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The driver tries to keep the car in the center of the lane. If the car is too near the left edge, this causes the driver to make a “corrective” right turn. If the car is near the right edge, a “corrective” left turn is made. Therefore, a quantity which decreases with increasing distance ΔL from the left edge may be considered as a stimulusSR producing the reactionRR of turning to the right. A similar situation holds for the distance ΔR from the right edge. When the car is in the center of the lane, ΔL = ΔR andSR=SL, the stimuli are equal. We thus have here a situation analogous to the one studied by H. D. Landahl in his theory of psychophysical discrimination. In general a reactionRR (resp.RL) will occur only ifRR−RL≥h* (resp.RL−RR≥h*) whereh* is a threshold. Applying Landahl’s theory to this situation, we find thath* determines the distance from the edge, at which a corrective turn is made. This distance is not constant, but a function of the speedv of the car. The requirement that a corrective turn should be madebeforre the car runs off the road leads to an expression for the maximum safe speed. Because of the transcendency of the equations involved, closed solutions cannot be obtained. It is, however, shown that the expression for maximum safe speed, given in a previous paper (Bull. Math. Biophysics,21, 299–308, 1959), is a rough first approximation to the expressions found now.

In this paper, "point," "line," and "incident upon," are undefined terms. The phrases, "point is on a line," "point is incident upon a line," "line is incident upon a point," "line is on a point," are all to be considered synonymous. We will say that a line I intersects a line k (at a point P) if and only if P is on both I and k. Two lines are parallel if and only if there is no point which is on both. The following axioms will be used: AXIOM I. If P and Q are distinct points, there is exactly one line on P and on Q. AXIOM II. If I is a line, there is at least one point not on 1. AXIOM III. If I is a line and P is a point not on 1, there are exactly m distinct lines on P (m ?2) which are parallel to 1. AXIOM IV. There is at least one line with exactly n points on it, n _2. The entire set of points and lines whose existence is postulated by these axioms (for given m and n) will be called a geometry. Other work done with an axiom system containing the Axiom III as here stated is not known to this investigator, but Szamkolowicz [3 ] has reported on an equivalent system, and similar systems have been studied [2], [4]. As they are here stated, the axioms may or may not be consistent, depending on the values assigned to m and n. For n =2, their consistency for any m is established by the existence of a model constructed by A. N. Milgram [4], and for n = 3, m = 4, their consistency is shown by a model described by Abraham Barshop [I ]. This paper will demonstrate their incon sistency for m=2 and n>2.

In this paper a class of branching processes applicable to populations reproducing by some asexual means or by a simple selfing system of mating is studied. The paper is divided into three parts. In part one the mathematical model is introduced, part two is a mathematical analysis of the model, and in part three concrete applications and examples are given. Many of the proofs of the theorems in part two are omitted but will appear in a subsequent issue of theBulletin.

Introduction. This paper is mainly concerned with the characterization problem for bounded spectral operators (see [5]). We restrict the discussion to scalar type operators with real spectrum, which we call pseudo-hermitian operators (p. h.). The criteria are given in terms of certain properties of the exponential group generated by the operator. The simplest criterion of this kind is for Hilbert space: a bounded linear operator S is p.h. if and only if || eitS| ? M < oo for all real t. This statement is false for Banach spaces, and even for reflexive Banach spaces. In the latter case, we give additional necessary conditions on the group eitS, which together with its uniform boundedness are both necessary and sufficient for S to be p.h. Another useful criterion is the following: a bounded linear operator S on a reflexive Banach space is p.h. if and only if for everyf in L1(R) (R the real line), we have II fRf(t)e itsdt || _ M || f l,,, where the norm on the left is the operator norm, f is the Fourier transform off and || * is the sup norm. Unlike the general theorems in [5] about the spectrality of an operator, the characterizations obtained in this paper are easily applied to analytically given operators on concrete Banach spaces. An example is discussed in ?4. Applications are discussed at the end of ?5. ?2 deals with an improvement of a theorem of Foguel [7] about the resolutions of the identity of sums and products of commuting spectral operators. ?1 contains preliminaries. The characterization problem for spectral operators is discussed in ??3 and 5.

In a previous paper (Ten Hoopen and Reuver, 1965) the selective interaction of two independent sequences of stimuli in time, called excitatory and inhibitory, was considered. An inhibitory stimulus annihilated the next excitatory stimulus. The interval distribution of the distorted excitatory sequence was derived. This concept has been used (Ten Hoopen, 1966a) to characterize unusual spike interval distributions of nerve cells, when it is assumed that every undeleted excitatory stimulus provokes a response. In this case the assumption is open to doubt as it ignores an elementary property of nerve cells, viz., the temporal and spatial summation of excitation. It is generally accepted that a certain minimum amount of excitation is necessary to cause a response. In this note a generalization called the deletion model is proposed, which removes the above drawback. Suppose excitatory stimuli arrive according to a recurrent process and accumulate at the nerve cell. As soon as n of these have arrived a response occurs and the process starts again. Inhibitory stimuli arrive according to a recurrent process also, independently of the other sequence. If in the course of summation of the excitatory stimuli an inhibitory stimulus arrives, all accumulated excitation is wiped out, and summation starts anew. The probability density function of the responses, denoted by p(t), is derived in Section 2. If incoming excitatory stimuli arrive more or less regularly, p(t) has a multi-modal character. This type of interval distribution has frequently been observed in analyzing neurophysiological data. 409

A multiple decision approach to the problem of ranking populations according to their population means has been formulated by Bechhofer (1954). A single-sample solution to this problem was presented by him for the case of normal populations with known variances. In the present paper the case of normal populations with unknown equal variances is considered. A two-sample procedure is proposed as a solution to the latter problem; this procedure is of the type used by Stein (1945) for obtaining a test of Student's hypothesis with power independent of the variance. Tables are available which enable the experimenter to apply this procedure with little computational effort to the ranking of two or three populations with unknown but equal variances. The same tables can be used when the variance ratios are arbitrary and known. Graphic comparisons of the expected sample size are made for the single-sample and two-sample procedures.

The Brauer-Speiser theorem asserts that the Schur index of a real-valued complex irreducible character of a finite group is either 1 or 2. In this paper we present a brief proof of this result. From this it follows that the K-central nonttrivial division algebra components of group algebras over a real algebraic number field K are quaternions. THEOREM (BRAUER-SPEISER [2], [5]). Let X be a real-valued irreducible complex character of thefinite group G. If X(l) is odd, then the Schur index mQ(X) of X over the raltionals Q equals 1. If X(l) is even, then mQ(X) equtals 1 or 2. PROOF. Let H be the direct product of G with itself and let Go-G be the diagonal of H. We identify G writh Go. Let S be an irreducible complex representation of G affording X and let SW'1 denote the representation of H obtained by taking the outer tensor product of S with itself. S(2) is an irreducible representation of H [4, p. 189]. Denote the clharacter of S(2) by x(2) so that X(2)(gl, g2) =X(g1)X(g2). Thus X2a1 G=X'. Let p be the 1-character of G. By Frobenius reciprocity and the orthogonality relations, (pH, X('))H = (p, X(2) 1 G)a = (p, X')o 1, since x-= Since the representation of H affording p" can be realized in Q, it follows from the theory of the Schur index that mQ(X(2)) 1 [4, ?70]. Let K = Q(X) and let A be the simple component of the group algebra K(G) such that S is an absolutely irreducible constituent of the representation of G afforded by a minimal left ideal of A. Siince K(H) -iK(G)OKcK(G), we see that S(2) is an absolutely irreducible constituent of the representation of H afforded by a minimal left ideal of A(2) =A 0KA. Since mnQ(X(2)) 1, it follows that A(2) is a complete matrix algebra over K [4, ?70]. Tlhus A has expoinent equal to 1 or 2 in the Brauer group of K. Since K is an algebraic number field, the exponent of A equals the index nQ(X) of A. (See [1].) The BrauerSpeiser theorem now follows from the fact that mQ(X) divides X(1) [4, Theorem 70.12]. Received by the editors November 25, 1969. AMS Subject Classifications. Primary 2080; Secondary 1640. Key W1ords and Phrases. Schur index, Brauer group, exponent. I This work was done under the sponsorship of NSF Grant GP-13164/Paige.

Vajda's paper in this volume has suggested to the author the following problem:Let A be a n × m matrix, m≦ n, let B be a m × n matrix, and let M = I – AB, where I is the unit matrix of order n. Given A, to find B such that of the n latent roots of M′M, k are unity, and the remaining n — k are zero.

This paper studies maximal quotient rings of semiprime P. I.-rings; such rings are regular, self-injective and satisfy a polynomial identity. We show that the center of a regular self-injective ring is regular self-injective; this enables us to establish that the center of the maximal quotient ring of a semiprime P. I.-ring R is the maximal quotient ring of the center of R, as well as some other relationships. We give two decompositions of a regular self-injective ring with a polynomial identity which enable us to show that such rings are biregular and are finitely generated projective modules over their center.

The application of the multimoment integral method to the study of the base pressure behind a supersonic vehicle is examined. The primary purpose is to understand the exact nature of the Crocco-Lees critical point, which provides a uniqueness condition in the problem. The analysis is carried out mainly in Poincare phase space, where the singularities of the differential equations are investigated. Two singular curves are found. The one which is physically meaningful for the near wake flow is located downstream of the rear stagnation point. This singular curve consists of saddle points which yield “wake” solutions, and focal points or saddle-foci. Only those saddle points which yield “wake” solutions correspond to Crocco-Lees critical points. Thus, the integration of the differential equations should be started from a saddle point both for the upstream and the downstream solutions. The current analysis then requires that one deal with only a one-parameter (freestream Mach number) family of solutions, rather than the two-parameter (freestream Mach number and Reynolds number based on thickness of the viscous shear layer at the rear stagnation point) family of solutions obtained in previous works. Finally, this paper clarifies anomalous details of previous numerical investigations carried out for the near wake by Webb, Golik, Vogenitz, and Lees.

This paper considers the testing of serial correlation coefficients when the underlying distribution is negative exponential. The first-order serial correlation coefficient is shown to be an asymptotically most powerful test criterion for certain plausible alternative hypotheses. The mean of its distribution, assuming independence, is known and a bound and estimate for its variance are obtained. These are compared with sampling results of D. R. Cox. The distribution of the serial correlation coefficient is proved to tend to normality for all underlying distributions with a finite fourth moment. The loss of power resulting from prior transformations of the variates is calculated. Finally, attention is drawn to Ogawara's modification of the first-order serial correlation coefficient which is asymptotically most powerful and for which the mean and variance are known exactly.

Let C be the ring of all real valued continuous functions on a completely regular topological space. This paper is an investigation of the ideals of C that are intersections of prime or of primary ideals.C. W. Kohls has analyzed the prime ideals of C in [3 ; 4] and the primary ideals of C in [5]. He showed that these ideals are absolutely convex. (An ideal I of C is called absolutely convex if |f| ≦ |g| and g ∈ I imply that f ∈ I.) It follows that any intersection of prime or of primary ideals is absolutely convex. We consider here the problem of finding a necessary and sufficient condition for an absolutely convex ideal I of C to be an intersection of prime ideals and the problem of finding a necessary and sufficient condition for I to be an intersection of primary ideals.

In [3], Niven proved that for any positive integer k, the density of the set of positive integers n for which (n, (φ(n))≤k is zero (where φ is the Euler to tient function). In this paper, we prove a related result—namely if k and j are any positive integers, then the density of the set of positive integers n for which (n,σj(n))≤k is zero (where σj(n) is the sum of the jth powers of the positive divisors of n). We will borrow from Niven’s technique, but we must make some crucial modifications.

Presented to the Society, September 7,1948; received by the editors December 28, 1948. 1 This paper was written at the Institute for Numerical Analysis of the National Bureau of Standards, with the financial support of the Office of Naval Research of the U. S. Navy Department. * G. Szego, On an inequality of P. Turdn concerning Legendre polynomials, Bull. Amer. Math. Soc. vol. 54 (1948) pp. 401-405. 256

The purpose of this paper is to determine when a symmetric element, regular with respect to other symmetrics, is regular in the ring. This result is true for simple rings, for prime rings with either Goldie chain condition, and for semiprime Goldie rings. Examples are given to show that these results are the best that can be hoped for. The structure of a ring with involution satisfying the property that no two nonzero symmetric elements annihilate one another was determined for 2-torsion. free rings by Lanski in [4], and in general, by Herstein in [2]. It happens, at least for semiprime rings which are 2-torsion-free, that this assumption on the symmetric elements implies that each nonzero symmetric element is regular in the whole ring. It is our purpose here to determine conditions sufficient to imply that any single symmetric element which annihilates no nonzero symmetric element is regular in the whole ring. Throughout this work, R will denote a ring with involution * and S = r e RIr*= rj, the set of symmetric elements of R. To avoid confusion, we may write S(A) for the symmetric elements of some ring A, with involution. We shall say that an element y is regular in S to mean that yt = 0 implies t = 0, for t E S. We are interested in determining when an element y e S, which is regular in S, must also be regular in R. Of course, we cannot hope to draw such a conclusion unless R is semiprime. A counterexample can be found easily by taking R @ N, for N a 2-torsion-free ring with trivial multiplication, and setting x* = -x for x e N. It is also easy to find examples where R has 1, and so N is not a direct summand. Let R = F[x, y. z]/(z2, zx, zy), for F a field with char F , 2, and set z*-z, x y and y*= x. One condition on R which allows us to conclude that elements regular in S are regular in R is given in our first observation. Received by the editors September 7, 1973. AMS(MOS) subject classifications (1970). Primary 16A28; Secondary 16A12, 16A46, 16A40, 16A38. (1) This work was supported by NSF Grant GP 38601. Copyright

A centered right -polygon (synonyms: -operand, -system) is called right compressive if and and left compressive if and . Here is the congruence on the semigroup called the kernel of the -polygon which is defined as follows: .The intersection of the kernels of all right (left) compressive -polygons is called the right (left) compressive radical of . In this paper we study compressively semisimple and compressively radical semigroups.

It is common to represent taxonomic hierarchies of related objects (such as similar plant or animal species or languages of the same family) by rooted trees with labelled terminal vertices which represent the objects. The multivariate data comparing numerous characteristics of the objects is first reduced to indices of similarity (or more often of dissimilarity) between each pair of objects. These are used to classify the objects into groups which are then depicted on a tree. This paper shows that an unrooted tree with labelled terminal vertices may provide a better representation of the relationships between the objects because the similarity indices are required to conform to fewer restrictions. Also for a given number of terminal vertices, there are fewer unrooted than rooted trees so that studies using probability distributions of trees or seeking the most suitable tree to represent the data are more practicable.

In this paper we give a general expression for operators on a Hilbert space which are similar to an operator of a certain type such as a contraction, a unitary operator, a dissipative operator, an operator with nuclear imaginary component, etc. Theorem 1 covers these types and many others. Theorem 2 is concerned with those operators whose spectrum lies in the open upper half-plane. By an operator we always mean a bounded linear operator on a Hilbert space !$. As usual H > 0 means that H is a positive operator, i.e., (Hx, x) 2 0 f or all x E &. We write H > 0 to indicate that H is positive and invertible. The spectrum of an operator B is denoted bY 4B).

This paper investigates a generalization to E4 of the notion of toroidal decomposition of E3. A certain type of this kind of upper semicontinuous decomposition is shown to be shrinkable and hence yield E4 as its decomposition space.

The main purpose of this paper is to show that X ω {X^\omega } is normal if and only if (1) X n {X^n} is normal for each n, and (2) X ω {X^\omega } is countably paracompact. Furthermore, X ω {X^\omega } is perfectly normal if and only if X ω {X^\omega } is hereditarily countably paracompact. Also, the compact Hausdorff space X is metrizable if and only if X 3 {X^3} is hereditarily countably paracompact.

In this paper we develop a multi-step method of order nine for obtaining an approximate solution of the initial value problemy'=f(x,y),y((x0)=y0. The present method makes use of the second derivatives, namely,\(y'' = \frac{{\partial f}}{{\partial x}} + f(x,y)\frac{{\partial f}}{{\partial y}}\) at the grid points. A sufficient criterion for the convergence of the iteration procedure is established. Analysis of the discretization error is performed. Various numerical examples are presented to demonstrate the practical usefulness of our integration method.

A blocking set S in a projective plane Π is a subset of the points of Π such that every line of Π contains at least one point of S and at least one point not in S. In previous papers [5; 6], we have shown that if Π is finite of order n, then n + √n + 1 ≦ |S| ≦ n2 – √n (see [6, Theorem 3.9]), where |S| stands for the number of points of S. This work is concerned with some applications of the above result to nets and partial spreads, and with some examples of partial spreads which give rise to unimbeddable nets of small deficiency.In the next section we re-prove a well known result of Bruck which states that if N is a replaceable net of order n and degree k then k ≧ √n + 1, and show how this bound can be improved if n = 7, 8, or 11.

1. Let X be a locally compact Hausdorff space, E a (real) locally convex, complete, linear topological space, and (C*(X, E), j) the locally convex linear space of all bounded continuous functions on X to E topologized with the strict topology (. When E is the real numbers we denote C*(X, E) by C*(X) as usual. When E is not the real numbers, C*(X, E) is not in general an algebra, but it is a module under multiplication by functions in C*(X). This paper considers a Stone-Weierstrass theorem for (C*(X), 3), a generalization of the Stone-Weierstrass theorem for (C*(X, E), (), and some of the immediate consequences of these theorems. In the second case (when E is arbitrary) we replace the question of when a subalgebra generated by a subset S of C*(X) is strictly dense in C*(X) by the corresponding question for a submodule generated by a subset S of the C*(X)-module C*(X, E). In what follows the symbols Co(X, E) and Coo(X, E) will denote the subspaces of C*(X, E) consisting respectively of the set of all functions on X to E which vanish at infinity, and the set of all functions on X to E with compact support. Recall that the strict topology is defined as follows [2]: DEFINITION. The strict topology (,B) is the locally convex topology defined on C*(X, E) by the seminorms

If a group G is presented in terms of generators and relations, then the classical Reidemeister-Schreier Theorem [1] gives a presentation for any subgroup of G . If G is a free product of groups G α each of which is presented in terms of generators and relations, then the main result of this paper is a presentation for any subgroup H of G , which shows the nature of H as a free product of certain subgroups of G . This result is a generalization of the celebrated Kuros Theorem [2]. It also includes the Reidemeister–Schreier Theorem and the Schreier Theorem [1] which states that any subgroup of a free group is free.

Let be an abelian variety imbedded in projective space, and let be an induced invertible sheaf on . In this paper explicit bounds are determined for the difference , where is the Neron-Tate height and is the Weil height.Bibliography: 5 titles.

The object of this paper is two-fold: first to collect together the known facts about combinatorial cobordism in general, and then to calculate the groups for the first 8 dimensions. As in (29), we shall denote the unoriented and oriented cobordism groups in dimensionnbyand Ωn, and will distinguish the combinatorial from the differential case by affixesc, d.

Let A be an m × n matrix of complex numbers. Let Aτ and A* denote the transpose and conjugate transpose, respectively, of A. We say A is diagonal if A contains only zeros in all positions (i, j) with i ≠ j. In a recently published paper [4], M.H. Pearl established the following fact: There exist real orthogonal matrices O1 and O2 (O1 m-square, O2 n-square) such that O1AO2 is diagonal, if and only if both AA* and A*A are real. It is the purpose of this paper to show that a theorem substantially stronger than this result of Pearl's is included in the real case of a theorem of N.A. Wiegmann [2]. (For other papers related to Wiegmann's, see [l; 3].)

Some general experimental procedures are suggested to test a previously developed theory of motivation interactions which gives a quantitative description of factors in motivation. The experiments are devised with a view of varying simultaneously the “positive” and “negative” terms of a postulated satisfaction function which is supposed to depend on the subject's “effort” and “remuneration.” If the output of effort on the part of the subject reaches a steady state for a given set of conditions, this output is taken to be the optimum output under those conditions. In other words, it is supposed that the satisfaction function is “maximized.” From the equations describing the derivatives of the satisfaction function with respect to different variables, the “biological constants” of the individual can be computed and his behavior predicted under a variety of other conditions. In particular, on the basis of previous papers, it should be possible to predict the behavior of two “cooperating” and “sharing” individuals.

A particular case is represented by dF(x) =f(x)dx, where f(x), summable and >0, is the "probability density" or "law of distribution" for x. The expression f -'.xsdF(x) is called the "sth moment" of the distribution, s taking values 0, 1, 2, The Second Limit-Theorem, which was the starting point of this paper, can be stated, with A. Markoff,t as follows: If a sequence of laws of probability Fkjx) (k = 1, 2, * is such that they admit moments of all orders, and if

A well-known procedure for testing for serial correlation is to plot out the sample path of the cumulated periodogram and to compare the resulting graph with the Kolmogorov-Smirnov limits. The paper considers small-sample aspects of this procedure when the periodogram is calculated from the residuals from least-squares regression. It is shown that for a test against an excess of low-frequency relative to high-frequency variation in the errors of the regression model, a pair of lines can be drawn on the graph such that if the path crosses the upper line the hypothesis of serial independence is definitely rejected, while if the path fails to cross the lower line the hypothesis is definitely accepted. In the intermediate case the test is inconclusive. Similar procedures are given for tests against an excess of high-frequency variation and for two-sided tests. To facilitate the tests a table of significance values of the appropriate modified Kolmogorov-Smirnov statistics is given. A further test based on the mean of the ordinates of the cumulated periodogram is considered. It is shown that bounding significance values are easily obtainable from significance values of the mean of a uniform distribution.

On account of its simplicity, its immediate applicability to many functions already in existence, and its suggestiveness with regard to generalizations of known functions or the creation of new, the algebra in question merits the somewhat detailed exposition in Part I. As its use is perhaps best seen from an example, we have sketched briefly in Part II the outlines of a new theory of the relations between the Bernoullian and Eulerian functions of any arguments and any ranks, showing that by means of the algebra all necessary computations are reduced to a minimum. These functions are of two variables, of which one is complex and the other, the rank, a positive integer. The algebra establishes a simple isomorphism between the theory of relations between the functions and the like for the ordinary sine and cosine. (It is necessary here to distinguish between ordinary and umbral circular functions; the nature of the distinction appears presently.) In a paper which I expect to publish later there is a more detailed application of this algebra to certain new functions, suggested by the algebra, of three variables, of which two are complex and the third a positive integer. For unit values of one of the complex variables these functions degenerate to the Bernoullian and Eulerian functions of Part II; for unit values of the other, they become certain polynomials, of considerable importance in the arithmetical theory of quadratic forms, discussed on several occasions by Hermite, Weierstrass and, more recently, by Bulygain and Gruder. In this application there are simple isomorphisms with the circular, the hyperbolic and the elliptic functions. In a third paper, to appear shortly, I have shown how the algebra gives at once the complete theory of the relations between the functions of Spitzer, which include as special cases the Bessel coefficients. Here there is simple isomorphism with the exponential function. By a generalization of the exponential function, Emxm+;/r(m+v), in which x, v are complex, I have shown in a paper not yet published that the algebra is readily applicable to the Bessel functions. The algebra therefore is of considerable utility. Its chief use, how

A locally convex space E is said to be an s-space [1] if every closed linear map of E onto a barrelled space is open. The aim of this paper is to replace the B-complete spaces in the closed graph theorem of A. P. and W. ROBERTSON [7] by s-spaces. Previous work of PTAK [5] and PERSSON [4] implies that B-complete spaces and t-polar spaces are s-spaces. Thus our result includes that of ADASCH [2] who generalized Robertson's theorem, taking t-polar spaces instead of B-complete ones.

In this work we study flat modules over commutative noetherian rings under two kinds of restriction: that the modules are either submodules of free modules or that they have finite rank. The first ones have nicely behaved annihilators: they are generated by idempotents. Among the various questions related to flat modules of finite rank, emphasis is placed on discussing conditions implying its finite generation, as for instance, (i) over a local ring, a flat module of constant rank is free, and (ii) a flat submodule of finite rank of a free module is finitely generated. The rank one flat modules already present special problems regarding its endo- morphism ring; in a few cases it is proved that they are flat over the base ring. Finally, a special class of flat modules—unmixed—is discussed, which have, so to speak, its source of divisibility somewhat concentrated in the center of its endo- morphism ring and thus resemble projective modules over flat epimorphic images of the base ring. Introduction. This paper is an attempt to lay a somewhat limited framework for the study of flatness on modules over noetherian rings. As already for abelian groups the general description of the torsion-free modules is practically hopeless, we shall not be too demanding here and shall concentrate in those aspects of the theory which best reflect the nature of the ring and the universal algebra involved. In this vein, after some preliminaries on the rank of flat modules and a generaliza- tion of a theorem of (4) on how well the finite generation of a high nonzero exterior power of a module may disclose the finite generation of the module itself, we discuss : annihilators of flat submodules of free modules, which, in general, have a good behavior in comparison to annihilators of arbitrary flat modules; various con- ditions disguising the finite generation of a flat module; flat modules of rank one and its endomorphism rings—which are, in many cases, flat epimorphic images of the base ring; a discussion of a class of flat modules—unmixed modules—where the source of divisibility lies, in general, in the complement of an affine subschema of the spectrum of the ring. The paper concludes with a list of unsolved problems whose solutions might yield new methods or, more hopefully, uncover other

1. Preliminaries Throughout the remainder of this paper, X will denote a completely regular Tl-space, while all functions will be complex valued unless it is explicitly stated otherwise. We shall let C(X) denote the space of all continuous functions on X, B(X) denote the space of all bounded functions on X, and Bo(X) denote the subalgebra of B(X) consisting of those elements in B(X) which vanish at infinity. Further, let Cb(X)= C(X)nB(X) and Co(X)= C(X)nBo(X). ¥

Let X be a metric space and let G(X) be the space of homeomorphisms of X with the compact open topology. Theorem. If G(X) is locally compact, then it is zero-dimensional. Some examples are also given of metric spaces X for which C(X) is compact. Introduction. Let Ibea metric space and G(X) be the set of all homeomorphisms of X onto X endowed with the compact open topology. In (9, Theorem 7) de Groot showed that algebraically G(X) may be any group. In (7) Cook showed that G(X) may be topologically a Cantor set with X a metric continuum. In (5) Brechner showed that G(X) may be algebraically and topologically any countable infinite product of finite cyclic groups with X a one- dimensional Peano continuum. The main theorem of this paper shows that the zero-dimensionality of the examples in (5) and (7) is an essential feature. We prove that if G(X) is locally compact, then it is zero-dimensional. In contradistinction to this result it is known that G(X) may be one-dimensional or infinite-dimensional (6). The principal tool in our proof is the work of Beck in (3). A function space result is also proved in this paper which improves some of the results in (4) and (8). This result is then used in the proof of the main theorem. In the last section of the paper it is shown that for any cardinal a, G(X) may be topologically and algebraically Z2 where Z2 is the discrete group with two elements. Preliminaries. Throughout the paper let X and Y be regular Pi topological spaces and let Yx denote the set of all continuous func- tions from X into Y. For a set A EX and BEY let (A, B) be the set of all functions/G Yx such that/(^4) CP- We topologize Yx by taking as a subbase all sets of the form (K, V) where KEX is compact and VE Y is open. The topology so defined is called the compact open topology or c-topology as in (10, p. 257). Of course G(X)EXX, but G(X) may not be a topological group in the c-topology. In (2) Arens

It is the purpose of this paper to show that Axiom A2 in Bourbaki's axiomatic system for set theory can be replaced by the weaker statement that every term x defines a set { x of which x is the only element. All references in the paper are to [1], and the terminology and notation of [1] are used. We place ourselves into a theory with specific signs = and , in which Si through S8 are schemas, and Al, A4 and the statement (1) (yx) Coll, (y = x) are axioms. The theory may contain other specific signs, schemas and axioms. Axiom (1) allows us to define a set {x} =Ev (y=x). Criterion C51 (p. 65) then can be proved. It follows that there is a set 4 such that (Vx)(xE4) is true. Sets {I } and, using A4, $(I({ }) can then be constructed. It follows that 4 Ez$({I J), {I} ({4 }), and 4'-x {I4. Let now R in S8 (p. 64) be the relation (x = u andy-4) or (x= v andy= {y}). The relation

Many results of the theory of Riemann surfaces derive only from the properties of the sheaf of harmonic functions on these surfaces. It is therefore natural to try to extend these results to more comprehensive structures defined by means of a sheaf of continuous functions on a topological space which should possess the main properties of the sheaf of harmonic functions on a Riemann surface. The aim of the present paper is to generalise some known results from the theory of Riemann surfaces to spaces endowed with sheaves satisfying Brelot’s axioms [2], which we call harmonic spaces. In order to do so we had to introduce and to study the maps, associated in a natural way with this structure, called harmonic maps; they replace the analytic maps between Riemann surfaces. In this general frame we reconstruct the whole theory of Wiener compactification as well as the theory of the behaviour of analytic maps at the Wiener boundary.

The principle of duality for a wide class of distributed parameter systems is developed in this paper. The necessary and sufficient conditions for the primal control problem are utilized to derive the dual and converse dual theorems of the dual problem. An example of the temperature variation in a slab is given to show the application of the theory.

In this paper we study differentiable orientation preserving involutions on S n§ with non-trivial k dimensional fixed set. Define a cobordism relation between involutions T o and T~ by requiring that there exist an involution T on S n+k x I whose restriction to the ends yields T O and T~. The set of cobordism classes becomes a group d(n+k,k) by taking connected sum about a fixed point. We assume throughout that 5

Several authors have recently shown that Brownian motion with continuous paths on [ 0 , 1 ] [0,1] can be expanded into a uniformly convergent (a.s.) orthogonal series in terms of a given complete orthonormal system (CONS) in its reproducing kernel Hilbert space (RKHS). In an earlier paper we generalized this result to Gaussian processes with continuous paths. Here we obtain such expansions for a Gaussian random variable taking values in an arbitrary separable Banach space. A related problem is also considered in which starting from a separable Hilbert space H H with a measurable norm | | ⋅ | | 1 || \cdot |{|_1} defined on it, it is shown that the corresponding abstract Wiener process has a | | ⋅ | | 1 ||\cdot |{|_1} -convergent orthogonal expansion in terms of a CONS chosen from H H .

Equations are derived for the flow of an anisotropic fluid in a tube. It is argued that these provide a model for arterial blood flow. Particular attention is paid to the effect of radial differences in hematocrit. Sequels to this paper (Bull. Math. Biophysics,29, 565–574; forthcoming, 1967) will respectively demonstrate possible wall-directed forces on the erythrocyte and enlarge on the physiological consequence of hematocrit variations. The present article develops the basic equations and explores the possible role of anisotropic effects in blood flow.

The paper provides a short survey of the actual state of the theoretical exterior ballistics. In particular the fundammental equations defining the motion of a rotating projectile are given in a form suitable for numerical calculation. Then, using the argument that a projectile possesses full rotational and also mirror summetry, the explicit development of the force and moment components is given in terms of the linear and angular velocities. Finally this development is compared with the experimentally known aerodynamic coefficients and stability derivatives.

hold for almost all n. Let .3 (G) be the set of all lattices of G with topology defined by limit of lattices and let A (G) be the group of all topological automorphisms of G.2 Clearly A (G) acts continuously on .3 (G). In [3], Chabauty raised the question whether A (G) r, with the induced topology from .3 (G), is homeomorphic to A (G)/N(r) where N(r) = {c C A(G): (r) = r}. The answer in general is negative and a counterexample was presented in [8.b]. It is the purpose of this paper to study some properties of the limit of subgroups and to establish Chabauty's conjecture in case that G is a connected semi-simple Lie group.

The paper contains some remarks on the problems treated by Hammersley(9). With the aid of a new and simple method the exact solutions of Hammersley's problems and the asymptotic behaviour of the solutions are found. The problems in question are: A Poisson process is transformed into a new process by a Type II counter, and this new process is transformed again by a Type I counter. The distribution of the number of events occurring in each transformed sequence is to be determined.

Introduction. Serre has asked [5, p. 243] whether projective modules over the polynomial ring K[XI, * * *, X"], K a field, are free, and Seshadri has proved that this is so when n =2. More precisely, he showed that every R[X]-projective module 'comes from" one over R if R is either a principal ideal domain [6] or the coordinate ring of a nonsingular affine curve over an algebraically closed field [7]. By modifying Seshardri's argument in [6] we show here (Theorem 2.4) that R may be any Dedekind ring(2). Moreover we provide an example, due to S. Schanuel, which shows that some assumption like integral closure on R, is, in general, necessary for this conclusion. Problems of the above type can often be reduced to showing that a projective module is a direct sum of modules of rank 1, and this led us to ask for what integral domains R this is true for all (finitely generated) torsion free modules. Our main result (Theorem 1.7) gives an essentially complete solution to this problem. With mild assumptions on R, a necessary and sufficient condition is that every ideal in R can be generated by two elements. Thus these integral domains are very close to, but need not be, Dedekind rings. We assume a familiarity with the basic techniques of commutative algebra as developed in Zariski and Samuel [9]. Homological algebra is appealed to in two discrete instances; one is a reference in the proof of Proposition 1.5, and the other is the proof of Lemma 1.9. The latter result, elementary as it is, seems to defy any proof which does not either use, or essentially reconstruct, the functor Ext'. Minor portions of this work were present in the author's doctoral dissertation. He gratefully records his indebtedness to Professor Kaplansky, who communicated a computation which suggested one of the essential ideas in the proof of Theorem 1.7. 1. Torsion free modules. Throughout this paper 'module" means "finitely generated module," and "ring" means "commutative ring." If M is anRmodule PuR(M) denotes the least number of elements required to generate M, and ,.t*(R) =sup PR (@) where 2t ranges over all finitely generated ideals of R. We call R local if it has a unique maximal ideal.

In a previous paper (Landau, 1951) it was shown that a society with a dominance relation would rarely tend to be close to the hierarchy in structure if dominance is determined solely by the inherent characteristics of the members. Here we consider the effects of other factors, due to social rank or to the outcome of previous encounters which affected dominance.

1. SUMMARY A number of statistical procedures involve the comparison of a 'regression' mean square with a 'residual' mean square using the normal-theory F distribution for reference. The use of the procedure for the analysis of actual data implies that the distribution of the meansquare ratio is insensitive to moderate non-normality. Many investigators, in particular Pearson (1931), Geary (1947), Gayen (1950), have considered the sensitivity of this distribution to parent non-normality for important special cases and a very general investigation was carried out by David & Johnson (1951a, b). The principal object of this paper is to demonstrate the overriding influence which the numerical values of the regression variables have in deciding sensitivity to non-normality and to demonstrate the essential nature of this dependency. We first obtain a simple approximation to the distribution of the regression F statistic in the non-normal case. This shows that it is 'the extent of non-normality' in the regression variables (the x's), which determines sensitivity to non-normality in the observations (the y's). Our results are illustrated for certain familiar special cases. In particular the well-known robustness of the analysis of variance test to compare means of equal-sized groups and the notorious lack of robustness of the test to compare two estimates of variance from independent samples are discussed in this context. We finally show that it is possible to choose the regression variables so that, to the order of approximation we employ, non-normality in the y's is without effect on the distribution of the test statistic. Our results demonstrate the effect which the choice of experimental design has in deciding robustness to non-normality.

Consider the equation f(ξ η, x, y) = 0. If definite values of ξ and η be taken, and x and y be current co-ordinates, f(ξ η, x, y) = 0 is represented by a curve in the plane of xy. If all the possible values of ξ and η be taken in turn, f(ξ η, x, y) = 0 will be represented by a doubly infinite family of curves in the xy plane. Similarly if all possible definite values of x and y be taken in turn and ξ and η be regarded as current co-ordinates, f(ξ η, x, y) = 0 will be represented by a doubly infinite family of curves in the ξη plane. Since the first and the second doubly infinite families of curves represent the same equation, it may be expected that some geometrical correspondence exists between them. It is the object of the present paper to investigate this correspondence.