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In a recent paperf entitled A note on the linear fractional group, Todd obtained an abstract definition for the group LF(2, 2) in terms of n+2 generators. Apparently he gave no consideration to the question of the independence of the defining relations, for they can be considerably simplified. First, in view of the condition RSi = Si+iR (which is the same as i£S0^~* = Si), the three generators U, R and S0 are sufficient to generate the entire group. If we give a definition in teims of these three generators alone, the relations

In [6] Higman and N eumann showed that the Frattini subgroup @(A * B) of the ordinary free product of nontrivial groups A and B is trivial and they asked (implicitly) whether or not the Frattini subgroup of a generalized free product G = A *H B of groups A, B with amalgamated subgroup H is always to be found inside H. Whittemore [16] took the first steps toward answering this problem. Subsequently her results have been fairly substantially generalized in a series of papers (e.g., [I, 2, 4, 151). This paper first offers further evidence to support the conjecture that Q(G) < H in all cases and then turns to the problem of the location of the Frattini subgroup of HNN groups. One notable corollary of this latter result is Corollary 6.2. As yet, however, the authors have been unable to see if the results they have obtained are of any value in trying to settle the problem of Neuwirth as to whether the Frattini subgroup of a knot group is trivial1 In Section 2 we first show that the two results which have to date been central in the theory are both corollaries of another rather artifical-looking theorem (Theorem 2.1) but the full hypothesis of the statement of Theorem 2.1 is used in proving, in Section 3, that if G = A *H B and if H has the maximum condition on subgroups then D(G) < H. This result itself depends on the result, found in Section 4 that if A, B are countable and His an arbitrary (countable) group

Such equations have an interesting history. In art. 358 of the Disquisitiones [1, a], Gauss determines the Gaussian sums (the so-called cyclotomic “periods”) of order 3, for a prime of the form p = 3n + 1, and at the same time obtains the number of solutions for all congruences ax− by ≡ 1 (mod p). He draws attention himself to the elegance of his method, as well as to its wide scope; it is only much later, however, viz. in his memoir on biquadratic residues [1, b], that he gave in print another application of the same method; there he treats the next higher case, finds the number of solutions of any congruence ax − by ≡ 1 (mod p), for a prime of the form p = 4n + 1, and derives from this the biquadratic character of 2 mod p, this being the ostensible purpose of the whole highly ingenious and intricate investigation. As an incidental consequence (“coronodis loco,” p. 89), he also gives in substance the number of solutions of any congruence y ≡ ax − b (mod p); this result includes as a special case the theorem stated as a conjecture (“observatio per inductionem facta gravissima”) in the last entry of his Tagebuch [1, c]; and it implies the truth of what has lately become known as the Riemann hypothesis, for the function–field defined by that equation over the prime field of p elements. Gauss’ procedure is wholly elementary, and makes no use of the Gaussian sums, since it is rather his purpose to apply it to the determination of such sums. If one tries to apply it to more general cases, however, calculations soon become unwieldy, and one realizes the necessity of inverting it by taking Gaussian sums as a starting point. The means for doing so were supplied, as early as 1827, by Jacobi, in a letter to Gauss [2, a] (cf. [2, b]). But Lebesgue, who in 1837 devoted two papers [3, a,b] to the case n0 = · · · = nr of equation (1), did not succeed in bringing out any striking result. The whole problem seems then to have been forgotten until Hardy and Littlewood found it necessary to obtain formulas for the number of solutions of the congruence ∑ i x n i ≡ b (mod p) in their work on the singular series for Waring’s problem [4]; they did so by means of Gaussian sums. More recently, Davenport and Hasse [5] have applied the same method to the case r = 2, b = 0 of equation (1) as well as to other similar equations; however, as they were chiefly concerned with other aspects of the problem, and in particular with its relation to the Riemann

complex can be thought of as a complex associated with a map of R' -+ R, where R is a commutative ring, it seemed likely that there should be a complex associated with a map of Rm-* R'. In fact, as long as one is willing to go that far, why not look for some complex associated with an arbitrary map of modules over any commutative ring. In this paper, we define a complex associated with a map of modules. In ?1, we discuss this complex in complete generality. In ?2, we pretty much restrict our attention to maps of R' -* R', and establish many of the formal properties of the complexes attached to this map which will be needed in the rest of this paper as well as in subsequent ones. ??3 and 4 are included here to give an indication of how the notion of E-sequence may be generalized, and to show that over local rings this generalized notion of E-sequence is (as in the usual case) independent of order. In subsequent papers we shall investigate these more general ideas, relating them to each other (as in [1]) and also applying them to the case of the singular variety of a variety which is defined by the ideal generated by the minors of the Jacobian matrix. It might also be mentioned that the usual Koszul complex plays a role in studying the invariant factors [2] of certain special modules, and we shall subsequently show the connection between the homology groups of the complex associated with a map f: Rm-* R', and the invariant factors of the cokernel of f.

In an earlier paper we considered homogeneous polynomials ƒ and g whose degrees are relatively prime, and solved the Diophantine equation f(x)=g(y). These results are generalized in the present paper. The solutions are given in terms of arbitrary parameters and if the parameters are integral the solutions are also. We begin our discussion with the hypothesis that the functions J\X) J\Xll, * * j X\m, , Xnif , Xnm), g\X) = g\Xn, *, X\m, , Xni, • • • i Xnm) are polynomials with integral coefficients, homogeneous in each set of variables Xk\Xk2 * Xkm ; ƒ being of degree ak e 0, g being of degree /3k^0 in the sets XkiXk2 • • * xkm and dk = ak—Pk. We suppose further that integers Xt, jUt^O exist such that 2

1. S. Ramanujan, Congruence properties of partitions. Math. Zeit. vol. 9 (1921) pp. 147-153. Reprinted in Collected papers of Srinivasa Ramanujan, Cambridge, 1927, pp. 232-238. 2. S. Ramanujan, On certain arithmetical functions, Trans. Cambridge Philos. Soc. vol. 22 (1916) pp. 159-184. Reprinted in Collected papers of Srinivasa Ramanujan, pp. 136-162. 3. H. S. Zuckerman, Identities analogous to Ramanujan's identities involving the partition f unction, Duke Math. J. vol. 5 (1939) p. 89, equation (1.15).

The big Picard theorem states that a holomorphic mapping from a punctured disk D* = {z e C: 0 < I z I < 1} into the Riemann sphere P1(C) minus three points can be extended to a holomorphic mapping from the disk {z E C: I z I < 1} into P1(C). H. Huber extended this theorem to the case where the image space is a domain G of hyperbolic type (in the classical sense) of a Riemann surface R such that the closure of G in R is compact [9]. The purpose of this paper is to prove the following. A holomorphic mapping from the punctured disk into a compact hyperbolic manifold M can be extended to a holomorphic mapping from the whole disk into M. Part of our proof uses some of the arguments H. Grauert and H. Reckziegel introduced in their paper [7, p. 120]. Generalizing further, we shall consider the extension problem of a holomorphic mapping defined on a complex manifold minus an analytic subset.

Here sn x is an abbreviation for sin (=rx/2). This paper gives other conditions for the validity of these identities. The previous conditions permitted qk to have various types of discontinuity. The present paper is concerned with smooth functions; however, the growth at 0 and oo is permitted to be greater than before. Theorems 1, 2, and 3 of the previous paper together with Theorems 2 and 3 of this paper form a fairly complete elementary theory of these identities. The proofs given here do not depend on the previous paper. The results of this paper hinge on the possibility of defining the Fourier sine transform for functions which do not vanish at infinity. Theorem 1 below shows that this is possible merely by employing summability. It is to be noted that Theorem 1 is not true, as it stands, for the cosine transform. For example, the cosine transform of any constant evaluated by such a definition would vanish. Hence the inversion formula could not apply. A theory of "generalized Fourier

Introduction. Comparatively few of the plane sextic curves are known. None of the papers written about them gives a complete treatment or a classification of these curves. The sextics appear more or less isolated in the different treatises. Apparently the methods employed produce only a very limited number of sextics. The projective method of deriving higher plane curves described by Dr. H. P. Pett i t in his Projective description of some higher plane curves allows a more systematical study of sextics and especially a direct construction point by point. I t is the purpose of this paper to give a projective description of some of the sextics that can be derived by this method, if conies are the base curves. The number of types of these curves which could be generated by this method is here limited to 354 by giving the triangle of reference special positions with respect to the base conies.

If one wants to investigate the properties and relations of homogeneous ideals in a commutative graded ring, one has as a model on the one hand the well known case of a polynomial ring and on the other hand the general commutative ideal theory. The case of a polynomial ring has been studied for the sake of algebraic geometry, and one of the methods was traditionally the passage to nonhomogeneous coordinates by choosinig a suitable hyperplane of infinity [5, pp. 750-755; 7, pp. 491-496]. On the other hand it appears that the homogeneous ideals of a graded ring form a system that is closed under the usual ideal operations, as is the system of all ideals of a commutative ring. Thus one may try to copy the whole ideal theory, but now for homogeneous ideals (and homogeneous elements) only. Samuel [4], Northcott [3] and Yoshida [6] have proved the elementary properties of homogen-eous ideals for a graded ring. In this paper we investigate how far the process of localization can be carried over to graded rings. The degrees in our graded ring are the integers; the case of a bigraded ring is each time treated as a corollary. In ?1 we summarize the elementary properties of homogeneous ideals. Having formulated and proved Lemma 1, all proofs become straightforward. In ?2 we study the localization (i.e., passage to a ring of fractions) with respect to a prime ideal or a finite set of prime ideals. Here we introduce the concept of a relevant prime ideal, as did Yoshida. In ?3 we discuss the transition to a nonhomogeneous ring by choosing a hyperplane of infinity. This may be called localization in the sense of the Zariski topology. By a hyperplane of infinity we mean simply a homogeneous element I of degree one, which is not nilpotent. The corresponding nonhomogeneous ring can be obtained in two ways, namely as R/(l-1)R, but also as follows: Let [1] be the multiplicatively closed subset of R, consisting of all powers of 1. Then the ring of fractions R[z] is again a graded ring. The zero-degree subring R[z]o of R[l] is our nonhomogeneous ring, i.e., R/(l-1)R-R[z]o. As elements of degree one may happen to be

1. This paper announces results concerning the cohomology groups H*(N, N) where A*" is in a certain class of finite-dimensional nilpotent Lie algebras over a field k and T is an abelian Lie algebra faithfully represented as a maximal diagonalizable algebra of derivations of N; we shall refer to such an iV as a T-algebra. The additional hypotheses to be placed on the pair N, T are inspired by the case when J is a Cartan subalgebra and T+N=B is a Borel subalgebra of a complex semisimple Lie algebra. In that case Kostant has shown [2] that H%N, N)=0 for i^.2 and the authors applied this result in [3] to conclude that H*(B, B) = 0. (A similar argument shows H*(P9 P ) = 0 for P parabolic.) Here we are concerned with the relations between the vanishing ofH%N, N), especially for i=2 , and the structure of the algebras N. Let W denote the set of weights of T in N. If dirn(r)=dim(7\T/A)=m then the subset of W arising from the induced representation of T on N/N has precisely m elements, say {a1? • • • , am}. Every a e Wthen has a unique representation a = 2 ^a* with each c? a nonnegative integer and ct 0. For such an a we call the sum (in Z) 2 i the height of a and denote it by |a|. For a in W, denote by Aa the weight space for a in N. DEFINITION. A T-algebra is called positive if (i) dim(70=dim(A7iV), (ii) N is graded by the heights of the weights, i.e., if N(j) = Q)\al=j Na then [N(j)9N(k)]cN(j+k). REMARK. Condition (ii) is superfluous in characteristic 0. However, in characteristic p>0 it has such consequences as N=0 for r > ( / > l ) d i m ( r ) .

In his fundamental paper on a posteriori probability,* Bayes considered a certain event M having an unknown probability p of its occurring in a single trial. In deriving his a posteriori formula he assumed that all values of p are equally likely, and he recommended this assumption for similar problems in which nothing is known concerning p. In the corollary to proposition 8 he derives the value r/n\ 1

Theorems 1 and 2 will be contained in a forthcoming paper by the author. Theorem 1 follows from work of Ray [6] and Neveu [4], [S] but a new proof will be presented. Theorem 2 is a consequence of a result already announced in Williams [7]. The terminology and notation are as in Dynkin's book [ l ] . Let £ be a countable set and let {pa(t): i,jCzE; t^O} be a transition function on E with the properties

This paper is primarily concerned with fibre mappings into an absolute neighborhood retract. Theorem 3 is a converse of the covering homotopy theorem; it characterizes fibre mappings (into a compact ANR) as mappings for which the covering homotopy theorem holds. Theorem 4 is Borsuk's fibre theorem; the proof which I present here is new. It seems to me that this theorem is a promising tool in function-space theory. Also I think that it furnishes conclusive justification for the generality of the Hurewicz-Steenrod definition of a fibre space. In fact, a fibre space of the type constructed by Borsuk's theorem almost never has a compact base space and almost never has its fibres of the same topological type. The common denominator of the proofs of Theorems 3 and 4 is a property which I call local equiconnectivity. Local equiconnectivity is a strengthened form of local contractibility and a weakened form of the absolute neighborhood retract property (Theorems 1 and 2). Definitions and notations are those of FS. I. Let A be the diagonal subset ]C&e#(^ &) °f 5 X 5 . I shall call the space B locally equiconnected (or, to be specific, (U, F)-equiconnected) if there are neighborhoods U and F of A and a homotopy X in B between the two projections of U which does not move the points of A and which is uniform with respect to V. Precisely: (1) X*(eo, fa) is defined for all (&0, fa) £ Uy (2) Xo(6o, &i)=*o

The purpose of the present paper is to give a new geometrical construction for the following affine diferencial invariants of a space curve: geometrical elements associated to an ordinary point of a space curve; geometrical characterization of the affine fundamental trihedral of a space curve at an ordinary point; geometrical interpretation of the affine curvature and geometrical interpretation of the affine torsion

The generalized Schoenflies theorem of M. Brown [2], [3] can be restated in the following way: If S~ is the equator of S, then any locally flat embedding ƒ : S~—>S can be extended to a homeomorphism Ft S-»S . The purpose of this paper is to show that, if n^ 5, the extension F can be constructed in a controlled manner; in particular, if ƒ:S~^»5 is close to the inclusion embedding, then F: S-*+S can be chosen to be close to the identity homeomorphism. Consequently if, ƒ, g :S'-^S are locally flat embeddings, n^ 5, and ƒ is close to g, then there is a homeomorphism H: S—^S which is close to the identity such that

1. A number of papers discussing the spectral decomposition and eigenfunction expansion for partial differential operators appeared in the last few years. Browder [l], [2], [3], [4], Gârding [5] and Mautner [12 ] proved the existence of an abstract eigenfunction expansion for elliptic partial differential operators. In 1953 A. Ya. Povzner [13] considered the detailed spectral decomposition of — A-\-q(x). This was completed by T. Ikebe [6] who used the theory of wave operators as developed by Kato [8] and Kuroda [10], [ll]. In this note we investigate an eigenfunction expansion for the operator P(D)+q(x) where P(D) is a linear homogeneous elliptic partial differential operator with constant coefficients. Detailed proofs of the results will appear elsewhere. 2. The Euclidean w-space will be denoted by Rn or Mn with ele­ ments x=(xu • - • , xn) or k=(ki> • • • , kn) respectively. ff(x)dx de­ notes integration with respect to Lebesgue measure. We set d — Dj = for 1 ^ j S nidxj Let P(x) be a homogeneous elliptic polynomial, i.e. P(x)jâqx| 2p where 2p is the order of P(x). Then P(D) =P(Dh • • • , Dn) is a linear homogeneous elliptic partial differential operator. All through this note we assume that 4p>n. It is well known that P(D) can be ex­ tended to a selfadjoint operator P(D) in Lz(Rn). Let g(x)GC2[„/2] with q{x) — 0{\x\~n~h) for some h>0. Then by Theorem 1 of [ll], P(D)+q(x) is a selfadjoint operator in L

Let G be a locally compact Abelian group. By the group algebra, L(G), of G we mean the Banach algebra of all (equivalence classes of) complexvalued functions on G integrable with respect to the Haar measure on G, with convolution as multiplication. By the measure algebra, M(G), of G we mean the Banach algebra of all finite complex-valued regular Borel measures on G, with convolution as multiplication. In this paper we give a characterization of those commutative Banach algebras which are the group algebras of locally compact Abelian groups, and we give a characterization of those commutative Banach algebras which are the measure algebras of locally compact Abelian groups. Our results have been announced already, without proofs, in [13]. We will now give a brief outline of our results. In ?1 we define an abstract complex L-space, and, in extension of Kakutani's theorem [11] on the concrete representation of abstract real L-spaces, we show that any abstract complex L-space is isometrically isomorphic to complex L'(X, m) for some measure space (X, m). Let B be any Banach space and f any nonzero linear functional on B. Let P(f) = {x: f(x) = l/fIl lxix }. Then P(f) is a cone in B and so defines a partial order on B in the usual way. In ?2 we study this phenomenon briefly, with particular attention to the case in which B under this order becomes an abstract complex L-space. Now let A be any complex Banach algebra, with multiplication denoted by *, and let f be a multiplicative linear functional (m.l.f.) on A. We say that f is L'-inducing if, under the order defined by P(f ), A is an abstract complex L-space, and if two other hypotheses are satisfied. In ?3 we see

Here we present the principal ideas and results of [5] with some indications of proof. The general notion of uniformity is difficult to harvest; nonetheless, various offshoots of it have borne fruit in all fields of mathematical logic. In this paper we introduce the notion of measure-theoretic uniformity, and we describe its use in recursion theory, hyperarithmetic analysis, and set theory. In recursion theory we show that the set of all sets T such that the ordinals recursive in T are the recursive ordinals has measure 1. In set theory we obtain all of Cohen’s independence results [1,2]. Solovay [8,9] has extended Cohen’s method by forcing statements with closed, measurable sets of conditions rather than finite sets of conditions; in this manner he exploits the concepts of forcing and genericity to prove: if ZF is consistent, then ZF + “there exists a translation-invariant, countably additive extension of Lebesgue measure defined on all sets of reals” + “the dependent axiom of choice” is consistent. Solovay’s result is also a consequence of the notion of measure-theoretic uniformity.

Let z=f(x, y) be an arbitrary surface S, in the sense that f(x, y) is an arbitrary one-valued real function of the real variables x and y. By the grade of a segment joining two points A and B, we understand the absolute value of the tangent of the angle which AB makes with the :ry-plane. The point A = (£, rj, f ) of the surface S is said to be of bounded grade—or S is said to be of bounded grade at the point (£, rj)—if the grade of AB is bounded for all B = (x, y, z) of 5 at a sufficiently small "horizontal" distance [{x — %) + (y — rj)] from A. HA does not satisfy this condition, S is said to be of unbounded grade at (£, rj). It is the object of the present paper to prove the following theorem, which identifies the aggregate—for the totality of arbitrary surfaces—of sets of points of unbounded grade with the aggregate of sets of type Go.f

This paper sketches the proofs that (1) an automorphism of a complete filtered ring is a limit of successive approximations, (2) given an nth order approximate automorphism, there is an obstruction to prolonging it to an (n + l)st order approximation, the obstruction lying in a certain 2nd cohomology group, and (3) the mapping which sends an nth order approximate automorphism to its obstruction is a crossed homomorphism from the multiplicative group of nth order approximate automorphisms to the (additive) 2nd cohomology group containing the obstructions. The rings in question need not be associative: we tacitly assume that there is given a "category of interest," 6, in the sense of [l ] (which may be, in particular, the category of associative, Lie, or commutative associative rings), and "ring" and "morphism" are meant relatively to

Manifolds with holonomy groups in SO(n) are the oriented Riemannian manifolds. Only general results may be obtained about the topology of this large class. The cohomology of Riemannian manifolds with holonomy groups in U(n) (Kahler manifolds), has been extensively studied (see [3], [6], [13]). The existence of compact Riemannian manifolds with holonomy groups in SU(n) or Sp(n) is not known for n =A 1. Hence, for the general groups, the most interesting cases left are those manifolds whose holonomy groups form subgroups of Sp(n) x Sp(l). These manifolds are called quaternionic manifolds. In the first part of this paper (??1-3), a decomposition analogous to the Hodge Decomposition for Kaihler manifolds is given for quaternionic manifolds (Theorem 3.5). Using a theorem of Chern, we get an increasing sequence of Betti numbers (Theorem 3.6). In the last part (??4 and 5), we define a quaternionic pinching. Using it, we give a quaternionic analogue (Theorem 5.5) to Klingenberg's Kahler pinching in [7] and [8].

0. Introduction. This note announces some new methods in algebraic topology, based on the results of [1], [2] and [3]. The relative lifting problem [5, p. 415] is fundamental to that subject. It includes the extension problem, the retraction problem, the lifting problem, the section problem, the relative section problem and the computation of [X, Y] problem, among its particular cases. These problems (excluding the last one) are usually taken to concern just the existence of extensions, retractions, etc. ; we use the terms in the wider sense, to include both the existence question and the homotopy classification question, for the respective extensions, retractions, etc. We will : (i) prove that many cases of the above problems (including the computation of the absolute and relative cohomology groups of the total space of a fibration, and the computation of certain homotopy groups) are equivalent to "parallel problems", involving the restricted fibered mapping projection (pq;a); (ii) give some examples of solutions of parallel problems. Our solutions include one lifting problem and two extension problems. The extension results have some immediate consequences; they give new derivations for the exact cohomology sequences of Serre and Wang. Gysin's sequence is derived separately. Future papers will discuss these and other applications in detail. Our argument is valid in several convenient categories, including the category of i-spaces [4], [2], [7] and the category of quasi-topological spaces [6], [1], [2]. This category of I-spaces, for a definition see [2, p. 276], contains the usual category of Hausdorff fc-spaces (= compactly generated spaces = Kelley spaces) as a subcategory.

Introduction. Let X be a finite dimensional complex Euclidean space, and let T be a linear operator acting on X. The Jordan decomposition theorem states that T has a unique decomposition T= S + N, where S = f(T)Z dE(z), E is a spectral measure supported by the spectrum a(T) of T, and N is a nilpotent operator commuting with S. Our main result (Theorem 2.1) is a generalization of the Jordan theorem for operators with real spectrum to infinite dimensional reflexive Banach spaces. We consider operators T satisfying the growth condition leitTI = O(jtjk) for some integer k ? 0 and all real t. In ?1, we construct the "Jordan manifold" for T, on which T is shown to have a unique Jordan decomposition, if the spectrum (which is real because of the growth condition) has Lebesgue measure zero (?2). Related results are described in ?2. The theory is illustrated by examples in ?3. This work is clearly related to Dunford's theory of spectral operators. However, the latter is needed as a prerequisite only for Theorem 2.12. The standard reference is [1], [2]. Many thanks are due to Professors H. Furstenberg and C. A. McCarthy for discovering an error in the original version of this paper.

A homeomorphism h of En or Sn onto itself is stable if 3 homeomorphisms &i, hi, • • • , hm and nonvoid open sets Z7i, Ui, • • • , Um such that h = hmhm-i • • • hi and hi\ Ui = I for i = l , 2, • • • , m. All orientation preserving homeomorphisms of En or Sn are stable provided n = 1, 2, or 3. There is no example known in any dimension of an orientation preserving homeomorphism which is not stable. In fact, the conjecture that all orientation preserving homeomorphisms of En or 5 n are stable is equivalent to the annulus conjecture (see [3 ]). I t is known that any homeomorphism of £3 onto itself can be approximated by a piecewise linear one (see [2] or [o]). The purpose of this paper is to announce that if n ^ 7 and h is a stable homeomorphism of En or Sn onto itself, then h can be approximated by a piecewise linear homeomorphism, and also, in the case of En, by a diffeomorphism. The set of all homeomorphisms on En or Sn forms a group under composition and the subset of stable homeomorphisms forms a normal subgroup. The stable group on Sn is simple while the stable group on En is not. Due to this fact, there is a shorter proof in the case of Sn than in the case of Eny and it is this proof which will be outlined here. The author thanks John Stallings for his assistance.

In the classical Lie theory it is shown how to construct a differential equation invariant under a given group, and how to solve an equation when a group leaving the equation invariant is known. However, little is said about the problem of determining the group for a given differential equation, which is by far the most interesting problem. In the present paper, necessary and sufficient conditions for the existence of an infinitesimal contact transformation leaving a given equation invariant are determined along with the general form of the characteristic function of the group. It will also be shown how to reduce, by a proper change of variables, the infinitesimal contact transformation to a point transformation. This enables one to solve the transformed differential equation by Lie's methods. Passing back to the original variables, a new differential equation is obtained which combined with the original equation gives its solution in parametric form. Let

Two point sets are mutually separated if they are mutually exclusive and neither of them contains a limit point of the other. A point set M is said to be not connected or connected according as M is or is not the sum of two non-vacuous mutually separated point sets. A subset X of a connected point set M is said to be a cutting of M, or is said to cut M, provided the set of points M—X is not connected; X is called an irreducible cutting of M provided X cuts M but no proper subset of X cuts M. If A and B are subsets of a connected point set Af, the subset X of M is said to separate A and B in My or to cut M between A and B, provided that M~X is the sum of two mutually separated sets Ma(X) and Mb(X) containing A and B respectively. In this paper the ordinary notation of point set theory will be employed, for example, X = X + X ' , whereX is the set of all limit points of the set X, K c H means that the set K is a subset of the set H, K H means the set of points common to K and Hy F(R) denotes the boundary of the set i?, etc. In addition, if X is a cutting of a connected point set M then, unless otherwise stated, the equation M—X — Ma(X) + Mb(X) is to be interpreted as meaning that M—X is the sum of two mutually separated sets Ma(X) and Mb(X) which contain the sets A and B respectively. 1. Cuttings of Connected Sets in General. The point sets considered in this section are assumed to lie in a separable metric space which will be denoted by 5. It is apparent from the proofs of some of the theorems, however, that they hold in even more general space.

Introduction. This paper is an announcement of a study on the theory of the topological variation of the phase space of one-parameter families of vector fields (ordinary differential equations, flows). This theory, sometimes called bifurcation theory, has been developed since H. Poincare from several points of view; see, for example, [l]> [z]i [3]» W> [5]. Here, we will be mainly interested in a collection of one-parameter families of vector fields which has the following properties: (a) it is large with respect to all the families, and (b) its elements exhibit a topological variation which is amenable to simple description. Collections with properties (a) and (b) are currently called "generic" and were introduced in the qualitative global analysis of differential equations by M. Peixoto [o], S. Smale [8], and I. Kupka [ l l ] . See S. Smale [9] for a thorough monography on this field. The geometry of the set 2 of structurally stable vector fields and the study of "generic" one-parameter families of vector fields are closely related. A vector field is structurally stable if its phase space does not change topologically under small perturbations; a oneparameter family of vector fields exhibits: the simpler a phase-space topological variation, the larger the intersection it has with 2 , or equivalently, the smaller the intersection it has with its complement —the set of nonstructurally stable vector fields. The importance of the set of nonstructurally stable vector fields for the study of the topological variation of the phase space of vector fields was noticed by A. Andronov and E. Leontovich. In [12] they defined the concept of first-order structural stability as a possible guide to pursue such study. In compact two-dimensional manifolds, the only case considered here, the set of first-order structurally stable vector fields is an imbedded Banach manifold of class C and codimension one of the Banach manifold of vector fields; see [13]. This fact, although (as follows from [14]) not sufficient to describe completely the "generic"

In this paper is presented a partial list of formulas giving the orders of the ruled hypersurfaces and ruled surfaces whose rulings are incident with given curves and surfaces in 4-space. If the number of incidences is equivalent to six simple conditions, the number of lines having these incidences is finite. The formulas herein presented are analogous to those giving the orders of the surfaces whose rulings are lines satisfying three conditions and the number of lines satisfying four conditions in 3-space.* Some of these formulas are obvious, others require proof. Those that are not herein included are left out for future consideration. We shall let h be the number of apparent double points, p the deficiency, of a given curve C; H the number of apparent double points, P the deficiency, of a 3-space section c of a given surface F; and t the number of apparent triple points of F or the number of lines through a given point meeting F three times, f First consider the hypersurface F" whose lines satisfy four conditions. For a line to meet a given curve once is equivalent to two conditions. Hence the °o lines cutting across two given curves Cs C* form a hypersurface whose order is obviously

In this paper we study von Neumann rings of operators which are the strong closures of increasing sequences of type (IJ) factors. These von Neumann rings will be said to be hyperfinite. Murray and von Neumann showed that all hyperfinite factors of type (II,) are isomorphic [10]. We investigate the analogous question for type (III) hyperfinite factors and arrive at the conclusion that there are uncountably many non-isomorphic type (III) hyperfinite factors. We intend to study hyperfinite rings using techniques from the theoryof C*-algebras. Every hyperfinite ring contains, and is generated by, a norm separable C*-algebra which is the norm closure of an increasing sequence of type (IJ) factors. These algebras, called uniformly hyperfinite (UHF) algebras, have been defined and studied by J. Glimm [4]. We consider each hyperfinite ring as the weak closure of a UHF algebra. In ? 2 we show how hyperfinite factors can be characterized by states, positive linear functionals of norm one, of UHF algebras. Implicit in this characterization is the well known connection between cyclic *-representations of a C*-algebra and states of the algebra (see [12]). We derive necessary and sufficient conditions that a state o of a UHF algebra A, characterize a factor representation of X (Theorem 2.5). In ? 3 we study the question: when are hyperfinite rings isomorphic. We will show (Theorem 3.7) that if H1, and 112 are *-representations of a UHF algebra X, on separable Hilbert spaces, then the hyperfinite rings, {JH,(W()}" and {J12(X)}"', are isomorphic if and only if there is a *-automorphism q of 2 such that the representations x l1(x) and x 11(x) = 12(p(Pv)) are quasiequivalent. It will follow from this result that the group of *-automorphisms of a UHF algebra X acts transitively on the pure state space of W-. Combining the results of ?? 2 and 3 we will derive a necessary and sufficient condition, expressed in terms of states of a UHF algebra, that two hyperfinite factors be isomorphic.

About forty-five years ago Professor T. H. Hildebrandt wrote a paper (Hildebrandt [l ]) on integrals related to and extensions of the Lebesgue integral. At that time the Lebesgue integral, first invented in the preceding decade, had already displaced the Riemann integral from the monarchial position that it had occupied for over a genera­ tion. However, the subject was still new enough so that it was pos­ sible for Professor Hildebrandt to give an intelligible account of each one of the generalizations or extensions of the Lebesgue integral then extant, and to finish the job within a reasonable space of time. To do this today for the whole field of integration, covering all topics ade­ quately, and still remaining within the bounds of a one hour talk, would be an impossible task. I have therefore chosen to confine my­ self to one particular line of integration theory. I shall pass by all developments of new types of integral for the sake of increasing the generality of the process of integration. Likewise, I shall pass by all research designed to increase our knowledge of already existent forms of integrals. The research that I intend to report on is extremely di­ verse in nature, but all the developments that I shall mention have one aspect in common. In each case a mathematician needed an integration process to attain some goal, found that none of the tradi­ tional types of integration did exactly what he wanted and proceeded to invent an integration method capable of producing the results that he wished. A common feature of the integrals about which I wish to speak is that they are all what might be called "second growth" integrals. Probably none of them would have been thought of at all if the man who defined it had not been thoroughly familiar with an assortment of standard integration procedures, and a number of them in fact cannot even be defined without using concepts that themselves would never have arisen if it had not been for the development of the theory of the Lebesgue integral. This means that the one-time dominance of the Riemann integral has not been replaced by a similar dominance of any other integral. Competence in this field demands familiarity with a large assortment of concepts connected with procedures that might be called integration. A compact example of the sort of theory that I am speaking of Presidential address delivered before the Annual Meeting of the Society, January

1. Definitions. A transformation group (G, X) consists of a group G acting on a topological space X to form a group of homeomorphisms of X onto itself. I t will be understood throughout this paper that G is finite. For a given transformation group or "action" (G, X) and subset HQGy we denote by F(H\ G, X) the fixed-point set of H—that is, the points x such that hx = x for h gx are of class C. When i = 0 we shall drop the manifold condition on X; every action is then of class C°. A differentiable action is a enaction. (G, X) is orthogonal if X is a euclidean sphere or an open submanifold of a euclidean space and the trans