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

reply to the letter to the editor by C.Nicolazzo et al.: "circulating cell-free DNA and circulating tumor cells as prognostic and predictive biomarkers in advanced non-small cell lung cancer patients treated with first-line chemotherapy"

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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 $$u" + q(t)g(u) = 0, \quad t \in [0, T],$$ where $g: [0, +\infty[\, \to \mathbb{R}$ is positive on $\,]0, +\infty[\,$ and $q(t)$ is an indefinite weight. Complementary to previous investigations in the case $\int_0^T q(t) < 0$, we provide existence results for a suitable class of weights having (small) positive mean, when $g'(x) < 0$ at infinity. Our proof relies on a shooting argument for a suitable equivalent planar system of the type $$x' = y, \qquad y' = h(x)y^2 + q(t),$$ with $h(x)$ a continuous function defined on the whole real line.Comment: 17 pages, 3 figure