Quantitative Singularity Theory for Random Polynomials

Motivated by Hilbert’s 16th problem we discuss the probabilities of topological features of a system of random homogeneous polynomials. The distribution for the polynomials is the Kostlan distribution. The topological features we consider are type-W singular loci. This is a term that we introduce and that is defined by a list of equalities and inequalities on the derivatives of the polynomials. In technical terms a type-W singular locus is the set of points where the jet of the Kostlan polynomials belongs to a semialgebraic subset W of the jet space, which we require to be invariant under orthogonal change of variables. For instance, the zero set of polynomial functions or the set of critical points fall under this definition. We will show that, with overwhelming probability, the type-W singular locus of a Kostlan polynomial is ambient isotopic to that of a polynomial of lower degree. As a crucial result, this implies that complicated topological configurations are rare. Our results extend earlier results from Diatta and Lerario who considered the special case of the zero set of a single polynomial. Furthermore, for a given polynomial function p we provide a deterministic bound for the radius of the ball in the space of differentiable functions with center p⁠, in which the W-singularity structure is constant

The effects of vitamin D supplementation on signaling pathway of inflammation and oxidative stress in diabetic hemodialysis: A randomized, double-blind, placebo-controlled trial

Objective: This study was carried out to determine the effects of vitamin D supplementation on signaling pathway of inflammation and oxidative stress in diabetic hemodialysis (HD) patients. Methods: This randomized double-blind placebo-controlled clinical trial was conducted among 60 diabetic HD patients. Subjects were randomly allocated into two groups to intake either vitamin D supplements at a dosage of 50,000 IU (n = 30) or placebo (n = 30) every 2 weeks for 12 weeks. Gene expression of inflammatory cytokines and biomarkers of oxidative stress were assessed in peripheral blood mononuclear cells (PBMCs) of diabetic HD patients with RT-PCR method. Results: Results of RT-PCR indicated that after the 12-week intervention, compared to the placebo, vitamin D supplementation downregulated gene expression of interleukin (IL)-1β (P = 0.02), tumor necrosis factor alpha (TNF-α) (P = 0.02) and interferon gamma (IFN-γ) (P = 0.03) in PBMCs of diabetic HD patients. Additionally, vitamin D supplementation, compared to the placebo, downregulated gene expression of transforming growth factor beta (TGF-β) (P = 0.04), protein kinase C (PKC) (P = 0.001), and mitogen-activated protein kinases 1 (MAPK1) (P = 0.02) in PBMCs of diabetic HD patients. Although not significant, vitamin D supplementation let to a reduction of nuclear factor kappa B (NF-kB) (p = 0.75) expression in PBMCs isolated from diabetic patients compared to the placebo group. There was no statistically significant change following supplementation with vitamin D on gene expression of interleukin (IL)-4, IL-6, and vascular endothelial growth factor (VEGF) in PBMCs of diabetic HD patients. Conclusions: Overall, we found that vitamin D supplementation for 12 weeks among diabetic HD patients had beneficial effects on few gene expression related to inflammation and oxidative stress. © 2018 Haddad Kashani, Seyed Hosseini, Nikzad, Soleimani, Soleimani, Tamadon, Keneshlou and Asemi

The effects of vitamin D supplementation on signaling pathway of inflammation and oxidative stress in diabetic hemodialysis: A randomized, double-blind, placebo-controlled trial

Objective: This study was carried out to determine the effects of vitamin D supplementation on signaling pathway of inflammation and oxidative stress in diabetic hemodialysis (HD) patients. Methods: This randomized double-blind placebo-controlled clinical trial was conducted among 60 diabetic HD patients. Subjects were randomly allocated into two groups to intake either vitamin D supplements at a dosage of 50,000 IU (n = 30) or placebo (n = 30) every 2 weeks for 12 weeks. Gene expression of inflammatory cytokines and biomarkers of oxidative stress were assessed in peripheral blood mononuclear cells (PBMCs) of diabetic HD patients with RT-PCR method. Results: Results of RT-PCR indicated that after the 12-week intervention, compared to the placebo, vitamin D supplementation downregulated gene expression of interleukin (IL)-1β (P = 0.02), tumor necrosis factor alpha (TNF-α) (P = 0.02) and interferon gamma (IFN-γ) (P = 0.03) in PBMCs of diabetic HD patients. Additionally, vitamin D supplementation, compared to the placebo, downregulated gene expression of transforming growth factor beta (TGF-β) (P = 0.04), protein kinase C (PKC) (P = 0.001), and mitogen-activated protein kinases 1 (MAPK1) (P = 0.02) in PBMCs of diabetic HD patients. Although not significant, vitamin D supplementation let to a reduction of nuclear factor kappa B (NF-kB) (p = 0.75) expression in PBMCs isolated from diabetic patients compared to the placebo group. There was no statistically significant change following supplementation with vitamin D on gene expression of interleukin (IL)-4, IL-6, and vascular endothelial growth factor (VEGF) in PBMCs of diabetic HD patients. Conclusions: Overall, we found that vitamin D supplementation for 12 weeks among diabetic HD patients had beneficial effects on few gene expression related to inflammation and oxidative stress. © 2018 Haddad Kashani, Seyed Hosseini, Nikzad, Soleimani, Soleimani, Tamadon, Keneshlou and Asemi