324 research outputs found
Eliciting implicit assumptions of proofs in the MIZAR Mathematical Library by property omission
When formalizing proofs with interactive theorem provers, it often happens
that extra background knowledge (declarative or procedural) about mathematical
concepts is employed without the formalizer explicitly invoking it, to help the
formalizer focus on the relevant details of the proof. In the contexts of
producing and studying a formalized mathematical argument, such mechanisms are
clearly valuable. But we may not always wish to suppress background knowledge.
For certain purposes, it is important to know, as far as possible, precisely
what background knowledge was implicitly employed in a formal proof. In this
note we describe an experiment conducted on the MIZAR Mathematical Library of
formal mathematical proofs to elicit one such class of implicitly employed
background knowledge: properties of functions and relations (e.g.,
commutativity, asymmetry, etc.).Comment: 11 pages, 3 tables. Preliminary version presented at the 3rd Workshop
on Modules and Libraries for Proof Assistants (MLPA-11), affiliated with the
2nd Conference on Interactive Theorem Proving (ITP-2011), Nijmegen, the
Netherland
Vortex density models for superconductivity and superfluidity
We study some functionals that describe the density of vortex lines in
superconductors subject to an applied magnetic field, and in Bose-Einstein
condensates subject to rotational forcing, in quite general domains in 3
dimensions. These functionals are derived from more basic models via
Gamma-convergence, here and in a companion paper. In our main results, we use
these functionals to obtain descriptions of the critical applied magnetic field
(for superconductors) and forcing (for Bose-Einstein), above which ground
states exhibit nontrivial vorticity, as well as a characterization of the
vortex density in terms of a non local vector-valued generalization of the
classical obstacle problem.Comment: 34 page
Relationship Between Vitamin D and Cardio-Metabolic Biomarkers Among Saudi Postmenopausal Women
Vitamin D deficiency is prevalent worldwide, and in Saudi Arabia in particular. There is growing evidence that hypovitaminosis D is involved in the pathogenesis of cardiovascular diseases. We determined concentrations of serum 25 hydroxy 25(OH) vitamin D in relation to several metabolic biomarkers including total cholesterol (TC), low density lipoprotein-cholesterol (LDL-C), high density lipoprotein-cholesterol, triglycerides (TG), atherogenic index (AI), glucose, C-reactive protein (CRP), adiposity, and blood pressure in a cross-sectional analysis in 300 Saudi postmenopausal women. Participants completed a detailed questionnaire and fasting blood samples were collected. Vitamin D deficiency was common, affecting 89% of individuals. Higher serum 25(OH) vitamin D levels were consistently found among subjects with no prevalent cardiovascular risk factors (p>0.05) except for those subjects with serum CRP level ≥3mg/dl, HDL-C <1.04mmol/L, AI≥5, exercising ≥3times/week, and those with 4 or more pregnancies. Hypovitaminosis D was inversely correlated with DBP (r=-0.118, p=0.042), TC (r=-0.165, p=0.004), TG (r=-0.119, p=0.040), LDL-C (r=-0.138, p=0.017), AI (r=-0.125, p=0.031), and veiling type (r=-0.127, p=0.028). No significant impact of hypovitaminosis D on CRP, levels of which were similar among vitamin D sufficient and deficient subjects. However, hypovitaminosis D was significantly related to dyslipidemia and diastolic blood pressure in a group of Saudi postmenopausal women
Positive solutions to indefinite Neumann problems when the weight has positive average
We deal with positive solutions for the Neumann boundary value problem
associated with the scalar second order ODE where is positive on and is an indefinite weight. Complementary to previous
investigations in the case , we provide existence results
for a suitable class of weights having (small) positive mean, when
at infinity. Our proof relies on a shooting argument for a suitable equivalent
planar system of the type with
a continuous function defined on the whole real line.Comment: 17 pages, 3 figure
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