Search

near a point solution xi ai (i 1, 1 *, n) when certain conditions are satisfied. In the present paper these conditions are replaced by less stringent ones; the methods of proof of certain existence theorems are very similar to those employed in the first paper, and these proofs are given here in an abbreviated forin. In addition, the asymptotic properties of certain trajectories are discussed by an extension of the methods of Bohl. On account of the more complicated form of certain quadratic forms which occur here, it has been convenient to leave undetermined certain constants which are determined explicitly in the first paper; this procedure, together with the algebraic method of transforming the canonical equations, reduces to a small amount the results common to both papers.

In the above-named paper (this JOURNAL, vol. 77 (1955), pp. 895-913), Theorem 6. 4 should have been stated only for rings of zero characteristic: the argument for the case of prime characteristic breaks down in the last formula line on p. 911. This involves jettisoning Theorem 6. 5 (the implied "proof " of which depended on applying Theorem 6. 4 to homomorphs which cannot be guaranteed to have zero characteristic, even when the given ring has). However, the writer has no evidence that Theorem 6. 4 as stated or Theorem 6. 5 is actually false. And in any case, since the valid part of the proof of Theorem 6. 4 establishes the weaker conclusion [Xm, ynp?] = 0 without characteristic hypothesis, the remarks following Theorem 6. 4 still hold good, provided that we modify the final parenthetical clause so as to read "which would at any rate imply that every weak K-ring R with kc, m, n satisfying (a) or (b) has R/J commutative."

If (aik) a ==a(t), where 0 ? t 1, then a C R is neither necessary nor sufficient for a C S (cf. the paper quoted in the title, which will be referred to as loc. cit.). On the other hand. if a C L means (as usual, with L =LI) that y e R holds for the norm, -y = I a J, of a also, then, according to a criterion which goes back to Bocher, a C L is sufficient for a C S (for references to the implication L > S, and for generalizations of L > S derived from L =>S itself, cf. loc. cit.). There is, however, something unsatisfactory in the standard criterion, L =>S. In fact, this criterion fails to contain the fact that, as readily seen from a quadrature, a C R is sufficient (and necessary) for a C S in the scalar case, n1 1. It is therefore of initerest that L > S can be improved to the following criterion: If ao (t) denotes the diagonal matrix the diagonal elements of which are those of a(t), then aO C R and a -aO L together are sufficient for a C S. It will be clear from the following proof that (and in which maIiner) this extended criterion can be combined with the criteria giveni loc. cit. First, it follows from L ) S by a known application of the method of the variation of constants (cf. vol. 68 (1946), p. 200, of this Journal), that if 3 =, /B(t) is any continuous matrix the real part of the trace of which has an indefinite integral posessing a lower bound (> -oo), then /3 C S and a -,B C L suffice for a C S. Since the trace condition is certainly satisfied if ,/ C R, it follows, by choosing / =ao, that the italicized assertion is proved

There are many examples of the use of topological methods to prove the existence of solutions to equations which arise in analytic problems. On the other hand, there have not been many applications of topology to the study of algebraically defined equations. The only well-known example is the topological proof of the Fundamental Theorem of Algebra (as, for instance, [5] p. 166-172). This paper presents topological proofs of the existence of solutions to certain classes of algebraically defined equations. As Hopf pointed out long ago, the natural setting for the use of topological methods in "algebraic" problems is the theory of H-spaces. Accordingly, the equations we consider will be defined on H-spaces. In the course of the work, I will describe a more-or-less unified theory which encompasses the few topological results concerning the existence of solutions to algebraically defined equations which are known to me. In Section 1, we consider n simultaneous words in n variables defined on an H-space and find a sufficient condition for their solution when the H-space is a compact, connected manifold without boundary. This section also contains the main theorem of the paper (Theorem 1), from which all other results are derived. Section 2 is concerned with the open cone on an H-space. We propose the open cone as a useful generalization, in H-space theory, of the real division algebras, and we extend the result of Section 1. Section 3 introduces the concept of a "bi-H-space," essentially the open cone on an H-space together with enough additional structure so that we can define "polynomials" on it. We present examples to show that such objects are reasonably common and prove an appropriate form of the Fundamental Theorem of Algebra for bi-H-spaces.

1. Introduction. In this paper, we consider spectral problems for lnultiplier transforms defined on the sequence spaces IP, 1 C p oo. We collect the basic and elementary facts about these operators in ? 2. We show that to every such operator T on IP there corresponds a unique (a. e.), bounded, measurable function TA(0) defined on the interval [0, 1]. The converse is not necessarily true, except for p =- 2. In ? 3, we characterize a wide class of functions TA (0) for which the spectrum of the corresponding multiplier transform T is the essential range of the function. This means that the spectrum of any such operator does not change when it is considered as acting different IP. However, the point spectrum may change as p changes. In ? 4, we give some theorems which lead to a characterization of the point spectrum of certain classes of operators in terms of sets of mnultiplicity for IP. We also give an example which shows the change of point spectrum as p changes. In so doing, we partially answer, in the negative, a question posed by P. llartman [2]. We close the paper in ? 5 with two theorems which are closely connected with the problem of spectral synthesis in lo.

Group-theoretic constructions like the one in Corollary 2 of the preceding paper have been used by several authors as a tool for the theory of abstract groups.* In this paper we shall show how the lemmas, proved by these authors on group constructions of that type, can be proved by means of simple topological reasonings on 2-dimensional complexes in the plane. The resulting proofs for those lemmas are shorter and of much clearer construction than the original proofs, but nevertheless liot essentially different. It is this 2-dimensional method of proof and not any original result which justifies the publication of this paper. The lemmas in section 1 of this paper give the connection between abstract groups and 2-cells, and are reminiscent of Dehn's theory of the " Gruppenbild." Their proof is only sketched. In section 2 the lemmas proved by the authors cited are restated in a generalized form in 3 theorems with 2 corollaries. Nowhere in this paper is a restriction placed on the number (power of infinity) of any set of generators or of relations used.

where Z is a multiplier for G. This series, which will converge in this case if q > 1, represent an automorphic form of dimension q. (Cf. Lehner [4], Chapter III) }. The purpose of the present paper is to extend Knopp's Lemma to the case q > 1. Knopp's proof involves substantial use of the Ber's spaces of automorphic forms. Since the character of these spaces in the case q ? 2 and 1 1 at every stage of Knopp's proof, the proof turns out to be valid. However, verfication of two of the resulting statements requires techniques far outside Knopp's paper. We begin by reproducing IKnopp's proof of the Main Lemma after which

A set of normal orthogonal functions {χ} for the interval 0 5 x 5 1 has been constructed by Haar†, each function taking merely one constant value in each of a finite number of sub-intervals into which the entire interval (0, 1) is divided. Haar’s set is, however, merely one of an infinity of sets which can be constructed of functions of this same character. It is the object of the present paper to study a certain new closed set of functions {φ} normal and orthogonal on the interval (0, 1); each function φ has this same property of being constant over each of a finite number of sub-intervals into which the interval (0, 1) is divided. In fact each function φ takes only the values +1 and −1, except at a finite number of points of discontinuity, where it takes the value zero. The chief interest of the set φ lies in its similarity to the usual (e.g., sine, cosine, Sturm-Liouville, Legendre) set of orthogonal functions, while the chief interest of the set χ lies in its dissimilarity to these ordinary sets. The set φ shares with the familiar sets the following properties, none of which is possessed by the set χ: the nth function has n−1 zeroes (or better, sign-changes) interior to the interval considered, each function is either odd or even with respect to the mid-point of the interval, no function vanishes identically on any sub-interval of the original interval, and the entire set is uniformly bounded. Each function χ can be expressed as a linear combination of a finite number of functions φ, so the paper illustrates the changes in properties which may arise from a simple orthogonal transformation of a set of functions. In § 1 we define the set χ and give some of its principal properties. In § 2 we define the set φ and compare it with the set χ. In § 3 and § 4 we develop some of the properties of the set φ, and prove in particular that every continuous function of bounded variation can be expanded in terms of the φ’s and that every continuous function can be so developed in the sense not of convergence of the series but of summability by the first Cesaro mean. In § 5 it is proved that there exists a continuous function which cannot be

Introduction. In this paper, we study unitary and uniformly bounded representations of the group G= SL(2,K) of two by two matrices of determinant 1 over a locally compact, totally disconnected, non-discrete field K. We obtain results similar to those obtained by Kunze and Stein in [8], [9] for SL(2,R) and SL(n,C) and by the author in [13] for the universal covering group of SL(2,R). The unitary representations with which we are primarily concerned are the principal series and complementary series of irreducible unitary representations of G. We also deal with the so-called "special representation" which has no direct analogue in the classical situation. These representations have recently received the attention of several authors ([2], [3], [4], [11], [12], [16]). We also note the occurrence of the "special representation" in the paper of Ihara [6] (pp. 229-230). In ? 1, we summarize the basic results of harmonic analysis on K which we require for the rest of the paper. Some of these results are well-known and others are taken from [15] where a more detailed treatment may be found. In ? 2, we discuss the principal series of representations of G. These representations are indexed by the unitary characters on K*. After summarizing the standard construction of these representations in L2 (K), we show that the principal series can also be realized as operators in L2(CU), where CT is the analogue of the unit circle in the complex plane. More precisely, if L = K ( Vr) is a ramified quadratic extension of K, then CT is the kernel of the norm from L to K. Here v denotes a generator of the valuation ideal in K. This is similar to the original construction of Bargmann [1] for SL(2,R) which was also utilized by the author in [13]. In this picture, it turns out that, for any character on L* (not necessarily unitary), we obtain a representation of G by bounded operators on L2(CT). The group CT is isomorphic to the subgroup of G defined by

This paper extends to boundary problems the analysis of the powers of an elliptic system of operators given in [6]. The results depend on the analysis of the resolvent given in [7], and this paper is a continuation of that one. We will think of that earlier paper as part I of this one, and not repeat definitions, theorems, notations, and so on here, but simply refer to the earlier paper. For instance, formula 1(29) refers to formula 29 of [7]

Introduction. In previous papers 1 [2, 3, 4], we have developed a method for the study of those properties of a subspace V4l (1 ? n _m 1) imbedded in any enveloping space V. which remain unchanged under conformal transformations of V,m. The pictorial interpretation of the analytic apparatus described in these papers will appear elsewhere. The present paper applies some of these general rsults to a discussion of various classes of surfaces V2 in Euclidean 3-space R3 which have a simple conformal character. The results also apply to conformally equivalent surfaces in any conformally Euclidean 3-space. The reader is referred to [4] for the notation used as well as for a statement of the assumptions concerning the class of the functions which occur in this paper.

1. A definition of a modular form of positive dimension has been given in a paper by Rademacher and the author.2 In that paper we found the Fourier expansions of those forms F(r) which belong to the full modular group and which have only polar singularities at the parabolic point T == i when measured in the uniformizing variable x e2wriT. In the present paper we shall also restrict ourselves to functions which belong to the full group and which have only polar singularities at -r == io, but in the definition of a modular form we shall omit the restriction that F(T) be analytic in the upper half-plane and shall merely assume that F(7-) has, as singularities in the fundamental region,3 at most a finite number of poles and possibly a polar singularity at ioo. This problem of determining expansions of modular forms having poles in the upper half-plane was partially considered by Hardy and Ramanujan.4 However they considered only forms o-t positive inltegral dimension which have no singularities at the parabolic points. The generalization to forms of real positive dimension presents no difficulties. We have only to introduce the roots of uiiity E(a, b, c, d) and e21iaL in the transformation formulas (1. 11) and (1. 12) below, and to carry them through the analysis. However in order to take care of forms having singularities at ioo we have to evaluate certain integrals which Hardy and Ramanujan were able to eliminate by means of simple estimates. This part of the work is contained in sections 3, 4, 5, and 6. In this paper we shall consider most of the integrals in the r-plane rather than in the x-plane, where x e2w1r. The original path owr of section 2 is taken in the i-plane so that we may avoid the poles of Fl(T) which are easier to treat there than in the x-plane. After this point much of the work could

The asymptotic development * for the irregular integrals of a homogeneous linear differential equationa has been obtained by both Horn t and Love. t Horn has published several papers on the case where the roots of the chlaracteristic equation are all distinct, while Love has taken up the case where the roots of the characteristic equation are unrestricted as to their order of multiplicity, including the case of distinct roots as a special case. In this paper we shall consider the non-homnogeneous equation where the roots of the characteristic equation are distinct, and follow, at tlle outset, the method employed by Dini? in his researches on linear differential equations. In Section I the two theorems stated and proved by him will be generalized so as to applv to the non-homogeneous equation and combined in one theorem. In Section II we shall determine a particular solution of the given equation. To this end we shall make use of the researches of Love t in the homnogeneous linear differential equation by employing his solutions. The particular solution thus obtained of the non-homogeneous equation will be in the form of quadratures. The determination of the asymptotic development of the particular integral found in Section II will form the content of Section III.

the theory of complex analytic mappings of complex manifolds and the classical study of value distributions is the study of the "size" of the image of a complex analytic mapping. We give in this paper a treatment, from a purely differential-geometric viewpoint, of complex analytic mappings of a Riemann surface (= one-dimensional complex analytic manifold) into a compact Riemann surface. In the case when the first Riemann surface is a compact one with a finite number of points deleted, we derive defect relations which generalize the classical relations of Nevanlinna-Ahlfors. In a subsequent paper we will consider the case when the first Riemann surface is a compact one with a finite number of points and a finite number of disks deleted. The paper is written for differential geometers, so that concepts currently in use in differential geometry are freely used and a minimum of function theory will be required. The explicit models of the Gaussian plane or the unit disk are avoided.

Q-?) 0ij. This problem has been completely solved in a series of papers [1, 3, 9, 10, 15] when each j if of type I, and partial results have been obtained in the general case [4, 13]. In order that 2 be of type I it is necessary that each Nj is of type I [11], hence that case is completely taken care of in the quoted papers. Similarly, if 2 is finite each 2?j is finite, hence a result of Bures [4, Thm. 4.3] gives necessary and sufficient conditions in order that 2 be finite. If one of the gj is of type III then 2 is of type III [11]. Therefore, what is left is to give necessary and sufficient conditions in order that 2 be semi-finite under the assumption that each ?j is semi-finite. This will be done in the present paper: the result is then quite analogous to that when each 2?j is a type I factor. Our proof differs from that in the type I situation in that we avoid most of the measure theoretic arguments of Moore [9] and some of the manipulations with infinite series by him and Takenouchi [15]. Instead we use some powerful results of Tomita [17] and Takesaki [16] on the modular automorphisms of von Neumann algebras induced by faithful normal states. For our purposes the simplest defintion of modular automorphisms is given by Takesaki [16, Thms. 13. 1 and 13. 2]. Let 2 be a von Neumann algebra and p a faithful normal state of S. p is said to satisfy the KMS boundary condition with respect to a strongly continuous one parameter *-automorphism group at, -oo 0 such that for each pair of operators A, B in 2 there is a function F(z) holomorphic in the strip 0 < Im z < 8 with boundary values, F(t)-1p (gt (A)B) and F(t + iP) p (Bgt (A) ). Let now p be a faithful

Introduction. It is well known that the number p(n) of unrestricted partitions of a positive integer n can be expressed by a convergent series (see [9]). In this paper we shall be concerned with the number p (n; a, M) of partitions of n into positive summands congruent to ? a modulo Mi, where a, M are integers with M ? 2. This partition function p (n; a, M) has been treated for certain special values of M; namely, M 2 by Hua [2], M==5 by Lehner [5], M=6 by Niven [8], and M = p (p a prime > 3) by Livingood [6]; a convergent series representation of p (n; a, M) being obtained in each case. The main object of the present paper is to derive a convergent series for p (n; a, M) in which M assumes general values. We may suppose without loss of generality that 0 1 and d I n, it reduces to p (n/d; a/d, M/d). Further we remark that the case M =2 (partitions into odd summands) is equivalent to the case M = 4, i. e., p (n; 1, 2) -p (n; 1, 4). Consequently it suffices to consider the case M ? 3. We apply the Hardy-Ramanujan method with modifications due to Rademacher [10]. In order to utilize this method, however, it is necessary to solve the following two subsidiary problems: The first problem is to find a suitable transformation equation for the generating function of p (n; a, M). The usual method of contour integration seems to be rather complicated in our case. However, we can show, in a simple way, the existence of the transformation equation, the proof of which being based on a certain functional equation recently obtained by the author [3]. This will be done in Section 1 of this paper. The second is to get a non-trivial estimate for a certain exponential sum which is introduced as a consequence of the transformation equation. We shall

Given any associative ring A we can form, using its operations and its elements, two new rings. These use the elements of A and the addition as defined in A, but new multiplications are introduced to render them rings, albeit not necessarily associative rings. The first of these, the Lie ring AL of A uses a multiplication defined by [a, b] =ab ba for any a, b e A where ab is the ordinary associative product of elements in A. The second of these, the Jordan ring of A, A', has its multiplication defined by ao bab + ba for any pair of elements a, b in A. Being defined in a manner so decidedly dependent on the associative product of A, it is natural to expect that an intimate relationship should exist between the structure of these two new rings and that of A. In this paper we study one phase of this relationship, namely the connection between the ideal structure of A as an associative ring with the ideal structure of AL and A' as Lie and Jordan rings respectively. To be more specific, we investigate how simplicity of A as an associative ring reflects into analogous properties of AL and A,. When we say that U is an ideal of Al, or, equivalently, when we say that U is a Jordan ideal of A, we mean that U is an additive subgroup of A and that for any xeU and any yeA, xoy=xy+yx is an element of U. We similarly define Lie ideals of A and ideals of AL. Although the main results of this paper deal with the case in which A is a simple ring, many of the other results do not require the assumption of simplicity in order to remain valid; so, unless otherwise stated, we make no assumption of simplicity for A.

Introduction. In this paper we shall be concerned with two problems. The first is a problem of Littlewood [1] in classical Fourier analysis concerning a lower bound for the LI norm of certain exponential sums. The second is the problem of determining all the idempotent measures on a locally compact abelian group. This second problem we shall solve entirely and thus complete a line of investigation begun by ilelson [3] and Rudin [5]. The problem of idempotent measures is related to the question of describing all homomorphisms of the algebra l' (G) into the algebra L' (H) where G and H are two locally compact abelian groups. We shall treat this problem in a subsequent paper. As will be explained, the problem of Littlewood is closely connected to the problem of idempotent measures, and though the first is stated on the circle group the method of proof will be indispensable for the analysis on the more general class of abelian groups. We now state the problem of Littlewood. Consider an exponential sum

Let i) and i)' be symmetric domains, which we express in the form = G/K, i)'= G'/K', where G (resp. G') is a connected semi-simple linear Lie group and K (resp. K') is a maximal compact subgroup of G (resp. G'). By a holomorphic imbedding of ) into 2Y, we shall always mean a (not necessarily injective) holomorphic mapping p of i) into i)' which comes from a homomorphism, denoted again by p, of G into G' such that p (K) C K'. Our previous paper [11] was concerned with the determination of all such imbeddings for the case where i)Y is a Siegel space.1 In the present paper we shall study the correspondence under p of the boundary components of 0 and of 2' in their natural compactification. In particular, in the case where G and G' are (the connected components of) algebraic groups defined over Q and p is (the restriction on G of) a rational homomorphism defined over Q, it will be shown that the rational boundary components of i) are mapped into the rational boundary components of V'. Let further r (resp. 1) be an arithmetic subgroup of G (resp. G') such that p(r) C F'. Denote by i)Z* (resp. i)'*) the union of i) (resp. 0Z)') and all the rational boundary components, endowed with a suitable topology. By a recent result of Baily and Borel [1], the quotient spaces 9]* =-r\ * and 9'* =-rF\ '* carry a natural analytic structure, by which they become compact normal analytic spaces imbeddable into a projective space. Our principal aim is to prove the following

Introduction. There has arisen, in recent years, a large literature upon infinite systems of non-linear implicit and differential equations.' The methods wrhich are employed have become classical in the treatment of finite systems. However these methods (employed in the treatment of finite systems) may be extended only to infinite systems of very special type, restricted by inequalities, which for many reasons are not fulfilled in possible applications of the method of infinitely many variables to various classical problems in analysis. Thus the theorems which have been proved by me a few years ago2 are the only ones to my knowledge which have found an application to concrete problems, in particular to the problems of Celestial Mechanics or the Calculus of Variations. I shall give here a comprehensive review of these questions without giving any essential extensions of my results as set forth in my previous papers and without assuming any previous knowledge of the theory of infinitely many variables. The applications will be excluded and the reader referred in this connection to some earlier papers appearing in the Mathemcbtische Annalen and in the Mathematische Zeitschrift. A characteristic application can be found in the paper of Martin appearing in this number of the Journal. The infinite system which I shall treat is composed exclusively of power series (and not of more general functions) of the infinite sequence of variables (as is indeed always the case in concrete applications). My problem was to introduce a method of treatment in which the power series are not subjected to inequalities which would not be fulfilled in concrete applications but which would be necessary in the classical modes of treatment. On page 252 there will be given a number of examples, of most important character, which will serve to illustratewhy the usual methods must fail. Inasmuch as the nature of the problems appears most clearly in its application to the complex space of Hilbert, the present paper will be restricted in its treatment to this space. It is of course possible by using the principles of General Analysis to further extend the methods here set forth. There is no difficulty in using other spaces ' than that of Hilbert. In the articles cited above also spaces other than that of Hilbert are treated; cf. the above mentioned paper of Martin. 241

In the paper [5], we considered harmonic integrals on local product manifolds, that is, manifolds having two families of submanifolds in complementary dimensions, such that locally they look like the product of two euclidean spaces. The metric was assumed to be such that this local product could be taken in the sense of Riemannian manifolds. The results obtained were such as to suggest that analogous theorems could be proved if we assunmed only one family of submanifolds, with a suitable choice of metric. This is indeed the case. We shall show in ? 4 that on compact manifolds, the cohomology of base-like differential forms (defined in ? 2) is isomorphic to the harmonic space of a certain semi-definite Laplacian (defined in ?3). The metric is assumed to be bundle-like in the sense of [6] ; that paper mav be referred to for examples of foliated manifolds possessing such a metric. In ? 5, we discuss the meaning of our harmonic integral theorem for these examples. 1. Definitions. By a manifold I1, we mean a C- differentiable manifold; topologically, it is a connected, orientable, separable, locally euclidean Hausdorff space. We shall assurme given on M (of dimension n) a C- completely integrable q form ?, that is, a locally decomposable, non-zero q form

This note an or dinary differential equations, though it does nlot presuppose the paper on partial differential equations which precedes it in this volume (pp. 525-541), can be thought of as an appendix to that paper, since it contains (among other things) a justification of the choice of the region of analyticity in Theorem 1 loc. cit. Consider the solution w ww(z) of the differential equation

and asymptotic formulas for q(n), the number of partitions of a positive integer n into positive odd summands. The necessary transformation equations as well as estimates of the magnitude of certain sums of roots of unity were obtained by using essentially the same procedures as Lehner [4], while the circle dissection method of Rademacher [6] was employed for the integration. In the present paper we shall impose an additional restriction on the partitions, namely that the summands be distinct. That is, we wish to find a convergent series and asymptotic formulas for Q(n), the number of partitions of a positive integer n into odd and unequal parts. Q (n) also represents the number of self-conjugate partitions of n (see pp. 278-9 in [3]). The investigation is parallel to that in [2], and free use will be made of the results obtained in the earlier paper whenever they are applicable. 2. The transformation equations. Several generating fuLnctions, each convergent in the interior of the unit circle, will be needed. We list these here. 00 00

In a preceding paper * the additional field produced by a system of conductors in an impressed electromagnetic field (EOePt, HWePt) has been represented in terms of surface integrals, which are equivalent to certain integrals of MacDonald t in the applications to be made. In a calculation which was made of the resonance wavelength of a long wire it appeared that the integral representing H could be taken as the basis of a numerical calculation of the current distribution, giving a fair check with experimental results. The usual formulas of alternating current theory correspond, however, to the representation of E, and the problem reduces to the numerical solution of an integro-differential equation. In the present paper the analysis of the problem of a straight wire in an impressed field has been developed and extended to include the theory of the loaded vertical antenna, and the inverted L antenna; the latter is briefly discussed in the special case in which the height is small ,compared to the length of the antenna. The theory is based on the method, well-kniown in the theory of integral equations, of assuming an expansion of the unknown function in a series of mutually orthogonal functions J, (s) which satisfy the boundary conditions at the open ends of the antenna. If the boundary conditions at the surface of the wire are written downi, a set of linear eq-aationis canl be constructed for the unknown Fourier coefficients of the current. It is shown that in the case of the loaded vertical antenna the functions Jn can be so chosen that the principal diagonal terms in these equations dominate the others; if the antenna wire is of small diameter, a good approximation is obtained with a small number of equations. The assumption often made in practice I that a single current component J1 gives an adequate approximation can therefore be partially justified analytically. It is shown that when this is possible, the impedance of the loaded antenna may be defined; from this impedance the fundamental period of the antenna and the radiationi resistance can be calculated.?

In a paper in 1932, W. L. Ayres1 suggests the question: If K is any finite subset of a Peano space M which is a subset of the Euclidean plane, and if no two poinits of K may be separated 2 by the omission of any two points of IMl, then does there exist a simple closed curve J such that M D J D K? Ayres answers the question in the affirmative in the same paper for K = 3 points, and in later unpublished work he discovered an example which answers the question in the negative for K ? 11 points. In this paper we prove that the answer to Ayres' question is in the affirmative for K ? 5 points and give examples to show that it is in the negative for K > 5 points. We also show that the same results are obtained provided the condition " no two points of K may be separated " is changed to read " no two points of M may be separated." Examples are constructed to show that in certain Peano spaces a finite set K can not be joined by im (where m is any given integer) mutually exclusive simple closed curves. The writer wishes to express his thanks to Professor W. L. Ayres for his helpful criticisms and generous assistance in the preparation of this paper.

1. In a joint paper with A. C. Schaeffer1 we discussed the following question: Let D be a closed domaina in the complex z-plane and z0 a fixed pointt of D. Let its consider all polynomtials f(z) of givez degree n forwhich f Jf(z) ? 1 in D and f(z0) is real. What is the largest possible value of !f (z) if f(z) is an arbitrary polynomial of the hinzd mentioned and z is arbitrary in D? Under certain conditions on D and zo the inequality I sf(z)! 0 clepends only oi 1D and zo. This theorem was well-known in the special case when D is a circle anld zO is ani in-terior point of D. fHowever Mr. P. Erd6s has called my attention to the fact that the best possible bound of j 3:tf(z) j is nlot known in this case. In the present paper I wish to determine the best possible bounid in case D is a circle ancd zo is the center of D. I prove the following