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F(k) ={xkj x F}; F,=F; Fk= (Fk_l,F). Let (u, v, 0)=u, (u, v, 1)=(u, v), (u, v, n)=((u, v, n-1), v). Then Sanov [3] proved that (1. 1) (u, v, apa _1) p-a E F(p#)Fapa+?, /3 a = 1, 2, where p is a prime. In this paper, (1.1) is proved for the cases a= 1, 2; 3 arbitrary. A slight generalization of these results is also proved. Sanov's proof involved an investigation of ideals in a Lie Ring. In this paper, Hall's Collection Process will be used. The method also yields other formulas, e.g. (1.2) (u, v, p2 1)P' (E F2p2_pF(p) (1.3) (u, v, pa+l 1)p~l C F2pa+l paF(pl), a = 1 2, and can be used to produce numerous formulas of a similar nature. The author hopes that some of these formulas and/or the method may be of use in solving other group-theoretic problems. The author was unable to use the method to prove (1.1) for a =3. Note that for a=f = 1, (1. 1) becomes (1.4) (U, v, p 1) E F(p)Fv+1. (1.4) plays an important role in the theory of the Restricted Burnside Problem.

In a recent paper [4]2 it is shown that lattices and trees have a common generalization. These systems will be called median semilattices. These semilattices are characterized below by means of segments (??2 and 4) and by means of betweenness (??3 and 4). In ?5 distributive lattices are characterized by means of medians with no assumption made as to the existence of universal bounds (compare Problem 66 of [1]). In another paper [5], median semilattices are characterized in terms of a binary operation and it is shown that these systems can be imbedded in distributive lattices.

In a series of papers [1; 2; 3; 4], K. T. Chen introduced and studied certain infinite series of numbers associated with paths in Euclidean n-space. These numbers were invariants under translations, and in [4] he proved that they uniquely characterize paths under translations. This paper's purpose is investigating similar path invariants for other transformation groups. Integral invariants are studied using prolongations in the context of C. Ehresmann's "jets" [5]. A general discussion of the ideas involved is given in ?1. ?2 is devoted to differential equations and some illuminating examples. We find in ?3 a sufficient condition that a group's invariants uniquely characterize paths. The theory also applies to pseudo groups having sufficiently regular behavior. All structures assumed C' (infinitely differentiable). 1. INVARIANTS

DEFINITION. A continuous curve every subcontinuum of which is a continuous curve will be called a perfect continuous curve.t In this paper, among other things, a study is made of this type of continuous curve. An important problem in analysis situs is to determine whether a given point set is arc-wise connected. It is known that a connected subset of a perfect continuous curve is not necessarily arc-wise connected.: Here, however, it is shown in Theorem 5 that, if a and b are two points of a connected subset N of a perfect continuous curve, then the set common to N and the arcs ab of N' is connected. And in Theorem 7 it is shown that if N' contains but a countable number of arcs ab then N contains an arc ab. ? A generalization of the problem as to whether there exists an arc joining two points in a given point set is the following: when do there exist in a given point set where q is a given positive integer, q arcs, distinct except for end points, joining two given points; or still a further generalization would be to ask when do there exist q such arcs joining two distinct closed point sets? This problem is considered in ?II of this paper. It might also be asked when do there exist q distinct arcs joining two distinct closed point sets? This problem is considered in ?I. I wish to express my thanks and to acknowledge my great indebtedness to Professor R. L. Wilder for his suggestions, criticisms, and constant encouragement.*

Introduction. This paper generalizes certain methods of decomposition of compact metric continua due to R. L. Moore [3; 4] and G. T. Whyburn [9; 10; 13]. While their methods yield acyclic continuous curves, hyperspaces are obtained here which are aposyndetic [1 ] continua. The concept of a con. tinuum being aposyndetic is a generalization of the concept of continuous curves and was introduced in 1941 by F. B. Jones. In compact metric continua, this idea is equivalent to Whyburn's notion of semi-locally-connectedness [14; 2]. A continuum M, i.e., a closed and connected point set, is said to be aposyndetic at a point p with respect to a point x provided that there exists a subcontinuum N of M and an open subset 0 of M such that M-xDNDODp. If M is aposyndetic at a point p with respect to each point x of M-p, then M is said to be aposyndetic at p. It is said that M is aposyndetic if M is aposyndetic at each of its points. In an early paper [10], Whyburn made use of connected cuttings of a compact metric continuum M to obtain a decomposition of M into an acyclic continuous curve. Later, he made use of nonseparated cuttings [13] to obtain a decomposition of a continuous curve into a nondegenerate acyclic continuous curve. Certain of these theorems concerning nonseparated cuttings are generalized in obtaining an aposyndetic decomposition. Moore [3] has obtained decomposition theorems by use of certain sets M(P) defined as follows: For each point P of a compact continuum M, let M(P) denote the set of all points X of M such that there do not exist uncountably many different points each separating P from X in M. He proved the following theorem: If M is a compact metric continuum and G is the collection of all point sets M(P) for all points P of M, then G is an upper semi-continuous collection of disjoint continua filling up M and G is an acyclic continuous curve with respect to its elements as points. The definition of the sets M(P) due to Moore may be generalized in the following manner: Suppose that M is a continuum and that p is a point of M.

In the new dynamical theory of economics there arises a very general problem which can be said to be a generalization of the Lagrange problem in the calculus of variations.4 It will not be necessary to consider the formulation of the corresponding economic theory here since I have already done this in another paper.? It would hardly be fair, however, to introduce the reader to a rather unusual mathematical situation without giving some hint as to its origin. It seems desirable, therefore, to give first a brief economic formulation of the problem whose mathematical aspects will be discussed in this paper. If there are two producers of an identical commodity C, manufacturing, respectively, amounts ul(x) and u2(x) of C per unit time, subject to the respective cost functions 41(ul, ui', U2, u1, U3, u f, x) and 42(ul, ul', U2, UI', U3, U3 , X), where u3(x) is the selling price of C at a time x, then the respective profits obtained during an interval of time xo < x < xi are

Swingle [7]1 has given the following definitions. (1) A continuum M is said to be the finished sum of the continua of a collection G if G = M and no continuum of G is a subset of the sum of the others.2 (2) If n is a positive integer, the continuum M is said to be indecomposable under index n if M is the finished sum of n continua and is not the finished sum of n +1 continua. Swingle has shown [7, Theorem 2] that if n is a positive integer and the continuum M is indecomposable under index n, then M is the finished sum of n indecomposable continua. The author has shown [2, Theorem 1] that if n = 2 and the continuum M is indecomposable under index n, and G is a collection of n indecomposable continua whose finished sum is M, then G is the only such collection. In the present paper, it is shown that for a compact continuum, this theorem holds for any positive integer n. Also, there is given a necessary and sufficient condition that a compact continuum be indecomposable under 'index n. An indecomposable continuum can be described as a nondegenerate continuum which is indecomposable under index 1. If n =1, then in order that a continuum M be indecomposable under index n, it is necessary and sufficient that M contain n+2 points such that M is irreducible about any n+1 of them.3 Swingle [7] has shown that it is impossible, in a certain manner, to generalize this theorem. Theorem 3 of the present paper might be considered a generalization of the necessary condition of the above theorem. However, it is easily seen that the converse of Theorem 3 is not true. Theorems 1-5 are proved on the basis of R. L. Moore's Axioms 0 and 18. Hence these theorems hold in any metric space.4

Introduction('). The present paper deals with a number of diverse topics, ranging from purely topological considerations, through a general theory of possible distributions of values of an analytic function, to more special theorems on simultaneous expansions of an infinity of analytic functions. No unifying principle is presented as an excuse for treating such a variety of subjects; but there is a slight sequence of argument running throughout. The parts of the paper were actually written in reverse order. The initial investigation (Part III) was started as an attempt to generalize, from n to infinity, a known theorem(2) on simultaneous expansions of n analytic functions. This generalization was found to depend on an affirmative answer to the following question on level curves of an analytic function: Given any sequence of points a1, a2, . , in the complex plane, which has the point at infinity as its only limit point, does there exist an analytic function with a level curve C such that C contains a distinct branch about each given point which separates that point from all the other points? This was, in turn, made to depend on a certain problem relative to the possible rate of growth of an integral function. It was shown long, ago by Poincare(3), Borel(4), and others that an integral function may be made to grow arbitrarily fast along the real axis or along other lines or curves extending to infinity. Our problem was to obtain an affirmative answer to the following related question: Does there exist a sequence of regions Si, S2, . , with ai interior to Si, such that, no matter how fast the sequence of numbers m1, M2, increases there will be an integraL function f(z) for which

In a previous paper (Lyness and Moler [ 1 ] [1] ), several closely related formulas of use for obtaining a derivative of an analytic function numerically are derived. Each of these formulas consists of a convergent series, each term being a sum of function evaluations in the complex plane. In this paper we introduce a simple generalization of the previous methods; we investigate the "truncation error" associated with truncating the infinite series. Finally we recommend a particular differentiation rule, not given in the previous paper.

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

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

then Ea, is said to be absolutely summable by Rieszian means of order k and type X, or summable R, X, k| . The case X), n is of particular interest in this paper. Summability R, n, k| has been shown by J. M. Hyslop [3] to be equivalent to absolute Cesaro summability of order k, or summability I C, k| . One of the principal results shown by Obreschkoff was the consistency of the j R, n, k means; that is, he showed that summability |R n, kI implies summability R, n, k' , where k'> k. In this paper we introduce a method of absolute summability based upon the (a, ,B) method of summability defined by Bosanquet and Linfoot [1]. Just as the Bosanquet-Linfoot method generalized Riesz's arithmetic mean (R, n, a), the method given here will generalize absolute Rieszian summability R, n, a I. DEFINITION 2. A series Ea, is said to be absolutely summable (a, ,B), or summable a, f31, where a >0 or ao=0, f3>0, if for each sufficiently large C

provided that a proper determination of the constant c1 be made. In the present paper I wish to consider the solutions of (1) in the vicinity of x = so, but under the restriction that p ( x ) and q ( x ) have the form ( 2)t The method of attack is essentially the same as that which I have employed earlier in the consideration of singular points.t In this special case the results may be given a more striking form. The last two theorems have no analoguesin my earlier paper and are susceptible of wide generalization.

In what follows we discuss certain inequalities involving harmonic polynomials of two and of three variables, that is, polynomials u(x, y) and u(x, y, z) which satisfy Laplace's equation uXX+u,,=O and uxx+uy+uZZ=0, respectively. The inequalities in question furnish bounds for these polynomials and for their derivatives under proper conditions. Of particular interest are inequalities of the type of S. Bernstein's theorem, as discussed by the second author in a recent paper [7 ] ('). The first part of the present paper deals with the two-dimensional and the second part with the three-dimensional case. Of fundamental importance throughout the paper is an interpolation formula which is stated and proved in ?2 of Part I. In fact the results of Part I may be regarded as systematic applications of this formula. Several inequalities of this part are generalizations and refinements of earlier theorems. Most of the problems of the second part are new. In an Appendix we consider a generalization of the main problem treated in Part II, and another problem which deals with ellipses and is only in loose relationship with the other topics considered in the present paper.

where f(x) is even, can be evaluated with surprising accuracy by means of the trapezoidal-sum formula. It is natural to anticipate that a more general result can be obtained when the integrand is not restricted to the form given above; the generalization was easily obtained by contour integration. The guiding idea which we employ in obtaining error-bounds is extremely simple: we express the error-term of an approximate integration by a contour integral and then choose a contour (among those which enclose the singularities of the integrand) on which the error-term is simply and easily bounded. Most often, it turns out that minimizing the absolute value of the error-term integrand is quite effective; and, in general, we have preferred methods for obtaining error-bounds which can be used and extended without undue expenditure of time by the staff of a computing laboratory. Subsequently, it was found that the technique of minimizing the integrand had been used earlier by Davis [2] in a paper whose contents somewhat overlap the contents of the present paper. Further trial showed that the guiding idea could be used to obtain useful error-bounds for Gauss-type formulae. These bounds are reported in Section 3. In the main, the paper is concerned with the Euler-Maclaurin formula and extensions

where c is a scalar matrix. It is well knownf that a relation of this type can not be satisfied by finite matrices. However, in the calculation of commutation formulas for polynomials in p and q no use is made of the fact that c is a scalar but merely that it is commutative with both p and q.% And there do exist pairs of finite matrices x, y of the same order such that xy—yx is not zero and is commutative with both x and y. Such matrices will be called quasi-commutative matrices and either may be said to be quasi-commutative with the other. In a certain sense the algebra of polynomials in a pair of quasi-commutative matrices is homeomorphic to the algebra arising in quantum mechanics. It is hoped to discuss such algebras in some detail in a later paper. In the present paper we shall make a brief study of quasi-commutative matrices whose elements belong to the complex number field. The concept of quasi-commutativity is an extension or generalization of commutativity, and as would be expected, some of the results obtained are generalizations of known theorems concerning commutative matrices. The problem of determining quasi-commutative matrices is that of finding matrices x, y, z (^0) which satisfy the equations

In his thesis, t Frechet gave a very beautiful generalization of the theory of point sets and of the theory of real valued functions of a real variable. His functions are real valued but the range of the independent variable is an abstract class -S of elements q . He secures his results principally through the medium of a properly conditioned distance function 8, a generalization of the distance between two points, which associates with each pair q1 q2 of elements a real non-negative number 8 (qi q2 ) . He is thus enabled to secure the more important theorems of point set theory and of real function theory, especially those relating to continuous functions and their properties. This theory of Frechet has excited considerable interest and has received much attention from mathematicians. Various contributions to its foundations and to its content have been made. In the present paper we follow the example of Frechet in assuming once for all that 8 (qq) = 0 and that 8 (qi q2 ) = 8 ( q2 q ) . In other words we assume that the distance from an element to itself is zero and that the distance between two elements is independent of the order in which they are taken. In the first part of the paper we give very simple conditions on systems ( ; 8) which are sufficient for many purposes and which, in the case of compact sets, we show to be equivalent, so far as limit of a sequence is concerned, to the voisinage and thus to the e'cart of Frechet.t The remainder of the paper is devoted to the theory of functions on the sets Z of systems (Z; 8). In terms of the conditions already mentioned we generalize the results of Frechet and Hahn as to the existence of non-constant continuous functions. We give two very

1. In this paper we intend to give a new proof and various generalizations of the following theorem due to Hahn :$ If If,(t) } is a sequence of summable functions in the interval J= (0, 1) and if lim. f Ef,(t)dt existsfor every measurable set E c J, then the indefinite integrals Fn(x) =fx fn(t)dt are equally absolutely continuous in J and therefore converge to an absolutely continuousfunction. The proof will be based on a theorem of Baire which has proved useful in many similar cases.? Incidentally there will be given a generalization of another theorem concerning sequences of functional transformations and published in a previous paper by the author.? 2. We shall denote by R the space of measurable characteristic functions in the interval J = (0, 1), i.e., functions which almost everywhere assume two values only, 0 and 1. The distance of two functions x(t), y(t) in R is defined by the formula

Introduction. In Cameron's paper [1] there appears a "rectangle formula" (there denoted by Theorem 1) by means of which the Wiener integral of F[x(.)], defined on the space of continuous functions on [0, 1 ] such that x(O) = 0, can be approximated by means of an n-fold Riemann integral provided F is sufficiently smooth and dominated by a suitable integrable functional. The formula employs a particular C.O.N. set of functions {ai(s) } 1 (the odd harmonic cosine functions.) A generalization of this formula appears in Cameron's unpublished notes and appears with proof in the author's [2], (there denoted by Theorem 2). The generalization allows for the use of an arbitrary C.O.N. set {ai(s) } of B.V. but suffers the defect that it is not known to hold except for bounded functionals. It is the purpose of this paper to show that if the {ai(s) }'s of this generalization are restricted to certain Sturm-Liouville sets of functions (among which sets is included the odd harmonic cosine functions), then the restricted generalization is applicable to suitably dominated (not necessarily bounded) functionals. Let w(s) >0 have continuous fourth derivative on [0, 1] and let {As} and {ai(s) } be the sets of characteristic numbers and normalized functions of the Sturm-Liouville problem: { [l/w(s) ]x'(s) }' +-Xa(s)=0, a'(O)=ca(1)=0 on [0, 1]. Let C be the space of continuous functions x(s) on [0, 1] such that x(O) = 0, let fli(s) =f'ai(t)dt and for x ()EC let ci = foea (s)dx (s): i = 1, 2, 3, * * (c, is of course a functional of x(.)). The main result of the paper is the following.

Under special conditions, it is shown how to adjust topological concordances so as to leave certain sets fixed. Also, a lemma is given which establishes the existence of pinched collars and bicollars under specified circumstances. These two results are then combined to show how to obtain nice extensions of certain homeomorphisms. If X and Y are topological spaces and ho and h1 are embeddings of X into Y, a topological concordance between ho and h1 is an embedding H:XXI-*YXI (1= [O, 1]) such that H(x, 0)=(ho(x), 0) and H(x, 1) = (hi (x), 1) for all xE X. If A CX, a topological concordance H between ho:X-Y and h1:XY is fixed on A if H(x, t) = (ho(x), t) forall (x,t)ELAXI. The following theorem generalizes the principal result of [4], i.e., Theorem 4.1 of [4]. In fact, the following theorem reduces to Theorem 4.1 of [4], when M is a point. THEOREM. Let Mm be an m-dimensional PL-manifold and let Qq, q-m > 3, be a q-dimensional PL-manifold. Suppose that M is (2m-q+2)-connected, dOM is (2m-q+ 1)-connected (or empty), and Q is (m+ 1)-connected. Let f: M->Int Q be a locally fiat embedding. If ho and h1 are topological homeomorphisms of Q onto itself which leave f(M) fixed and which are topologically concordant, then there is a topological concordance between ho and h1 which isfixed onf (M). REMARK 1. The above theorem can be generalized in a straightforward way to permit f to be certain allowable embeddings. We leave the calculation of the necessary connectivity conditions to the reader. REMARK 2. A generalization of the piecewise linear version of the theorem has been obtained in a joint paper with L. S. Husch [6] since the submission of this paper. Also, the piecewise linear version Received by the editors March 28, 1969. AMS subject classifications. Primary 5702, 5705, 5478, 5560, 5570.

Introduction. This paper is concerned with the convergence of a sequence of Perron integrals. Oscar Perron [4] considered the case of a sequence of uniformly convergent Perron integrable functions, and Bauer [1 ] extended Perron's work to functions defined in a space of n-dimensions. McShane [3] relaxed the condition of a uniformly convergent sequence of Perron integrable functions and stated necessary conditions for the limit of a sequence of Perron integrals to be the integral of the limit function. The following theorem is a generalization of the above. Throughout this paper integration is in the Perron sense. The Lebesgue integral is denoted by (2)f.

As a generalization of quasi-Frobenius rings, G. Azumaya [2] has investigated a ring with the property that every faithful left module is a generator' in the category of left mnodules, and he has proved that such a ring is left self-injective and a direct sum of indecomposable left ideals having minimal left ideals. In his proof, however, the existence of the faithful injective module plays an important role, while the injective module is not necessarily finitely generated,2 even if the ring is a left Artinian ring. Therefore, there naturally arises a problem: Is an Artinian ring quasi-Frobenius if every finitely generated faithful module is a generator? The purpose of this paper is to give an affirmative answer to this problem as a direct consequence from a more general result which is rather similar to that of Azumaya stated above and related to perfect rings introduced by H. Bass [3]. Throughout this paper we shall assume that the ring R has identity element 1 and all modules over it are unital. For a subset A of R we shall denote

In this paper we give a simple proof of the statement lim n → ∞ ∫ K n ( x , y ) f ( y ) d μ ( y ) = f ( x ) {\lim _{n \to \infty }}\int {{K_n}(x,y)f(y)d\mu (y) = f(x)} for μ \mu -almost all x x under weaker and more general assumptions than those of the usual Faddeev theorems.

Let T : M → M T:M \to M be a mapping of a metric space ( M , d ) (M,d) into itself. A mapping T T will be called a quasi-contraction iff d ( T x , T y ) ⩽ q max { d ( x , y ) ; d ( x , T x ) ; d ( y , T y ) ; d ( x , T y ) ; d ( y , T x ) } d(Tx,Ty) \leqslant q\max \{ d(x,y);d(x,Tx);d(y,Ty);d(x,Ty);d(y,Tx)\} for some q > 1 q > 1 and all x , y ∈ M x,y \in M . In the present paper the mappings of this kind are investigated. The results presented here show that the condition of quasi-contractivity implies all conclusions of Banach’s contraction principle. Multi-valued quasi-contractions are also discussed.

A well-known theorem of Jacobson asserts that a ring R R in which x n ( x ) = x {x^{n(x)}} = x for each x x in R R must be commutative. This paper gives a description of a ring with involution in which the above condition is imposed only on the symmetric elements. In particular, if R R is primitive, R R is either commutative or the 2 × 2 2 \times 2 matrices over a field, and, in general, any such R R is locally finite and satisfies a polynomial identity of degree 8.

A natural generalization of the notion of an even function (i.e., a uniform function of Z2) is that of a uniform function of any rational function of z. In this paper, we examine the following classes of functions and determine which functions in each class are uniform functions of non-linear rational functions: (1) periodic meromorphic functions; (2) Poincare's functions with rational multiplication theorems.t Among the periodic functions, we shall find in ?III that only the following possess the desired property

(1) (x, x, x) = O, (2) (x,y,z) = E(lr(x), ir(y), ir(z)), where for a in R, a/E exists in R and ir is in S3, the symmetric group on three letters. It is easily shown that E = ? 1 or (x, y, z) = E (y, z, x), where E2 + E -F 1 = 0. Throughout this paper an algebra (A, +, x, F) is defined as a set A, a field F and two operations such that (A, +, x) is a nonassociative ring, (A, +, F) is a finite-dimensional vector space and such that (ax) (3y) = (a3) (xy) if a, A are in F and x, y are in A. We shall, furthermore, find it convenient to define the following symbols

A higher dimensional generalization of the classical Bloch theorem depends in an essential way on the “boundedness” of the family of holomorphic mappings considered. In this paper the author considers two types of such “bounded” families and obtains explicit lower bounds of the generalized Bloch constants of these families on the hyperball in the space C n {{\mathbf {C}}^n} in terms of universal constants which characterize the families.

In my recent paper Generalized closed manifolds in n-space\ it was shown} that a compact point set B in Eny common boundary of (at least) two domains D and D\ which are respectively u.li-c.§ for OSi^j and O^i^n—j—3 (where » 2 > j ^ ( n 3 ) / 2 ) , and such that the Betti numbers p'(D)t p*(D), • • • , p~{D) are finite, is a g.c.(» —l)-m. This constituted a generalization of a former result|| to the effect that when n = 3, D and Dx are u.l.O-c, and p (D) is finite, B is a closed 2-manifold. In the present note I propose to show, as principal result, that the above conditions on the numbers p'+(D), • • • , p~{D) are irrelevant, and furthermore that it is immaterial whether we place the restriction as to finiteness on pi(D) or on p~~{D^, I t turns out that the only essential requirements are that the upper limits on the dimensions for which D and D\ are u.l.i-c. must total at least w —3, and that one of the domains have a finite Betti number as just stated. For the sake of brevity we make the following definitions. We shall understand without explicit statement hereafter that the imbedding space is En(n^3) (euclidean space of n dimensions).

Throughout this paper, R R will denote an arbitrary but fixed complete discrete valuation ring. We shall show that two reduced R R -modules which are direct sums of countably generated R R -modules are isomorphic if and only if they have the same Ulm invariants and the same basis type. This is a generalization of the celebrated Ulm and Kolettis theorem.

where J and N are positive integers, and the X's are real or complex constants. The present paper gives the chief results of a study of this problem. In order to add precision to the problem, and in order to exclude certain extreme cases which are of minor interest, it is assumed (1) that we do not have XoN=lX= * =XJN; (2) that we do not have XOn =X,= * *n =Xin:i$O for any n

In 1934, Priwaloff published a generalization of Littlewood's result, which turned out to be incorrect. When the domain under consideration is the unit circle, then Priwaloff's generalization consisted in allowing "non-tangential" approaches to the boundary of the disc. In 1942, during the course of his lectures on Subharmonic functions at Brown University, the late Professor J. D. Tamarkin discovered an error in Priwaloff's proof. Then in a letter to Tamarkin in 1943, Professor A. Zygmund described a counter-example to Priwaloff's result. In this paper, we present several generalizations of Littlewood's result (see ?3) as well as several counter-examples to Priwaloff's result (see ?4).

This article is an expansion of G. E. Forsythe’s paper "Von Neumann’s comparison method for random sampling from the normal and other distributions" [5]. It is shown that Forsythe’s method for the normal distribution can be adjusted so that the average number N ¯ \bar {N} of uniform deviates required drops to 2.53947 in spite of a shorter program. In a further series of algorithms, N ¯ \bar {N} is reduced to values close to 1 at the expense of larger tables. Extensive computational experience is reported which indicates that the new methods compare extremely well with known sampling algorithms for the normal distribution.

G. E. Forsythe, who edited the translation of Kantorovich's paper, included the following remark about this footnote: "It is not clear to me that Kantorovich's inequality really is a special case of that of Polya and Szego." Examining the relation between the two inequalities more closely we found that this remark is well justified and can be made even more specific in that the inequality of Polya and Szeg6 in the form (4) is a special case of the Kantorovich inequality

Throughout this discussion S will denote a compact connected topological semnigroup and E will denote the set of idempotents of S. The problem to be considered concerns a question posed by Professor A. D. Wallace. In [I], Wallace proves that if S has a left unit, if I is a closed ideal of S, and if L= D2 or if L is a closed left ideal of S, then Hn(S)_Hn(IQ_UL) for all integers n, where Hn(A) denotes the nth Alexander-Cech cohomology group of A with coefficients in an arbitrary but fixed group G. If S is assumed to have both a left zero and a left unit, then it follows that each closed left ideal L, of S is acyclic; that is, HP(L)= 0 for all p 1. A dual statement holds for closed right ideals if S has a right unit and right zero. A generalization of the case in which S has a left, right, or two-sided unit, is to require that S=ES, S=SE, or S=ESE, respectively, and Wallace has aslced: "If S has a zero, are closed right or left ideals of S necessarily acyclic in the more general situation?" [3]. A negative answer to this question is given here by way of examples, and a theorem is proved giviilg a necessary and sufficient condition for closed right ideals of S to be acyclic, assuming S=ESE and S has a zero. Following the proof of this theorem is an example of a semigroup not satisfying this condition. The above-mentioned example shows that even though S is acyclic, it is not necessarily true that all closed right ideals of S are acyclic. Trhus the question remains as to whether S is acyclic if S= ESE and S has a zero [3 ]. Wallace proves in [2 ] that for such a semigroup S, HI(S) = 0, however, an excample is given here of a semigroup S with zero, S=ESE and H2(S) ?G for all groups G, showing that this question also has a negative answer. Two further examples are included in this paper which show what can occur if one only assumes that S= SE, or S= ES. One example is of a semigroup S with zero, S= SE and HI(S)>--G for all groups G and the other is an example of a semigroup S with zero and left unit and S contains a closed right ideal R with HI(R)-G. DEFINITION. Let T be a semigroup, aC T and R(a) the closed right ideal of T generated by a. Then a is said to be right codependent on

Sp(a) = {fp(z) = f(zP)l/p : f(z) e S(a)}. Extremal problems for the coefficients in the power series expansions of functions in Sp(O) and powers of such functions were recently investigated by J. T. Poole [4]. In this paper we will show that these coefficient problems are equivalent to extremal problems for the coefficients in the power series expansion of functions in S(a), (a < 0). This alternative approach leads to a substantial generalization. We will also prove several theorems of the type commonly called "distortion theorems" for functions in the classes S(a). One such result is the: THEOREM. Let f(z) E S(a), -oo < a < 1. Then for each natural number n there is a point z = eio on the unit circle such that

In this note we retain the notation and terminology of our original paper [2]. Our purpose is to repair the statement and proof of Theorem 2.3 of [2], and using its generalization (Theorem A below) as a prototype, we can by use of an extension of Corollary 3.3 of [2] (Theorem B below) remove the weighty hypothesis of "large kernels" from Theorems 3.4, 3.5, 3.6 and Corollary 3.7 of [2]. This latter task derives some urgency from the conjecture of J. P. Jans [3] that an algebra with large kernels has finite module type. This conjecture, if true, would of course annihilate any importance of the above-mentioned results of [2] in its original form as far as the Brauer-Thrall conjecture is concerned(1). CORRECTION 1. The statement of Theorem 2.3 of [2] should read as follows

In this paper we give a description and analysis of a class of matricial difference schemes. This class of schemes is based in part on a generalization of the feature of classical numerical methods of being characterized by approximations at a single point in the complex plane. The schemes introduced here are effective for integrating stiff systems.

10. In this paper I wish to define a system of types of improper simple definite integrals, a system embracing in particular the four current types; of the theory of the general type I give at present merely the elements, the methods employed, however, being characteristic. The four current types are compared in ? 1 2-13o. By way of generalization of their diversities the new types arise (1lo-17o). As the desirable basis for the new types I propose (160) an extension of the notion of the proper simple definite integral; this involves likewise an extension of the notions of the four current types. In ? 2 I state in convenient notations a body of elementary properties of the general type of integrals. These properties with two definitional processes of induction developed in ?? 3, 4 serve as the basis for the definition in ? 5 of the system of types of improper integrals related to the (extended) type of proper integrals defined in 160.

Introduction. The primary concern of this paper is a generalization of the concept of expected or average value. We shall find all the set functions which resemble average values in certain ways and characterize in terms of an order structure alone those which are indeed of the form

with continuous coefficients on a t-interval I is said to be disconjugate on I if no solution (0-0) has n zeros on I. For n>2, most known results giving conditions which assure that (0.1) is disconjugate concern perturbations of un -=0 either for a fixed finite interval [as, e.g., in the theorem of de la Vallee Poussin (cf. [2, Exercise 5.3(d), p. 346])] or for large t. This paper deals with equations (1.1) which are perturbations of equations with constant coefficients, disconjugate on -00 < t < 00. As corollaries, we obtain theorems which are refinements of known results concerning perturbations of un -=0, but we do not obtain the "best" constants occurring in some of these results (n = 2). The proofs depend on the technique introduced in [5] for discussing asymptotic behavior of solutions of perturbed linear systems with constant coefficients (cf. [2, Chapter X]). This technique is based on suitable changes of variables and arguments which have been subsumed by general theorems of Wa2ewski [15], (cf. [2, pp. 278-283]). ??l and 2 use the simple Lemma 4.2, [2, p. 285]; ?3 requires a generalization given as Theorem (*) in an Appendix below. In addition to arguments from the theory of asymptotic integration, the proofs use a theorem of Polya [14] characterizing equations (0.1) disconjugate on an open interval I in terms of Wronskians of subsets of solutions of (0.1); for a generalization, see [2, pp. 51-54] (also obtained in [6]). Theorems I** and IV of Polya [14] show that no solution ( 0) of (0. 1) on an open interval I has n distinct zeros if and only if no solution (X0) has n zeros counting multiplicities; cf. also [13]. (A generalization of this last fact, when the linear family of solutions of (0.1) is replaced by an arbitrary (not necessarily linear) interpolating family of functions, is given in [1].) In ?4, it is observed that the results of the previous sections, together with theorems and methods of Lasota and Opial [8], give criteria for the existence of solutions of certain nonlinear boundary value problems.

where n denotes the degree of K over k. This inequality is well known when v and hence all the V are of rank one. The main object of this paper is to show that the general case can be treated very similarly to the rank one case. In II we give for valuations of arbitrary rank a theory of completions which does not seem to exist in the literature, although it is very easy and natural. This theory enables us to reduce the proof of (1) to the case where the ground field k is complete (see the corollary to Lemma 5 in II, ?8). However, for complete fields with respect to valuations of higher rank, Hensel's lemma is no longer true and therefore the usual proof of (1) for the complete case is no longer possible. For that reason we prove some kind of generalization of Hensel's lemma (corollary to Lemma 2 in II, ?5) which will serve us as well, together with some simple facts on the composition of valuations (III). One of the most important problems in valuation theory is to decide when equality holds in (1). If v is discrete and of rank one, there are two criteria known: the first criterion requires that K/k shall be separable. The second criterion requires that k is either absolutely or relative to a subfield K of v finitely generated and of dimension one('). We are able to generalize these two results to discrete valuation rings of arbitrary finite rank r in the following form: If v is of rank r there is a chain

I have given elsewhere a necessary and sufficient condition that certain categories of abstract groups of finite order be simply isomorphie with irreducible homogeneous linear groups in the domain of all real and complex numbers.t In the present paper I establish a two-fold generalization of this result by showing that the same condition applies to all groups of finite order and to an arbitrary domain. The proof for the necessity of this condition rests upon a conclusion drawn from Theorem II of SCHUR'S lMeue.Begrundung der Cheorie der Gr?ppencharaktere.t. Suppose that we have a hologeneous linear group S of finite order whose coefficients belong to n (an arbitrary finite field or an arbitrary domain) and which is irreducible in . If P is a given substitution on the same / variables

This paper deals with the infinitesimal transformation (2,1) of a family ( V m V_m ) of subspaces V m V_m in a Riemann space. In §§1–4 the transformation (2,1) is applied on internal objects of a V m V_m , while in §§5 and 6 the characteristic mixed tensors K a r … a 1 v K_{a_r\dots a_1}^v (cf. the equation (1,2; 4)) of a V m V_m are investigated with respect to (2,1). Finally, some applications of the theory are given in §§7 and 8. In particular the statement expressed by the equation (8,10)b is the generalization of the well known Levi-Civita result for m = 1 m = 1 , while the statement expressed by the equation (8,10)c generalizes the classical result (for m = 1 m = 1 , n = 2 n = 2 ) by Jacobi.

In this paper certain properties called symmetries are defined for links, and the problem of determining those links admitting a particular symmetry is attacked. The problems of symmetry are the generalization to links of the problems of amphichaerality and invertibility of knots which, because of Trotter's proof that there are noninvertible knots [13], are now fairly well under control. A link of two components is called interchangeable if it possesses a special type of symmetry, and certain invariants of interchangeability of a link are given and examined.

Fan has proven an inequality relating the arithmetic and geometric means of ( x 1 , ⋯ , x n ) ({x_1}, \cdots ,{x_n}) and ( 1 − x 1 , ⋯ , 1 − x n ) (1 - {x_1}, \cdots ,1 - {x_n}) , where 0 > x i ≦ 1 2 , i = 1 , ⋯ , n 0 > {x_i} \leqq \tfrac {1}{2},i = 1, \cdots ,n . Levinson has generalized Fan’s inequality; his result involves functions with positive third derivatives on (0, 1). In this paper, the above condition that requires 0 > x i ≦ 1 2 0 > {x_i} \leqq \tfrac {1}{2} has been replaced by a condition which only weights the x i {x_i} to the left side of (0, 1) in pairs, and Levinson’s differentiability requirement has been replaced by the analogous condition on third differences.

I. Introduction. A modified midpoint rule for the approximate calculation of weighted integrals fb p(x)f(x)dx, where p(x) > 0 is the weight function, has been recently proposed by Jagermann [1]. Although this formula reduces to the common midpoint rule in the particular case p(x) 1, in the general case of arbitrary weight functions the error does not vanish for all polynomials a + oSx. The purpose of this paper is to generalize the midpoint rule such that the formula is exact for polynomials of first degree and arbitrary weight function p(x) > 0. In view of practical calculations, the repeated midpoint rule is very useful because of its simplicity and small round-off error. Moreover, an error estimation does not require higher derivatives whose bounds are often not easy to obtain. For a comparison of the repeated midpoint rule to both Gaussian quadratures and "best" qualdratures we refer to Stroud and Secrest [2].

The Eakin-Nagata theorem says that if T T is a commutative Noetherian ring which is finitely generated as a module over a subring R R , then R R is also Noetherian. This paper proves a generalization of this result. However, the main interest is that the proof is very elementary and uses little more than the definition of “Noetherian".

The following is a generalization of a theorem due to Motzkin and Taussky [1] concerning matrices with property L. By definition, two matrices A and B have property L if any linear combination aA +,BB (where a and ,B are complex numbers) has as characteristic roots the numbers aXi+,3,ui where the Xi are the characteristic roots of A and theui are the characteristic roots of B both taken in a special ordering. It is shown in the above paper that if two hermitian matrices have property L, they commute. The following lemma may be noted

Strong summability, [a, #;p], for the BosanquetLinfoot (cx, f,) summability method is defined so that [cx, O;p] is identical to strong Rieszian summability, [R; cx,p]. The main result proved in this paper shows consistency in the sense that [cx, fl;p] summability implies [x',,f';qi summability, for a'>cx or cx'=aX, fl'>fl; and 1