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In an earlier paper, the author introduced the notions of N-solutions and IN-solutions of algebraic differential equations (ADE's). Here it is shown, in contradistinction to the situation for Cn solutions, that every N-solution of an ADE is an N-solution of an autonomous ADE. The corresponding result also holds for IN-solutions.

In a recent paper [1] MM. Arnal and Pinczon have classified all complex irreducible representations (ρ, V) of sl(2) having the property (P) that there exists a non-zero element x∈sl(2) such that ρ(x) admits an eigenvalue. It is the purpose of this note to demonstrate, by example, that there exist irreducible representations of sl(2) which do not have property (P). As usual, we consider sl(2) embedded in its universal enveloping algebra U and identify the representations of sl(2) and U.

Let (X, d) be a metric space and Ti(i=l, 2) be self mappings of X. The purpose of this paper is to investigate the fixed and common fixed points of Ti, when the pair Ti(i=l, 2) satisfies a condition of the following type:(1)

If M is a simply-connected 4-manifold with boundary, let D(M) denote its double MU∂M(-M). If M is closed, let D(M) denote M#-M. In either case, D(M) is a simply-connected 4-manifold of index zero, and so by a theorem of Wall [8], M#k(S2xS2) must be standard for k sufficiently large, where by standard we mean diffeomorphic to the connected sum of copies of S2 x S2 and S2×S2, the non-trivial S2 bundle over S2 (which is itself diffeomorphic to ℂP2#-ℂP2 [7]). In this paper we give abound on k, in the case where M has no 3-handles.

The purpose of this paper is to extend to almost periodic (a.p.) solutions a stability result on the periodic solutions of the autonomous equationx′ = F(x),(cf. Coppel [1], p. 82 or Coddington and Levinson [2], p. 323.)

The problem of finding bounds for the number h(n) of Hamiltonian circuits on the n-cube has been studied by several authors, (1), (2), (3). The best upper bound known is due to Larman (5) who proved that .In this paper we use a result of Nijenhuis and Wilf (4) on permanents of (0, 1)- matrices to show that for n≥5where τ, a and c are constants.

In this paper, we prove that a real valued bounded function, defined on a metric space and uniformly continuous is the uniform limit of a sequence of Lipschitzian bounded functions.As a consequence, a new criterion for the weak convergence of probabilities is given.

In a recent paper [3] Meir and Sharma introduced a generalization of the Sα- method of summability. The elements of their matrix, (ank), are defined by(1)where is a sequence of complex numbers. if 0 < αj < l for each j = 0, 1, 2,… then ank≥0 for each n = 0, 1, 2,… and k = 0,1,2,…

In this paper, we construct, for any open cover of a metric space, a partition of unity consisting of a family of uniformly continuous functions.

Let R be a ring such that, for each element a of R, there exists a positive integer n(a) > 1, depending on a, such that an(a)=a. Jacobson proved that such a ring R is necessarily commutative and the purpose of this paper is to give a proof of Jacobson’s Theorem that does not involve the use of the axiom of choice.We take this opportunity to point out that all rings are assumed to be associative and that nothing beyond elementary ring theory is assumed in the proof to follow. Such ring theory can be found, for example, in [1].

ABSTRA~. Let A be a finite dimensional algebra (not necessarily associative) over a field, whose automorphism group acts transitively. It is shown that K = GF(2) and A is a Kostrikin algebra. The automorphism group is determined to be a semi-direct product of two cyclic groups. The number of such algebras is also calculated. All algebras are assumed to be finite dimensional but not necessarily associative. If A is an algebra over a field K let Aut (A) denote the group of algebra automorphisms of A. We say that A has a transitive automorphism group if Aut (A) acts transitively on the non-zero points of A. An algebra A is said to be non-trivial if dim A > 1 and A2 Z 0. We show that if A is a non-trivial algebra with a transitive automorphism group then K = GF (2), A is a Kostrikin algebra and Aut (A) is the semi-direct product of two finite cyclic groups. THEOREM 1 : IfA is a non-trivial algebra with transitive automorphism group over a jield K then K = GF(2). PROOF: First assume that K is infinite. Let a, b E A \ (0). Then there exists an a E Aut(A) such that a(a) = b and this implies that aLaa-' = Lb where La and Lb indicate left multiplication by a and b respectively in A. That is, La and Lb are similar. But in particular, we may allow b = ha for any nonzero h E K. Now comparing the characteristic polynomials of La and L,, = hLa it is easy to show that La is nilpotent. Similarly Ra is nilpotent and so A is a special nil algebra as defined in [7]. It follows from Theorem 2 of the above paper that AZ = 0. Now assume that K is finite. Then Aut(A) certainly acts transitively on the one dimensional subspaces of A and so the results of Shult [5] imply that K = GF(2). DEFINITION: Let K = GF(2") and p be any faed element in K. Let : K X K += K be the map defined by x y = p(xy)'"-'. Let A(n, p) denote the algebra over GF(2) obtainedfrom K by replacing the usual multiplication in K by the map 0. We call A(n, p) a Kostrikin Algebra since these algebras were investigated by

The purpose of this note is to discuss the relationship between the concepts of approximate continuity and almost continuity of real functions of a real variable.Contrary to the papers of T. Husain [1], [2], [3], we shall show that there is little relationship between these concepts, that approximate continuity does not imply almost continuity without additional restrictions and we shall give one such restriction.

In [1] Demidovic considered a system of linear differential equationswith A(t) continuous, T-periodic, odd, and skew symmetric. He proved that all solutions of (1) are either T-periodic or 2T-periodic0 In [2] Epstein used Floquet theory to prove that all solutions of (1) are T-periodic without the skew symmetric hypothesis. Epstein's results were then generalized by Muldowney in [7] using Floquet theory. Much of the above work can also be interpreted as being part of the general framework of autosynartetic systems discussed by Lewis in [5] and [6]. According to private correspondence with Lewis it seems that he was aware of these results well before they were published. However, it appears that these theorems were neither stated nor suggested in the papers by Lewis.

Utumi has shown [3, Claim 5.1] that for a certain class of rings the associated maximal rings of right quotients are isomorphic to the endomorphism rings of modules over division rings. We shall prove a generalization of this theorem and then show how it is obtained as a corollary. The following proofs do not depend on Utumi's paper; instead, they make extensive use of results proved in [1]. The terminology and notations employed here are the same as in [1].I wish to thank Dr. B. Banaschewski for his suggestions and helpful criticism.LEMMA: If J is a left ideal with zero left annihilator in a ring R then a maximal ring of right quotients of R is also a maximal ring of right quotients of J.

Throughout this paper, (X, T, π) is a topological transformation group [1], L={x∊X:xt=x for some t∊{e}} and 0=X—L is nonempty; standard topological concepts are used as defined in [2].The problem to be considered here has been studied in [3] and [6]. In [3], X is assumed to be a compact metric space, and each t e T satisfies a convergence condition on certain subsets of X. Under these conditions, Kaul proved that if T is equicontinuous on 0, then the group properties of discontinuity, proper discontinuity, and Sperner's condition (see Definition 1) are equivalent.

In [1] we constructed a (G, σ)-space and determined a condition which is both necessary and sufficient for the (G, σ)-product of the vectors v1, v2, …, vn to be zero. The purpose of the present paper is to give a criterion for v1 Δ v2 Δ … Δ vn to be zero, in the particular case when V is a unitary space and the group G belongs to a special class of groups which we shall define below.

In an earlier paper [2], we raised the question of determining the minimum span of the kth derivative of a polynomial with real zeros having a given span. More precisely let πn, s denote the class of polynomials , with x1 ≤ x2 ≤ … ≤ xn, and the span σ(P) ≡xn - x1 = 2s (fixed).

In his paper ″Indexed systems of neighborhoods for general topological spaces″ (Amer. Math. Monthly 68, (1961), 886–893), A.S. Davis defined a hierarchy of what he called regularity axioms. The R1 -axiom is independent of both T0 and T1, but is strictly weaker than T2. In this note, we propose to study the properties of the spaces satisfying the R1 -axiom. In particular, we will show that in many well-known results, the hypothesis can be weakened from T2 to R1, which is part of our motivation in studying R1 -spaces.

In [4], A. Sade defines the singular direct product for quasigroups. In this paper we use the singular direct product to construct quasigroups satisfying the identity x(xy)=yx. In particular, we show that the singular direct product preserves the identity x(xy)=yx.

In this paper we give necessary and sufficient conditions that a continuous transformation f: A→A of a metric space A with the metric r should be a contraction with respect to an equivalent metric s. This is the solution of a problem stated by J. S. W. Wong [2].

In this note we state and prove the followingAny equiaffinity acting on the points of an n-dimensional vector space (n ≥2) leaves invariant the members of a one parameter family of hypersurfaces defined by polynomials p(xl…,xn)=c of degree m ≤n.The theorem, restricted to the real plane, appears to have been discovered almost simultaneously by Coxeter [4] and Komissaruk [5]. The former paper presents an elegant geometric argument, showing that the result follows from the converse of Pascal's theorem. The present approach is more closely related to that of [5], in which the transformations are reduced to a canonical form.

This note is to complement the earlier paper on the same subject (Canad. Math. Bull. 3 (1960), 41–57) in two points. The first part presents a simpler proof of the minimum property (cf. 1. c. section 3) of the orthogonal functions Ψv(x) (cf. 1. c. p. 46). In the second part we introduce another orthogonal system of sectionally linear functions χo(x), …, χn(x) which leads to a particularly simple interpolation formula. These functions appeared, mutatis mutandis, in the author's study on sectionally linear functions over an infinite range about which a report will be given elsewhere.

To each tournament Tn with n nodes n there corresponds the automorphism group G(Tn) consisting n of all dominance preserving permutations of the set of nodes. In a recent paper [3], Myron Goldberg and J. W. Moon consider the maximum order g(n) which the group of a tournament with n nodes may have. Among other results they prove that12

It has been conjectured that if p{z) is a polynomial of degree n then the product P{z)—p{z)p'{z)p"{z)...p~\z) has at least n+l distinct zeros unless p(z)=c(z—a). Professor P. Erdos who mentioned this problem in a lecture at the University of Montreal attributed it to Tiberiu Popoviciu. In the present paper we consider the special case where p(z) has only real zeros. Let the statement "f(z) is a constant multiple of g(z)" be abbreviated "/(z)

The purpose of this paper is not to solve a problem but to pose one that may be of some interest, depth, and consequence.Given that the positive integer n has the canonical representation n = Πhi=1 piαi the problem of finding the number F(n) = f(α1, α2, … αn) of ordered factorizations of n into positive nontrivial integral factors is equivalent to that of finding the number of ordered partitions of the vector (α1, α2, … αn) into nonzero vectors with nonnegative integral components.

For the Hermite polynomials Hn(z) defined by1it is easy to see from the Rodrigues formula that2where, as usual, Dz=d/dz.In recent papers ([1], [2]) Carlitz has proved the following formulae:34where, on the right-hand side of (4), the range of each summation is from i= 1 to i=k(k=1,2,…).

Let Er denote the rth elementary symmetric function on α1 α2,…,αm which is defined by1E0 = 1 and Er=0(r>m).We define the rth symmetric mean by2where denote the binomial coefficient. If α1 α2,…,αm are positive reals thenwe have two well-known inequalities3and4In this paper we consider a generalization of these inequalities. The inequality (4) is known as Newton's inequality which contains the arithmetic and geometric mean inequality.

The purpose of this paper is to determine the conditions under which the maximal ideals of the ring C*(X)—the bounded real-valued continuous functions on a completely regular Hausdorff space X—are closed under pointwise convergence of sequences. Whereas the maximal ideals of C*(X) are closed under pointwise convergence of nets if and only if X is compact, it is shown that a necessary and sufficient condition for their pointwise sequential closure is that X be pseudocompact (i.e. that all real-valued continuous functions of X be bounded).

The fundamental theorem of affine geometry is an easy corollary of the corresponding projective theorem 2.26 in Artin's Geometric Algebra. However, a simple direct proof based on Lipman's paper [this Bulletin, 4, 265−278] and his axioms 1 and 2 may be of some interest.Lipman's [desarguian] affine geometry G determined a left linear vector space L={a, b,…} over a skew field F. We wish to construct 1−1 transformations γ of G onto itself such that γ and γ-1 map straight lines onto straight lines preserving parallelism. Designate any point 0 as the origin of G. Multiplying γ with a suitable translation, we may assume γ0=0. Thus γ will then be equivalent to a 1−1 transformation Γ of L onto itself which preserves linear dependence. Since Γ-1 will have the same properties, Γ must also preserve linear independence.

Let (Ω, , P) be a probability space, and A1, A2… be a sequence of members of . The classical Borel-Cantelli problem is to determine the probability that infinitely many events Ak occur. The classical results may be found in Feller [2, p. 188]; while related work may be found in Spitzer [3, p. 317], and Dawson and Sankoff [1]. The latter works are generalizations of the Borel-Cantelli lemmas, taken in different directions.In this paper, necessary and sufficient conditions will be given for infinitely many events Ak to occur, with probability 1. A lower bound for the probability that only finitely many Ak occur, is developed. In addition, necessary and sufficient conditions that only finitely many Ak occur, with probability 1, are given.

It is shown in this paper that an immersion of a connected, closed n-manifold into another connected n-manifold is a diffeomorphism if and only if the induced homomorphism between the fundamental groups is surjective at some point. This is proved as a consequence of a more general assertion about topological spaces.

The following result is due to Wielandt [1, Lemma 2.9]: Let A, B, K be N-submodules of some N-module, where N is a zero symmetric near-ring. Then the N-module, Γ: = (A + K) ∩ (B + K) | (A ∩ B) + K is commutative. Using this result Wielandt obtained density theorem for 2-primitive near-rings with identity. Betsch [1] used Wielandt's result to obtain the density theorem for O-primitive near-rings. The purpose of this paper is to extend this result for loops.

Let X be a topological space and G a subgroup of the homeomorphism group H(X) with the topology of point-wise convergence. It is well-known that if X is uniformizable and G is equicontinuous with respect to a compatible uniformity then G is a topological group. In this paper we show that essentially this same result applies when X is only an R0-space (and hence in particular if X is T1 or regular). A corresponding result for regular spaces has been proved [2].

The purpose of this paper is to prove the following result:Theorem. Let T be a directed tree with k arcs and with no directed path of length 2. Then if G is any directed graph with n points and at least 4kn arcs, T is a subgraph of G.It would be appropriate to call T an antidirected tree or a source-sink tree, since every point either has all its arcs directed outward or all inward. As N. G. de Bruijn has noted (personal communication), such a linear bound in n cannot hold if T is replaced by any directed graph other than a union of such trees. The above theorem strengthens one of Graham [1], where an implicit bound of c(k)n is obtained, where c(k) is exponentially large. The proof we give here is also shorter. We first give two simple lemmas. Both are essentially due to ErdÄs, but it is not clear where either first appeared. Their proofs are easy, so we give them here for completeness; in neither case do we state quite the best possible result.

In a paper with the above title, T. M. Apostol and S. Chowla [1] proved the following result:Theorem 1.For relatively prime integers h and k, l ≤ h ≤ k, the seriesdoes not admit of an Euler product decomposition, that is, an identity of the form1except when h = k = l; fc = 1, fc = 2. The series on the right is extended over all primes p and is assumed to be absolutely convergent forR(s)>1.

The papers [2] and [3] study the function g(k, n), defined for integers k > 1 and n > 1 as the smallest r with the property that every integer is a sum of r kth powers mod n. This note identifies g′(k), defined as the maximum over all n of g(k, n), with the function Γ(k) studied by Hardy and Littlewood [1] fifty years ago in connection with Waring's problem.

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.

This paper constructs a class of examples to show that for torsion-free groups H with finite nilstufe v(H) = n < ∞ there can be divisible torsion groups D with v(H ⊕ D) - n + k for all k ≤ n + 1. This answers a question of Feigelstock. The construction is based on a proposition which bounds v(H ⊕ D) in terms of v(H) and rank (D).

AbstracrtThe Chinese Remainder Theorem states that if I and J are comaximal ideals of a ring R, then A/(I∩J)A is isomorphic to A/IA×A/JA for any left R-module A. In this paper we study the converse; when does A/(I∩J)A and A/IA×A/JA isomorphic imply that I and J are comaximal?

This note is intended as a commentary on a part of the paper [1] by Gulliksen, Ribenboim, and Viswanathan. It takes its inspiration from a colloquium talk by P. Ribenboim.Our aim is a partial generalization of the theorem of Cohen-Seidenberg (cf. [2], IX. 1, Propositions 9 and 10) to noncommutative algebras.Definition. Let A be a ring with 1 in the center of another ring B (with the same 1). B is said to be integral over A, if each element b ∊ B satisfies an integral equation

The following lemma, which appears as Lemma 4 in [5], was used to determine certain multicoherence properties of inverse limits of continua.Lemma. Let X denote the inverse limit of an inverse system {Xλ, fλμ, Λ} of compact Hausdorff spaces Xλ. If Xλ has no more than k components (where k < ∞ is fixed) for each λ ∊ Λ, then X has no more than k components.In this paper we give a set theoretic analogue of this lemma and an extension which was suggested to the author by Professor F. W. Lawvere. An application to inverse limits of finite groups is then given.

The Cauchy-Peano existence theorem (1) does not allow us to decide from the form of a given system of equations whether or not its solution can be continued for the infinite interval -∞ < t < ∞. Several sufficient conditions for such a continuation were given by A. Wintner in (2). The main result of his paper is the following theorem which is not proven.

Let A, B, denote two non-void finite complexes (= subsets) of the torsion free abelian group G,Let d(A),… denote the maximum number of linearly independent elements of A,… and let n = n(A, B) denote the number of elements of A + B whose representation in the form a + b is unique. In the preceding paper, Tarwater and Entringer [1] proved that n ≥ d(A).

This paper was inspired by [4]. The main result there suggests a close relationship between injective hulls of C* algebras as studied in [2] and [3] and rational completions as studied in [1]. We shall prove an analogue of Theorem 1 in [3] for rational completions. The latter theorem states that the injective hull of the algebra C(X) of all complex valued functions on the compact Hausdorff space X is the algebra B(X) of all bounded Borel functions modulo sets of first category.

We give an alternative construction of a Hausdorff compactification due to Fell [2]. We say that a space is compact if it has the Heine-Borel property, locally compact if each point has a fundamental system of compact neighbourhoods. The interesting spaces from the point of view of this paper, are the non-Hausdorjf ones since for locally compact Hausdorff spaces Fell's compactification is the usual one-point compactification. The motivation for the compactification comes from the theory of continuous fields of C*-algebras: the primitive spectrum of a C*- algebra A is a locally compact T0 space X and Fell [3] realizes A as an algebra of fields of operators over the compactification of X. This note is based on a discussion of the author with Professor Fell.

A Banach space X is said to be subspace homogeneous if for every two isomorphic closed subspaces Y and Z of X, both of infinite codimension, there is an automorphism of X (i.e. a bounded linear bijection of X) which carries Y onto Z. In [1] Lindenstrauss and Rosenthal showed that c0 is subspace homogeneous, a property also shared by l2, and conjectured that c0 and l2 are the only subspace homogeneous Banach spaces. In that paper no mention was made of subspaces of c0.

In this paper, an elementary proof is given of the following proposition:Theorem 1. If Qp is an arbitrary field of p-adic numbers, then it contains normal subfields Ln(2 ≤ n ≤ p) which have symmetric groups Sn as their respective Galois groups over Q, the field of rational numbers. Furthermore, each Ln may be chosen to be totally real.

Burgess [l] has pointed out that in the categories and (where . is the category of Ti. spaces), epi means onto. In this paper, Burgess1 technique will be used to show that epi has a different meaning in and that this meaning reduces to onto when the range is a T1 space.

In this paper we extend the results of an earlier note [1].Definition. Let E be an extension field of the rationals. A vector v = (b1, …, bn) in En is algebraic if each coordinate bi is algebraic over the rationals. A linear transformation T: En → En is algebraic if T(v) is an algebraic vector for every algebraic vector v.Definition. The degree of an algebraic linear transformation T, denoted by deg T, is the minimum of [K:Q] taken over all finite algebraic extensions K of the rationals Q such that T: Kn → Kn.

In their paper [1], Buseman and Glassco state that N(C, 7, 3) = 5 has been claimed but questioned. Schouten in [2] provides a classification of the orbits of ∧3V under the action of the group of automorphisms of ∧3V induced by automorphisms of V when dim V = 7.