Search

In order to apply our rather deep understanding of the structure of Lie groups to the study of transformation groups it is natural to try to single out a class of transformation groups which are in some sense naturally Lie groups. In this paper we iiltroduce such a class and commence their study. In Section 1 the inotioni of a l,ie transformation group is introduced. Roughly, these are grouips II of homeomorphismlls of a space X which admit a Lie group) topology which is stronlg enough to make the evaluation mapping (ht, x) ->h (x) of II X X into X continuous, yet weak enough so that H gets all the onie-parameter subgroups it deserves by virtue of the way it acts on X (see the definiition of admissibly weak below). Such a topology is uniquely determined if it exists and our efforts are in the main concerned with the questioni of wheni it exists anid how onie may effectively put one's hands on it wlheil it does. A niatural candidate for this so-called Lie topology is of course the compact-open topology for H. However, if one considers the example of a dense one-parameter subgroup H of the torus X acting on X by translation, it appears that this is not the general answer. In this example if we modifyv the compact-open topology by adding to the open sets all their arc componlents (getting in this way what we call the modified compact-open topology), we get the Lie topology of HI. That this is a fairly general fact is onie of our maini results (Theorem 5. 14). The latter theorem moreover shows that the reason that the compact-open topology was not good enough in the above example is connected with the fact that H was not closed in the group of all homeomorphisms of X, relative to the compact-open topology. Theorem 5. 14 also states that for a large class of interestinig cases the weakness condition for a Lie topology is redundant. The remainder of the paper is concerned with developing a certain criterion for deciding when a topological group is a Lie group and applying this criterion to derive a general necessary and sufficient conldition for groups of homeomorphisms of locally compact, locally connected finite dimensional metric spaces to be Lie transformiiation grouips. The criterion is remarkable in that local compactness is niot one of the assumptions. It states in fact

In a paper,* which discussed the imprimitive group of ordelr 5184 in S3, the existence of an imprimitive group of order 216 in S2 connected with the Hesse configurations was indicated. The purpose of this paper is to discuss this planar group and the geometry associated with it. Of special interest is a configuration of twelve triangles which possesses important properties. These are dealt with in Section II. In Section I is a study of four-fold perspective triangles. A new canonical form is used which leads to a ruler construction for them and to a compass construction for a Clebsch six-point. I. FOUR-FOLD PERSPECTIVE TRIANGLES.

1. Summary. In a preceding paper [7] there was presented a Galois theory, for a certaini kind of differenitial field extension called strongly normal. The Galois group of a strongly normal extension is enidowed with a structure very much like that of a group variety, as studied by Weil [14]. In the present paper the study of such Galois groups is renewed for the purpose of clarifyinig their conniection with group varieties on the onie hand and with strongly normal extensiolis on the other. Consider a differential field S of characteristic 0 with algebraically closed field of constants W, and a strongly normal extension A of 5; suppose given a universal extension 5* of X, and denote the field of constants of 5* by AV. For reasons of convenience we use the term "Galois group of & over 7" for the group of all (ipso facto strong) isomorphisms of & over 5, and not for its subgroup conisisting of all automorphisms of & over J. The field R*, being algebraically closed anid of finite transeendence degree over A, may be used as a uniiversal domain for algebraic geometry (Weil [13], ch. I, ? 1); it is the group varieties of this algebraic geometry, defilned over E and slightly generalized to permit group varieties which are reducible (i. e. which have more thani one comiponent), whieh we consider. By an "algebraic group" we miean either a Galois group or a group variety as above. Chapter I is primarily a study of " rational " homomorphisms of algebraic groups into algebraic groups; these seenm to be the homomorphismls appropriate to the consideratioin of algebraic groups. Ratioinal homomorphisms of group varities inlto group varieties, without restrictioin to fields of eharacteristic 0, were considered by Weil [14] (who omitted the adjective "rational"). Specializing the coneept of rational homomorplhism we obtaini that of birational isomorphisin. In Chapter II it is showni that every Galois group is birationally isomorphic to a group variety, and conversely that every irreducible group variety is birationallv isomorphic to a Galois group; this

In this paper we apply methods of G. Higman, outlined in [2], to determine on which 2-groups Q the Suzuki groups Sz(q) can act as a group of automorphisms with an element of order 5 acting fixed point freely. We find that Q is elementary abelian and we determine precisely the action of Sz(q) on Q. To state what this action is, we need some definitions. In his paper [3], Suzuki defines the group G(q), here denoted by Sz(q), as generated as a subgroup of GL(4,q) by a certain collection of matrices. We let M be a 4-dimensional vector space over GF(q) affording this representation of Sz(q). If N is a module over the group-ring GF (q) Sz(q) (here GF(q) is the field of q elements), then we say that the action of Sz(q) on N is natural if N is isomorphic as GF (q) -Sz(q) module to a direct sum of copies of M1. If N is an elementary abelian 2-group admitting Sz(q) as a group of operators, then we say that the action of Sz(q) on N is natural if N is isomorphic, as GF (2) Sz (q) module to a GF (q) Sz (q) module on which Sz (q) acts naturally. With this notation, the following is a precise statement of the main result of this paper.

then r\D is a compact complex manifold. The dimension of the vector space of r-automorphic forms of weight m has been determined in [9] by means of the Riemann-Roch theorem. Hirzebruch [10] has used the AtiyahBott fixed point theorem to eliminate condition ii). Not much is known about the dimension of the space of automorphic forms when r\D is not compact. Leslie Cohn [5] has applied the Selberg trace formula to a certain domain D = G/K and a discrete group r =-Gz that does not satisfy either ii) or iii). In this paper, the index theorem is used to determine the dimension of the space of r-automorphic forms of sufficiently large weight m for certain groups r that satisfy i) and ii) but not iii). The domain D in this paper is the same as the one considered by Cohn [5]. Here, now, is an outline of the procedure to be followed. Let D===SU(2,1)/S(U(2) X U(1)) and let r C S(2,1) be a discrete subgroup which acts freely on D. Suppose also that r\D has finite invariant volume. By the assumptions on r, r\D is a complex manifold with a cusps. If r is sufficiently nice, then it is possible to compactify rFD as a complex manifold by adding a torus to each cusp. One can then define a complex analytic line bundle Em-r\D such fhat dim HO(r\D, Q(Em)) is the dimension of the space of r-automorphic forms of weight m. Here Q(Em) denotes the

1. Recently Lehner [2]1 developed a convergent series for pi(n) and p2(n), the number of partitions of a positive integer n into summands of the form 51 ? 1 and 51 + 2 respectively. The purpose of this paper is to develop a similar series for p1(n), p2 (n), * (,-1)/2(n), the number of partitions of a positive integer n into summands ofthe form pl ? 1, pl ? 2, pl ? (p 1)72, p being a prime greater than 3. Here, as in the previous paper, we follow the method of Rademacher [5]. We consider the generating functions

Introduction. In this paper, we shall study the fundamental group of an algebraic variety V minus a subvariety W over an arbitrary ground field, the classical case being subsumed as a special case. This will be done via first studying finite alegbraic coverings of v with branch loci contained in W. Here in the introduction, we shall only approximately describe the situation and indicate some of the results. The finite galois groups over V of all tame (for definition see Section 2) finite galois coverings of V-isomorphic coverings being identified-with branch loci contained in W form an inverse system &'(V-W) of a special kind which we shall call a group tower. A group G is said to be a weak parent group of a group tower xr if G can be topologized so that the group tower of all continuous finite homomorphic images of G is isomorphic to 7r; if 7r is isomorphic to the group tower of all finite homomorphic images of G (i.e. if G is regarded as a discrete group), then G is said to be a parent group of 7r.1 The possible existence of a finitely generated parent group (or somewhat weaker: the possible existence of a finitely generated weak parent group) of r'(V -W) is the abstract analogue of the statement (Section 16) that in the classical case the topological fundamental group sr1(VW) is finitely generated; and hence if such a finitely generated parent (respectively, weak parent) group exists, we shall call it a tame fundamental parent (respectively, weak parent) group of V -W. Now one main result of this paper (Section 12) is that if V is nonsingular and simply connected, if W has only normal crossings and if the irreducible components of W move in linear systems of dimension greater than one, then denoting the number of these

In his paper "Concerning Simple Continuous Curves," * R. L. Moore applied the term simple continouou curve to any point set which is either an arc, a simple closed curve, an open curve or a ray, and gave various topological characterizations for a point set that falls within the general class of such curves, as well as for the individual types of curves.f The intent of the present paper is to supplement the work of Moore just cited, extending two of his results to more general spaces, and giving certain new definitions which the author believes may be of value.$ As Moore made clear in his paper, the requirement of boundedness in a definition of arc may introduce great difficulties in certain problems. It will be noted that none of the definitions of arc, simple closed curve, etc., given in the present instance makes use of this condition. Furthermore, the condition that the set in question be closed is eliminated in several cases-a feature that would seem to be of importance in problems concerning non-closed sets. To summarize briefly, before proceeding to the demonstrations: In ? 1, Moore's characterization of simple continuous curves as a class is extended to any euclidean space, and is then applied to establish a new characterization based on a certain type of set which the author has found useful in other connections. In ? 2 definitions are given of arc, the more general class of sets known as irredtcible connexes,? and of simple closed and quasi-closed curves. In this connection it is indicated how a simple proof may be obtained of the theorem ? that every interval of an open curve is an arc; also a

Introduction. In 1951, Zariski obtained as a special case of his connectedness theorems the fact that the total transform of a normal point by a birational transformation is connected [III]. In a recent paper, [IV], the connectedness of the total transform of a simple point by a birational transformation is proved without using the theory of Fholomorphic functions developed in [III]. In this paper another proof is obtained of this connectedness-theorem for a birational transformation at a simple point. In fact, a somewhat stronger result is proved, namely, the following. Chow has introduced the concept of linear connectedness, a set is said to be linearly connected if every pair of points can be connected by a sequence ofrational curves in that set. In this sense, the total transform of a simple point by a birational transformation is linearly connected.2 Furthermore, some applications are derived for specializations of a complete set of conjugates over a purely transcendental function field. The terminology and notations are from [I]. In conclusion, I want to express my warmest thanks to Professor A. Weil and Professor T. Matsusaka for their advice and interest during the preparation of this paper. I am especially indebted to Professor Weil for suggestions which improve the exposition of the proof of Theorem 1.

PART ONE. INTRODUCTION. In a former paper * the solution of the cooling problem for several media for the case in which the heat flows in parallel lines was shown to be reducible to the solution of a set of Volterra integral equations of the second sort. The purpose of this paper is to indicate the application of this method to the problem where the flow is along radii of a sphere. The arguments which are repetitions of those in the former paper will not be given, but reference will be given to that paper.

This paper deals with the system of curves each of which possesses the following property: A particle, starting from rest and moving along the curve, under the action of a given force, will reach a certain point, the center of tautochronous motion, in the same time from whatever point on the curve it starts. Friction is neglected and it is assumed that the actiing force depends only on the position of the particle. The first part of the paper deals with plane systems. It is assumed that within the region considered the rectangular components of the force, p (x, y), 4 (x, y), are uniform functions which possess first and second derivatives and do not vanish identically. In 1673 Huyghens found that when the acting force is gravity the tautochrones are cycloids with horizontal bases and concavity upwards. Since that time many extensions of the problem have been made and the tautochrones due to more complicated laws of force and resistance have been found. This paper, however, does not deal with the problem of finding the tautochrones due to a general positional force, but investigates the form of the differential equation and the derivable geometric properties of the system. This differential equation, which is of the third order and represents the total triply infinite system of tautochrones in the plane, is found in Article 1 from the well-known physical property of tautochrones, namely, that the tangential component of the acting force along the curve is proportional to the arcual distance of the moving particle from the center of tautochronous motion. The factor of proportionality must be negative for actual tautochrones, but, as it is eliminated in finding the differential equation of the system, the system includes virtual as well as actual tautochrones.

In other papers 1 a semi-group was defined as a set of elements closed ander an associative operation and for which the equivalence and the substitution postulates hold. tn the present paper we shall employ instead of the substitution postulate, the postulate that if A = B, then CA = CB and AC = BC for any A, B or C in the set, which we shall call the composition postulate.2 A gruppoid 3 is a semi-group with an identity element, that is, an E such that, AE = EA = A for any A in the set. A quasi-group is a semi-group such that from either of the relations

In Volume 3 of this Journal Sylvester [1] wrote the following postscript to his paper: "In a future communication I will prove very simply that if a prime number p = ef + 1, and e itself is a prime such that e -1 contains no odd square number, then every divisor, withoue exception (other than p), of the function whose roots are the periods of the primitive pth roots of unity, must be an eth power residue. If (e -1) contains any square number the proof still holds good, except as regards factors of such square, and there is no reason at present for supposing that the theorem may not be extended to the case of these excepted factors." A year later in Comptes Rendus [2] he returns to the problem of possible exceptional divisors (i. e. divisors which are not e-th power residues), states that they divide the discriminant, shows that they do not exist for cubic equations, complains (quite rightly) that the problem is not treated in Bachmann [3] and reserves it once more for future consideration. We have not been able to find any further reference to the problem in the works of Sylvester or elsewhere. It is completely ignored by the Smith or Hilbert Reports on the Theory of Numbers as well as the National Research Council Report on Algebraic Numbers [4]. All the above mentioned sources discuss at length Kummer's work on the period equation without pointing out that Kummer [5] had essentially a proof of the nonexistence of Sylvester's exceptional divisors some thirty-five years before Sylvester proposed the problem. In this paper we will give a proof proceeding along the lines outlined in Kummer [5] of the non-existence of such exceptional divisors, but we are at a loss to explain why Sylvester had to exclude divisors of e-1 which appear to the second power. His unpublished proof must have proceeded along quite different lines. More precisely we will prove

This paper may be considered as a preparation for later investigations. I shall prove some theorems and lemmas which are of general interest to the theory of modular functions of several variables. Some of the results were already known but it seemed desirable to find better proofs. The part about Poincare's series is closely connected with some recent work of H. Klingen [19], [20], [21]; a more detailed explanation about this connection may be found at the end of Chapter III, ? 3. The paper consists of three chapters, the contents of which is the following. The first chapter deals with the symplectic group. Some inequalities and the boundedness of certain expressions will be proved. The second chapter is devoted to Hilbert-Siegel's modular groups. First we define arithmetically certain equivalence classes and prove that there are only finite many such classes provided some definite expression is bounded. These results are needed to prove the convergence of Poincare's series. Then we deal with elliptic fixed points of Hilbert-Siegel's modular groups and improve some result obtained in [11]. At the end of the second chapter I prove that the quotient spaces of the generalized upper half-plane with respect to the full Siegel's modular group and the so called group of integral modular substitutions have the first homology group zero provided n > 1. In chapter three we show that under certain assumptions a HilbertSiegel's modular form of weight zero is constant. Then we investigate the convergence of certain integrals and in connection herewith the uniform convergence of Poincare's series. Finally some theorem about separation of variables through modular functions is deduced. This paper was partially supported by an N. S. F. grant.

Introduction. We shall consider a C2 action of the real line, T, on a compact, connected, two-dimensional manifold, M, of class C2. Thus a: T X l M is a transformation of class C2 such that tm = a (t, m) satisfies (t+s)m-=t(sm) and Om==m. Note that c2 vector fields on M give rise to such actions (see, fo rexample [5, p. 40]). A set, Q C 31, is called aminimal if it is closed, invariant under a (T12-Q), non-empty and contains no such proper subset. The purpose of this paper is to prove that any minimal set is either a ,xed point, a homeomorph of S1, or 11 of M and homeomorphic to T2. This theorem is an extension of work by Poincare [7], Bendixson [1], and Denjoy [2]. In particular, the analytical methods of Denjoy are modified, extended, and to some degree simplified. A theorem equivalent to this was stated by Haas [3, p. 297]. However, as Peixoto pointed out [6, p. 113], the accompanying proof was incorrect. The author wishes to acknowledge his indebtedness to Professors R. Sacksteder, E. Lima, and R. Ellis fortheir criticism and encouragement during the preparation of this paper.

In this paper we give a detailed exposition of some of the results announced in [18]. The primary motivation for this work is Witt's observation [31, Satz 7] that if F is a field of characteristic #72, his ring 1V(F) of classes of anisotropic quadratic forms may be written as Z[G]/K where G is an abelian group of exponent two (actually G =F*/F*2), Z[G] is the integral group ring of G, and K is an ideal of Z[G] generated by elements of the form gl + g2-g3g4 and 1 + g5 with gi in G. Of course these elements can be described more explicitly, but for our purposes the only information we need about K is that any homomorphism of Z[G] -> Z sends K to 0 or to an ideal of the form 2nZ. In this introduction we shall call any ideal of Z[G] with this property "admissible." In [26], Pfister proved certain structure theorems for IV(F) usilg his theory of multiplicative forms. His proofs were simplified in [11, 22, 23, 29, 30]. Harrison [11] and Leicht and Lorenz [22] gave important complements to Pfister's results concerning the ideal theory of WV(F). The main goal of our paper is to understand and generalize these structure theorems using standard techniques of commutative algebra. In [28] and [3], Scharlau and Belskii have introduced and studied Witt and Witt-Grothendieck rings for profinite groups. Their definitions generalize the usual notions of Wit.t rings, WV(F), and Witt-Grothendieck rings, K (F), of quadratic forms over fields of characteristic #72. These rings also have the form Z[G]/K with G an abelian group of exponent two and K an admissible ideal of Z[G]. Here we are mainly iinterested in another generalization of 1V(F) and K(F), namely the Witt rings, IV(C, J), and the Witt-Grothendieck rings, K(C,J) of classes of hermitian forms over a connected commutative semilocal ring C with involution J. Since J may be the identity these include the Witt and Witt-Grothendieck rings of classes of symmetric bilinear forms

Introduction. In a recent paper 1 it was shown that the problem of classification of metabelian groups of order pLn+,, which contain a given abelian group of order pfl as a maximal invariant abelian subgroup and have commutator subgroups of order pm is equivalent to the problem of classification of the matrices x1M+ x2M2 + --* + xkMk under projective transformations on the x's and elementary transformations on the square matrices M1, M2, * , Mk. The x's and the elements of the M's are of course numbers in a modular field as are also the coefficients of the transformations. The squareness of the matrices comes from the requirement that the commutator subgroup be of order pm. The situation may then be discussed in terms of the invariant factors of the matrix x1Ml + x2M2 + -* + XkM7 The argument of that paper still holds when the commutator subgroup is not of order pm and the matrices Mi are not square. In this case however we are deprived of the use of a well-developed theory of invariant factors. So far as I know the question of the conjugacy of two matrices of the above type under transformation on the x's and simultaneous transformations on rectangular M's has not been considered. It is our purpose to colnsider the groups which give rise to such matrices in the simple case where m = 4 and k = 2 and to use the results to obtain normal forms for the matrices. It will be convenient to interpret the matrices Mi as matrices of bilinear forms in which case the matrix above, which we shall denote hereafter as XlMl + X2M2, may be taken to represent a pencil of bilinear forms.

My second paper on metrical geometry * contains a characterisation of the n-dimensional euclidean space among general semi-metrical spaces in terms of relations between the distances of its points. In courses on metrical geometry at American universities I have considerably shortened and revised my original proofs and generalized the formulations by introducing the concept of congruence order. The following paper contains these new proofs. In the first part we prove that every semi-metrical space, each n + 3 points of which are congruent with n + 3 points of the n-dimensional euclidean space, is congruent with a subset of the n-dimensional euclidean space. This is expressed by saying that the n-dimensional euclidean space has the congruence order n + 3. In the second part we prove that each semi-metrical space containing more than n + 3 points each n + 2 points of which are congruent with n + 2 points of the n-dimensional euclidean space, is congruent with a subset of the n-dimensional euclidean space. This fact is expressed by saying that the R. has the quasi-congruence order n + 2. It is proved by a systematic study of those sets which contain exactly n + 3 points and are not corLgruent with n + 3 points of the n-dimensional euclidean space whereas each n + 2 of them are congruent with n + 2 points of the n-dimensional euclidean space. These sets are called pseudo-euclidean sets. By means of these results the problem is reduced to the question: under what conditions are n + 2 points congruent with n + 2 points of the R. and by what distance relations are the pseudo-euclidean (n + 3)-tuples characterized. These purely algebraic problems are solved in the third part.t

In a well-known paper, Hilbelt 1 has characterized the Euclidean and hyperbolic plane geometries by mere group and continuity axioms. He gets all the motions at once by requiring the existence of sufficiently many rotations. The present paper tries to point out how the existence of more and more translations gradually specializes the rather general metric it starts with to a Desarguesian geometry (Minkowskian or hyperbolic). We require our initial space to be a " Geradenraum " in Menger's terminology.2 The exact definition will be found in section 1. It is essentially a metric space with exactly one shortest line (s. 1.) of infinite length passing through two given points. We therefore call a Geradenraum an SL space. In order to be able to formulate the results, we must have the concept of an asymptote. If g is any shortest line, P a point not on g, and Q a point

The purpose of this paper is to determine the structure of some finite groups in which all Sylow subgroups of odd order are cyclic. This assumption on Sylow subgroups simplifies the structure of groups considerably, but the structure of 2-Sylow subgroups might be too complicated to make any definite statement on the structure of the groups. In this paper, therefore, we shall make another assumption on 2-Sylow subgroups, and our main result may be stated as follows. Let G be a non-solvable group of finite order. We assume that all Sylow subgroups of odd order are cyclic, and moreover that a 2-Sylow subgroup is either (a) a dihedral group, or (b) a generalized quaternion group. Then G contains a normal subgroup G1 such that [G: G1] ? 2 and G1 = Z X L, where Z is a solvable group whose Sylow subgroups are all cyclic, and L is isomor-phic with the linear fractional group LF (2, p) over the prime field of characteristic p in the case (a), and with the special linear group SL (2, p) in the case (b).

Whether K2 is a contracted functor is still open and more has to be known about (?) to answer that question. The present paper is a revised version of [7] and the author wishes to thank J. Milnor for several suggestions leading to a simplication of the presentation. In a further paper [8] the ideas of this paper will be extended to study K2 of twisted Laurent polynonmial rings and of integral group rings of amalgamated products of groups.

A universal central extension of a group G is a central extension, G, of G satisfying the additional conditions (i) G = [G, G], and (ii) every central extension of G is split. If, by analogy with topology, we call a group G connected if H1(G, Z) = 0 and simply connected if H1(G, Z) H(G, Z) = 0, the above conditions are equivalent to saying G is simply connected. In this language, a universal central extension of G is called a universal covering. Now suppose b is a reduced irreducible root system, A is a commutative ring with unit, and E (cD, A) is the elementary subgroup of the points in A of a Chevalley-Demazure group scheme with root system (. Define the Steinberg group, St ((, A), by generators and relations as in [19]. Then the main concern of this paper is to determine under what conditions E ((, A) is connected and St ((, A) is simply connected. When A is a field, these questions were posed and resolved in a well-known paper of Steinberg [19]. For the groups of type Al, 1 ? 2, they have been treated by Kervaire [10] and Steinberg [20]. The results of this paper were announced in [16]. Connectivity is discussed in Section 4. For groups of rank > 2, the main result is most suggestively stated "E ((, A) and St ((F, A)) are connected if and only if E ((, A/m) is connected for every maximal ideal m C A." Thus these groups are connected unless ( C= or G2 and A has a residue field with. two elements. The above statement is also true for groupsof rank 1, provided A is semi-local. In fact, a stronger result holds in the rank 1 case: E (A1, A) and St (A,, A) are connected if the ideal generated by {u2 _1, u C A*} is all of A. Although this implies that A has no residue field with 2 or 3 elements, I do not know whether these are, in general, equivalent conditions. Despite the relationship with the case of fields in the formulation. of these theorems, their proofs do not resemble the proofs for fields. They are based instead on careful exploitation of commutator relations in the groups of rank 2. To decide when St ((, A) is simply connected, it remains to determine when every central extension of St ((, A) is split. The answer, roughly

A continuum every subcontinuum of which is locally connected is said to be hereditarily locally connected. The principal contribution of the present paper is the establishing of the proposition that Every hereditarily locally connected, compact and metric continuum is a rational curve in the MengerUrysohn * sense, that is, each point of such a continuum M is contained in arbitrarily small neighborhoods having countable boundaries relative to M. This theorem was proved formerly f by the present author for subcontinua of the plane, but the demonstration in the present article is independent of the containing space and is therefore valid for subcontinua of any compact metric space. It is thus demonstrated that in the Menger-Urysohn classification of curves, the hereditarily locally connected continua form a distinct class occupying an intermediate position between the class of all regular curves ( continua each of whose' points is contained in arbitrarily small neighborhoods with finite boundaries) and the class of all rational curves. In other words, all regular curves are hereditarily locally connected, but-not conversely; and all hereditarily locally connected continua are rational curves, but not conversely. In the course of the demonstration of the proposition announced above, the author has found and used a number oi strong properties of hereditarily locally connected continua, each of which, incidentally, characterizes these continua among the compact metric continua. Proofs for these properties, together with some of their corollaries, form ? 2 of the present paper. In ? 3 there is given two lemmas of a general character which also are needed in the proof of the main theorem of the paper, given in -? 4. We shall employ the usual terminology and notation of the theory of sets. Our hypotheses' ordinarily concern a compact metric continuum M and its subsets, and in such cases we shall consider M as a space and shall speak of the open subsets of M as neighborhoods or open sets. If V is such a set, F(1) will denote the boundary of V, i. e., the point set 17 V. A component of a set X is a connected subset of K which is containedi in no other

In this paper we establish the following characterisation of a closed 2-cell (a set of points homeomorphic with a plane circle plus its interior): If C is a continuous curve (compact)* containing a simple closed curve J and at least one arc ab such that ab has its endpoints and these only on J,f and such that every arc spanning t J irreducibly separates I C, then C is a closed 2-cell (and J, of course, its boundary). We shall set about the proof in this fashion that we shall show that C J is topologically a euclidean plaine (therefore homeomorphic to the interior of a plane circle) and has every poinlt of J for limit point. We shall regard this as by far the major part of the argument, for we shall then know essentially that our point set is an open non-singular 2-cell bounded by a simple closed curve. At that point it is still conceivable that we have a singular 2-cell, the singularities being on the boundary. But it is as least intuitively clear that the condition that spanning arcs must separate can be invoked to rule out this situation. However, the methods of the paper do not lend to easy translation into combinatorial technique and we shall have to close the proof by carrying out a " mapping " of C onto a closed 2-cell. It is obvious from the nature of our conditions that there is little we can say about C, in the beginning at least, excepting through the point set J. The first thing we shall have to determine, and in the proof of this lies most of the novelty of this paper, is this that every spanning arc of C separates C into two components each of which contains a point of J. This is a consequence of the assertionl which we now prove that if xy is a spanning arc and A any component of C xy, A contains a point of J. For J (x + y) is

Introduction. In this paper we investigate branched coverings of compact complex manifolds and, in particular, we answer certain questions connected with the deformation theory of branched coverings. Let M and W be compact complex manifolds, 311 a branched k-sheeted covering of W. The Main Theorem of ? 4 gives sufficient conditions on M and W (expressed as cohomological conditions on W) for every small deformation of M to be a I-sheeted covering of W. Our methods apply to branched coverings of the following type: M is a compact submanifold of a projective line bundle, F#, over the compact manifold W; the bundle map F# -W makes M a branched covering of W. We show in ? 1 that cyclic coverings of W (and hence two-sheeted coverings) are of this type. Theorem 4. 1 gives a sufficient condition (involving cohomology groups on M and FP) that every small deformation of M is a branched covering of W. In ? 2 the spectral sequence for a fibre space is applied to express cohomology groups on F# in terms of cohomology groups on W. These results are used in ? 4 to reduce the condition of Theorem 4. 1 to a condition involving cohomology groups on W. If W= Pn, we obtain very explicit results (? 5 and ? 6) since strong results are available on cohomology groups on pn. In ? 7 we compute the number of moduli for cyclic coverings of P*, (n ?>2). Finally, in ? 6, we give some examples of branched coverings of P2 having small deformations which are not branched coverings of p2. Essential use is made of Theorem 1. 1 in which we show that a branched covering of a projective algebraic manifold is itself projective algebraic. We refer to the papers of Kodaira and Spencer (in particular [14] ) for those matters concerning deformatiois of complex structures. lirzebruch's book [10] will serve as a general introduction to vector bundles, Chern classes, duality, etc. By "vector bundle" we will always mean a fibre bundle whose fibre is Cq and whose structure group is GL (q, C). "Line bundle" will mean a vector bundle with q = 1. If B is a vector bundle on a manifold, (B)

In a previous paper 1 the exponential representation of canonical matrices was studied. These canonical matrices are automorphs of the normal form of a skew symmetric matrix. The corresponding problem, when the skew symmetric matrix is replaced by a symmetric or hermitiani matrix, is considered here. Three cases are treated: when the matrix is symmetric over the complex field., when the matrix is hermitian, and when the matrix is symmetric over the real field. Of these three the last is by far the most interesting. The methods employed are similar to those of the paper quoted above and, as in many cases the proofs are practically identical, they will not always be given in detail.

The list of the primitive groups of the first thirteen classes prepared by JORDAN has long been known to be defective in the case of class 12; in fact, JORDAN states explicitly that the calculations for that case, having been gone through but once, may very well contain errors.* In spite of the considerable advances in substitution theory in recent years, the complete a priori determination of all the groups of this class still involves a good deal of labor. In order therefore not to intrude too far upon the space of this journal, the writer avoids a redetermination of the primitive groups of class 12 on less than 21 letters.1 In a series of memoirs and in his lectures Professor MILLER has gone over this ground without use of the list of the groups of class 12. He has thus checked the results of MiSS MARTIN, of JORDAN, and of Mliss BENNETT on the degrees 18, 19, and 20, respectively, according to whiclh there is no primnitive gr'oup of class 12 on either of these three degrees. The method here followed will be much the same as that used by the author in treating the classes 6, 8, and 10, with the advantage of having at hand the processes employed in the paper entitled "On the Limit of the Degree of Primitive Groups." This last paper, as well as that "On Mlultiply Transitive Groups," and that "On the Order of Primitive Groups," will be used freely without specific reference.t In addition to the 25 primitive groups of class 12 of degree less than 18, there are four others, one of degree 27, two of degree 28, and one of degree 36.

Published accounts of properties of characteristic functions of third-order, irregular boundary value problems and of expansions in infinite series of such functions have been given by D. Jackson and Hopkinst and by the author.T In these papers were considered only cases which may be regarded as elementary in the sense that the characteristic functions were elementary functions. The present paper deals with the characteristic functions and the associated series of the differential system (1) d3u/dx3 + [p3 + r(x)]u = 0, u(0) = u'(0) = U(7) = 0. We shall suppose throughout that r(x) is a convergent power series in x3, and that, when the radius of the circle of convergence of this series is less than 7r, r(x) is continuous in the interval 0 ? x < w. Many of the methods and results obtained in this paper (but not all of them) can be carried over immediately to the case in which the differential equation is the same as the present one but the boundary conditions are of the type in Part I of the 1927 paper. The first part of this paper is devoted to a study of the characteristic functions, the second part to necessary conditions for the convergence of a formal series of these functions arising from a given function f(x), and the third part to a statement of conditions sufficient to ensure the convergence to f(x) of the formal series forf(x), and to the proof of such convergence.

In 1925 Menger f introduced the term "curve" for those compact connected spaces which have the property that for every point of the set and every e> 0 there exists a neighborhood of diameter less than e which contains the point and has a totally disconnected I boundary. A point of such a set is said to be regular if it is possible to choose these neighborhoods in such a manner that its boundary is finite, while it is of order n if for every e we can find the neighborhood of diameter less than e whose boundary consists of exactly n points. In a subsequent paper he shows that in a regular curve, i. C., a curse containing only regular points, there is corresponding to every point of order n a set of n arcs having the point in common and otherwise distinct.? It should be noted that every regular curve of Menger is also a continuous curve,? that is, it is conilected im kleinen at every poinlt. Of course there are continuous curves which are neither regular nor even curves in the sense defined above. In this paper is colisidered a plane continuous curve in which there are two points a and b which have the property that there are in the continuous curve exactly n arcs frorn, a to b which have no points other than a and b in comrnmon while it is impossible to find a similar set of n + 1 arcs from a to b. Under these hypotheses it is proved that there are exactly n points in the curve whose omission separates a from b in the curve. This theorem provides a sort of converse to the theorem of Menger mentioned. The precise converse of the theorem of this paper is an easy consequence of the theorem itself, and

In a previous paper 2 the writer defined the notion of a simply connected set and showed how it was useful in proving a number of separation theorems. Our purpose here is to extend this notion to multiple connectivity and derive some further results for the plane. The abbreviation S. C. will hereafter be used in referring to the paper mentioned above. Let n be a positive integer. A connected set M is said to be multiply connected of order n, or n-ply connected, if (1) for each pair of points A and B of M and any relatively closed subset L of M that separates A from B in M there exists a subset of L, consisting of n or less components, which separates A from B in M, and (2) there exists at least one pair of points A and B of M and a relatively closed subset L of M that separates A from B in M such that every subset of L which separates A from B in M has at least n components. This definition becomes the criterion for a connected set to be n-ply connected in the weak sense if " separates " is replaced throughout by " weakly disconnects." 3'

In the author's paper [1] there was an error in Lemma 5 which was pointed out by David J. Winter. The present paper gives a description of fixed-point-free automorphisms of algebraic tori and includes an appendix containing the necessary corrections to [1]. The reader is referred to [1] for results on Frobenius algebraic groups. We shall denote p the characteristic exponent of the ground field. A first attempt to describe the situation is in the case of a group H of fixed-point-free automorphisms of a torus T (i. e. H is an algebraic transformation group on T operating faithfully by fixed-point-free rational automorphisms).

Throughout this paper the space under consideration will be the Euclidean plane. The definitions given by R. L. Moore in [6] are essentially equivalent to the following, also due to Professor Moore. An arc t is said to spiral down on a point 0 provided there does not exist a finite sequence of rays 1 each starting from 0 such that the first has no point other than 0 in common with t, the last is a straight ray, and no two adjacent rays of the sequence have a point other than 0 in common. The arc t is said to go n steps toward spiraling down on 0 provided there does not exist a sequence of n rays satisfying the above conditions. An arc which spirals down on some point is called a spiral and a point set which contains no spiral is said to be spiral free. In this paper an example is given of a compact, totally disconnected, closed point set such that every arc containinig it spirals down on each of uncountably many points. In order that the compact, totally disconnected, closed point set ][[ should be a subset of a spiral free arc it is sufficient that the projection of M onto some straight line be totally disconnected and it is necessary and sufficient that M be a subset of the sum of the elements of a continuous and equicontinuous collection G of mutually exclusive spiral free arcs such that G, with respect to its elements as points, is totally disconnected. A number of related results are also obtained.

A point p of a compact metric continuum M is said to be a point of finite degree * of M provided that for each e > 0 there exists an uncountable family G of neighborhoods of p in Mll each of diameter < E anid each having a finite set for a boundary and such, that for any two neiborhoods U, V of G we have either U C V or V C U. Similarly, p is said to be of degree a provided ac is the least cardinal number such that for each e the neighborhoods of G can be chosen so that the boundary of each of them is of power < ac. (We shall be concerned principally with the case where a is an integer.) If X is a given subset of M, then p will be said to be,of finite degree or of degree ac in M relative to X provided that, in addition to satisfying the conditions previously stated, the neighborhoods of G can be so chosen that the boundary of each of them is a subset of X. These notions concerning the degree of a point should not be confused with the analogous but distinctly different notions concerning the order or index of a point as introduced by Menger and Urysohn. Both concepts will be used in this paper, but careful reading should enable one to avoid confusing them. A continuum, is said to be of finite degree provided each of its points is a point of finite degree. In this paper we shall show, among other results, that such continua may be identified with the continua previously studied by the author t which have the property that any subcontinuum contains uncountably many local separating points : of the given continuum.

This paper is a continuation of [13] which we refer to as I. In I we introduced some Hopf algebra techniques which together with the spectral sequence of [8] allowed us to deduce various results for stable Postnikov systems. Our objective in this paper is to obtain further results concerning llopf fibre squares and to determine the general form of the differential in the homology spectral sequence of a lopf fibre square. This task divides naturally into two parts. The first part (Sections 1-4) is concerned with the structure of the E2-term of the spectral sequence. More precisely suppose that A, B are homology Hopf algebras over a field k and f: B ->A is a morphism of Hopf algebras. We wish to determine the sructure of CotorA (B, k) as a Hopf algebra and the properties of the morphism

Recently, in a remarkable piece of work [4, 5] John Thompson has proved a result which implies as an immediate corollary the well-known Frobenius conjecture, namely that a finite group admitting a fixed-point-free automorphism (i. e., leaving only the identity element fixed) of prime order must be nilpotent. However, non-nilpotent groups are known which admit fixedpoint-free automorphisms of composite order. In all these cases one notices that the groups in question are solvable. Although the sample is rather restricted, it is not too unnatural to ask whether the condition that a finite group admit such an automorphism is strong enough to force solvability of the group. This question is related to another problem, which seems. equally difficult, which asks whether a finite group containing a cyclic subgroup which is its own normalizer must be composite. In the present paper we shall prove that a group G possessiilg a fixedpoint-f ree automorphism of order 4 is solvable. Although many of the ideas used carry over to the case in which 4 has order pq, and especially 2q, our key lemmas use the fact that 4 has order 4 in a crucial way. The proof depends upon a theorem of Philip Ilall which asserts that a finite group G is solvable if for every factorization of o(G) into relatively prime numbers m and n, G contains a subgroup of order m. We show (Lemma 7) that a group G which has a fixed-point-free automorphism of order 4 satisfies the conditions of H all's theorem. Once we k-now that G is solvable it is not difficult to prove that its commutator subgroup is nilpotent (Theorem 2). This -fact was also observed by Thompson. Graham Higman has shown [3] that there is a bound to the class of a p-group P which possesses an automorphism p of prime order q without fixedpoints. This does not carry over to automorphisms of composite order, for at the end of the paper we give an example due to Thompson of a family of p-groups of arbitrary high class each of which admits a fixed-point-free automorphism of order 4.

In the first four sections of the present paper there will be developed a method of obtaining decompositions of a continuum M into disjoint subcontinua by means of set-functions satisfying suitable conditions. If these functions are sufficiently restricted, the decompositions obtained will be upper semi-continuous, so that a decomposition space may be defined. The remainder of the paper is devoted to a study of the decompositions obtained when these set-functions are defined in various ways in terms of local separating points of M and cut points of M so that certain ones (and in some cases all) of the conditions considered earlier are satisfied. For example, it will be shown that if H is compact, then for each pEM, there exists a maximal subcontinuum CGi(p) of M containing p and containing only a countable number of local separating points of M; the sets C, (pi) are disjoint and the decomposition of M into sets CO(p) is upper semi-continuous; the decomposition space Ci is a regular curve every subcontinuum of which contains uncountably many local separating points both of C% and of M. Similarly for each p!M there exists a maximal subcontinuum C3(P) of M containing p and such that the local separating points of C,(p) are countable; furthermore, C3(p) is identically the sum of all totally imperfect connected subsets of M containing p; this decomposition need not necessarily be upper semi-continuous but it will be in case M is hereditarily locally connected (i. e., if every subcontinuum of 211 is locally connected), in which case the decomposition space C3 is hereditarily locally connected and contains no totally imperfect connected subset. A.nalogous results are obtained using punctiform instead of countable and likewise by using cut points in place of local separating points.

the object of investigation has the advantage of immediacy, the more convenient definition for the purposes of this paper, and the one given below, is in terms of the *-ring it generates. The exact relation between these two aspects of a binormal operator is the first problem to which we address our attention, and the desired formulation is stated in Theorems 1 and 2. The central result of the paper may be paraphrased as follows: We attach to a binormal operator four commuting normal operators (fixed polynomial functions of the operator and its adjoint) and show that except for comparatively uninteresting normal direct summands which each posseses, two binormal operators are unitarily equivalent if and only if these four functions of each are simultaneously. unitarily equivalent in pairs (Theorem 5). In the course of proving this it is necessary to obtain considerable information about the structure of a binormal operator, and in particular a sort of standard form is given for any such operator (? 9). Also we show that Dixmier's solution of the " two projection problem " [2] is subsumed under the theory of binormal operators in a natural and satisfactory manner.

where Skl O if k,7c/z and 1 if k7 = . Since the coefficients e;k are complex numbers, the group U depends upon (n + 1)2 real parameters. The geometry of Pn with the fundamental group U of transformations is called the Hermitian geometry (more precisely the elliptic hermitian geometry) and the space Pn itself is called an Hermitian space. The Integral Geometry in these spaces was initiated by Blaschke [2] who gave the densities for linear subspaces, without making applications to integral formulas. The case n = 2 was first considered by Varga [8] and later, in a more complete form, by Rohde [6]. In the present paper we generalize to the n-dimensional case some of the results which Varga and Rohde obtained for the plane. The main results we obtain are the following: We first determine the explicit form of the left invariant element of volume du of the group U. If Lr? is a fixed linear subspace of dimension r (throughout the paper we shall mean by dimension the complex dimension)

Professor J. Radon has called my attention to the fact that the presentation of the Legendre equation of a problem of Lagrange as a matrix differential equation of Riccati type, and the fundamental theorems on the integration of such an equation, are given in two earlier papers of his [" Uber die Oszillationstheoreme der konjugierten Punkte biem Probleme von Lagrange," Miinchner Sitzungsberichte (1927), pp. 243-257; "Zum Problem von Lagrange," Abhandlungen aus dem Mathematischen Seminar IHamburg, vol. 6 (1928), pp. 273-299]. In particular, the results of Sections 3 and 4 of the indicated paper of the author are contained in these papers of Radon.

In his previous paper [2],2 the author proved a criterion of an ample divisor 3 on a non-singular surface, i. e., a divisor X on a non-singular surface F is ample if and only if (X2) > 0 and X is arithmetically positive.4 In this paper he will prove a generalization of this result to a projective scheme over a field. Originally the author intended to prove this generalization only for a non-singular variety of any dimension, say n. However, in the course of the proof it became necessary to treat the problem on a variety of dimension n-1 with singularities, and then on a scheme of dimension n2. This is the reason why he finally decided to treat the problem on a general projective scheme right from the beginning. The author wishes to express his heartfelt thanks to Professor 0. Zariski and to D. Mumford. Without their encouragement and suggestions this work would not have been done. In this paper we shall make extensive use of notations, terminologies, and the results in Grothendieck's book, " flements de G-4ometrie Algebrique," which will be cited as [G]. Most of them will be used freely without any further explanations, except some less fundamental notions. We hope the readers will not find much inconvenience in this way.

The graphical representation of a group given by Cayley* leads to a diagram consisting of several lines of different colors, a so-called color-group, which affords a very clear insight into the structure of the group. Cayley himself applied his method only to groups of comparatively low orders, and it seems that the inethod has never been used for more complicated cases.t The purpose of the present paper is to show how readily Cayley's method can be applied to the construction and investigation of numerous groups of higher orders. In particular, the color diagrams for the rotation groups of the regular bodies can be arranged in such a way that they lend themselves miuch easier, at least in some respects, to a study of the groups concerned, than even the models of the regular bodies. The most prominent feature of these diagrams, to which their high degree of perspicuity is due, consists in the fact that their color lines do not intersect each other, so that the diagrams, when described on the sphere, constitute convex polyhedrons. I determine, in the first part of the paper, all the color-groups thus defined and show that, apart from two other cases, they are identical with the rotation groups of the regular bodies. In the second part I study in detail the connection between the rotation groups and the corresponding diagrams. The third part of the paper contains some extensions of the

the characteristic values of compact operators which may be of independent interest. These are established in Sections 1 and 2 and are applied in Section 3 to general operators with compact resolvent. Applications to spaces of distributions and to elliptic operators are given in Sections 4 and 5. The asymptotic formula for the eigenvalues of classical second order operators is due to Weyl [12]. His results were rederived by Carleman, using a powerful method which has been applied to more general elliptic problems by Pleijel, Browder, Garding, Agmon, and others; see [1], [7], and the references there. However, this method demands more regularity of the coefficients of the operator than seems necessary for validity of the formula. In an earlier paper [5] the author extended the formula to the Dirichlet problem under minimal assumptions on the coefficients, by approximation argument in the spirit of that of Weyl. The present paper establishes the result for coercive (differential) boundary value problems and for certain non-local

Introduction. In a recent series of papers [1], [2], [3], Zariski introduced the concepts of equisingularity and equisaturation. The main result of these papers was to define a series of equivalent criteria for a hypersurface to be equisingular along a submanifold of codimensions 1. In [4] Pham and Tessier extended this system of criteria via their introduction of absolute and relative equisaturation. Below we study the extensions of these criteria to a pure dimensional analytic set V with arbitrary embedding dimension . We retain the restriction on the dimension of the submanifold, and to simplify the exposition we assume that the submanifold is contained in the singular locus of V. In this case generalizations of Zariski's criteria give rise to three different notions of equisingularity; weak, strong, and residual. In this paper we restricted ourselves to discussions of weak and strong equisingularity. We show that strong implies weak but not conversely. In [18] we will show that residual also implies weak, and that strong and weak are distinct notions. In general neither implies the other. Section 4 contains a short description of the results in [18]. In sections 1, 2, and 4 we define these various types of equisingularity. The definitions are in terms of the dimensions of the tangent cones CG (V, p), i = 4, 5, introduced by Whitney in [7]. In [5] we showed that the dimensions of these cones determine the properties of branched coverings obtained from V via projection. TJsing the results and methods of [5] we divide our generalizations of Zariski's criteria as consequences of each type of equisingularity; the Whitney A, B conditions and the Jacobian criterion are weak, the discriminant criterion is strong, and the criteria involving resolution are residual. In the case of a hypers'urface all of the versions of equisingularity are identical so our results provide a new criterion for equisingularity in that case. In section 3 we show that the following are equivalent; strong equisingularity, absolute equisaturation, and relative equisaturation with respect

The notions of a point set M irreducibly connected about a point set K and basic set about which M is irreducibly connected were introduced and studied to some extent by H. M. Gehman.t The -present paper is an investigation of these same notions with the emphasis placed on non-closed sets.? The only spacial assumptions necessary are explicitly stated in the hypothesis of each theorem, in each case the set under consideration being considered as a general topological space without reference to any imbedding space. In general, only the notion of limit point is necessary, connectedness being defined in terms of limit point according to the usual Lennes-Iausdorff definition. In the case of Theorems 11-14 it is necessary to assume that the set under consideration is a " neighborhood " space in which it is true that if= P is a limit point of a point set M, and R is a neighborhood of P, then R contains infinitely many points of M. Otherwise, we need no such properties as are implied in denumerability axioms, covering theorems, etc. (Whenever references are made to theorems of other papers needed in proofs, it may be readily ascertained that such theorems hold true under our assumptions.) As pointed out below, "regular " is used in the sense of local connectedness (connected neighborhoods), and not in the sense in which it is usually employed, as, for instance, in metrization theory. As might be expected, however, where restrictions are placed upon the set about which the space is irreducibly connected, it turns out that the character of the space is considerably simplified (see, for instance, Theorems 8, 9 and its corollary, and 13, below). We have by no means exhausted the subject, and a number of problems may suggest themselves to the reader.

Introduction. In a previous paper,2 of which this is a continuation, topological properties of curve families which filled the Euclidean plane 7r, or a simply connected domain in r, were investigated. The families were assumed regular (i. e. locally homeomorphic to parallel lines) except at a possibly infinite collection of isolated singularities at each of which the family had the structure of a multiple saddle point; such families were called branched regular curve families. Further investigation of these families, in particular their relation to harmonic functions, is the aim of this paper. In what follows the definitions and theorems in [I] will be assumed, and the same notation will be used. In particular F, G will denote branched regular curve families filling the plane 7r, B will denote the set of singular points, R the domain 7r B in which F is regular, and so on. The Euclidean plane will be taken as a model for all simply connected domains. The principal result of [I] was to prove that any branched regular curve family F filling Xr can be given as the family of level curves of a function f (p) which is continuous on all of 7r and has no relative extrema. This generalizes a portion of [II] in which the same theorem is proved for a curve family without singularities in 7r. In this paper there are two main results: the first, proved in Section 1, is that F is actually homeomorphic to the level curves of a harmonic function; the second, proved in Section 2, asserts the existence of a decomposition of F into a countable collection of subfamilies of curves, each of which has the structure of the parallel lines y=constant of the upper half-plane. Such subfamilies will be called half-parallel, and this decomposition has consequences for the study of harmonic functions and analytic functions which will be mentioned below. These two results generalize

on £ n : x0w = ^, ^in, • • • , xnn = b, where the asterisk indicates a function defined on En (replacing the bold-face type in my former paper), *f(r) = *f(xrn) and A*/(r) = */(r + l ) * / ( r ) , every solution of this system goes over in the limit as n, the number of points in Eny becomes infinite, in such away t h a t X is completely^ subdivided to the corresponding solution, that is, the solution having the same initial values at x = a, of the differential system (1). The present paper shows that the conclusions of our former paper, stated above, are valid for all possible methods of defining the coefficients of system (2), so long as limn.*oo*^4;y(£) =Au(x), limn^oo*&(£) =]8i(ac), almost everywhere on X, and there exists a summable function G(x) on X such that \*Au(p) |, |*&(£) I

In a series of papers, published in the Transactions of the American Mathematical Society, the author has laid the foundations for a theory of ruled surfaces, based upon the consideration of the invariants of a system of ordinary linear differential equations. In the present paper, these considerations are extended to partial differential equations. The paper confines itself, for the most part, to a special case, and this for two reasons. In the first place, the complication of the more general cases is such as to make it appear unwise to attempt their treatment at present; moreover, the real essentials of the method appear just as well in a special case. In the second place, the special problern considered has its own intrinsic interest, as it leads to a theory of congruences. A whole geometry seems to be possible on this basis. The theory of curves, of surfaces, of complexes, etc., can all be built up, and have in part already been built up, by considerations which are of the same character as those discussed in this paper.

In two previous papers under the same title ([7], [8]), the writer has constructed a theory of eigenfunction expansions of a generalized PlancherelWeyl type for general classes of realizations of a partial differential operator L on an open subset G of El, the subnormal realizations in [7] and the decomposable realizations in [8]. These results were established without any restrictions upon G, the type of L, or on the smoothness or behaviour of the coefficients of L at the boundary of G or at infinity. Results of the same type were obtained for expansions in solutions of the equations (L CB) u = 0 for positive differential operators B. It is the purpose of the present paper to extend and generalize these results in two separate directions. In the first place, it is very difficult to determine in any concrete case whether a particular realization of a partial differential operator under specific boundary conditions falls within either of the two classes which we have studied, except in the relatively simple case in which L is symmetric. For this reason, we have begun the study elsewhere ([5], [61, [9] ) of the simplest of the basically non-symmetric cases, elliptic operators on unbounded domains under Dirichlet boundary conditions, but the results obtained at this level of generality do not as yet permit us to apply the theory as previously constructed. On the other hand, for the regular case on a bounded domain, the completeness of the eigenfunctions of the Dirichlet problem was proved by the writer in [4] by a method which extends to the whole class of regular variational boundary-value problems. It is of interest, therefore, to obtain some results on the completeness of the eigenfunctions of L for L singular but elliptic, without any assumption on the boundedness or smoothness in the large of its coefficients. In section 1, we show that if f is a distribution with compact support in G and if L satisfies certain simple local conditions at each point of G, then the orthogonality of f to each eigenfunction

In this paper the methods and notation of the theory of composition of the first kind, developed by V. Volterra and J. Peres,* are applied to certain fractional operators.t The concept of the " finite part" of an integral developed almost simultaneously by J. Hadamard and R. D'Adhemar I is employed to give a new definition of negative fractional orders of integration that is amenable to treatment as a very special case of composition. The symbolism already in use in the theory of composition is utilized with obvious advantage in the theory dealt with in this paper. Finally, definitions are given for the operators in the case of the complex variable, the concept of "finite part " being extended to contour integrals. 1. If a function F(x, y) can be put in the form

For a vector bundle E over a manifold V., denote by S (E) the sheaf of germs of local cross-sections of E. By using local coordinate we may consider linear differential equations on E. When V = Rn and E = V X Rm, the linear differential equations on E are linear differential equations on m unknown functions in n variables. The germs of solutions of a linear differential equation on E forms a subsheaf, say G, of S (E). The importance of such subsheaves G can be seen, for instance, in the recent works of Kodaira and Spencer in the deformation theory of pseudo-group structures. In the first half of the present paper, we study some of properties of such sheaves. Since it is very difficult to treat the general case, we restrict ourselves to the category of real analyticity and to the case where inconvenient degeneracy does not occur in the differential equations. In our argument these restrictions are necessary because in the last analysis we depend on Cartan's theorem of solvability of involutive real analytic differential equations. G together with its defining linear differential equation which is nondegenerate, is called a (S)-sheaf. Our main result is the existence of quotient (S)-sheaves of (S)sheaves by (S) -subsheaves. In the second half the writer gives concise description of how the above results can be applied to deformation theory of transitive continuous pseudo-group compact structures. Since our argument follows the more or less well known line developed in [3], details are omitted. It is noted that another approach, which is still within the same framework as ours but which is more differential geometric, is published in [5] by D. C. Spencer. In the present paper every notions are assumed to be in the category of real analyticity. So when we say manifolds or mappings for example, we mean always manifolds or mappings which are real analytic.