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By using arithmetic probability paper a normal distribution can be transformed into a straight line by plotting cumulative frequencies, expressed as percentages of the total frequency as the ordinate, against the mid-point of the class intervals as the abscissa. From such a line both the mean and standard deviation can be obtained. The mean is the point on the abscissa corresponding to the position where the line intersects the 5000 ordinate and the standard deviation is the distance along the abscissa lying between the mean and a point corresponding to either 15.870% or 84 13% on the ordinate. Conversely, if the mean and standard deviation are known, a distribution can be represented as a straight line. Recently, from an investigation into the movements of coarse fish (Stott 1961, 1967), data were obtained on the frequency with which recaptures were made and the distance they had moved from a release point which, when plotted in the above manner on arithmetic probability paper, produced sigmoid curves similar to that labelled AF in Fig. 1. The data for Fig. 1 are given in Table 1, and for the purpose of illustration they are hypothetical. Since it can be demonstrated that this type of curve results if two straight lines, such as XX' and YY' in Fig. 1, are compounded, it seemed reasonable to interpret the data from the investigation as being the resultant of two superimposed normal distributions of differing dispersions. This led to the postulate that the fish populations studied were divisible into two components on the basis of their observed mobility and indeed subsequent laboratory work confirmed this hypothesis (Stott 1967). In order to estimate the proportion and parameters of the mobile and static sub-populations observed in the field, a search was made for a technique to separate these components. Methods of separating polymodal frequency distributions have been described by Harding (1949), Cassie (1954) and Tanaka (1962), the two former using arithmetic probability paper. However, their solutions seem only to be of use if the amount of overlap between distributions is small and a similar limitation applies to the cases referred to by Cox (1966). It appeared, therefore, that with superimposed distributions the only approach possible would be one of trial and error based on the fact that the less dispersed normal distribution (XX' in Fig. 1) does not overlap the other one to a significant degree at the tails of the resultant sigmoid. By trial and error the shallower line (YY' in Fig. 1) could be obtained by progressively reducing the values of the frequencies between the points where it was obvious that the two distributions were 'superimposed (i.e. between B and E on line AF in Fig. 1) until the data from this common portion of AF plus that from the two tails, AB and EF would form a straight line when

to show that certain estimates by Nehari are sharp. The range of the complex parameter a for which these functions are schlicht in the unit disk can be treated as an example of a one-dimensional subspace in the Teichmfiller space of an annulus, and as such can be used to illustrate some of the ideas developed by Bers in [1]. In these remarks we carry out the details of this point of view. If, to fit into the framework, the setting is translated from the unit disk to the lower half-plane L, the corresponding family of mappings arise as solutions to the schwarzian differential equation { W, z} = ( for ( in the subspace spanned by the element l/z2 in B2(L, r), r a cyclic fuchsian group generated by a loxodromic transformation. The intersection of this subspace with the Teichmiiller space T(r) can be described explicity, and we write out formulas for the various mappings and group homomorphisms associated to each element 9 in or on the boundary of T(r). These elements are then interpreted, both qualitatively and quantitatively, as deformations of an annulus, and we point out a natural correspondence between this deformation space and the Teichmiiller space of a torus or a punctured torus. Unfortunately the correspondence with the space of a punctured torus, in the form that we would like, is not explicit enough, and we have left it at equation (11). A solution of this equation, explicit enough to answer the question posed there, would be of interest in that it would lead to the first non-trivial example of a Teichmfiller space in the setting of the preceding paper. The example does illustrate that a cusp on the boundary can be reached by "twisting" without "pinching," showing along the way that the expression in (8), shown by Bers in [1, ? 6] to have a universal upper bound, does not have a positive lower bound. Although the example itself is trivial, the deformations involved have analogues for general Riemann surfaces, where we might expect the qualitative behavior to be the same. The "twisting" and "pinching" of the annulus, for example, can be carried out near a boundary curve of any open surface; while the corresponding deformation of the torus

THE purpose of the following study is twofold: to examine the nature of the cycles in interest rates and to criticize the application of the periodogram method to economic statistics. As for the particular series under investigation, it may be said at once that no uniform periodicity is revealed. If we confine the analysis to those portions of the entire time interval which are fairly free of extreme deviations, we discover however a cyclical movement which conforms moderately well to the average found by the periodogram. This average cycle has a length of about 40 months, and its form may be described as that of a sine curve distorted so that the time interval from low to high considerably exceeds that from high to low. It is found that, because of the presence of a marked seasonal variation and of irregular extreme deviations in the economic statistical record, the blind application of the periodogram method is likely to yield results which are illusory. Even after the results are checked by the usual tests available for the criticism of the findings of a periodogram analysis, a thorough examination should be made of the actual cycles found by eliminating the seasonal variation from the original data. It is probable that the summation process involved in the periodogram method will be of considerable value in studies of the length and form of the cycles in those portions of a statistical record which are clearly free of irregular extremes, but it is very doubtful whether the attempt to analyze the form of the cycle by the Fourier method will assist in describing or understanding the cycle. The average form of the cycle is at best not very precisely determined, and its breaking up into a group of harmonic components of differing periods and amplitudes is certainly bewildering and probably furnishes no true insight into the nature of economic fluctuation.

This paper sets out a general criterion for the identifiability of a statistical system, based on Kullback's information integral. It is shown that the general identification problem is equivalent to a maximisation problem, or where parameter restrictions are present, a problem in nonlinear programming. The relationship of this criterion to that based on the information matrix of the underlying distribution is also exhibited. THE COLLECTION OF results on the identification problem in econometrics is by now assuming the proportions of an imposing edifice. It is, however, a little surprising to note that this structure has been growing upwards and, more recently, outwards, without a corresponding strengthening in the foundations. It is true that in the case of work on linear simultaneous equation systems (and this, with the work of Koopmans and Reiersol [3] and Chapters 1-4 of Fisher [1] in particular, has almost assumed the status of a "classical" line of development), these results are founded on a pretty secure rock; to wit, the identifiability of the reduced form in the absence of any singularities in the data matrices. Nevertheless, with the development of other wings on our edifice, it seems desirable to look to more basic things. The recent paper by Thomas Rothenberg [5] provided a welcome attack on this subject. The identifiability of a parametric system is approached via the nonsingularity of R. A. Fisher's "information matrix" evaluated at the true value of the parameter. The present note generalizes this approach by providing a simple criterion for identifiability, which not only affords an approach to global identification, but also makes no assumptions about the regularity of the underlying distribution. Rothenberg's basic theorem emerges as a simple corollary to this result. The approach has a natural relationship with estimation theory and also provides a straightforward method for delineating the subspace of observationally equivalent parameters in the case of underidentification.

THIS PAPER analyzes parametric variation for a general class of constrained static optimization problems. The problem treated is that of maximizing a real-valued objective function F(x, a) which depends on an n-dimensional vector of choice variables x and a q-dimensional vector of parameters a, subject to a constraint which requires that a real-valued constraint function g(x, a), dependent on the n variables and q parameters, equal a given value b. Section 1 establishes the existence of a generalized Slutsky equation everywhere on a subset of n + q dimensional Euclidean space except on a set of Lebesgue measure zero that is, almost everywhere (denoted a.e.) under concavity assumptions on the objective and constraint functions and nonsingularity of the relevant bordered Hessian. This is a new result; previous papers have treated only special cases. In Section 2 the restriction that the generalized matrix of substitution effects be symmetric is imposed, and the most general form of the objective and constraint functions satisfying this requirement is exhibited (linearity of the constraint function is not needed). It is further shown that the generalized matrix of substitution effects exhibits negative semidefiniteness in this case. Section 3 applies these results to the case of a consumer for whom prices enter the utility function. Section 4 applies these results to the Baumol producer maximizing sales subject to a profit constraint. The existence and basic properties (symmetry and negative semidefiniteness) of the generalized Slutsky equation a.e., even when a vector of parameters enters the objective function and the applications of these results to consumer theory and producer theory are the major results of this paper. The classic treatment of comparative statics and its applications in economics is that of Samuelson.2 Samuelson however did not develop the full properties of the generalized Slutsky system developed below. The problem treated in Section 3 has been treated by Kalman.3 His earlier results are extended in this paper. Finally we conjecture that the above properities of the generalized Slutsky system can be derived even with k nonlinear

Given a nonnegative Cl-function p(x) on a Riemannian manifold R, denote by BO(R) the Banach space of all bounded C2-solutions of Au = pu with the sup-norm. The purpose of this paper is to give a unified treatment of BO(R) on the Wiener compactification for all densities p(x). This approach not only generalizes classical results in the harmonic case (p 0), but it also enables one, for example, to easily compare the Banach space structure of the spaces BO(R) for various densities p(x). Typically, let ,8(p) be the set of all p-potential nondensity points in the Wiener harmonic boundary A, and Cp(A) the space of bounded continuous functions f on A with ft A 8(p) 0. Theorem. The spaces Bp(R) and Cp(A) are isometrically isomorphic with respect to the sup-norm. Throughout this paper R is an orientable Riemannian C?-manifold of dim > 2, and p(x) is a nonnegative Cl-function on R. Denote by Bp(R) the space of bounded C -solutions u on R of the elliptic equation Au = pu, where Au is the Laplacian of u on R. As one studies bounded harmonic functions on the Wiener compactification, the space B p(R) has been investigated on the so-called Wiener p-compactification (cf. Loeb and Walsh [2], Wang [91). However, their consideration restricts one to construct different compactifications for different densities p(x). The purpose of the present paper is to give a unified treatment of the spaces B (R) on the Wiener compactification R * for all densities p(x). p This approach, for instance, enables one to easily compare the linear space structure of the spaces B p(R) for various densities p(x). Typically, let ,8(p) be the set of p-potential nondensity points x in the Wiener harmonic boundary A (see below for its definition), and C p(A) the space of bounded Received by the editors May 31, 1973 and, in revised form, October 24, 1973. AMS (MOS) subject classifications (1970). Primary 30A48.

This paper deals with the Fourier expansion of the functiont n9O (,) '?-nl (,) 421l2 (Z) 4-t13 (z), hereafter denoted by [n0, n1, n2, n8, al, for all integral values of the n's satisfying no + n1 + n2 +n3 = s, It > 0. The problem has been treated previously in two ways. Among others, Biehler, Hermite, Appell and Bell have obtained expansions for specific values of the n's. Appell and Krause have developed general theory for the expansion of doubly periodic functions of the third kind in trigonometric series, but, as far as is known, this theory has been actually applied only to special functions with poles of low orders. Appell's method is here applied to the more general function given above. We find explicit expressions in the general cases, depending on the parities of no, n1, n2, and ng. 1. Appell's method. In a series of papers,+ Appell has discussed the expansion of doubly periodic functions of the third kind into trigonometric series. The following result, taken from these papers, forms the basic theory used. Let n=--oo AP (Z, y) = e2,Aniy qfinf(n-1) cot(Z -yn t), q | qe K 1< n==-00

Introduction. In two previous notes [9],1 I stated some results which, in a general way, may be expressed as follows: If the Taylor series which represents an entire function satisfies a certain gap condition (which depends only on the order of F(z)), the zeros of F(z) -f(z) are not exceptional with respect to the proximate order of F(z) by any meromorphic function f(z) 3 oX of lower order. On the other hand, Mandelbrojt (see for instance [3, Theorem XXV]) proves that an entire function represented by a Dirichlet series takes each value a 0 oc, except at most one, in any horizontal strip of width greater than a quantity which depends only on the order (R) and on the upper density of the sequence of exponents of the series. In the present note we shall prove some results closely related to those of my above mentioned notes and to that of Mandelbrojt just quoted. These results may briefly be stated as follows: If an entire function F(s) can be represented by a Dirichlet series satisfying certain gap conditions, the zeros of F(s) -a cannot be exceptional with respect to the proximate order (R) of F(s) in any strip of width greater than a quantity, determined by the order, for every value of a $ oX without exception. In ?1 we shall deal with functions of finite order (R); in ?2, we shall give results concerning functions of infinite order (R). I think it will be of interest to remark that the theorems of the type of the well known Hadamard's theorem (which asserts that a Taylor series satisfying a specific gap condition cannot be continued analytically outside its circle of convergence) together with the results concerning relationship between gap properties and the position of the Julia lines, and again together with the results of my above mentioned notes and those contained in one of my papers [10], enable us to state the following general principle (without pretending that it holds for every case). By means of gap conditions in the Taylor series the disappearance of the possibility of the existence of the exceptional cases can be affirmed. Therefore, the content of this paper may be regarded as a link

This is the third of a series of papers in these Transactions on hierarchies obtained by quantifying variables of recursive predicates. In Recursive predicates and quantifiers [10 ] variables for natural numbers (type 0) were quantified, and in Arithmetical predicates and function quantifiers [14] also variables for one-place number-theoretic functions (type 1). In the present paper we shall quantifyalsovariables for functions (i.e. functionals) of higher finite types (types 2, 3, 4, * * * ). A theory of recursive functions and predicates of variables of these types has not previously been developed. For the hierarchy results in the form that new predicates are definable by increasing the highest type of variable quantified('), or the number of alternations of the quantifiers of the highest type, it would suffice to extend the notion of primitive recursiveness to the higher types of variables. This is how we began the investigation in 1952(2). However there are situations where the question "What becomes of this theory for higher types?" calls for an extension of general and partial recursiveness. For example, before we can extend Post's notion of degree of unsolvability [25; 19], which proved fruitful in the study of hierarchies of number-theoretic predicates [5; 14; 16; 26], to predicates with type-1 variables, we must have a notion of relative general recursiveness for such predicates, which when uniform amounts to having general recursiveness for functions with type-2 variables. Accordingly we now give an extension of general and partial recursiveness. The treatment entails incidentally a somewhat new treatment of general and partial recursive functions of variables of types 0, 1. As will appear, many of the known results for types 0, 1 extend to the higher types, but not all. In this Part I we leave undiscussed various aspects of the subject on which work is in progress or completed which we hope to report in a Part II.

This paper takes off where the paper by Katti [1966] ended. In the 1966 paper, Katti used a criterion for flexibility and showed that there is a distribution called the log-zeroPoisson distribution (l.z.P.) which has more flexibility than each of the other distributions studied in that paper, including the negative binomial, the Neyman type A, and the Poisson binomial distributions. In this paper, the l.z.P. is fitted to the 35 sets of data given in Martin and Katti [1965] and the fits are compared with the fits given by the other well-known distributions. Some basic properties of the distribution are also discussed.

The developments of the present paper center around the observation that the ring B of bounded everywhere defined operators in Hilbert space contains non-trivial two-sided ideals.' This fact, which has escaped all but oblique notice in the development of the theory of operators, is of course fundamental from the point of view of algebra and at the same time differentiates 93 sharply from the ring of all linear operators over a unitary space with finite dimension number. As examples of two-sided ideals in 93 we may mention here the class of all operators A such that J(A), the range of A, has a finite dimension number, the class of all operators of Hilbert-Schmidt type,2 and the class fT of all totally continuous operators. Except for the ideal (0), every two-sided ideal in 93 contains the first ideal mentioned, and except for the ideal J3 itself, every twosided ideal in 93 is contained in the ideal f. Moreover, on the basis of the special spectral properties of the self-adjoint members of sJ, it is possible to characterize every two-sided ideal in ?B very simply in terms of the spectra of its nonnegative self-adjoint elements; for both the formulation and the proof of this result, which together with the facts mentioned above is discussed in ?1, the author is indebted to J. v. Neumann. The restriction of our attention to those ideals in if which are two-sided is basic for the points which we wish to develop; the two-sidedness compensates for the absence of commutativity in 93 in such a way as to permit the construction of quotient rings by the standard methods of abstract algebra.3 These rings, which are of course homomorphs of B with respect to addition and multiplication, are also homomorphs of. @ with respect to the operation *, and exhibit all of the formal properties of matrix algebras. This is established in ?2, and there also various properties of the associated congruences in gB are discussed. The remainder of the paper deals solely with the quotient ring @/{, where fJ is the ideal of totally continuous operators, and the associated congruence in i3. For essentially topological reasons, this is the only one of the quotient rings in

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 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

This paper considers a procedure, based on Anderson [1964], wherein the terminal decision of a sequential probability ratio test (SPRT) may be subsequently altered by a likelihood ratio procedure incorporating both the sequential observations and m additional observations. The procedure in this paper is concerned with sequential tests comparing two binomial parameters, and is motivated in the context of matched pair clinical trials. Three cases are considered: m is a known number, m has a binomial distribution, and m has a Poisson distribution. A computational formula is derived to evaluate the probabilities of the two types of error, and a set of charts is obtained to illustrate the application of the procedure.

(1.4) Q(n) ns r2s(n) __ (-))! (e) nO( s -> 3, M It=1 ~~(s-i!m where e is any positive integer and se (s) is 1, 0 or 2s according as Q is odd, e 2 mod 4 or e = 0 mod 4. For --1 this says that the function nl-sr28(n) is in the mean (on the average) equal to ITS/(s 1)! if s > 3. For other values of e the formula implies a weighted asymptotic average of the same function, the weight factors being re(n) for varying n. These are therefore suitable functions to smooth out on the average the irregularities of the function n1ls r2S(n). What is proved is that the irregularity is smoothed out in the limit as mn becomes infinite; some empirical evidence (not recorded in the paper) indicates that the smoothing effect probably is generally in evidence for rather small values of m. The paper also contains results similar to (1.4) concerning representations of integers as sums of an even number of triangular numbers and also

It is the purpose of this paper to demonstrate three theorems and two corollaries, whoseformal statement will be preceded by a brief explanation. The treatment is suggested bv recent papers of LEsEsGuE,t in which corresponding problems are discussed for the case of trigonometric series. A part of the work of LEBESGUE is reproduced at length in my thesis,: and where the developments of the present paper are closely parallel to those in the trigonometric case I shall occasionally refer the reader to one of those other presentations.

Carl Friedrich Gauss (1777-1855) is generally ranked with Archimedes and Newton as one of the three greatest mathematicians that ever lived. His work, in terms of its all-pervasive importance, its painstaking attention to detail, and its completely developed beauty, somehow reminds one of the work of Beethoven, his contemporary and compatriot. Gauss was the last of the truly universal mathematicians and scientists, whose realm embraced virtually all the domains of pure and applied mathematics, and astronomy, and theoretical and experimental mechanics, hydrostatics, electrostatics, magnetism, optics.... "Gaussian" as a modifier has been applied to a remarkable assortment of mathematical terms, and "gauss" is the universal unit for the intensity of magnetic force.Tord Hall's biography concisely presents the outer events of Gauss's life, but the emphasis is, as it should be, on the inner core of that life--the mathematical creations. These are unfolded in such a way as to be clearly understandable to readers of modest mathematical attainment, but more than that, such readers are given a proper sense of the intellectual excitement and aesthetic completeness of Gauss's achievement.Gauss's external life was fairly uneventful and conventional. His solid, conservative, burgherlike exterior served to mask and protect an incredibly fecund mental flux, as evidenced by a fifty-year period of unflagging creative output. During this time he was Director of the Astronomical Observatory in Gottingen and as such rarely gave formal lectures on purely mathematical subjects. In addition, because of his Olympian reserve and standoffishness from his colleagues, he was disinclined to present his discoveries informally; he chose instead to reveal them only when they were embodied in the form of perfectly developed papers, which can be regarded as integral works of art, finished and unutterable.This disposition prevented Gauss from adding still more illustrious discoveries to his credit: for besides those presented in the large body of his formal papers, a study of his journal and other posthumous papers reveals that he had achieved (but did not publish, because he had not developed the work up to his high standards of completeness and rigor) some of the most important results later obtained by Abel, Cauchy, Jacobi, and others. Gauss combined an acute sense of priority of discovery with distaste for public controversy, and made his claims discretely, in personal letters. Hall's account makes full use of these letters and the journal entries.In describing Gauss's work, the author carefully describes the problems Gauss set for himself, and the solutions he uncovered. Hall also outlines the mathematical approach or "style" of the proofs--the lines joining the problems and the solutions--when these are too involved to be presented in full form. The topics so discussed include (among others) the fundamental theorem of algebra, the 17-gon, geodesic triangles and Gaussian curvature, non-Euclidean geometry, elliptic functions, arithmetic residues, and the determination of Ceres' orbit, and the first workable telegraph, constructed in collaboration with Wilhelm Weber.

In the first part of this paper a special class of differential ideals(') is investigated. The results of this section are used in the following one to derive some structural properties of differential polynomials. The last part of the paper is devoted to a special differential ideal. With the help of some conventions of notation, more precise indications of the scope of our work may be given. Let IR. denote the ring of differential polynomials, with rational numbers for coefficients, in the unknown y. The special class of differential ideals studied in Part I is composed of those generated by yP, where p is a positive integer. These ideals are among the most simple ideals encountered in the theory of differential equations. Viewed as algebraic entities, however, they are by no means trivial. We denote the ith derivative of y by yi; R. thus appears as a polynomial ring with infinitely many indeterminates y, Yl, Y2, * . Since the Hilbert basis theorem does not hold on 'A, one would expect almost any ideal in R. to be unruly. By introducing order relations into fR we have been able to proceed despite the absence of the basis theorem and to obtain fairly comprehensive results concerning these differential ideals. In particular a simple criterion for determining the membership in such an ideal of an element of R, is obtained which plays a fundamental role in Part II. This second part establishes the abstract counterparts of some results of J. F. Ritt concerning essential manifolds which figure in the decomposition of a manifold into irreducible ones. It has been found possible to present results which cover situations not discussed by him. The differential ideal discussed in Part III is that generated by uv, where u and v are unknowns. Among other properties, it is shown that this ideal has no representation as the intersection or product of two differential ideals, whose manifolds are respectively u = 0 with v arbitrary, and v = 0 with u arbitrary. This result owes its interest to the fact that the manifold of the equation uv=O is evidently reducible into the union of the two manifolds just defined. In a narrow sense, this paper is independent of other literature; the argu

The subject of the present paper is closely related to the first paper of this series, [8], henceforth to be quoted as I. Our purpose is to generalize Lyapunov's second method to the case of conditional stability, simple or asymptotic. We shall show that the existence of (suitably generalized) Lyapunov functions with certain properties is essentially equivalent to the existence of bounded solutions of the inhomogeneous equation and also to certain kinds of behavior of the solutions of the homogeneous equation (equivalence of properties (Q1), (Q3), (Q') in the Introduction of I). Our results are closely related to recent theorems of Maizel' [6] and Krasovskii [5]. It does not seem possible, however, to attain the same degree of generality and neatness as in I, especially if the results are not restricted to the finite-dimensional case. Except for some functional-analytic lemmas in Section 2, the methods employed are essentially classical. The notation and results of I are freely used throughout, reference being made to them as, e.g., Theorem I. 5.3, formula I(2.3). Section 2 of this paper gives some additional information concerning the differential equation we are investigating and provides the definitions and properties of the generalized Lyapunov functions we propose to use. In Sections 3 and 4 we discuss Lyapunov's second method and converse results in the case of asymptotic and simple conditional stability, respectively.

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

The Pontrjagin Duality Theorem, known to be true for locally compact groups, asserts that the given group and the character group of its character group are isomorphic under a "natural" mapping. So-called "reflexive" linear topological spaces (l.t.s.) are equivalent to their second conjugate spaces, again under a "natural" mapping. This raises two questions: first, for what class of topological groups does the Pontrjagin theorem hold? Second, what relation, if any, is there between the two kinds of theorem when considered in linear spaces? The present paper answers the first question in part,. bly showing that in an extensive class of infinite-dimensional (and thus not locally compact) topological linear spaces the Pontrjagin theorem is valid, and the second by showing that in reflexive linear topological spaces the group-duality theorem is also true and, in fact, merely restates the reflexive property. The key observation which makes our results possible is very simple: if f is a linear functional on a real l.t.s. then the function x defined by: x(x) = exp (if(x)) for each x in the space, is a character. The author's proof of the main theorem as presented to the Society (abstract 209t, Dec. '48) leaned heavily on Arens' paper [1].' It has since been possible to make this investigation more self-contained, and in so doing, to generalize its principal result from holding for real reflexive Banach spaces to arbitrary locally convex real l.t.s. which are either reflexive (though not necessarily either complete or normable) or Banach spaces (though not necessarily reflexive). This has served also to settle the converse problem: since completeness and normability suffice to make the Pontrjagin theorem valid it becomes clear that reflexivity is not a necessary condition for, and so certainly not identical with, the group-duality theorem, though in the special case of reflexive spaces the two sorts of duality coincide. It should be pointed out that despite the fact that Arens' results are not too obviously or heavily leaned upon in this paper in its present form, the ideas in [1] supplied the clue and motive, as well as occasionally the method for the discovery of these theorems. The author is indebted also to J. L. Kelley and to Irving Kaplansky for helpful discussions, and to the latter especially for the essential ideas in the proof of Lemma 1.

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.

This paper deals with some applications of a general theory for the analysis of factorial experiments as reported by the authors in the June 1962 issue of the Annals of Mathematical Statistics. General expressions are given for the usual quantities associated with the analysis of variance for the cases where simple treatments or factorial treatment-combinations are applied to Randomized Blocks, Balanced Incomplete Blocks, Group Divisible designs, and a wide class of Kronecker Product designs. The main point of the new theory is that, for a wide class of the more practical designs, the complete analysis can be carried out almost by inspection of the normal equations, with no requirement for inverting the normal equations. The complete version of this paper is published in BIOMETRIKA, Vol. 50, Parts 1 and 2, June 1963.

fff(x, y, z, X, Y, Z) du dv, where X, Y, Z are the three jacobians of x, y, z with respect to u, v. In the present paper a similar investigation is conducted for a wider class of surfaces, including for example surfaces for which the derivatives x", etc., are summable together with their squares, and also surfaces z = z(x, y) which are absolutely continuous in the sense of Tonelli. Along with theorems concerning semi-continuity, we find that for our class of surfaces the Lebesgue area is given by the classical double integral. The consideration of this wider class of surfaces is desirable not merely for the sake of generalization, but also because of the fact that the important class of saddle surfaces bounded by a Jordan curve and of finite area is not included in the class of rectifiable surfaces, but is included (as will be shown in a later paper) in the class of surfaces for which x2, etc., are summable. Since this paper is intended as a sequel to the preceding one' on double integrals, we retain the notation and definitions of that paper, and request the reader to read the first portion of it as an introduction. The first part of the present paper is concerned with the definition of the class of surfaces considered and with the statement of the problem. In the second part we obtain some lemmas on approximation by polyhedra which enable us to reduce the problem to a simpler one. The third part contains the statement of the principal theorems of the paper. The fourth part is devoted in part to the presentation of certain sets of conditions on the functions x, y, z which are sufficient to assure us that the surface x = x(u, v), y = y(u, v), z =z(u, v) belongs to our class of surfaces, and in part to the indication of certain fragmentary extensions of our theorems.

T HE most frequently used estimating technique for applied economic research has been ordinary least squares (OLS). There are two theoretical justifications for its use. First, the Gauss-Markov theorem suggests that OLS estimators will outperform all other techniques in the class of linear unbiased estimators.1 Secondly, under the assumption that the error terms are normally distributed, OLS estimators can be derived as the maximum likelihood estimates. Frequently when OLS is introduced in elementary texts, the method of minimizing the sum of absolute deviations (MAD) is presented for comparison purposes, but rarely is it given serious consideration for applied uses.) Certainly one reason for such behavior stems from previous evaluations of OLS versus MAD estimators. Asher and Wallace (1963) found that the use of MAD meant " one should be prepared to give up considerable efficiency . " 3 More recently Glahe and Hunt (1970), suggested several estimators derived under the general minimization criterion. In contrast to the Asher-Wallace study, the Glahe-Hunt model was a two-equation linear simultaneous system. Their results show that neither form of absolute deviation estimator outperformed either OLS or two-stage least squares.4 The purpose of this paper is to suggest that in at least one aspect these comparisons have given OLS a differential advantage. That is, both of these studies employed errors for their hypothesized models which were drawn from normal distributions. Consequently the OLS estimators are both maximum likelihood and best linear unbiased estimators (BLUE) under such circumstances. Since recently published works by Zeckhauser and Thompson (1970), Fama (1965) and others have called into question the assumption of normally distributed error terms, attention has begun to shift to other alternatives.5 Blattberg and Sargent (1971) have examined three techniques including both OLS and MAD when the errors are drawn from a stable Paretian distribution. Their findings indicate that the MAD estimator ". . . performs sufficiently well that it deserves further study and elaboration." G Accordingly we have chosen to explore the relative merits of OLS and MAD for a single equation model whose errors are drawn from a double exponential parent distribution. This distribution was chosen because both estimators will exhibit theoretically desirable properties. OLS remains the BLUE estimator, while MAD is the maximum likelihood estimator. Furthermore, this distribution is one member of the power distribution suggested by Zeckhauser and Thompson as an alternative to the normal. This paper is divided into, three sections. The first describes the design of the experiments. Section II presents the empirical results and the last summarizes the primary findings of the paper.

The purpose of the present investigation is to extend the scope of the important and comprehensive results of McShane on the semi-continuity of double integrals in parametric form(1). The class of surfaces considered by McShane is very general, but we shall see that the conditions placed by him upon his surfaces can be greatly relaxed without hurting the validity of any one of his theorems. Our condition (see 1.19) is however not only less restrictive but also possesses a certain degree of finality. Indeed, we shall be able to prove that the class of surfaces for which our condition holds is precisely the class of surfaces which admit of a representation where the Lebesgue area of the surface is given by the usual integral formula (see 3.15, 3.19). The methods used in this paper are based on previous work by McShane in the calculus of variations and by the author on the area of surfaces. Since this work is scattered in a number of papers in various periodicals, it seemed advisable to attempt at this time a somewhat self-contained and systematic presentation, as well as to carry out various simplifications of detail suggested by a comparative study of the literature. In particular, the theory of the Lebesgue area of surfaces will not be presupposed. On the contrary, we shall find that the most advanced results of that theory are simple corollaries of ouir results on general double integrals.

This paper contains a proof that either all, or none, of the nontrivial solutions of the fourth order linear selfadjoint differential equation have an infinite number of zeros on a half line, provided that no nontrivial solution has more than one double zero on that half line. Throughout this paper, let Ly = (ry")" -(qy')'+py where r, q, and p are given real-valued functions, a?(o, oo) is given, r", q', pCC[a, oo), and r(t)>O for t_?a. A nontrivial solution to Ly=O is said to oscillate if its zeros in [a, oo) are unbounded. THEOREM 1. If no nontrivial solution to Ly = 0 has more than one double zero in [a, oo), then all the nontrivial solutions oscillate or none oscillate. W. Leighton and Z. Nehari [1, p. 367 ] obtain the same conclusion using the hypothesis that q(t) 0, p(t) > 0 for t > a. As they note, these assumptions imply that no nontrivial solution has more than one double zero. Some lemmas will be established before proving Theorem 1. Ly = 0 will be said to be 2-2 disconj ugate if no nontrivial solution has more than one double zero in [a, oo). Of course, this is a special case of the concept known as n-n disconjugacy. When r, q, and p are all constants, it can easily be shown that Ly =0 is 2-2 disconjugate if and only if rw4+qw2+p > 0 for all real numbers, w. For i = 1, 2, 3 and a < b < oo, let ybi designate the solution to Ly = 0, y(i)(b) = bij, j = 0, 1, 2, 3, where 6ij is the Kronecker delta. Denote the zeros of Yb3 by ,[ . . < -q(b .1 < nb 1)

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

Introduction. The idea of an integral has been extended by Radon, Young, Riesz* and others so as to include integration with respect to a function of bounded variation. These theories are based on the fundamental properties of sets of points in a space of a finite number of dimensions. In this paper a theory is developed which is independent of the nature of the elements. They may be points in a space of a denumerable number of dimensions or curves in general or classes of events so far as the theory is concerned. It follows that, although many of the proofs given are mere translations into other language of methods already classical (particularly those due to Young), here and there, where previous proofs rested on the theory of sets of points, new methods have been devised (see, for example, theorems 3(3), 3(4), 5(1)). Mooret has developed a theory of a similarly general nature, but restricts himself to the use of relatively uniform sequences. This concept is not used nor is it necessary in the following paper. We consider a group of elements p, which may be whatever we choose, and certain classes of functions f(p) of those elements so that to every element p of the group there exists a real number f(p) (which may be infinite in certain cases). To each function of a certain class there corresponds a real number S(f ) or I(f ) which is defined so as to satisfy certain conditions. S(f ) is a generalized Stieltjes integral, while I(f ) may be called the positive integral and the latter possesses correlates of nearly all the properties of the Lebesgue integral. It is shown that any I-integral is an S-integral while any S-integral is expressible as the difference of two I-integrals. Two symbols have been taken over from symbolic logic, namely those for logical sum and logical product. The concepts involved are used extensively by Young and by the author and the symbols have been introduced to save space and to clarify the reasoning. The reader is referred also to Hildebrandtj for references to an exten

The problem of Plateau or the proof of the existence of a surface of least area bounded by a given Jordan contour in Euclidean space is one of the classical problems of the Calculus of Variations. This problem has been studied at least since the time of Riemann but it was not until 1930 that satisfactory proofs for a general contour were given independently by Douglas [25]' and Rado [15]. However, before that time, considerable information was gathered and the problem was solved in many interesting special cases. For the literature of this period, we refer to the excellent report of Rad6 [38] on the problem of Plateau. Shortly after the appearance of the solutions of Douglas and Rado, McShane presented an interesting solution which was perhaps more along the lines of the straightforward "direct methods of the Calculus of Variations" (see [30]). In the past seventeen years the problem has been generalized by replacing r by several contours r1, * * *, rk and by prescribing the topological structure of the desired surface ([5], [19], [27], [28], 129], [33], 135], [41], [43]), or by replacing the condition that the surface be bounded by a given contour by the condition that only parts of the boundary be prescribed, the remainder of the boundary being merely restricted to lie along certain manifolds (see [18], [22], [23], [24]). Finally minimal surfaces which do not furnish even relative minima have been discussed (see [20], [21], [31], [32], [33], [34], [35], [39], [40], [42], [43]). The present paper generalizes the problem by replacing the underlying Euclidean space by a Riemannian manifold of considerable generality. The writer knows of only one paper in which the Plateau problem has been solved in any space other than Euclidean space, namely the recent paper by A. Lonseth [7] where the space considered is "hyperbolic space". However, S. Bochner [2] has studied the "harmonic" functions in a general Riemannian metric (of sufficient differentiability) but has not demonstrated their minimizing property in general. This work is, of course of great interest as it is still true in a Riemannian space (of sufficient differentiability) that if a minimal surface is mapped conformally on a plane region, then the representing vector is "harmonic" in Bochner's sense and also that a surface of least area (if of sufficient differentiability) is a minimal surface (in the sense of Differential Geometry). The present writer does not attempt to generalize all of the preceding results on the Plateau problem but restricts himself to the case of a surface of the type of a k-fold connected plane region bounded by k given contours. The space is a "homogeneously regular" (see Def. 3.1) Riemannian manifold 9) of class C'. Every compact, regular Riemannian manifold of class C' is seen to be homo

where j1, * *, j. is a permutation of 1, * * *, n. He arrives at the conclusion that the number of arbitrary constants in the solution of (1) does not exceed the greates sum (2). Now the number of arbitrary constants in the solution of a system is not a notion which is definite in advance. A system may have many families of solutions, each with its own arbitrary constants. Not every unknown need be determined by the system to within arbitrary constants. It seems safe to assume that before one can speak with completeness on the question of arbitrary constants, one will have to possess a theory for the decomposition of a system into systems of some normal type. Apparently no such theory exists at present for systems other than algebraic systems. It is thus not surprising that the argument on which Jacobi bases his conclusion, in the first of the above mentioned papers, should be whimsical in aspect. Without justification, Jacobi replaces the given equations by their equations of variation, which are linear. He then asserts without proof that, for his purpose, a linear system may be imagined to have constant coefficients. His treatment of linear systems with constant coefficients was completed by Chrystal.2 There are indications in Jacobi's second paper that his conclusion was actually based on a speculation as to the number of times which the given equations would have to be differentiated to permit the elimination of all unknowns except one. Investigations of the problem from this point of view were given by Nan

This paper will be concerned with some estimates on the lower bound of the dimension of the Euclidean space in which a compact Riemann manifold can be imbedded isometrically, if its curvatures satisfy certain conditions. Our basic geometrical idea is a very simple one. Denote by M a compact Riemann manifold of dimension n in an Euclidean space E of dimension n + N, the Riemann metric on M being induced by the imbedding. Let 0 be a fixed point of E. The distance OP, P e M, is a continuous function in M and attains a maximum at a point P0 e M, since M is compact. It is intuitively clear that M will be "concave toward 0" at Po, so that there will be some restrictions on the Riemann curvature of M at P0. If M is given abstractly, the imbedding will not be possible, if these restrictions are not fulfilled by the given Riemann metric at any of the points of M. Actually, however, if the difference N of the dimensions of M and E is greater than one, the implication of this geometrical fact on the Riemann curvature of M is not very simple. The question leads to algebraic problems which probably do not have simple answers. We propose to give in this paper a few conclusions which can be drawn. It should be mentioned that the above geometrical idea has been used by Tompkins1 to prove that a locally flat compact Riemann manifold of dimension n cannot be isometrically imbedded in an Euclidean space of dimension 2n 1. Among other things this theorem will be generalized and the invariants entering in the problem geometrically interpreted. As for the differentiability assumptions we suppose our manifold and the imbedding to be of class > 4.

This paper presents a method of evaluation of the moments of K,(t)/l,(t).iTwo pairs of expressions, each consisting of two series, are obtained according to the index being an even or an odd integer. The method is an extension of the method used by Watson. Values are tabulated to 12D for p = 0(1)2. In a recent paper [1], Roberts considered the computation of the integral (1 ) f dtk J dt (k _ 2v), where I, and K, are modified Bessel functions. He developed the integral into an 'asymptotic series' suitable for computation when k is a large integer. When k is otherwise a small integer, he integrated numerically the following equivalent integral by using Simpson's rule: (2) = k411i(t)dt (k > 2v). In particular, values are tabulated to 8S for v = 1 and k = 2(1)100. Some time ago, Watson [2] evaluated the integral in (2) when k is an even integer. He developed the integral into two series by employing a method based on a modification of Plana's summation formula. It is found that this method can be extended to the case when k is an odd integer. Furthermore, it is also found that the same integral can be developed into different series by a modification of the method. Altogether, two pairs of expressions are obtained according to k being an even or an odd integer. It is the purpose of this paper to present such results. Values of the integral are thereby evaluated to 12D for v' = 0(1)2. It is mentioned that a method analogous to the present one was used recently by the authors for the evaluation of two Howland integrals [3]. For convenience, the integral is redefined, together with a factor, as follows: k+1 (3) ~~~~~~2 kK,(t)_ 7r(k!) lo I,(t) ( v so that it tends asymptotically to unity as k tends to infinity. The equivalent integral is 2k+l1 C tk (4) L 1)! f kdt (k > 2v). 7r(k + 1)! 1I(V The following results are obtained for k > v: Received September 22, 1971. AMS 1969 subject classifications. Primary 6525.

In the case of the upper half-plane, or the unit disc in the theory of one complex variable, there is a simple classical way of constructing the Poisson kernel from the Szegb kernel function. This construction was successfully used in the more general cases of the four classical types of bounded symmetric domains by L. K. Hua [9], and in the case of wedge domains (i.e., tube domains over convex cones) by E. M. Stein, G. Weiss, and M. Weiss [18], and the author [11]. Recently S. G. Gindikin [4] determined the Szegb kernel function for a wider class of generalizations of the classical upper half-plane than wedge domains, the so-called Siegel domains of type ii introduced by I. I. PjatetskiiSapiro [16]. The purpose of the present paper is to study the Poisson kernel obtained by the classical construction for these domains. First, in ? 2, we consider the case of a general Siegel domain of type II. The main result here is Theorem 2.4 which states that the Poisson kernel is an approximate identity. This theorem contains as a special case the analogous theorem of [18], while its proof given here is considerably simpler than the special proofs in [18] and [11]. As applications of Theorem 2.4, we establish a few simple facts about HI-spaces on D. We do not pursue this investigation very far; the present paper is rather meant to lay the ground work for a future study of finer questions of analysis on generalized half-planes. In ? 3, we consider the case of a Siegel domain of type II which is symmetric in the sense of E. Cartan. It is known [16], [12] that every bounded symmetric domain has a realization of this kind. We prove that in this case the Poisson kernel is harmonic in the sense of the theory of symmetric spaces. This identifies our Poisson kernel with a kernel studied in more general contexts by R. Hermann [7] and H. Furstenberg [3]. In the case of symmetric domains, our results go farther than those in [3] and [7] in two directions. We obtain fairly strong results on the boundary behaviour of the Poisson integral (?? 2 and 4) and we find easily computable explicit formulas for the Poisson kernel (?5). In ? 4, we consider bounded symmetric domains in the canonical Harish

1. The permutability of two bounded self-adjoint operators T1 and T2 defined on a Hilbert space S& is defined in the most natural and obvious manner: We say that T1 commutes with T2 if T17T2 = T2T1 throughout S. If {E1(X) } and {E2(0-) } are the canonical resolutions of the identity of T7 and T2 respectively, then the following result is well known: A necessary and sufficient condition that T7 commutes with T2 is the validity of any one of the following conditions: (a) The product T1T2 or T2T1 is self-adjoint. (b) E1(X) commutes with E2(0-) for each X and o. If Ti is a bounded self-adjoint operator and T2 any (not necessarily bounded) self-adjoint operator, then one usually defines permutability as follows (cf. [2] p. 31 or [4] p. 404): T1 commutes with T2 if Ti T2 (7 T2T1 . That is, if and only if, whenever f is in the domain of T2, Tif is also in the domain of T2 and T71T2f = T2T1f. This definition is equivalent to (b). Therefore, the permutability of any two selfadjoint operators is usually defined by (cf. [2], p. 50). DEFINITION 1. Two self-adjoint operators T1 and T2 with the corresponding canonical resolutions of the identity {E1(X)} and {E2(0-) } respectively are said to be permutable (or T1 commutes with T2) if E1(X) commutes with E2(o-) for each X and a-. The objective of this note is to show that two self-adjoint operators T1 and T2 commute if there exists a self-adjoint operator T such that T ( 7 T1T2 or T a T2T71. For the standard results and techniques of general Hilbert space theory used in this paper the reader is referred to the various papers of J. von Neumann [3], [4] and the books by M. H. Stone [5] and B. v. Sz. Nagy [2]. 2. To elaborate on our statements in ?1 we first recall the definition of multiplication of unbounded operators. The product AB of two operators A and B is defined as follows: (AB)f is defined if and only if Bf and A (Bf) are both defined and is then equal to the latter. DEFINITION 2. We say that the operators S, T are C' if S and T are self-adjoint, D(T) z D(S) (D((T) denotes the domain of T) and for f, g E D((T), the inner product (Sf, Tg) is Hermitean symmetric in f and g. LEMMA 1. Suppose that S, T are C' and E is a projection operator which commutes with T and such that R(E) ( D((T) (R(E) denotes the range of E). Let Si and T1 be the restrictions of ES and T respectively to R(E). Then S1 and T1 are bounded self-adjoint operators and S1 commutes with T1 . PROOF. Let f, g E R(E). Since R(E) C D((T) C D(S), f and g belong to D(S) and (I-E)Sf and (I-E)Sg belong to S& E R(E). Therefore

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

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

Several exact sequences have been established recently which relate cobordism groups of various types [1], [3], [8]. These have all been established in the differential case. The main object of this paper is to establish the same results in the combinatorial case. We first define a combinatorial analogue to Thom's notion of t-regularity, and prove some basic lemmas on submanifolds representing double coverings (Part I). After this, we find that the same proofs are valid as in the differential case, so we treat the two cases together. This gives us an opportunity to collect together all the geometrical arguments used in the proofs of these theorems, which have hitherto been somewhat scattered in the literature. We establish the three exact sequences in Theorem 1; the proofs are so simple that one could axiomatise a category of manifolds in which they work. The proof would be valid for topological manifolds if only some corresponding version of Lemma 7 could be established in that case. We then give a proof, in the same order of ideas, that one of our maps is a derivation. The behaviour of the others with respect to multiplication is rather more complicated, and we hope to return to it in a subsequent paper. We shall assume known the definitions of combinatorial and of differential manifolds (which may have boundary). All manifolds occurring in this paper will be compact. A compact manifold without boundary is called closed. To avoid logical difficulties concerning sets, we may regard all manifolds as imbedded in a finite dimensional subspace of a given Hilbert space; however, it will be more convenient to state our constructions for abstract manifolds. We shall usually denote manifolds by the symbols: V, W,M,N.

In the geometry based on the infinite group of conformal transformations of the plane (or on the equivalent theory of analytic functions of one complex variable), two types of problems must be carefully distinguished: those relating to regions and those relating to curves or arcs. Two regions of the plane are equivalent when there exists a conformal representation of the one on the other, the representation to be regular at every interior point. The classic Riemann theory shows that all simply connected regions are equivalent, any one being convertible into say the unit circle. The difficulties connected with the behavior of the boundary (which may be a Jordan curve or a more general point set) have been cleared up in the recent papers of Osgood, Study, and Caratheodory, Logically simpler problems relating to curves or arcs have received very scant attention. Two arcs are equivalent provided the one can be converted into the other by a conformal transformation, the transformation to be regular at the points of the arcs, and therefore in some (unspecified) regions including the arcs in their interiors. The main problem hitherto discussed by the writer in his papers on conformal geometry is the invariant theory of curvilinear angles.t Such a configuration (which may be designated also as an analytic angle) consists of two arcs through a common point, both arcs being real, analvtic, and regular at the vertex.: In this theory it is necessary to distinguish rational and irrational angles. If 0 denotes the magnitude of the angle (invariant of first order), then when 0/uis rational there exists a unique conformal invariant

One first finds the Riemann function, which is the solution of a homogeneous adjoint equation subject to simple boundary conditions that are independent of the given boundary data. The solution of (1.1), for any appropriate boundary data, can then be obtained by evaluating a definite integral, where the Riemann function and the boundary data appear in the integrand. The main obstacle to the use of this method for numerical computation has been the difficulty in finding the Riemann function. It was pointed out in a 1947 paper by Cohn [2] that, aside from a(xi, X2) = constant and a(xi, X2) = k(k 1) * (xI + x2f2, which was treated by Riemann, there are very few cases where expressions for the Riemann function have been obtained. One can, of course, use the Picard method of successive approximations, but this is usually not practical. In this paper we derive a simple recurrence formula for the Riemann function, for the case where a(xi, X2) can be expanded in a Taylor series about the point where the solution is sought. We consider here the Riemann function for the operator L in the n dimensional analogue of (1.1), (1.2) L(u) = CnU/C1XOX2 ... **xn + a(xi, x2, X * *, )u = F (xi, x2, Xn)

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