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In the paper of the title [1], a number of problems are posed. Ne­ gative solutions of two of them (Problems 2 and 3) are derived in a straightforward way from a paper of L. Gillman and the present author [2]. Motivation will not be supplied since it is given amply in [1], but enough definitions are given to keep the presentation reasonably self­ contained. 1. A Hausdorff space X is said to satisfy (Qm), where m is an in finite cardinal, if, whenever U and V are disjoint open subsets of X such that each is a union of the closures of less than m open subsets of X, then U and V have disjoint closures. In particular, a normal (Hausdorff) space X satisfies (Q*,) if and only if disjoint open F.-SUbsets of X have disjoint closures. (For, an open set that is the union of less than ~, closed sets is a fortiori an F.. Conversely if U is the union of countably many closed subsets Fo> then since X is normal, for each n there is an open set Un containing Fn whose closure is contained in U. Thus U is the union of the closures of the open sets Un.) In Prob­ lem 3 of [1], it is asked if every compact (HaUSdorff) space satisfying (Qm) for some m>~o is necessarily totally disconnected, and it is re­ marked that this is the case if the first axiom of countability is also as­ sumed. If X is a completely regular space, let C(X) denote the ring of all continuous real-valued functions on X, and let Z(f)= {x EX: f(x)=O}, let P(f)={xEX:f(x»O}, and let N(f)=P(-f). As usual, let (IX denote the Stone-Cech compactification of X. If every finitely generated ideal of C(X) is a principal ideal, then X is called an F-space. The fol­ lowing are equivalent. ( i) X is an F-space. ( ii) If f E C(X), then P(f) and NU) are completely separated [2, Theorem 2.3]. (iii) If f E C(X), then every bounded g E C(X-ZU» has an ex­ tension g E C(X) [2, Theorem 2.6]. A good supply of compact F-spaces is provided by the fact that if X is locally compact and ,,-compact, then {lX-X is an F-space [2, Theo­ rem 2.7].

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 his previous paper [1] the author has stated a formula concering stochastic differentials with the outline of the proof. The aim of this paper is to show this formula in details in a little more general form (Theorem 6).

In a beautiful paper which appeared in 1939 ([4]), F. Gantmacher made a thorough study of automorphisms of semi-simple Lie algebras over the field of complex numbers. Among other things, he defined the index n(G i ) of a connected component G i of the automorphism group G = G(8) as the minimum multiplicity of the characteristic root 1 for elements of G i . The main purpose of this note is the determination of these indices. It is somewhat surprising that this does not appear in Gantmacher’s paper since all the methods for deriving the formula for index G i are available in his paper. The secondary purpose of this note is to extend Gantmacher’s theory to the case of Lie algebras over arbitrary algebraically closed base fields of characteristic 0. This can be done by using algebraic group concepts and techiques which are by now well known. Nevertheless, it seems worthwhile to carry out the program in detail since Gantmacher’s results give a real insight into the action of an automorphism in a semi-simple Lie algebra. For example, as we indicate, they can be used to give a new derivation and sharpening of theorems on fixed points which are due to Borel and Mostow ([1]).

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

An Artίnίan module can be characterized in terms of certain properties of its factor modules. A module Mis Artinian if and only if the following two conditions hold for M: (I) Every nonzero factor module of M contains a minimal submodule. (A) The socle of every factor module ofM is finitely generated. The dual to the factor module is the submodule. We state the dual of (I): (Π) Every nonzero submodule ofM contains a maximal submodule. We call a module with property (Π) a Max module and one with property (I) a Min module. Every Noetherian module is a Max module but not conversely. This paper investigates these generalizations of the Artinian and Noetherian conditions and the relationships among them. Throughout this paper M denotes a right module over an arbitrary

0. Introduction. Let M be a differentiable manifold of class C°°. By a Morse function / on M, we mean a differentiable function / on M having only non-degenerate critical points. A well-known topological result of Reeb states that if M* is compact and there is a Morse function / on M having exactly 2 critical points, then M is homeomorphic to an ?^-sphere, S (see, for example, [3], p. 25). In a recent paper, [4], Nomizu and Rodriguez found a geometric characterization of a Euclidean ^-sphere SaR in terms of the critical point behavior of a certain class of functions Lp, p e R , on M. In that case, if p e R, x e M, then Lp(x) = (d(x, p))\ where d is the Euclidean distance function. Nomizu and Rodriguez proved that if M {n ^ 2) is a connected, complete Riemannian manifold isometrically immersed in R such that every Morse function of the form Lp, p e R , has index 0 or n at any of its critical points, then M is embedded as a Euclidean subspace, R, or a Euclidean ^-sphere, S. This result includes the following: if M is compact such that every Morse function of the form Lp has exactly 2 critical points, then M = S. In this paper, we prove results analogous to those of Nomizu and Rodriguez for a submanifold M of hyperbolic space, H, the spaceform of constant sectional curvature —1. For p € H, x e M, we define the function Lp(x) to be the distance in H from p to x. We then define the concept of a focal point of (M, x) and prove an Index Theorem for Lp which states that the index of Lp at a non-degenerate critical point x is equal to the number of focal points of {M, x) on the geodesic in H from x to p. In section 2, we prove that a metric sphere S c H can be characterized by the condition that every Morse function of the form Lp, p e H, has exactly 2 critical points. In section 3, we give an example which shows that a result analo

The oscillatory or Apollonian packing in two dimensions is well known and is described for example in [13]. Recently we investigated the three dimensional osculatory packing of a sphere [4] However, the results of that paper indicate that, for N > 3, ΛΓ-dimensional osculatory packings are irregular and not invariant under inversion as is the case for N = 2 and 3. In this paper we introduce a class of packings which we call discrete packings, and produce some examples. This class is analogous to the class of lattice packings which appear in the theory of packings of equal spheres. We shall use the systems of polyspherical coordinates developed in [4]. Section 2 contains a description of these as well as the proofs of some additional results needed here. The idea of the separation Δ{X, Y) between two spheres X and Y will again play an important role. In § 3 we consider inversively generated configurations obviously generalizing the construction used in [4]. That is, we begin with a 'cluster' of (N + 2) disjoint spheres and by successive inversions replace the spheres one at a time with new spheres in such a way that the separations between the spheres in the new cluster are the same as for the initial cluster. In terms of polyspherical coordinates the necessary inversions are represented by matrices which preserve a certain indefinite quadratic form. Repetition of the process leads to a configuration of spheres in EN which may or may not be a packing, depending on the initial cluster. In § 4 we give sufficient conditions under which an inversively generated configuration is a packing. The conditions force the separations between the spheres in the configuration to lie in a discrete subset of the rational numbers, hence the name 'discrete packing'. In addition to the two and three dimensional osculatory packings, we give examples of discrete packings for dimensions 2, 3, 4, 5, and 9. We do not know yet whether such packings exist in all dimensions. The examples we have found are given in § 6. The packings described in § 4 are not in general osculatory; that is, the largest possible sphere is not generated at each step. However

In this paper it is shown that every strongly locally homogeneous Hausdorff continuum is locally connected. It is known that every connected representable space is homogeneous and that every locally connected representable space is strongly locally homogeneous; this paper investigates the problem of whether or not every connected representable space is locally connected. The discovery that homogeneity did not characterize the circle among planar continua motivated other classifications of continua in terms of the actions of their homeomorphism groups, and the pseudoarc remains the most tortuous proving grounds for testing such topological properties. Those homogeneity-like properties that are not possessed by the pseudo-arc appear to be closely related to local connectedness. Such properties include strong local homogeneity, representability, (strong) 2-homogeneity and countable dense homogeneity. In this paper it is shown that every strongly locally homogeneous Hausdorff continuum is locally connected. It is known that every connected representable space is homogeneous and that every locally connected representable space is strongly locally homogeneous; this paper investigates the problem of whether or not every connected representable space is locally connected. It follows from results of Ben Fitzpatrick, Jr. and Ralph Bennett that every connected locally compact separable metric representable space is locally connected. Arguments of Fitzpatrick and de Groot are used to show that if (X, τ) is a representable connected complete metric space such that τ does not have a base of totally disconnected sets, then (X, τ) is locally connected (and hence arcwise connected and strongly locally homogeneous). The authors do not know of even a homogeneous connected complete metric space with a base of totally disconnected sets; however, there exists a separable connected metric topology that has such a base. Let (X, τ) be a topological space. We let H(X) denote the group of all homeomorphisms from the space (X, τ) onto itself and let i denote the identity of H(X). If A c X, then A' = {he H(X): h\A = i I A) and if G is a subgroup of H(X), then G' = {x e X: g(x) = x for each geG}. If GaH{X) and AcX, then G(A) = {g(a):geG and aeA}. We write G(x) in room of G({x}). Throughout this paper all spaces are assumed to be Hausdorff.

The manifold in this paper is assumed to be connected differentiable of class C∞. Let Dr(M) and Ӿr(M) be the set of all diffeomorphisms and vector fields of class Cr on a manifold M with Whitney Cr topology, respectively. In [2], the concept of weak stability is defined. The definition is equivalent to the following ((2.1) of this paper); f∈Dr(M) or X ∈ Ӿr(M) is weakly (allowably) stable if and only if there is a neighborhood U of f or X in Dr(M) or Ӿr(M) such that for any (a suitable) g or Y ∈ U the set of all elements topologically equivalent to g or Y is dense in U, respectively. Here, f, g ∈ Dr(M) are said to be topologically equivalent if they are topologically conjugate and X, Y ∈ Ӿr(M) are said to be topologically equivalent if there is a homeomorphism mapping any trajectory of X onto a trajectory of Y preserving the orientations of the trajectories. Similarly, weak Ω-stability is defined for f and X.

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

A conjecture concerning characterization of blocks of representations of symmetric groups by means of Young diagrams, given formerly by one of the writers, was affirmed recently by R. Brauer and G. de B. Robinson jointly. Namely, it was proved in Brauer’s paper, relying on his general, and profound, theory of blocks in finite groups, that the number of p-blocks in a symmetric group of degree n, is exactly equal to the number of p-cores, or p-kernels in the terminology of NI, with n- Ip (l= 0, 1, 2,…) nodes, and that certain statements designated as (A), (B) and (C) would secure that two irreducible representations with a same p-core belong to a same block (whence the conjecture itself when combined with the estimate of the number of blocks). Then these statements (A), (B), (C) were proved actually in Robinson’s paper.

In a recent paper Tsuji [1952] has given some theorems which may be considered as a generalization to Fuchsian groups of the classical theorems of Blichfeldt [1914] and Minkowski in the geometry of numbers. In both cases, however, the generalization is restricted to circular domains. The object of this paper is to carry this generalization further. First, by considering more general groups than the Fuchsian groups; second, by considering a kind of domains more general than the circular domains https://doi.org/10.2969/jmsj/00720208

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.

It is well known that a hypersurface of an almost Hermitian manifold admits an almost contact metric structure naturally induced on it. The study of hypersurfaces of a Euclidean space and of a Kahlerian manifold on which the induced almost contact metric structure satisfies certain conditions has been started by one of the present authors [4, 5]. On the other hand Blair [1, 2], Goldberg [3], Ludden [1, 2], Yamaguchi [8] and the present authors [6, 9] started the study of hypersurface of an almost contact manifold and of submanifolds of codimension 2 of an almost complex manifold. These submanifolds admit, under certain conditions, what we call (/, g, u, v, λ) structure. An even-dimensional sphere of codimension 2 of an even-dimensional Euclidean space is a typical example, of a manifold which admits this kind of structure. In a previous paper [9], we have studied the (/, g, u, v, -structure and given characterizations of even-dimensional sphere. In the present paper, we study submanifolds of codimension 2 in an evendimensional Euclidean space which admit a normal (/, g, u, v, Λ)-structure. In § 1, we consider submanifolds of codimension 2 of an even-dimensional Euclidean space regarded as a flat Kahlerian manifold. In the next section, we deal with (/, g, u, v, -structure induced on a submanifold of codimension 2 of an even-dimensional Euclidean space. In § 3, we find differential equations which /, g, u, v and λ satisfy. § 4 is devoted to the study of relations between the structure equations of the submanifold and the induced (/, g, u, v, ^-structure. In § 5 we prove a series of lemmas which are valid for normal (/, g, u, v, λ}structures and in § 6 we study properties of the mean curvature vector of the submanifold with normal (/, g, u, v, -structure. In the last § 7, we study hypersurfaces of an odd-dimensional Euclidean space and determine all the hypersurfaces admitting a normal (/, g, u, v, Λ)-structure. Our main theorem appears at the end of §7.

In the previous paper [7], we have studied the relationship between the Riemannian connection of an n-dimensional Riemannian space M imbedded into the (n + k)-dimensional Euclidean space Rn+k and the canonical connection in the bundle Pn, k = O(n + k)/{l} Х O(k) over the Grassmann manifold Mn, k = O(n + k)/O(n) Х O(k).In the first half of the present paper, the relationship between the canonical connections in bundles and the invariant Riemannian connection on Mn, k will be discussed. We obtain the holonomy groups of these canonical connections.

In the previous paper [3] we have studied the prolongations of G-structures to tangent bundles of higher order. The purpose of the present paper is to study the prolongations of connections to tangential fibre bundles of higher order, and to generalize the results due to S. Kobayashi [1] for the case of usual tangent bundle —— in fact, the arguments in [1] will be, in a sense, more or less simplified and clarified by using the notion of tangent bundles of higher order. In addition, as a consequence of our results, we shall obtain the prolongations of linear (affine) connections to tangent bundles of higher order.

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

Suppose P(x, D) is a linear partial differential operator on an open set ~ contained in R ~ and that A is a closed subset of ~. Given a class ~(~) of distributions on ~, the set A is said to be removable for ~(~) if e ach /E ~(~), which satisfies P(x, D ) / = 0 in ~ A , also satisfies P(x, D)/= 0 in ~. The problem considered in this paper is the following. Given a class ~(~) of distributions on ~, what restriction on the size of A will ensure that A is removable for ~(~). We obtain results for Lroc (~) (p ~< ~ ) , C(~), and Lipa (~). The first result of this kind was the Riemann removable singularity theorem: if a function / is holomorphic in the punctured unit disk a n d / ( z ) = o ( H -1) as z approaches zero, then / is holomorphic in the whole disk. Bochner [1] generalized Riemann's result by considering the class ~(~) of functions f on ~ such that ](x)=o(d(x, A) -q) uniformly for x in compact subsets of ~, and giving a condition on the size of A which insures tha t A is removable for ~(~) (Theorem 2.5 below). Bochner's theorem is remarkable in that the condition on the size of A only depends on the order of the operator P(x, D). The theorem applies, therefore, to systems of differential operators, such as exterior differentiation in R n and ~ (the Cauchy-Riemann operator) in C n. The same can be said for the other results in this paper. The proof of Bochner's theorem provided the motivation for our results. I t is interesting to note tha t a very general result (Corollary 2.4) f o r / ~ (~) (due to Li t tman [7]) is an easy corollary of Bochner's work. Here the condition on the singular set A is expressed in terms of Minkowski content. In section 4 the case of Ll~oc(~) is studied again, and results in section 2 are improved by replacing Minkowski content with Hausdorff measure. In addition, the cases C(~)

The local theory of continuous (infinite) pseudo-groups of transformations was originated by S. Lie, and developed by himself, F. Engel, E. Vessiot, E. Cartan, etc. In the beginning, the definition was not clear and we can find several different definitions in the papers of pioneers. In 1902, E. Cartan introduced a definition using his theory of exterior differential systems and made an extensive study in his series of papers [1], [2], and [31 The writer will adopt his definition in this series of papers. A continuous pseudo-group of transformations is, roughly speaking, a collection of real (or complex) analytic homeo-morphisms of domains in a real (or complex) euclidean space, which is closed under the operations of composition and inverse, and which forms the general solutions of a system of partial differential equations.