for some real number X, the symbol V denoting the lattice least upper bound. Any ring R is regular [10] if for each xER there is an xaER such that xx°x = x. It is evident that every regular F-ring R contains a maximal bounded sub-F-ring R, the F-ring of all xER satisfying equation (1.1). The relationship between a regular F-ring and its maximal bounded sub-F-ring is analogous to that between th...

For an associative ring R, let P be an R-module with S = EndR(P). C. Menini and A. Orsatti posed the question of when the related functor HomR(P, −) (with left adjoint P ⊗S −) induces an equivalence between a subcategory of R M closed under factor modules and a subcategory of S M closed under submodules. They observed that this is precisely the case if the unit of the adjunction is an epimorphi...

A closed convex cone K is called nice, if the set K∗ + F⊥ is closed for all F faces of K, where K∗ is the dual cone of K, and F⊥ is the orthogonal complement of the linear span of F. The niceness property plays a role in the facial reduction algorithm of Borwein and Wolkowicz, and the question whether the linear image of a nice cone is closed also has a simple answer. We prove several character...

It is shown that if F\ and Fi are algebraically closed fields of nonzero characteristic p and F\ is not isomorphic to a subfield of F2 , then F\ does not embed in the skew field of quotients 0Fj of the ring of morphisms of the additive group of F2 . From this fact and results of Evans and Hrushovski, it is deduced that the algebraic closure geometries G(K¡/Fi) and (7(^2/^2) are isomorphic if an...

In this article, we examine sets of lines in PG(d,F) meeting each hyperplane in a generator set of points. We prove that such a set has to contain at least 1.5d lines if the field F has more than 1.5d elements, and at least 2d− 1 lines if the field F is algebraically closed. We show that suitable 2d−1 lines constitute such a set (if |F| ≥ 2d−1), proving that the lower bound is tight over algebr...

In this article, we examine sets of lines in PG(d,F) meeting each hyperplane in a generator set of points. We prove that such a set has to contain at least b1.5dc lines if the field F has at least b1.5dc elements, and at least 2d− 1 lines if the field F is algebraically closed. We show that suitable 2d − 1 lines constitute such a set (if |F| > 2d − 1), proving that the lower bound is tight over...

For a proper continuous map f : M → N between smooth manifolds M and N with m = dimM < dimN = m + k, a homology class θ(f) ∈ H m−k(M ; Z2) has been defined and studied by the first and the third authors, where H ∗ denotes the singular homology with closed support. In this paper, we define θ(f) for maps between generalized manifolds. Then, using algebraic topological methods, we show that f̄∗θ(f)...

At 532 nm wavelength, optical properties of black carbon (BC) particles mixed with sulfate are computed by use of two morphological models, a closed cell and a coated aggregate model. For high BC volume fractions f, both models yield comparable results. As more sulfate is added, some of the optical properties diverge. The backscattering depolarization ratio δL is particularly sensitive to the m...

Abstract. In this work we find a closed form expression for matrix averages over the Gaussian ensemble. More precisely, given an n × n Hermitian matrix A and a continuous function f(x) we find a closed form expression for the expectation E(Tr(f(XAX∗))) where X is a Gaussian n × n matrix with complex independent and identically distributed entries of zero mean and variance 1. Taking f(x) = log(1...

We show that for a continuous map of a closed interval to itself, if the set of periodic points is closed, then every recurrent point is periodic. If, furthermore, the set of least periods of the periodic points is finite, then every nonwandering point is periodic. This answers a question of L. Block [Proc. Amer. Math. Soc. 67 (1977), 357-360]. Introduction. In [1] and [2], L. Block studied con...