We classify simply-connected, orientable, complete Willmore surfaces with vanishing Gaussian curvature. also study the cones in $${\mathbb {R}}^{3}$$ and give a classification. As an application, we show that for embedding $$f:{\mathbb {R}}^{2} \rightarrow {\mathbb , if its corresponding curvature is nonnegative image of Gauss map lies $$\big \{\theta \in {S}}^{2}: d_{{\mathbb {S}}^{2}}(\theta ...

To construct the rotation curve of Galaxy, classical Cepheids with proper motions, parallaxes and line-of-sight velocities from Gaia DR2 Catalog are used in large part. The working sample formed literature data contains about 800 estimates their age. We determined that linear velocity Galaxy at a solar distance is $V_0=240\pm3$~km s$^{-1}$. In this case, Sun to axis found be $R_0=8.27\pm0.10$~k...

We describe a new series of identities, which hold for certain general theta series, in two completely independent variables. We provide explicit examples of these identities involving the Dedekind eta function, Jacobi theta functions, and various theta functions of Ramanujan. Introduction Let z ∈ H = {x+ yi : x, y ∈ R, y > 0} and for each x ∈ R set q = exp(2πixz) and e(x) = exp(2πix). The Dede...

In a recent work PRD 106, L021501 (2022), new geometric approach is proposed to obtain the photon sphere (circular orbit) and black hole shadow radius. this approach, spheres radius are determined using geodesic curvature Gaussian in optical geometry of spacetimes. However, calculations (2022) only restricted subclass static spherically symmetric holes with spacetime metric $g_{tt} \cdot g_{rr}...

Previous studies indicate that 11,12,15-trihydroxyeicosatrienoic acid (11,12,15-THETA), an endothelium-derived hyperpolarizing factor in the rabbit aorta, mediates a portion of the relaxation response to acetylcholine by sequential metabolism of arachidonic acid by 15-lipoxygenase, hydroperoxide isomerase, and epoxide hydrolase. To determine the stereochemical configuration of the endothelial 1...

This study was performed to evaluate the alterations of glomerular filtration barrier characteristics following acute renal ischemia-reperfusion (I/R). Ischemia was induced in anesthetized rats by unilateral renal artery occlusion for either 20 or 60 min, followed by reperfusion during 20 or 60 min, respectively, with the contralateral kidney serving as control. Sieving coefficients (theta) wer...

It is known that if the Gaussian curvature function along each meridian on a surface of revolution $(\mathbb{R}^{2}, dr^{2} + m(r)^{2} d\theta^{2})$ decreasing, then cut locus point $\theta^{-1} (0)$ empty or subarc opposite (\pi)$. Such called von Mangoldt's revolution. A generalized Mangoldt For example, m_0(r)^{2} d\theta^{2})$, where $m_{0}(x) = x/(1 x^{2})$, has same structure as above and...

It is shown that if $A \subseteq \mathbb{R}^3$ a Borel set of Hausdorff dimension $\dim A \in (3/2,5/2)$, then for a.e. $\theta [0,2\pi)$ the projection $\pi_{\theta}(A)$ $A$ onto 2-dimensional plane orthogonal to $\frac{1}{\sqrt{2}}(\cos \theta, \sin 1)$ satisfies \pi_{\theta}(A) \geq \max\left\{\frac{4\dim A}{9} + \frac{5}{6},\frac{2\dim A+1}{3} \right\}$. This improves bound Oberlin and Ober...