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In recent papers, the authors have developed the technique of using Kan extensions to obtain extensions of homology and cohomology theories from smaller to larger categories of topological spaces. In the present paper, it is shown that the conditions imposed there to guarantee that the Kan extension of a cohomology theory is again a cohomology theory in fact also imply that the Kan extension commutes with stabilization. A construction, due to Adams, for completing a space with respect to a homology theory by using categories of fractions is generalized to triangulated categories, and it is shown that, for any family of primes P, the Adams completion of a space X with respect to the homology theory H ~ ∗ ( − ; Z P ) {\tilde H_ \ast }( - ;{{\mathbf {Z}}_P}) is the localization of X at P in the sense of Sullivan. Using this, the Kan extension of the restriction of a homology theory to the category of spaces having P-torsion homotopy groups is determined.

This paper constructs a minimal element in the partial order on the set of skew fields generated by a free algebra, and shows that the partial order contains a certain sub partial order. Examples of embedding free algebras in skew fields of heights one and two are also given. Let R be an integral domain (noncommutative) with identity. We will assume that the integral domain R is always embedded in some skew field (this need not be the case in general [2], [3], [6], [7]). There are many skew fields which contain R but in the commutative case there is only one field which is generated by R, namely the field of fractions. In the noncommutative case there may be several distinct skew fields generated by R [4, pp. 277]. For skew fields D1 and D2 generated by R, we say D1 > D2 if there exists a place from D1 to D2 which extends the natural isomorphism between the embeddings of R. This paper shows that this is a partial order on the set (P (we identify isomorphic embeddings) of skew fields generated by R and in the case where R is the free algebra on two generators we show that (P contains the subposet ? where C(x) is the unique maximal element [1, Theorem 27 ] of (P and Ki are distinct elements of (P with K2 minimal in (P. We also examine the height of an integral domain and give examples of embeddings of different heights of a free algebra in two skew fields. I. The partial order and the chain of domains Qi(K, ?(R)). DEFINITION. Let D be a skew field and 0 an isomorphism of R into D. R is fully embedded in D if the smallest sub skew field of D containing + (R) is D itself. We will denote a full embedding of (D, 4 (R)) by (D, q). DEFINITION. If D1 and D2 are two division rings we say 0 is a place from D1 to D2 if 0 is a homomorphism from a local subring S of D, onto D2 [local means the set of nonunits is an ideal ]. DEFINITION. For full embeddings (D, a) and (K, y) we say (D, a) > (K, Py) if there is a place q from D to K such that qV'(K) Doa(R) and f1 a (R) =,y-c'. Received by the editors October 9, 1970. AMS 1969 subject classifications. Primary 1646, 1615.

Our purpose is to extend Kolmogorov's theorem [5, Th. 10] on mutual subordination for univariate weakly stationary stochastic processes over the (discrete) group of integers to multivariate processes over any (Hausdorff) locally compact abelian (lca) group. This extension is given in Theorems (1.12) and (3.4) below. We shall lean heavily on the joint paper [10] on the decomposition of matricial measures, to which the present paper may be regarded as a sequel.

Introduction. Let H be a compact Hausdorff space, let E and F be locally convex topological vector spaces over the real or complex field where Fis Hausdorff. Let C(H, E) be the space of continuous functions from H into E with the topology of uniform convergence. The general problem is to find an integral representation theorem for a continuous linear transformation T from C(H, E) into F. The well-known Riesz representation theorem [3] gives a Stieltjes integral representation for T when H is a closed interval and E and F are the real numbers. Tucker [6] gives a Stieltjes' integral type representation for T in the case of H, a closed interval, and E and F, linear normed spaces. This paper was generalized by Uherka [7] to the case of H, a compact space, and E and F, linear normed spaces. Also see Swong's paper [5] in which H is a compact Hausdorff space and E and F are locally convex topological vector spaces. This last paper takes a fundamentally different approach from that of Tucker and Uherka, and Swong writes y'T as an integral where y' is in the topological dual F' of F. However, Swong is able to write T as an integral under various additional assumptions on E, F or T. The approach taken in this paper more closely follows that of Tucker and Uherka where a much smaller class of sets is employed in the definition of integral. The result is that the integral converges in the s00 topology, as compared to the weak topology in Swong's paper. And T is written as an integral.

In (1) Baer studied the following problem: If a torsion-free abelian group G is a direct sum of groups of rank one, is every direct summand of G also a direct sum of groups of rank one? For groups satisfying a certain chain condition, Baer gave a solution. Kulikov, in (3), supplied an affirmative answer, assuming only that G is countable. In a recent paper (2), Kaplansky settles the issue by reducing the general case to the countable case where Kulikov's solution is applicable. As usual, the result extends to modules over a principal ideal ring R (commutative with unit, no divisors of zero, every ideal principal).The object of this paper is to carry out a similar investigation for pure submodules, a somewhat larger class of submodules than the class of direct summands. We ask: if the torsion-free i?-module M is a direct sum of modules of rank one, is every pure submodule N of M also a direct sum of modules of rank one? Unlike the situation for direct summands, here the answer depends heavily on the ring R.

In a previous paper [1] Chang and the present author presented a system of infinite valued predicate logic, the truth values being the closed interval [0, 1] of real numbers. That paper was the result of an investigation attempting to establish the completeness of the system using the real number 1 as the sole designated value. In fact, we fell short of our mark and proved a weakened form of completeness utilizing positive segments, [0, a], of linearly ordered abelian groups as admissible truth values. A result of Scarpellini [8], however, showing that the set of well-formed formulas of infinite valued logic valid (with respect to the sole designated real number 1) is not recursively enumerable indicates the above mentioned result is the best possible.

1. Summary. Given below is a brief description of the main results of this paper: ?3. It is shown that the study of a Markov process X,(Q , v) where Q is a general state space, E a separable a-field of subsets of Q, and v a a-finite stationary measure for the process, can essentially be reduced to the study of a real-valued Markov process Yn(K, A, A) where K is a bounded interval or the entire real line, A the Borel sets of K, and A Lebesgue measure, stationary for the process. To show this a notion of isomorphism of processes is introduced. The main tool is the geometric-isomorphism theorem of Halmos and von Neumann. ?4. Processes Xn(R, A, A) are discussed where R is the real line, and A and A are as defined in ? 3. The selection of R rather than a bounded interval K is made because we are particularly interested in infinite stationary measures and the infinite case introduces a few technical difficulties. Treatment of the finite case is practically identical to that of the infinite case except that it is simpler. The basic result is that the process Xn(R, A, A) may be approximated in a certain way by processes (called k-processes) which are essentially Markov chains on a countable state space. The nature of the approximation is "weak" convergence of measures on function space. Such an approximation is useful because certain results difficult to prove directly for processes on a continuous state space may be easy to prove for Markov chains, and then carried over to the general process by using the k-processes. (Similar approaches have been used to approximate continuous time by discrete time processes; in this paper it is the state space rather than the time parameter which is discretized-time is discrete throughout.) Next, if the process is conservative and ergodic (that is, the shift T is; see [9] and Halmos' Lectures on ergodic theory) we derive a few conclusions about the nature of the k-processes. ?5. As an application of ?4, we give a probabilistic proof of Birkhoff's ergodic theorem. ?6. Using the isomorphism of ?3, we extend the results of the paper to processes Xn(Q, , v) for E separable.

In this paper, we obtain upper and lower bounds on the designed and actual distances of any sequence of extended primitive BCH codes of increasing lengths n and fixed rate R . The results of this paper are based on [1, ch. 12], which gives an exact expression for the rates of any sequence of extended primitive BCH codes of increasing length and fixed ratio of distance/length.

A method for testing and realization of threshold functions through classification of inequalities is suggested in the present paper. The standard procedure of testing and realization of threshold functions consists in solving a system of linear inequalities in which the unknowns are the different weights to be assigned to the variables of the functions. In the paper it is first shown that when the different inequalities involving the weights of the variables are represented in terms of their subscripts only, thon, depending on the number and on the sum of the subscripts appearing on either side, the entire set of inequalities of any function can be classified into nine distinct types. The sot of inequalities is next expressed in terms of the least weight and the other different incremental weights, the knowledge of which along with that of the types of inequalities, furnishes information regarding I-realizability of the function and on the assignments of values to the different weights for integ...

In a recent book, Parthasarathy provides limit theorems for sums of independent random variables defined on a metrizable locally compact abelian group. These results make heavy use of the metric assumption. This paper consists of a reworking of certain results contained in Parthasarathy to see what can be done with- out the metric restriction. Among the topics considered are: necessary and sufficient conditions for a limit law to have an idempotent factor; the relationship between limits of compound Poisson laws and limits of sums of independent random variables; and a representation theorem for certain limit laws. 1. Introduction. A recent book by Parthasarathy, (1), includes a number of limit theorems for uniformly infinitesimal arrays of measures which are defined on metrizable locally compact abelian groups. These theorems provide generalizations of the classical theorems for sums of independent real-valued random variables which appear in the treatise of Gnedenko and Kolmogorov, (2). The metric assumption is deeply involved in the proofs of the various results of (1) because of the extensive use of the Prokhorov Theorem (3) and the Shift- Compactness Theorem (4). These theorems appear as Theorem II.6.7 and Theorem III.2.2 respectively in (1). It can be shown (cf. (5)) that the assertions of these theorems are invalid on nonmetrizable locally compact abelian groups, even when the class of measures is restricted to regular Borel measures. This paper consists of a reworking of certain results of (1) to see what can be done without the metric assumptions. In essence, it is a study of probability on locally compact abelian groups without using the Prokhorov Theorem. It turns out that the familiar limit theorems still hold, with slightly different statements con- cerning the assumptions, and, of course, with rather different proofs. Three basic theorems are proved in this paper. The first basic theorem (Theorem 4.1) gives necessary and sufficient conditions for a limit law to have an idempotent factor. This theorem generalizes certain facts observed in (2) to hold for the real

Lattices of congruences are studied in section II.6 of Cohn [2]. Papers [5] and [6] by Munn deal with lattices of congruences on semigroups and conditions under which these lattices are modular. In [4] Lallement shows that the lattice of congruences on a completely 0-simple semigroup is semimodular, giving an alternative proof of the result, due to Preston [7], that the lattice of congruences on a completely 0-simple semigroup satisfies a certain chain condition which is a natural extension to arbitrary lattices of the Jordan-Dedekind chain condition for finite lattices. In this paper we show that this chain condition is also satisfied by the lattice of congruences on any semilattice, and we show that this lattice is upper-semimodular, and in fact satisfies the condition (1) if y >x/\y then xvy^x, for any lattice elements x and y. (For any elements a, b e L, a > b means that a 2; b and that a ^ b, and a > b {a covers b) means that a > b and there is no element xeL such that a > x > b. For a lattice L with a zero element 0, the atoms of L are those elements which cover 0.) The proofs are easy once we determine the atoms in the lattice of congruences on any semilattice.

Then the function f(z) has both a rational multiplication and a rational addition theorem. We propose to determine all such functions. Our problem is a generalization of one treated in a paper by Ritti1), in which all periodic meromorphic functions having a multiplication theorem were determined. This is equivalent to taking S(z) =z in (2). For this case, Ritt proved that \m\ ^ 1 ; and if \m\ > 1, then f(z) is a linear function of one of the functions eaz, cos(az+s), p(z+s); when gs = 0, g»2(z+j3); when g2 = 0, S>'(z+s) and j?3(z+j3). Here a is arbitrary while s is restricted to certain values. If \m\ =1, then it was shown that m is either —1 or a third, fourth or sixth root of unity. The forms for the function f(z) were also explicitly given in this case. Use will be made of the results quoted and also of the methods of Ritt's paper. We distinguish the following three cases : (A) m >1; (B) m ?1, with m not a rational root of unity; (C) m a primitive wth root of unity. For Case (A), which is the one usually considered in multiplication theorems, we can restate the problem in a way which shows its connection with the theory of a class of functions first systematically studied by Poincare(2). He provided an existence theorem for meromorphic functions satisfying (1) assuming that 7?(z) has a fixed point(3) a for which 7?'(a) has a modulus greater than unity. Thus our problem is equivalent to that of finding all Poincare functions having a rational addition theorem.

Let X 1, X 2,... be a sequence of independent, identically distributed random variables with P(X⩽0)=0, and such that pκ = ƒ0 ∞ x κ dP(x) u) for u⩾0. An alternate method of approximating Ψ(u, T) was presented in [10] by Olof Thorin and exemplified in [11] by Nils Wikstad. One of the purposes of this paper is to compare the two methods for two distributions of claims where the number of claims is a Poisson variate. The paper will also discuss the advantages and disadvantages of the two methods. We will also present a comparison of our approximate figures with the exact figures for the claim distribution

In § 1 we present conceptual material concerning the notion of a Gödel numbering of the partial recursive functions. § 2 presents a theorem about these concepts. § 3 gives several applications. The material in § 1 and § 2 grew out of attempts by the author to find routine solutions to some of the problems discussed in § 3. The author wishes to acknowledge his debt in § 2 to the fruitful methods of Myhill in [M] and to thank the referee for an abbreviated and improved version of the proof for Lemma 3 in § 2.In the literature of mathematical logic, “Gödel numbering” usually means an effective correspondence between integers and the well-formed formulas of some logical calculus. In recursive function theory, certain such associations between the non-negative integers and instructions for computing partial recursive functions have been fundamental. In the present paper we shall be concerned only with numberings of the latter, more special, sort. By numbers and integers we shall mean non-negative integers. Our notation is, in general, that of [K]. If ϕ and ψ are two partial functions, ϕ = ψ shall mean that (∀x)[ϕ(x)≃(ψx)], i.e., that ϕ and ψ are defined for the same arguments and are equal on those arguments. We consider partial recursive functions of one variable; applications of the paper to the case of several variables, or to the case of all partial recursive functions in any number of variables, can be made in the usual way using the coordinate functions (a)i of [K, p. 230]. It will furthermore be observed that we consider only concepts that are invariant with respect to general recursive functions; more limited notions of Gödel numbering, taking into account, say, primitive recursive structure, are beyond the scope of the present paper.

This paper is concerned with the problem of determining whether a set of sequences R \t', obtained by some given rule from a regular set of sequences R , is again a regular set. A number of such problems are solved in this paper and a basic technique is used which is easy to apply to problems of this type. This technique yields an explicit construction of the sequential machine which recognizes the sequences in R \t', if R \t' is a regular set. Among other things it is shown that: (i) the derivative of a regular set R with respect to any set of sequences W is a regular set, although a regular expression designating this set cannot in general be computed; (ii) the set of sequences obtained from a regular set R by removing “arbitrary halves” of the sequences in R and the set of these removed “halves” are both regular sets; (iii) the set of sequences, obtained from a regular set R by making no more than k changes in any m consecutive digits in sequences from R , is regular; (iv) the set of sequences which can be concatenated in one and only one way from sequences in R is regular.

In the preceding paper Bose, Shrikhande, and Parker give their important discovery of the disproof of Euler's conjecture on Latin squares. In this paper we show that their results can be strengthened to imply that N(n), the maximal number of pairwise orthogonal Latin squares of order n, tends to infinity with n. In fact there exists a positive constant c, such that N(n) > nc for all sufficiently large n.Our proof involves no new combinatorial insights, but is based entirely on a number-theoretical investigation of the following inequality due to Bose and Shrikhande.

Introduction. The subject matter of this paper belongs to the general theory of sets. The objectives here are the examinations of the structure of classes of sets which are closed under various finite and transfinite set operations and those transformations which preserve these operations. The major results of this paper are partial extensions of (a) the well known theory of a-algebras (e.g. [4]), (b) the work of E. Marczewski [1] on isomorphisms, (c) the work of R. Sikorski [2] on o-homomorphisms and (d) the work of A. Tarski [3] on fields of sets and set functions. Three classes of sets-the algebra, m-algebra and total algebra-as well as four transformations-the m-homomorphism, weak isomorphism, m-isomorphism, and total isomorphism-are studied. A class 3C of sets is an algebra if it is closed under finite set operations, i.e. addition of two sets and complementation. Analogically 3C is an m-algebra if X is closed under m-operations, i.e. complementations and addition of not more than m sets, where m is an arbitrary fixed cardinal number. X is a total algebra if it is closed under all operations, i.e. complementation and arbitrary addition. Two classes of sets, 3C and ?, are weakly isomorphic if 3C and ? considered as partially ordered by the relation of proper inclusion are similar. 3C and ? are m-isomorphic if they have the same properties from the point of view of m-operations; and, finally, X and ? are totally isomorphic if they have the same properties from the point of view of all operations on sets. The most important structure theorems concern the m-operations for mr_n&. The formulation of the corresponding finite and total structure theorems follow easily from the m-theorems and will be omitted except in special cases. The work is divided into two parts: (1) algebras of sets and (2) homomorphisms. In (1) the existence, composition and construction are treated; and, further, the relation between algebras and the "natural set units" is developed. In (2) are discussed some properties of m-additive, complementative transformations. Homomorphisms

Cyclically permutable codes have error-correcting properties which are invariant under arbitrary cyclic permutation of any of their code words. This paper summarizes the results of an empirical investigation of certain of these codes, which have parameters not covered by a previous paper of E. N. Gilbert. ^{1} These codes are thought to be nearly optimal. Estimates of the obtainable number of code words are given. The codes may be suitable for use in certain asynchronous multiplex communication systems.

and the sciences we are required to simulate random processes. The simulation is usually effected by a computer program which generates a non-random, deterministic sequence of numbers xl, x2, I * which is supposed to resemble a sequence of independent, random samples from the uniform probability distribution on the interval 0 ? x < 1. The purpose of this paper is to define some general properties of random sequences and to investigate certain deterministic sequences which have some or all of these properties. We shall ignore the limitation that a digital computer with a finite word-length and a finite memory, operating under a single stored program, can produce only sequences of limited precision which are ultimately periodic. This limitation is a kind of round-off error. We shall take as a model of a deterministic mechanism any of the stored-program digital computers now commonly used for scientific computation modified in a single respect: let the wordlength be infinite; let rational and irrational numbers x be recorded and computed with perfect precision. The fundamental problem approached in this paper is to construct an infinite, deterministic sequence xi which has every property shared by all infinite, random sequences of independent samples from the uniform distribution. Equidistribution is a first requirement of randomness. The sequence x" is equidistributed in 0 < x < 1 if, for 0 < a < b < 1

The notation which we use is that of a recent paper by one of us, and we quote results from that paper as they are required. It is known (see R, Theorem 1, for example) that the vector space k of all cusp-forms f(z) of even negative dimension – k (k ≥ 12), belonging to the full modular group Γ(1), possesses a finite basis of formswhere k is defined by (2·10) of R and the coefficients possess the following properties:for a prime p, where p is a positive integer.

This paper contains proofs of the closure of certain sets of rational functions in various spaces. Thus, for example, conditions are derived for the closure of the sequence (x2 + zv2)−1 in the space L2(0, ∞), and for the set\(\frac{{ct - z_v }}{{1 - \bar ctz_v }}\) in C(−1, +1). Analogous results are proved for other related sets of rational functions. Some of these results are new; others are new proofs of known theorems. The main point is that a uniform method is used throughout this paper. For a description of the method see article 1.

This paper relates to the Theory of Local Probability—that is, the application of Probability to geometrical magnitude. This inquiry seems to have been originated by the great naturalist Buffon, in a celebrated problem proposed and solved by him. Though the subject has been more than once touched upon by Laplace, yet the remarkable depth and beauty of this new Calculus seem to have been little suspected till within the last few years, when the attention of several English mathematicians has been directed to it, and results of a most singular character have been obtained. The problems on Local Probability which have been hitherto treated relate almost exclusively to points taken at random. The object of the present paper is to show how the Theory of Probability is to be applied to straight lines whose position is unknown, or, in other words, which are taken at random.

Seemingly disparate comparison theorems for nonselfadjoint elliptic equations have been established by C. A. Swanson and the author. The present paper establishes a class of comparison theorems involving an arbitrary vector valued function p(x). Special choices of p(x) yield both of the comparison theorems cited above. Several papers have recently concerned themselves with generalizations of Sturm's comparison theorem to nonselfadjoint elliptic equations. The first such theorem was due to Protter [1], and subsequent generalizations have been established by Swanson [2 ] and the author [3 ]. An interesting aspect of [2 ] and [3 ] is the seemingly disparate hypotheses which the principal theorems require. The purpose of this paper is to establish a class of comparison theorems for nonselfadjoint elliptic equations which contain the results of [2] and [3] as special cases. Our method consists of combining appropriate generalizations of techniques used by Picard [4] and Picone [5]. Let D denote a smooth bounded domain in Rn and let u(x) and v(x) be nontrivial real valued solutions of the elliptic equations ) a Eij(x)-+ 2 Z bi(x) + c(x)u = 0, -i=1 aXi axi i(l (x) (2) -E aij(x)+ 2 E Zfl(x) -+ y(x)v = 0. i, j= i 0xj d9xi i=1 d9xi It is assumed that the coefficients a,j(x), aij(x), bi(x) and fi(x) are of class C' in D while c(x) and y(x) are to be continuous in D7 (some of these requirements can be weakened somewhat as will be clear from the manipulations below). Furthermore (1) and (2) are to be uniformly elliptic in D. Received by the editors September 4, 1970 and, in revised form, November 16, 1970. A MS 1969 subject classifications. Primary 3511.

The object of this paper is to study the groups formed by the residue classes of a certain type of Kronecker modular system and some closely related generalizations of well-known theorems in number theory. The type of modular system to be studied is of the form 9) = (mn, mn-1, , Mnl, m). Here m, defined by (ml, m2, ***, Mk), is an ideal in the algebraic domain Q of degree k. Each term mi, i = 1, 2, *.., n, belongs to the domain of integrity of Qi = ( Q, xl, x2, * , xi), and is defined by the fundamental system ((1t/, t'2j), ***, i,t')). The various 1(i), j = 1, 2, *.., jI, are rational integral functions of xi with coefficients that are in turn rational integral functions of xl, x2, ... , xj_i, with coefficients that are algebraic integers in Q . In every case the coefficient of the highest power of xi in each of the 't') shall be equal to 1. We shall see later that the developments of this paper also apply to modular systems where the last restriction here cited is omitted, being replaced by another admitting more systems, these new systems in every case being equivalent to a system in the standard form as here defined. Any expression that fulfills all of the conditions placed upon each {() with the possible exception of the last one, we shall call a polynomial, and no other expression shall be so designated. This definition includes all of the algebraic integers of Q. Throughout this paper we shall deal exclusively with polynomials as here defined. The first part of this paper will contain the introduction with the necessary definitions and a discussion concerning the factoring of the system 9). The second section will then be devoted to setting up necessary and sufficient conditions that a set of residue classes belonging to 9) form a group when taken modulo 91. In the third section we shall study the structure of such a group with respect to groups belonging to certain modular factors of 9), besides

The nature of the eigenvalues of a square quaternion matrix had been considered by Lee [1] and Brenner [2]. In this paper the author gives another elementary proof of the theorems on the eigenvalues of a square quaternion matrix by considering the equation Gy = μȳ, where G is an n x n complex matrix, y is a non-zero vector in Cn, μ is a complex number, and ȳ is the conjugate of y. The author wishes to thank Professor Y. C. Wong for his supervision during the preparation of this paper.

In a previous paper (Crossley and Porter 1969) simple and explicit derivations were given for the first-order eigenvalue and eigenvector sensitivity coefficients for the Fundamental eigenproblem associated with the behaviour of linear dynamical systems governed by equations of the form [xdot] = Ax. In the present paper, similar techniques are used to derive the corresponding second-order eigenvalue and eigenvector sensitivity coefficients. The second-order theory is then applied to two examples in order to compare the results obtainable from this theory with the corresponding results obtainable from the first-order theory and from exact calculations.

In [3] Kemeny made the following conjecture: Suppose *Z is a nonstandard model of the ring of integers Z. Letand let F be the subgroup of those cosets ā which contain an element of infinite height in *Z. Kemeny then asked if the ring R = {a: ā ∈ F} is also a nonstandard model of Z. If so then Goldbach's conjecture is false because Kemeny also shows in [3] that Goldbach's conjecture fails in R.The papers [1] and [5] by Gandy and Mendelson show that R is not a nonstandard model of Z but we give here a simpler proof based on Mendelson's paper. Suppose R is a nonstandard model of Z. Then each positive number in R is a sum of four squares. Choose a in R so that a is a positive element of R of infinite height in *Z. Then since a is infinite in *Z, a − 1 is positive. Thus , xi ∈ R for i = 1, …, 4. Now each xi must be of the form ai + ni, where ai has infinite height in *Z and ni, ∈ Z.

1. Introduction. Finite non-Desarguesian projective planes have been known for all orders pf, where p is prime, n _ 2, and pn_ 9, except when p = 2 and n is a prime ? 5. In this paper a new class of projective planes is defined, having the orders 2n where n > 5 is not a power of two, thus establishing, in particular, the existence of non-Desarguesian planes of the missing orders. The new planes are coordinatized by semifields (sometines called division algebras, non-associative division rings, or distributive quasifields), which are algebraic systems satisfying the axioms for a field except with a loop replacing the multiplicative group. It is easy to show that finite proper semifields, i.e., semifields which are not fields, must have the orders pn where p is prime, n _ 3, and pn > 16. Proper semifields of these orders have been known to exist except as above, when p = 2 and n is prime; therefore the new systems show that proper semifields do exist for any order not excluded by simple arguments. A detailed treatment of the general theory of semifields and their relation to projective planes may be found in [3]. The manner in which these planes where discovered is perhaps as interesting as the planes themselves, since computers played a key role in the discovery. The author had received copies of two tables prepared by R. J. Walker (see [4]), which listed all commutative semifields of order 32 for which, if x is a generating element, x(x(x(x2))) = x + 1 or x2 + 1, respectively. There were 24 solutions in each table, and so it seemed plausible that a rule could be found yielding a correspondence between one set of solutions -and the other. A few hours of "cryptanalysis" did, in fact, result in the discovery of such a rule; and, since one of the solutions for the table x(x(x(x2))) = x2 + 1 was the field GF(32), the corresponding system for the other table could be written in a simple algebraic form [2]. After this, there was little difficulty showing that the same construction could be generalized to the construction of proper semifields of all orders 22k+1 with k > 1, and subsequent generalizations yielded the systems described in this paper. Therefore, we have an example in which

A classical question for modules over an integral domain is, "When is the torsion submodule t(A) of a module A a direct summand of AT' A module is said to split when its torsion submodule is a direct summand. Kaplansky has shown [-14] that if R is a Dedekind domain, then every module whose torsion submodule is of bounded order splits. The converse of this result has been shown by Chase [5]. Results of [15] and [3] show that every finitely generated module splits if and only if R is a Priifer domain. Finally, if every R-module splits, then Rotman has shown [19] that R must be a field. Recently, many concepts of torsion have been proposed for modules over arbitrary associative rings with identity. Almost all of these are special cases of "torsion theories" in the sense of Dickson [6]. Moreover, most of these torsion theories are hereditary (i.e., the submodule of a torsion module is torsion); and hereditary torsion classes are classes of negligible modules associated with a topologizing and idempotent filter of left ideals in the sense of Gabriel [12] (also see [17]). Some recent papers ([4], [7], and [11]) have dealt with splitting results for specific hereditary torsion theories over certain commutative rings. The main purpose of this paper is to continue the investigation of the splitting properties of a torsion theory of modules over a commutative ring. Some characterizations for the splitting of modules, whose torsion submodules have bounded order, are obtained (see definition of bounded order below and Theorems 2.2 and 4.6). In particular, these results generalize the abovementioned theorems of Chase [5] and Kaplansky [14]. Our results show that the splitting of modules whose torsion submodules have bounded order frequently forces non-zero torsion modules to have non-zero socles. This increases our interest in the smallest hereditary torsion class 6e containing the simple modules. The class ow has previously been used in the study of commutative Noetherian rings and (left) perfect rings (e.g., see [7] and [10] and their references). For a commutative ring R, we show that 6e(A) is a summand of each R-module A if and only if non-zero R-modules have non-zero socles. This generalizes the main results of [7] and [11]; moreover, in case R is a Dedekind domain, our result coincides with the above result of Rotman. We also examine the properties of a splitting hereditary torsion theory of modules over a local ring R (unique maximal left ideal). We show that if R possesses a non-trivial torsion theory (~--, #-) such that every finitely generated module splits, then R is an integral domain, and (Y-, ~ is Goldie's torsion

One reason for changing the representation of a game is to make it similar to a previously solved game. As a definition of similarity, people have proposed homomorphism-like structures. Two such structures are discussed in this paper and it is proven that they "preserve" winning strategies. They are incomparable in their strength and areas of applicability; i.e., neither of them is a special case of the other. The games to which these homomorphisms have been applied are positional games and decomposable games. The reason for concentrating on these two classes of games is that powerful methods for playing these games are known. For motivation, these methods are briefly described in the paper. The two homomorphisms discussed in this paper effectively extend the methods for playing positional and decomposable games to a much larger class of games. For several specific games which are neither positional nor decomposable, it is shown how they can be played as though they were positional or decomposable by using the homomorphisms.

In this paper the group of homeomorphisms of an arbitrary topological manifold is considered, with either the compact-open, uniform (relative to a fixed metric), or majorant topology. In the latter topology, a basis of neighborhoods of the identity is given by the strictly positive functions on the manifold, a homeomorphism being in the neighborhood determined by such a function if it moves each point less than the value of this function at the point. The main result of the paper is the proof of the local contractibility of the group of homeomorphisms in the majorant topology. Examples are easily constructed to show that this assertion is false for the other two topologies for open manifolds. In the case of a compact manifold the three topologies coincide. In conclusion a number of corollaries are given; for example, if a homeomorphism of a manifold can be approximated by stable homeomorphisms then it is itself stable.

A. Baker's theorems on linear forms in the logarithms of algebraic numbers imply information on the arithmetic nature of definite integrals of rational functions. This paper pro vides a convenient formulation of these implied results. 1. In his monograph on transcendental numbers, C. Siegel [3, p. 97] observed that the theory of transcendental numbers at that time (1949) was as yet unable to determine the arithmetic nature of such integrals as rl dx I / r\ Jo +X = _ 3 log2++/) More recently (1966), A. Baker mentions that his theorems on linear forms in the logarithms of algebraic numbers [1] imply the transcendence of the above integral. Baker's results have led to further "folk theorems" on the arithmetic nature of certain integrals but no general statement appears to have been formulated. In this paper we establish a convenient criterion for determining the arithmetic nature of such definite integrals of rational functions as we are able to deal with by virtue of currently established theorems. THEOREM. Let P(z), Q(z) denote polynomials with algebraic coefficients and with no common polynomnialfactor. Denote by a1, a2, an the distinct zeros of Q(z) and by ri, r2, * * *, rn the residues respectively at the poles of the rational function P(z)/Q(z). Further let P be some contour in the complex plane for which the definite integral (' P(z) dz r Q(z) exists, and suppose that r is either closed, or has endpoints which are algebraic or infinite. Received by the editors February 20, 1970 and, in revised form, October 5, 1970. AMS 1969 subject classifications. Primary 1032.

This paper considers the probabilities of first emptiness of two dams in parallel, both subject to a steady release at constant unit rate, and fed by a discrete additive input process such that unit inputs are always directed to the dam with lesser content. The problem is equivalent to that of the single dam fed by two ordered inputs, and a recurrence relation for the probabilities of first emptiness in this process is obtained. Equations for the generating functions of the probabilities are derived, and a formal solution to these is given. A more convenient method of evaluating probabilities of first emptiness is found by reducing the process to an associated occupancy problem; it is shown how the probabilities of first emptiness for Poisson inputs are then obtained by a rapid computational procedure. The paper concludes with a general formulation of the problem when the times of arrival for two ordered non-negative inputs of random size form a Poisson process.

In this paper we shall derive a concise formula for the number of Euler graphs on n labelled nodes and k edges. An Euler graph is a connected graph in which every node has even valency, where by the valency of a node is meant the number of edges which are incident with that node. Throughout most of the paper we shall be dealing with graphs whose nodes have even valencies but which may or may not be connected. For convenience we shall refer to these graphs as Euler graphs, although the usage is not, strictly speaking, correct. We shall impose the condition of connectedness in § 4.

Consider a sequence of independent random variables $x_1, x_2, \cdots, x_k, \cdots$ with mean 0 and variance $\sigma^2_k$. Let $S_n = (x_1 + \cdots + x_n)/s_n$ where $s^2_n = \sigma^2_1 + \cdots + \sigma^2_n$. The classical forms of the central limit theorem state that, with certain assumptions, the distribution function $F_n(x)$ approaches the Gaussian distribution $\Phi(x) = \frac{1}{\sqrt{2\pi}} \int^x_{-\infty} e^{-u^{2/2}} du.$ Berry [1] and Essen [3] have studied the behavior of $M_n = \underset{-\infty < x < \infty}\sup |F_n(x) - \Phi(x)|$ and in their main theorems have obtained bounds on $M_n$ which involve the moments of $x_k$ through the third. More generally consider a system of random variables $(x_{nk}), k = 1, 2, \cdots, k_n; n = 1, 2, \cdots$ such that for each $n$, the variables $x_{n1}, \cdots, x_{nk_n}$ are independent. Let $S_n = x_{n1} + \cdots + x_{nk_n}$ and again let $F_n(x)$ be the distribution function of $S_n$. From a well known theorem of Khintchine [5] it follows that if the random variables $x_{nk}$ are infinitesimal (i.e., $\lim_{n \rightarrow \infty} \max_{1 \leqq k \leqq k_n} P\{|x_{nk}| > \epsilon\} = 0$ for every $\epsilon > 0$) then the class of possible limiting distributions of $F_n(x)$ coincides with the class of infinitely divisible distributions. Let $F(x)$ be any infinitely divisible distribution function and let $M_n = \sup_{-\infty < x < \infty} |F_n(x) - F(x)|$. In this paper we obtain bounds on $M_n$ in the case where $F(x)$ and the $x_{nk}$ have finite second moments. It is shown that under necessary and sufficient conditions for $F_n(x)$ to approach $F(x)$, the bounds on $M_n$ obtained approach zero as $n$ becomes infinite. Throughout the paper, given the system $(x_{nk})$ we shall let $F_{nk}(x), \varphi_{nk}(t), \mu_{nk}$, and $\sigma^2_{nk}$ be the distribution function, characteristic function, mean, and variance respectively of $x_{nk}$, and $F_n(x), \varphi_n(t), \mu_n$, and $\sigma^2_n$ have the same meaning for the random variable $S_n$.

In [2] J. D. Weston proves that given any separable Banach space Y, there exist a normed linear space X and a compact one-one linear operator which maps the conjugate space X' onto a subspace dense in Y. It is the purpose of this paper to solve the following problem. Given normed linear space X and Banach space Y, under what conditions does there exist a one-one compact linear map from X onto a subspace dense in Y? Necessary and sufficient conditions for such an operator to exist are given (cf. (C) of the theorem below). We first introduce some notations. Suppose X and Y are normed linear spaces. Then 63(X, Y) (resp., ,c(X, Y)) is the space of all bounded (resp., compact) linear maps from X to Y. 630(X, Y) (resp., aCO(X, Y)) is the set of all one-one maps in 63(X. Y) (resp., 3C(X, Y)). 63d(X, Y) (resp., aCd(X, Y)) is the set of all maps in 63(X, Y) (resp., 3C(X, Y)) with range dense in Y. 63,d(X, Y) =630(X, Y)n63d(X, Y), and aCo,d(X, Y)=aCo(X, Y)nC\d(X, Y). Finally, 0 is the void set. If X is a normed linear space and A is a subset of X', then A is total if and only if for each x $0 in X there exists an x' in A such that x'x$0. The following preliminary remarks are easily verified. The first of these gives alternative ways of stating a condition arising prominently in the rest of the paper. (i) If X is a normed linear space, then X' contains a countable total subset if and only if X' is separable with respect to the w* topology and also if and only if X' contains a total separable linear subspace. (ii) If X is a separable normed linear space, then each of the conjugate spaces X' and X" contains a countable total subset. (iii) If X and Y are normed linear spaces, 6(30(X, Y) $0, and Y' has a countable total subset, then X' has a countable total subset. DEFINITION. Suppose X is a Banach space, and suppose Xk is in X and ek is a real number for k=1, 2, * 1. Then {Xk}l iS {ek}lindependent if and only if (a) IIEkXk II < oo and

Let f be a real-valued function defined and finite on sets from a family F \mathcal {F} of bounded measurable subsets of Euclidean n-space such that if T ∈ F T \in \mathcal {F} , the measure of T is equal to the measure of the closure of T. An earlier paper [Trans. Amer. Math. Soc. 145 (1969), 439-454] considered the questions of finiteness and boundedness of the upper and lower regular derivates of f and of the existence of a unique finite derivative. The present paper is an extension of the earlier paper and considers the summability of the derivates. Necessary and sufficient conditions are given for each of the upper and lower derivates to be summable on a measurable set of finite measure. A characterization of the integral of the upper derivate is given in terms of the sums of the values of the function over finite collections of mutually disjoint sets from the family.

Recently a great deal of progress has been made in the study of groups of automorphisms of operator algebras and states invariant under the automorphisms. We are then given a C*-algebra Q!, a group G of automorphisms of GY satisfying certain requirements and want to draw conclusions about the invariant states and the fixed point algebra in LY under G. The present paper is concerned both with this problem and the converse one. In the latter case we are given a state p of a C*-algebra O? and want to find a group G of auto- morphisms of G! with respect to which p is invariant, such that the above mentioned theory is applicable. In some cases we find a natural abelian subalgebra 9 of OZ, which is the fixed points of a group G and a canonical G-invariant projection @ of U onto 9 such that p = (p ) 9) 0 @. Then we can prove theorems on integral decompo- sition of states. The main portion of this paper rests largely on the results of KovPcs and Sziics [13]. Given a von Neumann algebra 9 and a group G of automorphisms (i.e. *-automorphisms) of 9, we say .J% is G- finite if there is a family 5 of normal G-invariant states of 9 such that if A is a nonzero positive operator in 9 then for some p in F, p(A) # 0. We shall for simplicity say that

In recent years, problems related to the characterization of the normal distribution from the property of stochastic independence of linear functions of independent random variables have been investigated by various authors. The most general result in this direction is that obtained independently by Darmois [4] and Skitovich [14], who proved that if there exist two linear functions $$X = \sum^n_{j = 1} a_jx_j \text{and} Y = \sum^n_{j = 1} b_jx_j\quad\text{with} a_ib_j \neq 0$$ $(j = 1, 2, \cdots n)$, such that they are stochastically independent, where $x_1, x_2, \cdots, x_n$ are $n$ independently (but not necessarily identically) distributed proper random variables, then each $x_j$ is normally distributed. Their methods of proof are similar in nature, both being based on the use of characteristic functions, without any assumption about the existence of moments. The same theorem has been proved independently by Basu [1], under the assumption that the random variables are identically and independently distributed and have finite moments of all orders. This result is also obtained independently by Lukacs and King [11], under the assumption that the random variables are independently (but not necessarily identically) distributed, each having finite moments up to order $n$, and by Linnik [10], under the assumption that the random variables have only finite variances. The special case of this theorem for $n = 2$ was proved earlier independently by Kac [6], Gnedenko [5], and Darmois [3], without any assumption about the existence of moments. Thus we see that the problems on the consequences of stochastic independence of linear statistics have been exhaustively studied. Hence the question that naturally arises is whether similar investigations into the distribution laws of the random variables are possible under the assumption of a suitable type of stochastic dependence of the linear statistics. In this direction, the author [8] has derived a characterization of the stable law with finite expectation from the property of linearity of regression of one linear statistic on the other for the case $n = 2$. The author [7] has also obtained some characterizations of the normal distribution from the consequence of the linearity of the multiple regression of one random variable on several others, when the variables have the linear structural setup as in the case of the bi-factor theory of Spearman. For the formulation of the problem investigated in the present paper, we require a precise definition of the terms conditional distribution, linearity of regression, and homoscedasticity. Let $F(x, y)$ and $F_0(x)$ denote respectively the distribution function of the two-dimensional random variable $(x, y)$ and the marginal distribution function of $x$. Then we define the conditional distribution function of $y$ for fixed $x$ by $F_x(y)$ so that it satisfies the relation \begin{equation*}\tag{1.1}\int^x_{-\infty} F_\xi(y) dF_0(\xi) = F(x, y).\end{equation*} In the present investigation, the following assumptions on the distributions of the random variables will be made: Assumption 1. The conditional distribution of $y$ for fixed $x$ as defined in (1.1) is assumed to exist, wherever needed. Assumption 2. Each of the random variables concerned has a finite second moment. This assumption allows us to take derivatives of the characteristic functions of the corresponding random variables up to and including the second order. Assumption 3. Without any loss of generality in the proof, it is assumed that the expectation of each of the random variables concerned is equal to zero. The role of these assumptions is to ensure the existence of the expectation and the variance of the conditional distribution of $y$ for fixed $x$, which may be defined respectively as \begin{align*}\tag{1.2}E_x(y) &= \int^\infty_{-\infty} y dF_x(y), \\ \tag{1.3}V_x(y) &= \int^\infty_{-\infty} \{y - E_x(y)\}^2 dF_x(y) \\ &= S_x(y) - \{E_x(y)\}^2, \end{align*} where $$S_x(y) = \int^\infty_{-infty} y^2 dF_x(y).$$ In this case, the regression of $y$ on $x$ is said to be linear, if the relation \begin{equation*}\tag{1.4}E_x(y) = \beta x\end{equation*} is satisfied for all $x$, except for a set of probability measure zero, as the expectations of both the random variables $x$ and $y$ are already assumed to be zero. The constant $\beta$ in equation (1.4) is called the coefficient of regression of $y$ on $x$. Similarly the conditional distribution of $y$ for fixed $x$ is said to be homoscedastic, if the conditional variance $V_x(y)$ as defined in (1.3) is a constant $\sigma^2_0$ not involving $x$. Thus if the regression of $y$ and $x$ is linear and given by $\beta x$ and the conditional variance of $y$ for fixed $x$ is a constant $\sigma^2_0$, we have the relation \begin{equation*}\tag{1.5}S_x(y) = \sigma^2_0 + \beta^2x^2\end{equation*} to be satisfied for all $x$, in addition to the condition (1.4). For simplicity in notation, throughout the present paper we shall use the term the conditional distribution of $y$ for fixed $x$ is L.R.H. ($\beta, \sigma^2_0)$ meaning thereby that the regression of $y$ on $x$ is linear and given by $\beta x$ and that the conditional variance of $y$ for fixed $x$ is $\sigma^2_0$, being free of $x$, i.e., equivalent to the statement that both the relations (1.4) and (1.5) are simultaneously satisfied. In the following sections we shall derive some characterizations of the normal distribution from the property of linearity of regression and homoscedasticity of suitable linear statistics. The main theorem (Theorem 3.2) is given in Section 3. The proof of this theorem uses as a starting point a very simple set of necessary and sufficient conditions for the linearity of regression and homoscedasticity (Lemma 2.1) and finally a theorem of Linnik (Lemma 2.3) on an analytical extension of Cramer's theorem [2] on the normal law. Several important corollaries are deduced in the subsequent section.

This paper contains a complex version of Theorem 5.4 (Dominated Extension Theorem) from the paper “Structure in real Banach spaces” by E. Effros and the present author (to appear in Annals of Mathematics).

A class of decision problems concerning $k$ populations was considered in [1] and it was shown that a particular decision rule is the uniformly best `impartial' decision rule for many problems of this class. The present paper provides certain improvements of this result. The authors define impartiality in terms of permutations of the $k$ samples rather than in terms of the $k$ ordered values of an arbitrarily chosen real-valued statistic as in the earlier paper. They point out that (under conditions which are satisfied in the standard cases of $k$ independent samples of equal size) if the same function is a sufficient statistic for each of the $k$ samples then the conditional expectation of an impartial decision rule given the $k$ sufficient statistics is also an impartial decision rule. A characterization of impartial decision rules is given which relates the present definition of impartiality with the one adopted in [1]. These results, together with Theorem 1 of [1], yield the desired improvements. The argument indicated here is illustrated by application to a special case.

The weak sequential compactness of reflexive Banach spaces is used to explain the fact that certain ill-posed, linear problems become well-posed if the solutions are required to satisfy a prescribed bound. Applications are made to the computability of solutions of ill-posed problems associated with elliptic and parabolic partial differential equations. 1. Introduction. In defining the concept of a well-posed problem, J. Hadamard (1) recognized that the existence and the uniqueness of the solution are not enough to ensure the computability of the solution. Numerical computation requires that the solution have stability, or continuous dependence on the data. F. John (2), (3), (4) and other authors have shown that, for certain problems whose solutions have existence and uniqueness but not stability, if the solutions considered are required to satisfy a prescribed bound, stability results. An extensive bibliography appears in the paper (5) by L. E. Payne. All these results depend on precise definitions of "continuous dependence on data" which are suitable for the particular problems discussed. Much of the work preceding the present paper contains not merely qualitative statements of stability, but quantitative inequalities satisfied in particular problems. The present paper contains only the qualitative observation that an elementary result, Theorem 1, on the weak topology of reflexive Banach spaces yields the stability of uniformly bounded solutions of a large class of linear functional equations. Several new applications are made, including a theorem on the stability of solutions of the final-value problem corresponding to the general initial-value problem discussed in the book (6) by R. D. Richtmyer. 2. Notation and Review. We shall use the letter T to stand for a bounded linear operator mapping a Banach space B1 into a Banach space B2. We suppose that the range TB1 is dense in B2, and that Tz = 0 only if z 0. If B is a Banach space, the notation B* represents the conjugate Banach space of bounded linear functionals f mapping B into the real numbers. If x E B and f E B*, the notation (x, f) represents the number which results when f is applied to x. If B** = B, the space B is called reflexive (7). Gilbert spaces are reflexive Banach spaces. If 1 < p < D, the space L' of measurable functions x(t) for which rb.

1. Introduction. We give here a solution of the isomorphism problem for finitely presented algebras in various classes of multiplicative systems. That is, we describe an algorithm for deciding whether two loops (quasigroups, groupoids, inverse property loops, etc.) given by finite sets of generators and defining relations are isomorphic. We give the details only for loops but with trivial changes, the procedure described here applies also to a number of other systems with a nonassociative multiplication which have been extensively discussed in the literature. The basic ideas used are those of an incomplete algebra [2 ; 4] and of a minimal incomplete algebra [6; 7]. With a finitely presented loop, we associate a finite minimal incomplete loop which freely generates the loop and we prove that two finitely presented loops are isomorphic if and only if the associated finite minimal incomplete loops are isomorphic. In other words, a finitely presented loop has (to within isomorphism) a unique finite minimal incomplete loop, which we call a basis of the loop, which freely generates the loop and, furthermore, a basis can be constructed in a finite number of steps from the generators and defining relations of the loop. Clearly, this solves the isomorphism problem for loops. In §2, we recall from [4] those properties of loops freely generated by incomplete loops needed in the rest of the paper. In §3, the crucial properties of minimal incomplete loops are derived and then used, in §4, to give a solution of the isomorphism property for loops. In the final section, we discuss the changes which have to be made to apply the ideas of this paper to the other multiplicative systems mentioned above. In addition, we obtain some results on the structure of finitely presented loops which follow from the properties of minimal incomplete loops. For example, a finitely presented loop which does not have a free loop as a free factor has only a finite number of automorphisms. Comparatively little is known about the isomorphism problem for various classes of algebras. It is solvable, of course, for finitely presented abelian groups and recently it has been shown to be unsolvable for finitely presented semigroups and groups. At the time the results of this paper were originally obtained [6], the author felt that similar methods could probably be applied to other varieties of algebras. In particular, it seemed plausible that the isomorphism problem could be solved, using the same ideas, for varieties satisfying the condition that incomplete

The topology of abstract 2-metric (area-metric) spaces has been the object of study in recent papers of Gähler (1) and Froda (2). The geometric properties of such spaces, however, have remained largely untouched since the initial work of Menger (3). As in ordinary metric spaces, a notion of 2-betweenness, or interiorness, can be easily defined in 2-metric spaces. In abstract metric spaces the betweenness relation is characterized among all relations defined on each triple of points of every metric space by six natural properties (4, pp. 33-40; 5). The purpose of this paper is to prove a similar theorem characterizing the relation of 2-betweenness in 2-metric spaces.

In an earlier paper, Maurer and Rhodes (1965) proved the following result: Let G be a finite simple nonabelian group and f: G" --> G a map of the set of n-tuples of elements of G into G. Then there is an integer m and an m-tuple t ( t I , t 2 , . . . , t m ) of elements of G 0 { 1 , . . , n } such that for all g = (g l , . . . , g , ) C G ~, f ( g ) --E ( t , g) , where E ( t , g) is the product gl p . . . g~' C G of elements g t C G defined by letting for eachj C {1, . . . , m} g j = t~ if tj C G org j = gtj i f t~ C {1, --,n}. While this result is of some intrinsic algebraic interest, the motivation behind the result was its application to automata theory. I t is the purpose of this paper to explore this connection, in particular, we set forth here a new method for the generation of boolean functions. Currently, the maior method used for that purpose is one developed by Shannon which, while well-known, can require in hardware realizations circuitry too cumbersome for some applications. The alternative method set forth here involves the use of a fixed simple nonabelian group machine (which can be realized in hardware by n flip-flops, where 2" exceeds the order of the group involved) and, for each boolean function, a storage sequence, the length of which depends upon the boolean function and the group machine used. A few examples will convince the reader that there is little correlation between the length of storage sequences and the amount of circuitry necessary to reMize the corresponding functions

This paper is a continuation of two papers [4], [5] and brings out the solution of the ballot problem in its general form.In [5], Narayana has considered a generalised occupancy problem which can be viewed as a problem in compositions of integers. In what follows, we use the definitions of [6].

The importance of multi-dimensional t-distributions has been indicated in a series of the papers on successive process of statistical inferences by T. KITAGAWA [1] •--[3] and some of numerical considerations have been given by C. W. DUNNETT and M. SOBEL [11 and [2]. In 1953 T. KITAGAWA [4] and [5] gave three different formulations of empirical functions and discussed statistical inferences concerning concurrence-functions and their derivatives. Among others the uses of n-dimensional t-distributions were indicated in his formulation of the empirical functions of the type AD verifying to be fundamentally important. The object of the present paper is to provide for various auxiliary tables which are indispensable for the uses of ndimensional t-distributions in this sense. We shall, however, restrict ourselves to the special case when the following three conditions are satisfied: (i) the n points of observations 0 < tl< t,< ••• < t„ < 1 is equi-distributed ; (ii) the numbers of observations are equal, i.e., r, r2= ••• r„= r; (iii) c),(t,; n) } (1, = 0, 1, 2, •-• , n —1) is the orthogonal normalised polynomial system associated with ti In view of Theorems 1. 1 and 1. 2 in T. KITAGAWA [4] our tasks of constructing auxiliary tables were scheduled in the following steps

that this question is intimately connected with the existence of certain kinds of simple radical rings; it is this connection which we wish to emphasize in this paper. We define a left valuation ring to be a nonzero ring R in which, for every x, y in R, there exists a u in R such that either x = uy or y = ux. A right valuation ring is defined similarly. (A valuation ring in the sense of [1] above is a left and right valuation ring, in the sense of this paper, without divisors of zero.) There do exist left valuation rings which are not right valuation rings. An example can be given modifying [2, ?6, p. 219]. The construction is, however, omitted. We now prove some results valid for left valuation rings to lead up to the connection with simple radical rings. The lesser ones are called simply results, the more important ones theorems. RESULT 1. The left ideals of R are linearly ordered by inclusion.

In a foundational paper on the theory of automata A. W. Burks and H. Wang (1957) conjectured that a certain complexity measure involving the size of the strong components of a logical net formed a hierarchy for net behavior. This conjecture was established by Rhodes and Krohn. In this paper a strengthened version of the conjecture is proved by establishing that any logical net can be interpreted as a series-parallel composition of nets associated with its strong components. Some properties of the periodic behavior of machines, shown to be preserved under simulation and composition operations, are used to complete the proof. The relationship of this approach to algebraic proofs of series-parallel irreducibility is discussed.

Let N be a finite set of n elements. A collection ﹛S1, S2, … , Sm﹜ of subsets of N is called a determining collection if an arbitrary subset T of N is uniquely determined by the cardinalities of the intersections Si ⋂ T, 1 ≤ i ≤ m. The purpose of this paper is to study the minimum value D(n) of m for which a determining collection of m subsets exists.This problem can be expressed as a coin-weighing problem (1; 7).In a recent paper Cantor (1) showed that D(n) = O(n/log log n), thus proving a conjecture of N. J. Fine (3) that D(n) = o(n). More recently Erdös and Rényi (2), Söderberg and Shapiro (7), Berlekamp, Mills, and Leo Moser have independently found proofs that D(n) = O(n/log n).