The Impact of Early Positive Results on a Mathematics and Science Partnership: The Experience of the Institute for Chemistry Literacy Through Computational Science

After one year of implementation, the Institute for Chemistry Literacy through Computational Science, an NSF Mathematics and Science Partnership Institute Project led by the University of Illinois at Urbana-Champaign’s Department of Chemistry, College of Medicine, and National Center for Supercomputing Applications, experienced statistically significant gains in chemistry content knowledge among students of the rural high school teachers participating in its intensive, year-round professional development course, compared to a control group. The project utilizes a two-cohort, delayed-treatment, random control trial, quasi-experimental research design with the second cohort entering treatment one year following the first. The three-year treatment includes intensive two-week summer institutes, occasional school year workshops and year-round, on-line collaborative lesson development, resource sharing, and expert support. The means of student pre-test scores for Cohort I (η=963) and Cohort II (η=862) teachers were not significantly different. The mean gain (difference between pre-test and post-test scores) after seven months in the classroom for Cohort I was 9.8 percentage points, compared to 6.7 percentage points for Cohort II. This statistically significant difference (p\u3c.001) represented an effect size of .25 standard deviation units, and indicated unusually early confirmation of treatment effects. When post-tests were compared, Cohort I students scored significantly higher than Cohort II and supported the gain score differences. The impact of these results on treatment and research plans is discussed. concentrating on the effect of lessening rural teachers’ isolation and increasing access to tools to facilitate learning

Some ground-state expectation values for the free parafermion Z(N) spin chain

We consider the calculation of ground-state expectation values for the non-Hermitian Z(N) spin chain described by free parafermions. For N=2 the model reduces to the quantum Ising chain in a transverse field with open boundary conditions. Use is made of the Hellmann-Feynman theorem to obtain exact results for particular single site and nearest-neighbour ground-state expectation values for general N which are valid for sites deep inside the chain. These results are tested numerically for N=3, along with how they change as a function of distance from the boundary.Comment: 17 pages, 4 figures; extra reference

Exceptional Points in the Baxter-Fendley Free Parafermion Model

Certain spin chains, such as the quantum Ising chain, have free fermion spectra which can be expressed as the sum of decoupled two-level fermionic systems. Free parafermions are a simple generalisation of this idea to $Z(N)$-symmetric clock models. In 1989 Baxter discovered a non-Hermitian but $PT$-symmetric model directly generalising the Ising chain, which was much later recognised by Fendley to be a free parafermion spectrum. By extending the model's magnetic field parameter to the complex plane, it is shown that a series of exceptional points emerges, where the quasienergies defining the free spectrum become degenerate. An analytic expression for the locations of these points is derived, and various numerical investigations are performed. These exceptional points also exist in the Ising chain with a complex transverse field. Although the model is not in general $PT$-symmetric at these exceptional points, their proximity can have a profound impact on the model on the $PT$-symmetric real line. Furthermore, in certain cases of the model an exceptional point may appear on the real line (with negative field).Comment: 17 pages, 6 figures, Figure 3 has been changed to use the convention that the quasienergies have arguments in the range (-pi/N, pi/N], + minor change