On Gromov's Waist of the Sphere Theorem

The goal of this paper is to give a detailed and complete proof of M. Gromov's waist of the sphere theorem.Comment: 34 pages, 1 figur

Prediction of HBF-0259 interactions with hepatitis B Virus receptors and surface antigen secretory factors

Hepatitis B virus (HBV) is an etiological agent of viral hepatitis, which may lead to cirrhosis, and hepatocellular carcinoma. Current treatment strategies have not shown promising effect to date but various complications such as, drug toxicity-resistance have been reported. Study on newly discovered compounds, with minimal side effects, as specific HBV inhibitors is a fundamental subject introducing new biologic drugs. Here, we aimed to, by prediction, estimate interactions of HBF-0259 as a non-toxic anti-HBV compound on inhibiting the HBV through either interaction with the viral entry or HBsAg secreting factors using In Silico procedure. Molecular docking was performed by Hex 8.0.0 software to predict the interaction energy (Etot) between HBF-0259 and known cellular factors involved in HBV entry and HBsAg secreting factors. Hex 8.0.0 also employed to create protein–protein complexes. These interactions were then used to analyze the binding site of HBF-0259 within the assumed receptors by MGLTools software. Finally, the amino acid sequences involved in this interaction were aligned for any conservancy. Here, we showed that HBF-0259 Etot with CypA (–545.41 kcal/mol) and SCCA1 (499.68 kcal/mol), involved in HBsAg secretion and HBV integration, respectively, was higher than other interactions. Furthermore, HBF-0259 predicted interaction energy was even higher than those of CypA inhibitors. In addition, we claim that preS1 and/or preS2 regions within HBsAg are not suitable targets for HBF-0259. HBF-0259 has higher interaction energy with CypA and SCCA1, even more than other known receptors, co-receptors, viral ligands, and secretory factors. HBF-0259 could be introduced as potent anti-viral compound in which CypA and or SCCA1, as previously shown, are involved. © 2016 Indian Virological Societ

Positivity of the Cotangent Bundle of Complex Hyperbolic Manifolds with Cusps

Let $\overline{X}$ be the toroidal compactification of a cusped complex hyperbolic manifold $X=\mathbb{B}^n/\Gamma$ with the boundary divisor $D=\overline{X}\setminus X$. The main goal of this paper is to find the positivity properties of $\Omega^{1}_{\overline{X}}$ and $\Omega^{1}_{\overline{X}}\big(\log(D)\big)$ depending intrinsically on $X$. We prove that $\Omega^{1}_{\overline{X}}\big(\log(D)\big) \langle -r D \rangle$ is ample for all sufficiently small rational numbers $r >0$, and $\Omega^{1}_{\overline{X}}\big(\log(D)\big)$ is ample modulo $D.$ Further, we conclude that if the cusps of $X$ have uniform depth greater than $2\pi$, then $\Omega^{1}_{\overline{X}}$ is semi-ample and is ample modulo $D$, all subvarieties of $X$ are of general type, and every smooth subvariety $V\subset \overline{X}$ intersecting $\overline{X}$ has ample $K_{V}$. Finally, we show that the minimum volume of subvarieties of $\overline{X}$ intersecting both $X$ and $D$ tends to infinity in towers of normal covering of $X.$Comment: Minor change