The regularized visible fold revisited

The planar visible fold is a simple singularity in piecewise smooth systems. In this paper, we consider singularly perturbed systems that limit to this piecewise smooth bifurcation as the singular perturbation parameter $\epsilon\rightarrow 0$. Alternatively, these singularly perturbed systems can be thought of as regularizations of their piecewise counterparts. The main contribution of the paper is to demonstrate the use of consecutive blowup transformations in this setting, allowing us to obtain detailed information about a transition map near the fold under very general assumptions. We apply this information to prove, for the first time, the existence of a locally unique saddle-node bifurcation in the case where a limit cycle, in the singular limit $\epsilon\rightarrow 0$, grazes the discontinuity set. We apply this result to a mass-spring system on a moving belt described by a Stribeck-type friction law

Reduction of singularities of three-dimensional line foliations

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Low-dose estradiol and endothelial and inflammatory biomarkers in menopausal overweight/obese women

<p><b>Objective:</b> We aimed to investigate the effects of low-dose transdermal estrogen on endothelial and inflammatory biomarkers in menopausal overweight/obese women.</p> <p><b>Methods:</b> We recruited 44 menopausal women (47–55 years; body mass index 27.5–34.9 kg/m²) and divided them into estradiol (1 mg/day; <i>n</i> = 22) or placebo groups (<i>n</i> = 22). They were double-blinded, followed and treated for 3 months. At baseline and post-intervention, inflammatory biomarkers (hs-CRP, IL-1β, IL-6, MCP-1 and TNF-α) and of vascular injury (activated circulating endothelial cells, CEC-a) and repair (endothelial progenitor cells, EPC) were quantified. Resting CECs (CEC-r) were also assessed. Microvascular reactivity and vasomotion were analyzed by laser-Doppler flowmetry.</p> <p><b>Results:</b> Volunteers (51.8 ± 2.3 years; mean body mass index 31.5 ± 2.5 kg/m²) had been menopausal for 3 (range 2–5) years. After treatment, no changes were observed in the placebo group, while levels of CEC-r and EPC increased in the estradiol group. In this group, no changes in inflammatory biomarkers were observed but it required a lower cumulative dose of acetylcholine to achieve peak velocity during endothelial-dependent vasodilatation and there was increased endothelial-independent vasodilatation.</p> <p><b>Conclusions:</b> The short-term use of low-dose transdermal estradiol therapy in overweight/obese menopausal women increased markers of vascular repair and improved microvascular reactivity without changing the inflammatory biomarkers.</p> <p><b>Clinical trial registration:</b> NCT01295892 at <a href="http://www.clinicaltrials.gov" target="_blank">www.clinicaltrials.gov</a>.</p