Definability in the lattice of equational theories of semigroups

We study first-order definability in the latticeL of equational theories of semigroups. A large collection of individual theories and some interesting sets of theories are definable inL. As examples, ifT is either the equational theory of a finite semigroup or a finitely axiomatizable locally finite theory, then the set {T, T ^ϖ} is definable, whereT ^ϖ is the dual theory obtained by inverting the order of occurences of letters in the words. Moreover, the set of locally finite theories, the set of finitely axiomatizable theories, and the set of theories of finite semigroups are all definable. The research of both authors was supported by National Science Foundation Grant No. DMS-8302295     

Interactions of the directed plasma from a cathodic arc withelectrodes and magnetic fields

Curved magnetic ducts are frequently used to remove macroscopic-sized droplets from the plasma stream of cathodic vacuum arcs. The plasma of a cathodic vacuum arc in a magnetic filter is characterized by a strongly directional ion velocity (corresponding to 20-100 eV) and magnetized electrons. In the first section of this paper the effects of these features on the I-V characteristic curves of planar probes are identified and explained using a simple model. This is then used to interpret the interaction of the plasma with the walls of a biased quarter torus duct. Two small electrodes placed on the outer and inner sections of the curved duct wall show that the I-V characteristic is determined primarily by the electron-ion current balance at the wall on the outside of the curve. The application of a bias to a planar electrode on the outer wall section was found to give the same increase in throughput as a positive bias applied to the entire duct with the advantage of a much smaller electron current being drawn by the biasing power supply. The improvement in duct throughput achievable with positive-biasing of the duct wall was found to depend on both the configuration and strength of the magnetic field in the quarter torus filter. The plasma density profile and potential were unaffected by the application of the bias     

Measurement of the cascade time of ∑ in liquid helium

In a study of ∑⁻π⁺ production by stopped K⁻ in liquid helium, (19 ± 6)% of stopped ∑⁻ were observed to decay. This corresponds to a “cascade time” of 28 ± 8 psec.     

The Zig-Zag Property and Exponential Cancellation of Ordered Sets

This paper outlines a chapter in the theory of ordered sets that concerns decompositions of ordered sets using various natural operations introduced by Garrett Birkhoff in 1940 – chiefly, the operations of sum (disjoint union), Cartesian product, and exponentiation. This area has seen two brief, four-year flowerings of research activity during the past sixty-four years. Ivan Rival, working with several collaborators around 1978, obtained beautiful results on exponentiation which attracted other researchers to this topic. Altogether, about a dozen people contributed a chain of significant new results between 1978–1982. The subject then lay dormant until a major contribution by Jonathan D. Farley appeared in 1996. This pre-figured the solution, four years later, of a central problem about exponentiation of finite ordered sets that Birkhoff had formulated in 1942. The theory of operations on ordered sets is a beautiful chapter of mathematics. We concentrate on those portions that are most relevant for the solution of the Birkhoff problem, offering (hopefully) accessible proofs of many results, including some new results. We would like to see a new renaissance of activity in this area, which continues to offer many attractive open problems. This revised version was published online in June 2006 with corrections to the Cover Date.     

Toward an arithmetic of polynomials

Summary ForR a commutative ring, which may have divisors of zero but which has no idempotents other than zero and one, we consider the problem of unique factorization of a polynomial with coefficients inR. We prove that, if the polynomial is separable, then such a unique factorization exists. We also define a Legendre symbol for a separable polynomial and a prime of commutative ring with exactly two idempotents in such a way that the symbols of classical number theory are subsumed. We calculate this symbol forR = Q in two cases where it has classically been of interest, namely quadratic extensions and cyclotomic extensions. We then calculate it in a situation which is new, namely the so called generalized cyclotomic extensions from a paper by S. Beale and D. K. Harrison. We study the Galois theory in the general ring situation and in particular define a category of separable polynomials (this is an extension of a paper by D. K. Harrison and M. Vitulli) and a cohomology theory of separable polynomials.     

The Monitoring of Exponentially Weighted Forecasts

By considering equivalences between various forecasting systems, the exact stochastic process followed by the one-step-ahead errors may be found. This process, the error process, is important for any monitoring scheme, and is a function of the forecasting system and the underlying data process. The error process is obtained for the most general form of exponential smoothing systems used in optimal conditions. The statistical properties are derived. In particular, the approximate variance of Trigg's smoothed error tracking signal is obtained explicitly for several exponential smoothing systems, and a procedure is given for obtaining it numerically for any such system. The use of different smoothing constants in the forecasting system and the tracking signal is discussed and it is found that suitable choice of the tracking signal constant greatly improves the performance of the signal, making it more comparable with CUSUM schemes.     

Counting models in universal Horn classes

Definen _K (λ) to be either ω, or the number of non-isomorphic models inK having cardinality α, whichever cardinal is larger. This paper contains a proof that for a congruence modular variety ⋎ of algebras of countable similarity type, there are only six possible functionsn _⋎. It is also proved that ifn _K (λ)≠2^λ for some λ, andK is a universal Horn class of models for a countable language, thenK must satisfy two conditions, one of which is quite restrictive and requires that the members ofK are all in a certain sense Abelian. Presented by B. Jonsson.