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

Lie algebroid structures on a class of affine bundles

We introduce the notion of a Lie algebroid structure on an affine bundle whose base manifold is fibred over the real numbers. It is argued that this is the framework which one needs for coming to a time-dependent generalization of the theory of Lagrangian systems on Lie algebroids. An extensive discussion is given of a way one can think of forms acting on sections of the affine bundle. It is further shown that the affine Lie algebroid structure gives rise to a coboundary operator on such forms. The concept of admissible curves and dynamical systems whose integral curves are admissible, brings an associated affine bundle into the picture, on which one can define in a natural way a prolongation of the original affine Lie algebroid structure.Comment: 28 page

Dynamics of reflection of ultracold atoms from a periodic 1D magnetic lattice potential

We report on an experimental study of the dynamics of the reflection of ultracold atoms from a periodic one-dimensional magnetic lattice potential. The magnetic lattice potential of period 10 \textmu m is generated by applying a uniform bias magnetic field to a microfabricated periodic structure on a silicon wafer coated with a multilayered TbGdFeCo/Cr magneto-optical film. The effective thickness of the magnetic film is about 960 nm. A detailed study of the profile of the reflected atoms as a function of externally induced periodic corrugation in the potential is described. The effect of angle of incidence is investigated in detail. The experimental observations are supported by numerical simulations.Comment: 15 pages, 11 figure

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

Factor Substitution and Unobserved Factor Quality in Nursing Homes

This paper studies factor substitution in one important sector: the nursing home industry. Specifically, we measure the extent to which nursing homes substitute materials for labor when labor becomes relatively more expensive. From a policy perspective, factor substitution in this market is important because materials-intensive methods of care are associated with greater risks of morbidity and mortality among nursing home residents. Studying longitudinal data from 1991-1998 on nearly every nursing home in the United States, we use the method of instrumental variables (IV) to address the potential endogeneity of nursing home wages. The results from the IV models are consistent with the theory of factor substitution: higher nursing home wages are associated with lower staffing, greater use of materials (specifically, physical restraints), and a higher proportion of residents with pressure ulcers. A comparison of OLS and IV results suggests that empirical studies of factor substitution should take into account unobserved heterogeneity in factor quality.

Isolation and characterization of 8 microsatellite loci for the ‘‘killer shrimp’’, an invasive Ponto-Caspian amphipod Dikerogammarus villosus (Crustacea: Amphipoda)

Dikerogammarus villosus is a freshwater amphipod of the Ponto-Caspian origin recognized as one of the 100 worst alien species in Europe, having negative impact on biodiversity and functioning of the invaded aquatic ecosystems. The species has a wide ecophysiological tolerance and during the last 20 years it has rapidly spread throughout European inland waters. In consequence, it presents a major conservation management problem. We describe eight polymorphic microsatellite loci developed for D. villosus by combining a biotin-enrichment protocol and new generation 454GS-FLX Titanium pyrosequencing technology. When genotyped in 64 individuals from two locations, the loci exhibited a mean diversity of 4.87 alleles per locus (2–13). The mean observed and expected heterozygosities were, respectively, 0.439 (0.091–0.844) and 0.468 (0.089–0.843). Gametic disequilibrium was not detected for any pair of loci. The microsatellite markers will be a valuable tool in assessing the demographic processes associated with invasion of the killer shrimp from a genetic point of view

New developments in geometric mechanics

We review the concept of a graded bundle, which is a generalisation of a vector bundle, its linearisation, and a double structure of this kind. We then present applications of these structures in geometric mechanics including systems with higher order Lagrangian and the Plateau problem.Comment: 16 pages, conference proceedings "Geometry of Jets and Fields" (Bedlewo, 10-16 May, 2015