Richard Mercer

Work in progress - the WSU model for engineering mathematics education

This paper summarizes progress to date on the WSU model for engineering mathematics education, an NSF funded curriculum reform initiative at Wright State University. The WSU model seeks to increase student retention, motivation and success in engineering through application-driven, just-in-time engineering math instruction. The WSU approach involves the development of a novel freshman-level engineering mathematics course EGR 101, as well as a large-scale restructuring of the engineering curriculum. By removing traditional math prerequisites and moving core engineering courses earlier in the program, the WSU model shifts the traditional emphasis on math prerequisite requirements to an emphasis on engineering motivation for math, with a just-in-time structuring of the new math sequence. This paper summarizes the development to date of the WSU model for engineering mathematics education, including a preliminary assessment of student performance and perception during the initial implementation of EGR 101. In addition, an assessment of first-year retention results is anticipated in time for the conference

Isometric isomorphisms of Cartan bimodule algebras

Abstract This paper is dedicated to proving a single result: that isometric isomorphisms of Cartan bimodule algebras can be extended to ∗ -isomorphisms of the generated von Neumann algebras. If M is a von Neumann algebra with Cartan subalgebra A , then a (Cartan) bimodule algebra is simple a σ-weakly closed subalgebra S of M which contains A ; S generates M if there is no proper von Neumann subalgebra of M containing S . The key idea is that a theorem of Muhly, Saito, and Solel concerning isomorphisms of maximal subdiagonal algebras can be extended to this much wider class of algebras if we restrict our attention to isometric isomorphisms. Although the algebras are not assumed to be hyperfinite, a finite-dimensional result of Davidson and Power lies at the leart of the proof. What makes the use of finite-dimensional techniques possible is the existence of “sufficiently many” finite subequivalence relations of an arbitrary countable measurable equivalence relation.

General quantum measurements: Local transition maps

On the basis of four physically motivated assumptions, it is shown that a general quantum measurement of commuting observables can be represented by a “local transition map,” a special type of positive linear map on a von Neumann algebra. In the case that the algebra is the bounded operators on a Hilbert space, these local transition maps share two properties of von Neumann-type measurements: they decrease “matrix elements” of states and increase their entropy. It is also shown that local transition maps have all the properties of a conditional expectation of a von Neumann algebra onto a subalgebra except that their range is not restricted to the subalgebra. The notion of locality arises from requiring that a quantum measurement may be treated classically when restricted to the commutative algebra generated by the measured observables. The formalism established applies to observables with arbitrary spectrum. In the case that the spectrum is continuous we have only “incomplete” measurements.

Transition Maps and Locality

This is a summary of some of the results obtained in [1]. We will deal with quantum measurement theory and the notion of a quantum stochastic process, and the close relationship between the two. The unifying theme will be the concept of a local transition map (to be defined below). We will present the claim that local transition maps are generalized quantum measurements, and state results justifying this claim. Local transition maps in a different context will serve as propagation maps for quantum stochastic processes, leading to the result that quantum stochastic processes (as defined here) necessarily contain quantum measurements. For details and further references on this material, see [1].