mizar-items: Exploring fine-grained dependencies in the Mizar Mathematical Library

The Mizar Mathematical Library (MML) is a rich database of formalized mathematical proofs (see http://mizar.org). Owing to its large size (it contains more than 1100 "articles" summing to nearly 2.5 million lines of text, expressing more than 50000 theorems and 10000 definitions using more than 7000 symbols), the nature of its contents (the MML is slanted toward pure mathematics), and its classical foundations (first-order logic, set theory, natural deduction), the MML is an especially attractive target for research on foundations of mathematics. We have implemented a system, mizar-items, on which a variety of such foundational experiements can be based. The heart of mizar-items is a method for decomposing the contents of the MML into fine-grained "items" (e.g., theorem, definition, notation, etc.) and computing dependency relations among these items. mizar-items also comes equipped with a website for exploring these dependencies and interacting with them.Comment: Accepted at CICM 2011: Conferences in Intelligent Computer Mathematics, Track C: Systems and Project

Modular classes of skew algebroid relations

Skew algebroid is a natural generalization of the concept of Lie algebroid. In this paper, for a skew algebroid E, its modular class mod(E) is defined in the classical as well as in the supergeometric formulation. It is proved that there is a homogeneous nowhere-vanishing 1-density on E* which is invariant with respect to all Hamiltonian vector fields if and only if E is modular, i.e. mod(E)=0. Further, relative modular class of a subalgebroid is introduced and studied together with its application to holonomy, as well as modular class of a skew algebroid relation. These notions provide, in particular, a unified approach to the concepts of a modular class of a Lie algebroid morphism and that of a Poisson map.Comment: 20 page

Jacobi-Nijenhuis algebroids and their modular classes

Jacobi-Nijenhuis algebroids are defined as a natural generalization of Poisson-Nijenhuis algebroids, in the case where there exists a Nijenhuis operator on a Jacobi algebroid which is compatible with it. We study modular classes of Jacobi and Jacobi-Nijenhuis algebroids

Service-Learning Pedagogy, Civic Engagement, and Academic Engagement: Multiple Bidirectional Relationships in College Freshmen

This study begins to unravel the multiple bidirectional relationships between service-learning pedagogy and civic and academic engagement attitudes and behaviors. A quasi-experimental, nonequivalent comparison group pre- and post-test design was used with a sample of 300 first- semester freshmen participating in either a service-learning-based learning community or a learning community without service-learning. Participants completed a pre-test at the beginning of the semester measuring high school civic and academic engagement behaviors and attitudes and a post- test at the end of the semester measuring the same variables based on their first semester in college. Students with higher civic engagement attitudes and behaviors prior to college were more likely to take a service-learning course than students with lower civic engagement attitudes and behaviors. Students in service-learning were more likely to participate in community activities than students not participating in service-learning. Finally, within the service-learning groups, students who were more academically engaged had higher academic and civic attitudinal engagement at the end of the course. Students who were more civically engaged were more likely to see lower costs of helping to themselves; they did not change in terms of their beliefs about the community’s needs. This study replicates and extends previous research to demonstrate that there are multiple bidirectional relationships among these variables that need to be taken into account in research and practice.

Trapping atoms on a transparent permanent-magnet atom chip

We describe experiments on trapping of atoms in microscopic magneto-optical traps on an optically transparent permanent-magnet atom chip. The chip is made of magnetically hard ferrite-garnet material deposited on a dielectric substrate. The confining magnetic fields are produced by miniature magnetized patterns recorded in the film by magneto-optical techniques. We trap Rb atoms on these structures by applying three crossed pairs of counter-propagating laser beams in the conventional magneto-optical trapping (MOT) geometry. We demonstrate the flexibility of the concept in creation and in-situ modification of the trapping geometries through several experiments.Comment: Modifications: A) Reference I. Barb et al., Eur. Phys. JD, 35, 75 (2005) added. B)Sentence rewritten: We routinely capture more than 10^6 atoms in a micro-MOT on a magnetized pattern. C) The magnetic field strengths are now given in Teslas. D) The second sentence in the fourth paragraph has been rewritten in order to more clearly describe the geometry and purpose of the compensation coils.E) In the seventh paragraph we have rewritten the sentence about the creation of the external magnetic field for the magnetic-domain patterning. F) In the ninth paragraph, we clarify the way to shift the trap center. G) Caption of Fig. 4 changed. H) We have modified paragraph 12 to improve the description on the guiding of the trap center along a toroidal pattern. I) The last two sentences of the manuscript have been rewritte