On the Existence of Towers in Pseudo-Tree Algebras

A tower in a Boolean algebra (BA) is a strictly increasing sequence, of regular order type, of elements of the algebra different from 1 but with sum 1. A pseudo-tree is a partially ordered set T such that the set T↓t = {s ∈ T:s < t} is linearly ordered for every t ∈ T. If that set is well-ordered, then T is a tree. For any pseudo-tree T, the BA Treealg(T) is the algebra of subsets of T generated by all of the sets T↑t = {s ∈ T:t ≤ s}. The main theorem of this note is a characterization in tree terms of when Treealg(T) has a tower of order type κ (given in advance).     

Pseudo-trees and Boolean algebras

We consider Boolean algebras constructed from pseudo-trees in various ways and make comments about related classes of Boolean algebras.     

The inverse electromagnetic scattering problem for a partially coated dielectric

We use the linear sampling method to determine the shape and surface conductivity of a partially coated dielectric infinite cylinder from knowledge of the far field pattern of the scattered TM polarized electromagnetic wave at fixed frequency. A mathematical justification of the method is provided based on the use of a complete family of solutions. Numerical examples are given showing the efficiency of our method.     

Linear grain growth kinetics and rotation in nanocrystalline Ni

We report three-dimensional atomistic molecular dynamics studies of grain growth kinetics in nanocrystalline Ni. The results show the grain size increasing linearly with time, contrary to the square root of the time kinetics observed in coarse-grained structures. The average grain boundary energy per unit area decreases simultaneously with the decrease in total grain boundary area associated with grain growth. The average mobility of the boundaries increases as the grain size increases. The results can be explained by a model that considers a size effect in the boundary mobility.     

An atomless interval Boolean algebra A such that $ \mathfrak{a} (A) $ < $ \mathfrak{t} (A) $

For any Boolean algebra A, is the smallest cardinality of an infinite partition of unity in A. A tower in a Boolean algebra A is a subset X of A well-ordered by the Boolean ordering, with but with is the smallest cardinality of a tower of A. Given a linearly ordered set L with first element, the interval algebra of L is the algebra of subsets of L generated by the half-open intervals [a, b). We prove that there is an atomless interval algebra A such that . Received January 21, 2002; accepted in final form March 13, 2002.     

Discrete compactness and the approximation of Maxwell's equations in

We analyze the use of edge finite element methods to approximate Maxwell's equations in a bounded cavity. Using the theory of collectively compact operators, we prove -convergence for the source and eigenvalue problems. This is the first proof of convergence of the eigenvalue problem for general edge elements, and it extends and unifies the theory for both problems. The convergence results are based on the discrete compactness property of edge element due to Kikuchi. We extend the original work of Kikuchi by proving that edge elements of all orders possess this property.

     

The spectrum of partitions of a Boolean algebra

The main notion dealt with in this article is