Abstract. We define and study a notion of discrete homology theory for metric spaces. Instead of working with simplicial homology, our chain complexes are given by Lipschitz maps from an n-dimensional cube to a fixed metric space. We prove that the resulting homology theory satisfies a discrete analogue of the Eilenberg-Steenrod axioms, and prove a discrete analogue of the Mayer-Vietoris exact sequence. Moreover, this discrete homology theory is related to the discrete homotopy theory of a metric space through a discrete analogue of the Hurewicz theorem. We study the class of groups that can arise as discrete homology groups and, in this setting, we prove that the fundamental group of a smooth, connected, metrizable, compact manifold is isomorphic to the discrete fundamental group of a 'fine enough' rectangulation of the manifold. Finally, we show that this discrete homology theory can be coarsened, leading to a new non-trivial coarse invariant of a metric space.
Objective: A comparison of efficacy, safety and tolerance of a twice-daily application of calcitriol 3 μg/g ointment with dithranol cream. Methods: The study was an 8-week, prospective, randomised, open, parallel-group trial with 114 patients. Subjects received either 3 μg/g calcitriol ointment (twice daily) or 0.25–2% dithranol cream (once daily for 30 min). Results were measured using global improvement, global severity, PASI, quality of life (QOL) and Psoriasis Disability Index scores. Safety was determined from reports of adverse events and blood chemistry analysis. Results: At final assessment, calcitriol and dithranol were comparably efficacious. Skin irritation was reported by 5% of calcitriol and 72% of dithranol patients. Patients rated QOL and overall acceptability of calcitriol more highly. Conclusions: Twice-daily calcitriol ointment (3 μg/g) is equally effective as a once-daily, short-contact dithranol regimen. However, calcitriol is better tolerated than dithranol, and provides a better QOL and a greater patient acceptability.
We investigate double transitivity of Galois groups in the classical Schubert calculus on Grassmannians. We show that all Schubert problems on Grassmannians of 2-and 3-planes have doubly transitive Galois groups, as do all Schubert problems involving only special Schubert conditions. We use these results to give a new proof that Schubert problems on Grassmannians of 2-planes have Galois groups that contain the alternating group. We also investigate the Galois group of every Schubert problem on Gr(4, 8), finding that each Galois group either contains the alternating group or is an imprimitive permutation group and therefore fails to be doubly transitive. These imprimitive examples show that our results are the best possible general results on double transitivity of Schubert problems.
This report describes student performance in a state-level initiative that provided first-year college coursework in chemistry to high school students. Upon successful completion of the coursework, students received both high school and college credit. In this initiative, high school teachers team taught college-level chemistry courses in collaboration with a university chemistry professor on high school campuses within proximity to the university. High school student performance was measured using an ACS standardized exam in general chemistry as well as common midterm exams and was compared with performance of traditional college students using the same assessment instruments. Course completion rates were also compared. In most cases, statistical analysis of student performance data indicates that high school students participating in the dual-credit courses developed content knowledge equivalent with measured norms for college students yet had a higher rate of course completion, suggesting that this dual-credit initiative was a viable option for increasing accessibility to college chemistry. Furthermore, broad insights were gained for addressing and potentially improving student success and course completion in response to recent shifts in funding formulas within higher education.
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