All rings considered will be commutative with identity. By a graded ring we will mean a ring graded by the non-negative integers.

A ring R is called a π-ring if every principal ideal of R is a product of prime ideals. A π-ring without divisors of zero is called a π-domain. A graded ring (domain) is called a graded π-ring (-domain) if every homogeneous principal ideal is a product of homogenous prime ideals. A ring R is called a general ZPl-ring if every ideal is a product of primes. A graded ring is called a graded general ZPl-ring if every homogenous ideal is a product of homogeneous prime ideals.

In 1948, R. Frucht [5] proved that, given a finite group G, there are infinitely many connected cubic graphs X such that the automorphism group Aut X is isomorphic to G. In a letter, Professor Frucht has proposed the problem, whether in addition X can be required to be hamiltonian. One of the aims of the present note is to answer this question affirmatively.

THEOREM 1.1. Given a finite group G there are infinitely many finite hamiltonian cubic graphs Y such that Aut Y ≌ G.

In fact, we prove the following:

THEOREM 1.2. Given a finite cubic graph X having no component isomorphic to K4, there exists a hamiltonian cubic graph Y such that Aut Y ≌ Aut X and |V(Y)| = 6|V(X)|.

This implies 1.1 by the theorem of Frucht [5] mentioned above.

This paper studies dualities (or contravariant category equivalences) between two categories of R-right and S-left modules which are finitely closed) that is, closed under submodules, factor modules and finite direct sums. Omitting the requirement that the categories contain all finitely generated modules from the classical Morita situation provides a generalization which substantially increases the number of such dualities.

We prove that a duality between two finitely closed categories A and B of modules is representable if and only if A and B consist of linearly compact modules. This encompasses work of Mueller ([7], [8]) for Morita dualities and of Goblot ([5], [6]).

For a non-principal Dirichlet character χ modulo q,

Let the Pólya-Vingradov inequality asserts that M(x) < q1/2 log q see [7]. in the opposite direction it is a trivial consequence of lemma 1 below and 1. Parseval's identity that if χ is primitive modulo q, then

We show that on average the latter of these estimates is the more precise.

THEOREM 1. For any real k > 0

where the summation is over all non-principal characters modulo q.

THEOREM 2. For any k > 0,

The de Bruijn—van Aardenne Ehrenfest— Smith—Tutte theorem [1] is a theorem which connects the number of Eulerian dicircuits in a directed graph with the number of rooted spanning arborescences. In this paper we obtain a proof of this theorem by considering sequences over a finite alphabet, and we show that the theorem emerges from the generating function for a certain type of sequence. The generating function for the set of sequences is obtained as the solution of a linear system of equations in Section 2. The power series expansion for the solution of this system is obtained by means of the multivariate form of the Lagrange theorem for implicit functions, and is given in Section 3, together with a restatement of the theorem as a matrix identity.

Cet article contient une classification détaillée des symétries des C*-algèbres uniformément hyperfinies. Soient une symétrie de ; on associe à θ un entier généralisé N(θ), un réel Tr (θ) ∈ [0, 1], et on a les résultats suivants:

Il existe une symétrie unique à conjugaison près qui possède une suite centrale (Un) d'unitaires vérifiant ‖θ(Un) + Un‖ → 0(n → ∞). Cette symétrie est la seule dont l'algèbre des points fixes est uniformément hyperfinie.

Use the two element subsets of κ, denoted by [κ]2, as the edge set for the complete graph on κ points. Write CP(κ, µ, v) if there is an edge coloring R: [κ]2 → µ with µ colors so that for every proper v element set X ⊊ κ, there is a point x ∈ κ ∼ X so that the edges between x and X receive at least the minimum of µ and v colors. Write CP⧣(K, µ, v) if the coloring is oneto- one on the edges between x and elements of X.

Peter W. Harley III [5] introduced CP and proved that for κ ≧ ω, CP(κ+, κ, κ) holds to solve a topological problem, since the fact that CP(ℵ1, ℵ0, ℵ0) holds implies the existence of a symmetrizable space on ℵ1 points in which no point is a Gδ.

Let R be a 2-torsion free simple artinian ring with involution*. The element u of R is said to be unitary if u is invertible with inverse u*. In this paper we shall be concerned with the subalgebras W of R over its centre Z such that uWu* ⊆ W, for all unitaries u of R. We prove that if R has rank superior to 1 over a division ring D containing more than 5 elements and if R is not 4-dimensional then any such subalgebra W must be one of the trivial subalgebras 0, Z or R, under one of the following extra finiteness assumptions: W contains inverses, W satisfies a polynomial identity, the ground division ring D is algebraic, the involution is a conjugate-transpose involution such that D equipped with the induced involution is generated by unitaries.

This paper establishes the following combinatorial result concerning the automorphisms of a modular lattice.

THEOREM. Let M be a modular lattice and let G be a finite subgroup of the automorphism group of M. If the sublattice, MG, of (common) fixed points (under G) satisfies any of a large class of chain conditions, then M satisfies the same chain condition. Some chain conditions in this class are the following: the ascending chain condition; the descending chain condition; Krull dimension; the property of having no uncountable chains, no chains order-isomorphic to the rational numbers; etc.

In two previous papers [1, 2] we investigated the zeros of certain arithmetic functions. Using units of cubic fields, we also succeeded to construct, almost by accident, and as a by-product so to speak, entirely new and comparatively complicated combinatorial identities. In an interesting paper combinatorialist L. Carlitz [10] proved those identities in an elementary way. In a/m, we had to prove that the units used were fundamental ones.

Encouraged by these results, we took a closer look at this method that had led to the construction of combinatorial identities. Since the latter are such an important tool in mathematics, we thought it would be “einer Messe wert“ to generalize these results and lay the theoretical foundations of a new method for the construction of highly sophisticated combinatorial identities.

This paper is a continuation of earlier work [6], in which we studied the existence and the stability of solutions to the infinite system of nonlinear differential equations

(1.1)

i = 1, 2, …. Here s is a nonnegative real number, Rs = {t ∈ R: t ≧ s}, and denotes a sequence-valued function. Conditions on the coefficient matrix A(t) = [aij(t)] and the nonlinear perturbation were established which guarantee that for each initial value c= {ct} ∈ l1, the system (1.1) has a strongly continuous l1valued solution x(t) (i.e., each is continuous and converges uniformly on compact subsets of Rs). A theorem was also given which yields the exponential stability for the nonlinear system (1.1).

A connection between the theory of quadratic forms defined over a given field F, and the space XF of all orderings of F is developed by A. Pfister in [12]. XF can be viewed as a set of characters acting on the group F×/ΣF×2, where ΣF×2 denotes the subgroup of F× consisting of sums of squares. Namely, each ordering P ∈ XF can be identified with the character

defined by

It follows from Pfister's result that the Witt ring of F modulo its radical is completely determined by the pair (XF, F×/ΣF×2).

Section 1 of this paper contains all the work which deals exclusively with generalizations of Hadamard matrices. The non-existence theorem proven here (Theorem 1.10) generalizes a theorem of Hall and Paige [15] on the non-existence of complete mappings in certain groups.

In Sections 2 and 3, we consider the duals of (Hanani) transversal designs; these dual structures, which we call (s, r, µ)-nets, are a natural generalization of the much studied (Bruck) nets which in turn are equivalent to sets of mutually orthogonal Latin squares. An (s, r, µ)-net is a set of s2µ points together with r parallel classes of blocks. Each class consists of s blocks of equal cardinality. Two non-parallel blocks meet in precisely µ points. It has been proven that r is always less than or equal to (s2µ – l) / (s – 1).

Let X be a real Banach space. According to von Neumann's famous geometrical characterization X is a Hilbert space if and only if for all x, y ∈ X

Thus Hilbert space is distinguished among all real Banach spaces by a certain uniform behavior of the set of all two dimensional subspaces. A related characterization of real Lp spaces can be given in terms of uniform behavior of all two dimensional subspaces and a Boolean algebra of norm-1 projections [16]. For an arbitrary space X, one way of measuring the “uniformity” of the set of two dimensional subspaces is in terms of the real valued modulus of rotundity, i.e. for

The space is said to be uniformly rotund if for each 0 we have .

Higher dimensional quasigroups (a set Q with a cancellative, n-ary operation 〈 〉, ([2]) have been studied by T. Evans ([3], [4]), A. Cruse [1], C. C. Lindner ([10], [11]) and also by many others under the guise of magic cubes, Graeco-latin cubes, etc. Conjugates or parastrophes have been discussed by S. K. Stein [18], A. Sade [17] and more recently by C. C. Lindner and D. Steedley in [14], where it is shown that ordinary quasigroups exist of every order ≧ 4 with a prescribed number of distinct conjugates. It is suggested that the problem be extended to n-ary quasigroups.

Let L be a finitely generated field extension of a field K of characteristic p ≠ 0. By Zorn's Lemma there exist maximal separable extensions of K in L and L is finite dimensional purely inseparable over any such field. If ps is the smallest of the dimensions of L over such maximal separable extensions of K in L, then s is Wiel's order of inseparability of L/K [11]. Dieudonné [2] also investigated maximal separable extensions D of K in L and established that there must be at least one D such that L ⊆ Kp–∞(D) (such fields are termed distinguished). Kraft [5] showed that the distinguished maximal separable subfields are precisely those over which L is of minimal degree. This concept of distinguished subfield has been the basis of a number of results on the structure of inseparable field extensions, for example see [1], [3], [5], and [6].

The problems considered here arose in connection with the interesting use by Loeb [8] and Anderson [1], [2] of Loeb's measure construction [7] to define measures on certain topological spaces. The original problem, from which the results given here developed, was to identify precisely the family of sets on which these measures are defined.

To be precise, let be a set theoretical structure and * a nonstandard extension of , as in the usual framework for nonstandard analysis (see [10]). Let X be a Hausdorff space in and stx the standard part map for X, defined on the set of nearstandard points in *X. Suppose, for example, µ is an internal, finitely additive probability measure defined on the internal subsets of *X.