the usual tensor product. The differential on K 0, 0 M is modified using a twisting cochain 4 in Hom(K, A). Now suppose p: E-*X is a fiber space (essentially of the Hurewicz type) with fiber F. Then C(X) is a D.G.A. coalgebra, C(QX) a D.G.A. algebra, and C(F) a D.G.A. C(QX)-module. Brown's main theorem states that there is a twisting cochain 4 in Hom(C(X), C(QX)) and a chain equivalence 41: C(X) X C(F) -> C(E). In another recent paper [1], Barratt, Gugenheim, and Moore define the twisted cartesian product of two simplicial sets (semi-simplicial complexes). If X and F are simplicial sets and G is a simplicial group acting on F, the twisted cartesian product of X and F, X XT F, coincides with the usual cartesian product except that the initial face is modified in terms of a twisting function r: X -G. It is proved in [1] that any simplicial fiber space p: EX with fiber F can, for the purposes of algebraic topology, be replaced by a twisted cartesian product X XT F. (The group G and the twisting function r: X >G are shown to exist.) Considering these two results, one might expect that an analogue of Brown's theorem could be proved for twisted cartesian products, explicitly defining the twisting cochain in terms of the twisting function(2). This is in fact done in Part I of this paper. In Part II, the explicit form of the twisting cochain is used to investigate fiber bundles over spheres. The homology and cohomology Wang sequences are derived and some partial results obtained describing the behavior of the maps in the cohomology Wang sequence with respect to cup products. I would like to express my gratitude to Professor Saunders MacLane for his patient assistance and encouragement during the preparation of this

On certain character sums

Abstract: where x(f(x)) = (f(x) | p), have been studied. The connection of ?P2(Q) and {13(Q) with the representations p=a2 +b2 (a=1 (mod 4)) and p=s2 +3t2 (s 1(mod 3)) of an odd prime p was established byjacobsthal [4] and von Schrutka [6]. It follows from the results of Jacobsthal that 4)2(1) has the value -2a or 0 according as p=a2+b2 or pla2+b2, and from the results of von Schrutka that ,13(1) has the value -2s or 0 according as p=s2 +3t2 or p :S2+3t2. (An elegant development of these particular results is given in [3].) If p=8k+1 =c2 +2d2 (c=(-1)k+l (mod 4)), Whiteman [7] has shown that 44(1) = 4c. However, if p = 8k+3 = c2+2d2, 4)4(1) = 0. We shall establish a theorem for primes of the form c2 +2d2 entirely similar to the results of Jacobsthal and von Schrutka mentioned above, and indicate some results obtained for primes of the form u2+5v2. If p is a prime of the form a2+b2 (b even) and Q is a quadratic nonresidue of p, Jacobsthal showed that 4)2(Q) =? 2b, but was unable to remove the ambiguity in sign for any specific choice of the nonresidue Q. E. Lehmer [5] has determined the sign of (12(2) if p is a prime of the form 8n+5. We shall determine the sign of 4)2(-3) if p is of the form 12n+5, and for these same primes, obtain a congruence defining b similar to one given by Stern for primes of the form 8n+5. 2. Three lemmas. If GF(pm) denotes the finite field of pm elements (p prime), then the following lemma is evident.

Rings of integer-valued continuous functions

Abstract: During the past twenty years extensive work has been done on the ring C(X). The pioneer papers in the subject are [8] for compact X and [3] for arbitrary X. A significant part of this work has recently been summarized in the book [2]. Concerning the ring C(X, Z), very little has been written. This is natural, since C(X, Z) is less important in problems of topology and analysis than C(X). Nevertheless, for some problems of topology, analysis and algebra, C(X, Z) is a useful tool. Moreover, a comparison of the theories of C(X) and C(X, Z) should illuminate those aspects of the theory of C(X) which derive from the special properties of the field of real numbers. For these reasons it seems worthwhile to devote some attention to C(X, Z). The paper is divided into six sections. The first of these treats topological questions. An analogue of the Stone-Cech compactification is developed and studied. In ?2, the ideals in C(X, Z) are related to the filters in a certain lattice of sets. The correspondence is similar to that which exists between the ideals of C(X) and the filters in the lattice of zero sets of continuous functions on X. This theory provides a characterization of those ideals of C(X, Z) which are intersections of maximal ideals. ?3 is concerned with the space of maximal ideals in C(X, Z). In ?4, some existence theorems for maximal ideals are proved. The residue class fields of C(X, Z) modulo maximal ideals are studied in the last two sections. It turns out that those of prime characteristic are trivial: the integers modulo the characteristic. The residue class fields of characteristic zero are distinctly nontrivial. In ?5, the cardinality of such fields is investigated. The main result is that they are always uncountable. In ?6, the algebraic properties of the zero characteristic residue class fields are examined. It is shown for example that these fields are always quasialgebraically closed. As we noted above, very little has been published concerning the ring C(X, Z). Nevertheless, a considerable number of "folk theorems" exist in the subject. One of our objectives in writing this paper is to get these results

Cohomology of Lie triple systems and Lie algebras with involution

Abstract: A Lie triple system is a subspace of a Lie algebra closed under the ternary composition [[xy]z]; equivalently, it may be defined as the subspace of elements mapped into their negatives by an automorphism of order two (an involution) in a Lie algebra; or, finally, a Lie triple system may be defined by a set of identities. Lie triple systems were first noted by E. Cartan in his studies on totally geodesic submanifolds of Lie groups and on symmetric spaces [l] (see also [lO]): Lie triple systems are related to totally geodesic submanifolds in the same way that Lie algebras are related to analytic subgroups, and the symmetry in a symmetric space gives rise to the involution in the Lie algebra. Lie triple systems were studied from the algebraic point of view by Jacobson [6; 7] and Lister [9], the latter giving a complete structure theory including the classification of the simple finite-dimensional systems (in characteristic zero), the Levi decomposition, and (a special case of) the first Whitehead lemma. Simpler axioms were given by Yamaguti [14], who has also studied these and more general systems [15; 16]. In this paper we introduce cohomology groups for Lie triple systems and show that the usual interpretations (in terms of derivations and factor sets) hold for the first and second cohomology groups. In particular we obtain the two Whitehead lemmas (in characteristic zero). Further, these cohomology groups fit into the theory of [2 ] : they are the cohomology groups of a supplemented associative algebra (very closely related to the universal associative algebra of the Lie triple system). The groups may briefly be described as follows: if T is any Lie triple system and M a T-module we can construct the universal Lie algebra LU(T) of T and the "standard extension" M, which is an LU(T) module; an involution a operates on both LU(T) and Ms such that the elements mapped into their negatives are T and M respectively, and cr([/, «])= [cr(/), o(n)] for IELU(T), nEM.. Now (in a more general situation) if L is any Lie algebra with involution o and N any L-module also with involution 0, then the base field, U the universal associative algebra of L) ; we assume further that the characteristic is not 2, so that Hn(L, N) is a direct sum of two subspaces invariant under cr: say H"+(L, N), H1(L, N). We show that H\(L, A0 = Extx($, AO