The following two theorems are proved. If R is an Archimedean conducive integral domain, then R is quasilocal and dim(R) ≤1. If each overring of an integral domain R has ascending chain condition on divisorial ideals, then the integral closure of R is a Dedekind domain. Both theorems sharpen results already known in the Noetherian case. The second theorem leads to a strengthened converse of the Krull-Akizuki Theorem. We also investigate the effect of restricting the hypothesis in the second theorem to the proper overrings of R.

The purpose of this note is to give some characterizations of collectionwise normality by means of the normality of certain product topological spaces; they are a consequence of a theorem proved by M. Katětov, but they appear to be new.

We construct, under MAcountable, a countably compact topological subsemigroup of which is not a group, hence a counterexample for the Wallace problem. We also show that there is no p-compact counterexample for the Wallace problem, answering a question of D. Grant. Finally, we show that—in some sense—our counterexample for the Wallace problem constructed under MAcountable cannot be done in ZFC.

This paper studies the integration of inclusion of subdifferentials. Under various verifiable conditions, we obtain that if two proper lower semicontinuous functions $f$ and $g$ have the subdifferential of $f$ included in the $\gamma $-enlargement of the subdifferential of $g$, then the difference of those functions is $\gamma $-Lipschitz over their effective domain.

In this paper we consider the equation

(1.1) (r(t)y′(t))′+p(t)f(y(t)) = 0

under the conditions

((H0): the real valued functions r, r′ and p are continuous on a non-trivial interval J of reals, and r(t)>0 for t∈J;

and

(H1):f:R→R is continuously differentiable and odd with f'(y)>0 for all real y. We also consider the equation

(1.2) y″(t)+m(t)y′(t)+n(t)f(y(t)) = 0

under the conditions (H1) and

(H2): the real valued functions m and n are continuous on a non-trivial interval J of reals.

In an article P. E. Himelwright and J. E. Williamson [3] proved a theorem on 1-factorability of Cartesian product of two graphs. With a very short proof we prove a more general theorem which immediately implies their theorem as a corollary. We will follow the notations and definitions of [1], [2] and [3].

Some equationally defined classes of loops are identified and characterized among a class of loops which are isomorphic to all of their loop isotopes.

A two dimensional system of differential equations with a finite number of discrete delays is considered. Conditions are derived for there to be no stability switching for arbitrary such delays.

In Theorem 4 of [5] Stone stated that the theory of Boolean rings was "mathematically equivalent" to the theory of Boolean spaces without, however, properly defining the phrase "mathematically equivalent". It is the main purpose of this note to establish a precise reformulation of Theorem 4 in [5]. This is accomplished by introducing special classes of maps between Boolean lattices, Boolean rings and Boolean spaces respectively, and showing the categories arising in conjunction with these maps to be equivalent in the sense of Grothendieck [2]. Thus the notion of equivalence of categories will replace the phrase "mathematically equivalent" in [5]. In addition the well-known axiomatic characterization of meet and complementation of Boolean lattices with unit is discussed in analogous terms.

It is well known that certain results such as the Radon-Nikodym Theorem, which are valid in totally σ -finite measure spaces, do not extend to measure spaces in which μ is not totally σ -finite. (See §2 for notation.) Given an arbitrary measure space (X, S, μ) and a signed measure ν on (X, S), then if ν ≪ μ for X, ν ≪ μ when restricted to any e ∊ Sf and the classical finite Radon-Nikodym theorem produces a measurable function ge(x), vanishing outside e, with

It is shown that a suitable conformai change of the metric in the leaf direction of a transversally oriented Riemannian foliation on a closed manifold will make the basic component of the mean curvature harmonic. As a corollary, we deduce vanishing and finiteness theorems for Riemannian foliations without assuming the harmonicity of the basic mean curvature.

In this article, we establish a new atomic decomposition for $f\,\in \,L_{w}^{2}\,\bigcap \,H_{w}^{p}$, where the decomposition converges in $L_{w}^{2}$-norm rather than in the distribution sense. As applications of this decomposition, assuming that $T$ is a linear operator bounded on $L_{w}^{2}$ and $0\,<\,p\,\le \,1$, we obtain (i) if $T$ is uniformly bounded in $L_{w}^{p}$-norm for all $w-p$-atoms, then $T$ can be extended to be bounded from $H_{w}^{p}$ to $L_{w}^{2}$; (ii) if $T$ is uniformly bounded in $H_{w}^{p}$-norm for all $w-p$-atoms, then $T$ can be extended to be bounded on $H_{w}^{p}$; (iii) if $T$ is bounded on $H_{w}^{p}$, then $T$ can be extended to be bounded from $H_{w}^{p}$ to $L_{w}^{2}$.

A ring $R$ is called quasi-Baer if the right annihilator of every right ideal of $R$ is generated by an idempotent as a right ideal. We investigate the quasi-Baer property of skew group rings and fixed rings under a finite group action on a semiprime ring and their applications to ${{C}^{*}}$-algebras. Various examples to illustrate and delimit our results are provided.

A well-known theorem, due to R. H. Bruck ([4], p. 398), is the following:

If a finite projective plane of order n has a projective subplane of order m < n, then either n = m2 or n > m 2+ m.

In this paper we prove an analagous theorem concerning affine subplanes of finite projective planes (Theorem 1). We then construct a number of examples; in particular we find all the finite Desarguesian projective planes containing affine subplanes of order 3 (Theorem 2).

The local class group of a Krull domain A is the quotient group G(A) = CI(A)/Pic(A). A Krull domain A is locally factorial if and only if G(A) = 0. In this paper, we characterize the Krull domains for which G(A) is a torsion group. We evaluate the local class group of several examples and finally, we explain why every abelian group is the local class group of a Krull domain.

A class of generalized Marcinkiewicz integral operators is introduced, and, under rather weak conditions on the integral kernels, the boundedness of such operators on ${{L}^{p}}$ and Triebel–Lizorkin spaces is established.

Let p*(n) denote the number of product partitions, that is, the number of ways of expressing a natural number n > 1 as the product of positive integers ≥ 2, the order of the factors in the product being irrelevant, with p*(1) = 1. For any integer if d is an ith power, and = 1, otherwise, and let . Using a suitable generating function for p*(n) we prove that

A definition of an isometric shift operator on a Banach space is given which extends the usual definition of a shift operator on a separable Hilbert space. It is shown that there is no such shift on many of the common Banach spaces of continuous functions. The associated ideas of a semi-shift and a backward shift are also introduced and studied in the case of continuous function spaces.

Langlands has conjectured the existence of a universal group, an extension of the absolute Galois group, which would play a fundamental role in the classification of automorphic representations. We shall describe a possible candidate for this group. We shall also describe a possible candidate for the complexification of Grothendieck's motivic Galois group.

Lower bounds are given for the independence ratio in graphs satisfying certain girth and maximum degree requirements. In particular, the independence ratio of a graph with maximum degree Δ and girth at least six is at least (2Δ − 1)/(Δ2 + 2Δ − 1). Sharper bounds are given for cubic graphs.