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In the first two sections of this paper we provide a brief account of the Cayley–Dickson algebras and prove that the radius on these algebras is given by the Euclidean norm. With this observation we resort to three related topics: a variant of the Gelfand formula, stability of subnorms, and the functional power equation.

The purpose of this paper is to study a semilinear two point boundary value problem of resonance type in which the nonlinear perturbation may grow linearly. A significant improvement of a recent result due to Cesari and Kannan is given.

Suppose X X is a metric space, and Y Y is a complete metric space. In this paper a sufficient condition is given to insure that a sequence of continuous functions from X X into Y Y converge to an embedding from X X into Y Y .

Every power partial isometry (p.p.i.) in the Calkin algebra lifts to a p.p.i. in B(4). An element u in a C* algebra is a power partial isometry (p.p.i.) if ul is a partial isometry for every integer n > 1. This notion was introduced by Halmos and Wallen [5], and they characterized all p.p.i.'s in B(?). In a penetrating study [6], Herrero classifies p.p.i.'s up to their unitary orbits and up to unitary equivalence modulo the compacts. In this note, we show that every p.p.i. in the Calkin algebra lifts to a p.p.i. in B()). This is done by using a few straightforward computations involving the theory of C* extensions. Some of the results of [6] follow from this method as well. In this paper, Hilbert spaces are always separable. The ideal of compact operators will be denoted by K. The canonical quotient map of B()) onto the Calkin algebra B())/K is denoted by ir. Two elements 5 and t of the Calkin algebra are (strongly) compalent ( -t) if there is a unitary U in B()) so that t = ir(U)s7r(U*). They are weakly compalent ( -w t) if there is a unitary u in B())/K so that t = U5U*. For p.p.i.'s, these two notions may differ. Given a nuclear C* algebra A, Ext A denotes the group of extensions of K by A modulo compalence [1, 3, 8] and Extw A denotes the quotient of Ext A modulo weak compalence. We use little more than the fact that when 0 J -J :/,7-*0 is an exact sequence, and Ext J = 0 = Ext AI/J, then Ext A = 0. The first step is to classify p.p.i.'s up to algebraic equivalence by computing C*(T) for all p.p.i.'s. That is, given two p.p.i.'s S and T, when is there a C* isomorphism p of C*(S) onto C*(T) such that o(S) = T? For convenience, this will be written S T. Let Nk be the nilpotent Jordan cell of order k acting on a k-dimensional Hilbert space, and let S be the unilateral shift. If a belongs to No U {w}, let A(a) be the direct sum of a copies of A acting on 3((a), the direct sum of a copies of 91. By [5], every p.p.i. T represented on a Hilbert space can be decomposed as

In this paper, we prove that normal.zed harmonic maps from R2 or R2\{O} into H2 are just geodesics on H2 and that the quasiconformal harmonic maps from R2 into H2 are constant maps. We prove also that the only solution to Aa = sinh a on R2\{O} is the zero solution. INTRODUCTION Harmonic map is a common generalization of minimum submanifolds, harmonic functions and nonlinear a-models ([3], [4]). We have a better understanding of harmonic maps from compact manifolds ([3], [5]), whereas about harmonic maps from noncompact manifolds we know rather little. Hence, it is very interesting to know harmonic from R2 into H2 ([4]). Here, we give a description of harmonic maps from R2 into H2 under some additional assumptions. Theorem 1. The only normalized harmonic map (o from R2 or R 2\{} into H is geodesic, i.e., under suitable coordinate system, o(x,y) = y(x), where y:R H2 is a geodesic. The concept of normalized harmonic map is introduced in [7] and [8]. It is known that normalized harmonic maps from Q c R2 into H2 exist locally ([7]). Our theorem shows however, that the global problem is quite different from the local one. (The definition of normalized harmonic map will be given in ?2.) The proof of the theorem is reduced to the discussion of the solution to the Sinh-Laplace equation: Aa = sinh a on R2 or R2\{} . The local version of the reduction is given by [7], and a global version will be given in ?2. Received by the editors October 4, 1988. 1980 Mathematics Subject Classification (1985 Revision). Primary 58E20. (3 1990 American Mathematical Society 0002-9939/90 $1.00 + $.25 per page

Introduction. It has been shown in a previous paper [311 that every algebra A with radical R, such that AIR is separable, is a homomorphic image of a certain maximal algebra which is determined to within an isomorphism by A/R, the A /R-module (two-sided) R/R2, and the index of nilpotency of R. Furthermore, some indication was given of how the structure of maximal algebras can be determined in simple cases. Here, we wish to give a further illustration by describing a rather wide class of maximal and primary algebras whose structure will be shown to resemble that of crossed products, in certain respects. In fact, we shall impose a certain normality condition and then trace the consequences of a few simple facts of the noncommutative Galois theory. An algebra B over the field F, with radical R, is called primary if it has an identity element and if BIR is simple. As is well known,2 B is then isomorphic with a Kronecker product FmXC, where Fm denotes the full matrix algebra of degree m over F, and where C is completely primary, in the sense that it has an identity element and that the quotient of C by its radical is a division algebra over F. We are concerned with primary algebras B for which this division algebra (which is determined to within an isomorphism by B) is normal over F, in the sense of the noncommutative Galois theory.3 This will be the case if and only if the center Z of BIR is a separable normal extension field of F and every automorphism of Z over F is induced by an automorphism of BIR. A completely primary algebra C with radical S will henceforth be called quasinormal if C/S is normal over F. If 4 is an isomorphism of Fm X C onto B then 4 maps the radical Fm XS onto the radical R of B, and B/R is isomorphic with Fm X C/S. Therefore, if C is quasinormal over F, then B/;R is automatically separable over F. By [3], B has then a maximal related extension B* which is evidently primary. Moreover, it is easily seen that B* FmXC*, where C* is the maximal related extension of C, and is quasinormal over F. Finally, the natural extension to B* of a homomorphism of C* onto C is a homomorphism of B* onto B. From these facts it is evident

Introduction. A topological group (D is said to act on a topological space R if the elements of (M are homeomorphisms of R onto itself and if the mapping (o, p)->of(p) of (5XR onto R is continuous. Familiar examples include the rotation group acting on the Cartesian plane and the Euclidean group acting on Euclidean space. The set @(p) (that is, the set of all a(p) where orEz) is called the orbit of p. If p and q are two points of R, then (i(p) and @(q) are either identical or disjoint, hence R is partitioned by the orbits. The topological structure of the partition becomes an interesting question. In the case of the rotations of the Cartesian plane we find that, excising the singularity at the origin, the remainder of space is fibered as a direct product. A similar result is easily established for a compact Lie group acting analytically on an analytic manifold. In this paper we make use of Haar measure to extend this result to the case of a compact Lie group acting on any completely regular space. The exact theorem is given in ?3. ??1 and 2 are preliminaries, while in ?4 we apply the main result to the study of the structure of topological groups. These applications form the principal motivation of the entire study,' and the author hopes to develop them in greater detail in a subsequent paper.

It was proved by Hurewiczl that a compact space which is both LC1 and lc" is LCD. In the present paper the corresponding result for locally compact spaces is proved, (a) for uniform local connection, and (b) for relative local connection.2 The extension of Hurewicz's theorem to locally compact spaces is included in (b). The main difficulty in extending Hurewicz's methods is that his "Satz 6," on the passage from e-homotopy to true homotopy, cannot be carried over to locally compact spaces without substantial modification, even when uniform local connection is assumed. To overcome this a stronger form of the 1cP and LCP conditions is used, namely (for l1C), the existence of a function t(5, x) such that, given a compact set F in the neighbourhood U(x, t(5, x)) of any point x, there is a compact subset F' of U(x, 5) such that every g-cycle in F bounds in F', for 0_ q ? p; and analogously for LCP. It is shown that these are equivalent to the ordinary 1cP and LCP properties in locally compact (metric) spaces.

The theorem was established and proved in order to bridge a gap in the proof of a classical theorem on surfaces z(x, y) of nonpositive Gaussian curvature. Theorem 1 might, however, be of interest in other respects. The completed proof of the differential geometric theorem is given in the subsequent paper.-The special case of Theorem 1 needed for the completion is that where the boundaries of the open set Q and of the open and connected sets Q'CQ have only one or two points in common. None of these points need, of course, be accessible from Q'. Theorem 1, however, ascertains that each of these points is accessible from Q. The conclusion of the theorem remains valid if the open set Q'CQ, instead of being connected, is merely supposed to be connected to the vicinity of C. We call an open set Q' connected to the vicinity of a closed set S if there is a neighborhood N* of S (open set containing S) with the following property. For every neighborhood N of S there is a Jordan arc whose interior lies in Q' and whose end points lie in N and on the boundary of N*, respectively. The generalized Theorem 2 is found to be more easily proved than Theorem 1. It is convenient to extend the notion of accessibility of a closed set S from an open set Q to arbitrary closed sets (not necessarily parts of the boundary of Q).

It is obvious that the coefficients An, of the series development (1) determine completely the function F(r, cos 0, 4). Thus the sequence of coefficients determines the analytic character of F, the locations and nature of singular points, and so on. It seems therefore natural to look for and extract those properties of a sequence of coefficients Any which most readily yield relevant information. In the case of harmonic functions of two variables, the problem reduces immediately to that of detecting the singularities of an analytic function given by its Taylor development. This last mentioned problem, also known as the Hadamard problem, has occupied the attention of mathematicians for a considerable period and a great many results have been obtained. When dealing, however, with harmonic functions of three variables, the problem becomes much more difficult and it seems impossible to apply directly the methods developed in the theory of functions of one complex variable. We have, however, at our disposal the method of integral operators introduced by S. Bergman, which enables us to represent harmonic functions in three variables by means of an integral operator on a function of a complex variable. This representation makes it possible to carry over certain results of the well known theory of functions of one complex variable into the relatively little developed theory of harmonic functions of three variables. The present paper is to be considered as a part of an extended program of study of harmonic functions given by the de

Presented to the Society, October 30, 1948; received by the editors September 13, 1948 and, in revised form, February 9, 1949. 1 The author is greatly indebted to the referee, who made a very careful review of the paper, correcting some errors and elarifying a number of ambiguities. He also supplied footnote 2. 2Actually p can be given explicitly. It is (23. 111+1)/24 for a even and (13* 11 +1)/24 for a odd. 3 Cf. footnotes 2 and 5 in J. Lehner, Ramanujan identities involving the partition function for the moduli IIa, Amer. J. Math. (1943) pp. 492-520. This paper will be referred to hereafter as I.

()a n (n)[y(n) +1]* [y(n) + kn -1J (3 ) = E k=-1k+2k+-k. . ,kZ,k$O n1 kni Thus the partition k = lki+2k2+ +Ikt of k into ki ones, k2 twos, and so on, is weighted in a certain way by the product shown, and various partition functions are obtained by choosing the function 'y(n) appropriately. For example, if 'y(n) =1 for all n, then ay(k) becomes the unrestricted partition function; if 'y(n) = n for all n, then a,,(k) becomes the plane partition of Wright [2];' if 'y(n) =1 for rth powers and 0 otherwise, then a.,(k) is the number of partitions of k into rth powers; and so on. The purpose of this paper is to establish the asymptotic formula

The objective of this paper is to give an alternative derivation of results on bounded analytic functions recently obtained by Ahlfors [1] and Garabedian [2].1 While it is admitted that the main idea to be used is more in the nature of a lucky guess than of a method, it will be found that the gain in brevity and simplicity of the argument is considerable. As a by-product, we shall also obtain a number of hitherto unknown identities between various domain functions. The basic problem treated in the above-mentioned papers is the following generalization of the classical Schwarz lemma: Given a finite schlicht domain D of connectivity n (n> 1) in the complex zplane and a point r in D, to find a function F(z) with the following properties: (a) F(z) belongs to the family B of analytic functions f(z) which are single-valued and regular in D and satisfy there If(z) I _ 1; (b) I F'(r) I > lf'(r) J, where f(z) is any function in B. Evidently, it is sufficient to solve this problem for any domain D' which is conformally equivalent to D. In particular, we may therefore assume, without restricting the generality of what follows, that D is bounded by analytic curves. It was shown by Ahlfors that F(z) yields a (1, n) conformal mapping of D onto the interior of the unit circle and that n-I of the n zeros of F(z) coincide with the zeros of a single-valued function h(z) which is regular in D with the exception of a simple pole at z = and satisfies -'ih(z)dz>O on the boundary r of D; the nth zero of F(z) is located at zx=. It was subsequently noticed by Garabedian that the function h(z) can be written in the form h(z) = F(z)q(z) where q(z) (Z )-2 is regular in D and that the extremal property of F(z) can be deduced in a very elegant manner from the resulting inequality

A widely used theorem of analysis asserts that a uniformly bounded, equicontinuous family of functions has a compact closure in the space of continuous functions. This lemma, variously attributed to Arzela, Escoli, Montel, Vitali, and so on, is of importance in the theory of integral equations, conformal mapping, calculus of variations, and so on. In recent years the lemma has been generalized by S. B. Myers [1 ].I A part of his results may be formulated as follows; If a topological space X is either (a) locally compact, (b) satisfies the first axiom of countability, and if Y is a metric space, then a family F of continuous functions from X to Y is compact (in a suitable topology) if and only if (1) F(x) = UfEFf(x) is compact for all xCX, (2) F is closed, (3) F is equicontinuous. The main purpose of ?1 of this paper is to characterize compact sets of functions when Y is any regular topological space. The problem is therefore to find a condition to replace equicontinuity, which no longer makes sense. We obtain such a characterization which holds for an easily described class of spaces X which includes both locally compact and first countable. In ?2 these results are applied to obtain a sort of duality theorem for the ring of real-valued continuous functions, R(X), on a space X. Namely, it is shown that, under quite general conditions, the space X is homeomorphic with the space H(X) of continuous homomorphisms from the ring R(X) onto the real numbers, where R(X) and H(X) are given the "compact-open" topology.

Introduction. Let G be a locally compact, connected topological group (satisfying the second countability axiom). Let G* be a compact space which contains a dense subset G' homeomorphic to the space G and is such that G*-G' is totally disconnected. Then, Freudenthal has proved [1, Satz 1X, p. 277]1 that the set G*-G' consists of at most two distinct points. Actually, Freudenthal's theorem even for topological groups is more general than here stated, and this theorem is an application to group spaces of a wider theory of "ends" of topological spaces. However, we shall quote only so much of Freudenthal's results as are necessary to this paper. It will be convenient to regard G' as identical with G so that G is topologically imbedded in G*. We shall call a locally compact, connected group G two-ended if a G* exists such that G* G consists of two distinct points. The simplest example of such a group is the additive group of reals. Other examples are afforded by the direct product of this group and any compact connected topological group; it is likely that these are the only examples. The principal objective of this note is the following theorem.

In this paper a definition is proposed for the radical of a non-associative ring. Our results are somewhat similar to those given for algebras by Albert in [3],1 but the difficulties that arose in the earlier theory from absolute divisors of zero have been overcome. With slight modifications, the present proofs are applicable to algebras. A non-associative ring 9? is an additive abelian group closed under a product operation with respect to which the two distributive laws hold. Multiplication on the right (left) by a fixed element xCj? determines an endomorphism Rx (L.) of 9? as an additive group. For x, yCIR

Introduction. Frobenius [1]' derived expressions for the characters of a few very simple classes of Sm, the symmetric group on m things. Here we give formulas for some more complicated classes. The method is quite general. A recurrence formula, due to Murnaghan, is used. The need for the formulas arose from general considerations of nuclear binding forces.2 Some applications will be given in another paper.

The Wedderburn theorem that every finite division ring is commutative has been extended by several authors [1].1 Vandiver, in his paper The p-adic representation of rings [2 ] conjectured the following generalization of Wedderburn's theorem: every finite, non-commutative ring contains an element which is a divisor of zero and is not in the centrum. In this paper we give a short and simple proof of this conjecture. We also exhibit one generalization of it which was pointed out to us by the referee.

The purpose of this paper is to rephrase a conjecture about simple groups into the language of linear algebra. Let G be a group of finite order o(G). Then by rF we shall mean the group ring of G over a field of characteristic p (for instance the integers modulo p). We shall denote the radical of rF by N,. If p = 0 or p o(G), then it is known that Np=(O); and if p|o(G), Np (O). We now consider the following two assertions: (A) If G is a simple group of odd order, o(G) is a prime. (B) If G is a group of odd order o(G), then for some prime p, p[ o(G), we can find a gCG, g1, such that g-1CNp. The theorem which we propose to prove is

the parameters X, p are replaced by a, f3, the resulting system will be denoted by (1; a, ,B). The present paper will present a study of the algebraic integrals of the system (1; X, 1). The equations (1; X, 1) are, excepting for a change of variables, those of the motion of a solid about a fix point, in a uniform field of force, when the ellipsoid of inertia relative to the fix point is of revolution and the baricenter of the solid belongs to the equatorial plane of the same ellipsoid. Theorems 3 and 4 are original. The demonstration of Theorem 3, presented by R. Liouville [1],1 accepted by P. Burgatti [2], is incomplete. In fact, it is implicitly admitted that one of the polynomials belonging to the rational integral possesses terms independent of yI, Y2, y3. The method of demonstration of Theorem 4 was inspired from a paper of Husson [3 1. For the sake of simplicity of nomenclature x will be used instead of (X1, X2, X3) and y instead of (yI, Y2, y3) wherever possible. Let us, "ab initio," make the following two observations about the system (1;X, 1): (a) The differential equations (1; X, 1) partake of the following particular property of homogeneity: they are not altered when the x variables are multiplied by a constant k, the y variables by k2; and t, by k-1.

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 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).

where f(x) is an arbitrary function in the class E,. Some functions (for example, f(x) =x/p, which leads to the representation of a number as a decimal to the base p) when employed in the algorithm (A) yield one-one correspondences between real numbers and sequences of integers mod p. On the other hand, any function, for example, whose graph has more than one point in common with any of the straight line segments connecting the points (j, 0) and (j+1, 1), j=0, 1, * * *, p-I, will obviously lead to a correspondence which is many-one. We shall denote by Ep* the subclass of Ep consisting of those functions which in the algorithm (A) give rise to one-one correspondences. The present paper contains very simple characterizations of those correspondences between real numbers and sequences of integers mod p which can be obtained by applying the algorithm (A) with functions from the classes E * and Ep -E * respectively. By means of these characterizations it is possible to settle two of the problems raised by Everett and to give an answer (albeit not a completely satisfactory one) to a third, namely that of characterizing the class Ep* itself.

where the symbol on the right denotes the smallest y such that A (X, y) = 0, under the assumption that there is such a y for each g. Kleene showed that this definition of general recursive function is equivalent to Herbrand-G6del metamathematical definition.2 In this paper we shall be concerned with the mathematical (as opposed to metamathematical) aspects of the theory of general recursive functions. Starting from the definition stated, we shall investigate the possible restrictions on the defining schemes. Part of the results obtained are already known from the work of Kleene, but we go further in this direction than Kleene did. No previous knowledge of general recursive functions is assumed in this paper. It will be convenient to have a logical symbolism to express the conditions that appear in applications of the u-rule. We shall use: A (for every), V (there exists), A (and), V (or), (not), -* (if then), *-+ (if and only if). The following equivalences will be useful

The set M is closed if M=uM. +(M) is the least ordinal number t for which the set utM is closed, that is, uEM=ut+1M. G(P, u) is the set of all ordinal numbers +(M) where MCP. E. Cech3 has posed the following problem: What are necessary and sufficient conditions on a set H of ordinals in order that there exist a topology u in a countable set P for which H=G(P, u)? V. Jarnik4 solved the generalization of the problem for a set of arbitrary cardinal number s&; his topology, however, is not an additive one, as it satisfies only the axiom of monotony. The present paper contains a solution to this problem when u is to be additive; this solution is stated in Theorem 1. The proof is carried through for a class of cardinals including No. First we have to prove the following lemma.

This note extends to Euclidean n-space En for n ?2 and to the generalized Euclidean space E, certain investigations undertaken in [1]I with reference to E2. Among other things, we shall show that, given any absolutely convergent series ECk of elements in a Banach space, there is a subseries Ecj* of maximum modulus; that in En for n ?2 this maximum is greater than p Ck , where the "best possible" value for p is J(n/2)/{27,rl12r[(n+1)/2] }; and that in Ezz4 the corresponding quantity p is zero. We shall also obtain certain related results in En and in Eoo for series ,bk with Z bkI = CC . The value of p in E. is not new, although its inaccessibility seems to have been ignored. It was obtained in 1938 by A. E. Mayer [2] for finite series, and again in 1943 by H. Hadwiger [3] as a corollary to a general theorem on "direction-functionals." Since many of our present proofs are strictly parallel to proofs in [1 ], we shall make our symbols conform as closely as possible to those in that paper, shall use freely the theory of multi-dimensional analytic geometry, and shall generally omit proofs where the corresponding proofs in [1] are sufficiently suggestive. (Many of these proofs can be constructed from those in [I] by merely replacing the variable angle q by a variable unit vector and the element of integration d? by the element of n-dimensional solid angle.) We shall denote components of vectors by superscripts on the symbols for the vectors themselves, let x represent a variable unit vector, use the summation convention with respect to repeated superscripts (thus xib4 signifies Zixib), use abbreviations such as BkI bkl and (x, bk) -arccos (xib'/Bk), and insert a "(+)" before a summand or integrand to denote summation or integration over positive values only. Unless we specifically assert that any particular discussion relates to En or to E,x, it is to be assumed to relate to both. Let 2 be the class of absolutely convergent series, finite or infinite, Eak (denoted alternatively by A) the class of nonzero vectors in En (n2!:2) or in E,,, and Eaj (denoted alternatively by S) the general subseries of >ak for fixed Eak. Let e be the class of series Ebk (denoted alternatively by B) of nonzero vectors, in the same space, for which EJ bk = oo, and Zbj' (denoted alternatively by T) the general subseries of Ebk for fixed Zbk.

admit no solution +(x) in LP', p'=p/(p1), except 40. A complete account of the present status of the problem is given in Segal's paper and it is unncessary to review the facts here. It is the purpose of this paper to point out the close connection between this problem and a certain uniqueness problem for trigonometric integrals. A point set S on the real axis is said to be of type q if the conditions

In a recent paper (Square roots in locally euclidean groups, Bull. Amer. Math. Soc. vol. 55 (1949) pp. 446-449), A. M. Gleason proved that, in a locally euclidean group G which has no small subgroups, there exist neighborhoods M and N of the neutral element e such that every element in M has a unique square root in N. The author clearly considered this result to be a step towards proving that, in such a group, some neighborhood of e is entirely filled by oneparameter subgroups of G. We shall establish here that he was justified in his expectation. As a matter of fact, our present proof requires only the local uniqueness of the square root, not its existence. This local uniqueness is established in the second part of the proof of Theorem 4 in Gleason's paper; and it may be observed that this part of the argument is independent of the assumption that the group be locally euclidean: it works equally well under the weaker assumption that the group be locally compact. We therefore have the following result

In a recent paper E. J. McShane [3]2 has given a theorem which is the common core of a variety of results about Baire sets, Baire functions, and convex sets in topological spaces including groups and linear spaces. In general terms his theorem states that if 7 is a family of open maps defined in one topological space X1 into another, X2, the total image 7(S) of a second category Baire set S in X1 has, under certain conditions on 7 and S, a nonvacuous interior. The point of these remarks is to show that his argument yields a theorem for a larger class than the second category Baire sets. From this there follow sligh4ly stronger and more specific versions of some of his results, including his principal theorem, as well as a proof that if S is a subset of a weak sort of topological group and S contains a second category Baire set, then the identity element lies in the interior of both S-1S and SS-1. There is also at the end an extension of Zorn's theorem on the structure of certain semigroups. In a topological space X let the closure and interior of a set E be denoted by E* and EO and the null set by A. For any set S let I(S) - U[GIG open, GnS is first category] and II(S)=X-I(S), and let III(S) be the open set II(S)0?I(X-S). By a fundamental theorem of Banach [2], SnI(S)* is first category and hence S is second category if and only if II(S)O7A. From these we note that if N is a non-null open subset of III(S), then N-S is in the first category set (X - S)G'iI(X - S), and NnQS cannot be first category since N is non-null open and disjoint with I(S). This gives us the following lemma. LEMMA 1. For any non-null open subset N of III(S), the sets N- S

1. Minimal functions. We are concerned with continuous functions whose domains are open subsets of the plane P and whose ranges are contained in P. In this paper the word mapping is used to designate such a function. By a disk we mean a subset of P which is a closed topological 2-cell. If S is a simple closed curve in P, we denote by S* the disk formed by taking the union of S and the interior of S. Let S be a simple closed curve in P and let f be a mapping whose domain contains S*. We say that f is minimal on S* if f(S*) Cg(S*) for every mapping g whose domain contains S* and which is such that f IS=gI S (that is, which is such that f(x) =g(x) for each xES). We define f to be minimal if f is minimal on each disk contained in the domain of f. In this paper we prove that a light mapping is open if and only if it is minimal. 2. Winding numbers. We make use of the concept of winding number or topological index. If f is a mapping whose domain contains a simple closed curve S and p GP -f(S), then we denote by W(f, S, p) the winding number of f on S with respect to p. Intuitively, as a point x travels once around S in a counter-clockwise direction, W(f, S, p) is the net number of revolutions that the vector from p to f(x) makes about p (a revolution being positive if made in a counter-clockwise direction and being negative if made in a clockwise direction). The following facts concerning winding numbers are well known and are assumed.

In this paper the following result will be proved. Let f(w) be an analytic function of w for IwI 1 and C is the unit circle wI |1, is given by 01( Ia1, p) if 1 _p 1 + Ia, 02(1a I, p) if p _lI+ Ia, where

1. Let G be a connected locally compact group and let G' denote the closure of the commutator subgroup of G. G' is called the derived subgroup of G. Consider the derived series of G, that is, the sequence of subgroups Go, G1, , G ... where Go=G and Gn+?=G'. Each Gn is connected and this sequence becomes stationary at some finite stage,2 that is, for some n, G, =G,+ We define G, to be the "core" of the group. The purpose of this paper is to prove the following

Received by the editors June 10, 1950. 1 This work is an offshoot of investigations carried out under the auspices of the Office of Naval Research, Contract N9-ONR90,000. 2 B. Riemann, Ueber die Anzahl der Primzahlen unter einer gegebenen Grbsse, Monatsberichte der Preussischeni Akademie der Wissenschaften (1859, 1860) pp. 671680. 3A. Hurwitz, Zeitschrift fur Mathematik und Physik vol. 27 (1882) p. 95. Throughout this paper, x =exp (s log x), the logarithm being real for x>0. R. Lipschitz (J. Reine Angew. Math. vol. 105, pp. 127-159) has used the thetafunction transformation device to derive a functional equation for a general type of zeta-function, but his results do not appear to include ours. 6 Whittaker and Watson, Modern analysis, p. 475.

Introduction. In many cases it is practically impossible to solve an initial value problem for a partial differential equation exactly although it can be proved that the exact solution does exist and is uniquely determined. Therefore the partial differential equation is very often replaced by a difference equation which is easier to solve and which furnishes an approximation to the solution of the original problem. Three questions arise immediately. (a) If x and t are the independent variables, does the mesh ratio r (r=At/(Ax)2 in the parabolic case) have any influence on the convergence or stability of the approximate solution? (b) Which one of the difference equations leading to the differential equation will furnish a good approximation? (c) How do the initial values of the problem for the difference equation have to be chosen in order to furnish a good approximation? The author believes that the first question has been overemphasized while the importance of the second and third has not been fully realized. This is primarily due to the fact that, in the paper by O'Brien, Hyman, and Kaplan []1 where von Neumann's test of stability is introduced into the literature, it is erroneously stated that a positive answer to von Neumann's test is necessary and sufficient for convergence. As an example it is pointed out in their paper that the numerical solution of Richardson [2] for the problem of finding the temperature in a slab with faces at temperature zero does not converge to the exact solution because von Neumann's test shows instability for all r> 0.2

By considering elements whose principal traces are zero instead of nilpotent elements, this paper exhibits an ideal ZD91 called the liberal, which is a generalization of the radical and which, in fact, reduces to the radical when and only when 2191 is separable. An extension of Theorem 1 is obtained in which there is always a decomposition of type (1) for the liberal of an arbitrary alternative algebra. The associative case of this paper is a portion of the author's doctoral dissertation written under Professor C. C. MacDuffee. 2. The first liberal. Let t= E#;ej be a general element [2, p. 112] of an alternative algebra with basis e1, e2, * * , en over an arbitrary field 0. Then T(t), the negative of the coefficient of the second highest power of X in the minimum polynomial m(t, X) of {, is called the principal trace of t. When the indeterminates {i are replaced by elements xi of 0 in T(t) we get the principal trace T(x) of x = Exiei. The quantity T(x) is independent of the basis of Wf. We now wish to prove the following lemma.

Introduction. The purpose of this paper is to give a necessary and sufficient condition in terms of the ordinary curvatures of a curve for it to lie on any real central hyperquadric in a Euclidean space 9n. This problem was solved for the hypersphere in 9% by Karel Havllcek in his paper Contact des courbes et des hypersphAres dans un espace euclidean a' n dimensions.-Courbes spheriques, which appears in Casopis pro P6stovanl Matematiky a Fysiky vol. 72 (1947). First we find a necessary and sufficient condition in terms of socalled hyperquadric curvatures for a curve to lie on a given real central hyperquadric. Then by means of invariant functions in ?3 we find the solution of our problem.

The purpose of this paper is to present solutions to certain problems in ergodic theory suggested by Einar Hille in his book Functional analysis and semi-groups [1]. 1 Let T(t) ( > 0) be a semi-group of linear bounded transformations on a complex Banach space X to itself. Following Hille, we say that T(t) is ergodic at infinity (or at 0) if it has a generalized limit of some sort when t-*oo (or 0+). We define the Abel mean

Liapounoff2 established in 1940 that the range of a countably additive finite measure with values in a finite-dimensional real vector space is bounded and closed and in the nonatomic case convex. A simplified proof of this result was given by Halmos' in 1948. The aim of the present paper is to extend this result to the following case. Let pit, 1 0 there exist a a>0 such that g*(E)

where p(a, b) means the distance between the points a and b. Their Theorem 4 shows that if E is a finite-dimensional Euclidean space or a real Hilbert space, then for any e-isometry of E into itself, there is an actual isometry of E into itself such that the distance between the images of any point under these two transformations is less than 10e. Their treatment does not concern bounded spaces. The purpose of this paper is to show that the type of k uniformity which Hyers and Ulam obtained in their complete spaces cannot be obtained in general even in very special bounded spaces, but that an -q, e type relation holds for any compact metric space.

Let S be a set of points in the Euclidean plane E2. It is our purpose to establish a necessary and sufficient condition that a simply connected1 closed set S be arcwise convex. In order to do this precisely, the following notations and definitions are used. NOTATION. The line determined by two distinct points x and y in E2 is denoted by L(x, y). We designate the open line segment joining x and y by xy, and the corresponding closed segment by [xy]. The two closed half-planes having L(x, y) as a common boundary are designated by Ri(x, y) and R2(x, y). The boundary of a set K is represented by B(K), and H(K) denotes the convex hull of K. The complement of S is denoted by C(S). DEFINITION 1. A set SCE2 is said to be unilaterally connected if, for each pair of distinct points x and y in S, there exists a continuum2 Si CS which contains x and y, and which lies in one of the closed halfplanes determined by L(x, y). DEFINITION 2. A set SCE2 is said to be arcwise convex if each pair of points in S can be joined by a convex arc lying in S. (A convex arc is one which is contained in the boundary of its convex hull.) In a previous paper [1 ]3 the author studied the complements of both arcwise convex sets and unilaterally connected sets. The theorem below establishes another intimate connection between these two concepts. I am indebted to the referee for the following lemma which simplifies the proof of the theorem.

is an extension of m. Recall that X, Y, Z are naturally embeddable in X-, Y--, Z-resp. Moreover, certain properties, such as associativity, when m has them, are transmitted to m*** (this makes sense only when Y=Z=X). On the other hand, the transmission of commutativity (which makes sense when Y=X) was left open, and will be considered in this paper. This question of commutativity can be generalized as follows. If m satisfies 1.1-1.3, one can define the transposed operation

A simple closed curve is the simplest example of a compact, nondegenerate, homogeneous continuum. If a bounded, nondegenerate, homogeneous plane continuum has any local connectedness property, even of the weakest sort, it is known to be a simple closed curve [1, 2, 3].1 The recent discovery of a bounded, nondegenerate, homogenous plane continuum which does not separate the plane [4, 5] has given substance to the old question as to whether or not such a continuum must be indecomposable. Under certain conditions such a continuum must contain an indecomposable continuum [6]. It is the main purpose of this paper to show that every bounded homogeneous plane continuum which does not separate the plane is indecomposable. NOTATION. If M is a continuum and x is a point of M, U. will be used to denote the set of all points z of M such that M is aposyndetic at z with respect to x.2 It is evident that U. is an open subset of M.

The exterior of a bounded closed point set b in E will mean the unbounded region of the complementary set E b. The remainder of E-b, if not vacuous, will be called the interior of b. (A) As a corollary to the above theorem, c is intersected by any simple arc with one end point interior and one exterior to c. This paper contains an elementary constructive proof of the Jordan-Schoenflies Theorem, motivated by the belief that such a proof should be presented at a fairly early stage to students of topology and analysis. To that end, it is desirable that the argument be disassociated from conformal mapping theory and be accomplished by methods as elementary as possible.

ordered systems are given. We shall use some of the notation and definitions of [3]. For the sake of convenience we list here some of those definitions and symbols that will be employed. By an ordered system is meant a nonempty set R of elements in which a reflexive binary relation r = r' is defined. Unless otherwise specified, an italic capital letter always will denote an ordered system in the sequel. A subsystem T of R is a subset of elements of R with the order relation in T imposed by that in R. The expressions and symbols maximal element, greatest element, ascending chain condition, isomorphic, >, and so on, will have their usual meanings (see [2], for example). The symbols V and A will be used in denoting least upper bound (l.u.b.) and greatest lower bound (g.l.b.) respectively. The symbols 0 and I will denote the bounds of bounded ordered systems. The term number will mean partially ordered set. The symbol S>R will mean that R is isomorphic to a subsystem of 5. If for each element r in R, Sr is an ordered system, the ordered sum over R of the systems Sr (denoted by ~Y+rSt) is the system P where the elements of P are the ordered pairs (r, s) with r in R and s in 5r, and (r, s) = (r', s') means that r > r' or else r = r' and s = s'. If all Sr = S, we write R o S for ^rS?. The ordered product over R of the Sr (denoted by IJa-SV) is the system P where the elements of P are the functions/ defined on R such that f(r)ESr, while /^/' means that if f(r) ?af'(r), there exists r'^r such that/(r') >f'(r'). We list several results in [3 ] that are used in proofs in this paper. [3,2.2] states that ^,rSt>R and ^,RSr>St for every element t

2. The improper integral defined as the limit of a sum. We first consider the case where ak = 1 (k =1, 2, 3, * * * ). A fundamental difficulty not encountered in the previous paper is apparent from the definition of .fof(x)dx. Bromwich and Hardy have considered this problem [2]. As they point out, the ordinary definition of this improper integral is by means of a double (repeated) limit; and in order to replace this limit by the single limit

and rP, Q? are the symmetric and skew-symmetric parts respectively of Lj [i, p. 231]. In case Qk=0, the Ln reduces to a symmetrically connected space An in which (1.3a) defines the infinitesimal affine collineations [2, p. 125], so that motions in an Ln can be considered as generalizations of such collineations of an An [3]. In a previous paper [3] we obtained all (symmetric) two-dimensional spaces A2 admitting real continuous groups of affine collineations, these being obtained by solving (1.3a) for the I", the ti being known and obtained from Lie's classification [4, pp. 71-73, 379-380] giving all real continuous groups Gr in two variables. In ?2 we carry through a like procedure to obtain all linearly connected L2 admitting complete groups of motions. This will involve the solution of the

Received by the editors March 17, 1951. 1 A. Ostrowski, Sur la determination des bornes inferieures pour une classe des determinants, Bull. Sci. Math. (2) vol. 61 (1937) pp. 19-32. The fact that, if all a,T are less than 1, A does not vanish is due to Hadamard and others. 2 A. Ostrowski, Ueber die Determinanten mit siberwiegender Hauptdiagonale, Comment. Math. Helv. vol. 10 (1937) pp. 69-96. I use this opportunity to mention the following misprints in this paper: p. 70, 1. 9, f.a., read f h,,,| instead of h..; p. 73, formula (13), read I h,,,f instead of h. and E' ,,; instead of y>1 p. 73, 1. 4, f.b., read 1881 instead of 1899; p. 76, the right-side product-sign in the formula (18) is to be dropped; p. 86, in the formula (11, 1) read on the left side m,zyp, instead of y,, and on the right side M instead of 1; p. 96, 1. 7, f.b., read S1/S2 instead of S2/Si.

Introduction. In this paper we shall describe an effective procedure to determine in a finite number of steps whether two abstract algebras A' and A" of finite order have: (i) the same set of laws, (ii) the set of laws of one of them is included in the set of laws of the other, (iii) distinct sets of laws, (iv) overlapping sets of laws. The procedure is a generalization of certain results obtained by the author in case of truth-tables.' Some parts of the paper On the structure of finite algebras by G. Birkhoff2 will be presupposed, namely: (i) the definition of an abstract algebra A = (T, F) and of its order (?2, pp. 433-434), (ii) the definition of the direct product of abstract algebras (?7, pp. 437-438), (iii) the definition of an abstract algebra of species 1, and of a uniform operator (?8), (iv) the definition of a function 4 of rank n associated with the speciesl, (?9, Definition 2), (v) the idea of a substitution t of elements of an algebra for each primitive symbol of a function 4 and of the resulting value t(4) of the function (?9, p. 439), (vi) the definition of a law of an algebra (?9, Definition 3), (vii) the idea of a law of a set of algebras (?9, p. 439), (viii) the theorem that the set of laws of any aggregate of algebras A is the same as that of the direct product A of all the A (Corollary 2, p. 440). The notation used by Birkhoff in the paper referred to will be followed without further explanation. L(Ao) will stand for the set of laws of any algebra Ao.

One of the aims in the study of almost periodic functions on a group is to generalize the Fejer summation process, whereby the Fourier series may be summed to yield the function. This has been achieved partially by Bochner-von Neumann [1] and Maak [2; 3].J But in [3] Maak says "In general it is not possible to give a summation procedure which may be successfully applied to all almost periodic functions on the group." This paper is intended to show that we may indeed find a generalized Fejer kernel and a summation process, which may be applied equally to all almost periodic functions on a group. We prove this for the case of complex-valued almost periodic functions on an arbitrary group. The procedure also works for functions with values in a linear topological space, but we shall not concern ourselves here with the more general case. In order to proceed to the heart of the matter, we shall assume the definitions and results of [1] and [4]. We shall denote an irreducible, normal, unitary representation [4, Definitions 9 and 10] of a group G by D. The image of xCGis thus a square matrix D(x) = (Dp, (x)) where p, a = 1, * * *, sD. Each Dp,, is an almost periodic function on G [4, Theorem 19]. Let f be a complex-valued almost periodic function on G. The Fourier coefficients of f are defined by

1. The present note contains several remarks on an earlier paper by the author [2].1 In Chapter IV, ?4, which deals with the question of when we can have equality of modules for a triply-connected domain and a proper subdomain, the last sentence was added in proof. This accounts for the apparent disparity between it and the preceding one. In order to justify this statement we observe first that equality of the modules for triply-connected domains implies equality of the modules for the hexagons into which the domains are divided by their lines of symmetry. Thus by Theorem 5 and the remarks preceding Theorem 2a in [2], the value of al a2 :a3 in question must give rise to a degenerate case for each domain. The canonical domains will be of type 2, 4, or 5, the cases rising being those indicated on p. 344 in [2]. For the triply-connected domains there is a symmetric mapping of the one into the other characterized by the type of the canonical domains in question. We observe that for the functions v defined by equations (1), (2), (3) on pp. 339-340 in [2] if we set (d?/dz)2 = Q(z) and choose the constant C appropriately, then Q(z)dz2 is a positive quadratic differential of the triply-connected domain. The zeros of this quadratic differential lie at * and its symmetric point in cases (1), (2), and at z*, z** in case (3). If we count boundary zeros with half their multiplicity, the total multiplicity of the zeros is 2 in each case (as also would follow from the general theory). Indeed these give all the positive quadratic differentials of the triply-connected domain (see [3 ]). In the canonical domain of type 2 the point A! in Fig. 3 corresponds to such a zero. In the canonical domain of type 4 the points corresponding to zeros are A and As (for the case drawn) and in the canonical domain of type 5 they are A5 and A6 (for the case drawn). The cases where equality of modules may occur are those illustrated in Fig. 5. They correspond to the following situations in order: (i) The subdomain is obtained from the original domain by producing slits out from the zeros of Q(z)dZ2 into the domain along the curves on which Q(z)dz2 >0 (corresponding to the vertical segment through A! in the domain in Fig. 3), these slits are symmetric with respect to the line of symmetry; (ii) the subdomain is obtained from the original domain by