GENUS MINIMAL k-NOIDS AND SADDLE TOWERS IN

For each $k\geq 3$ , we construct a $1$ -parameter family of complete properly Alexandrov-embedded minimal surfaces in the Riemannian product space $\mathbb {H}^2\times \mathbb {R}$ with genus $1$ and k embedded ends asymptotic to vertical planes. We also obtain complete minimal surfaces with genus $1$ and $2k$ ends in the quotient of $\mathbb {H}^2\times \mathbb {R}$ by an arbitrary vertical translation. They all have dihedral symmetry with respect to k vertical planes, as well as finite total curvature $-4k\pi $ . Finally, we provide examples of complete properly Alexandrov-embedded minimal surfaces with finite total curvature with genus $1$ in quotients of $\mathbb {H}^2\times \mathbb {R}$ by the action of a hyperbolic or parabolic translation.

Evaluating underlying causative factors for earthquake-induced landslides and landslide susceptibility mapping in Upper Indrawati Watershed, Nepal

The main objective of this study is to understand the overall impact of earthquake in upper Indrawati Watershed, located in the high mountainous region of Nepal. Hence, we have assessed the relationship between the co-seismic landslide and underlying causative factors as well as performed landslide susceptibility mapping (LSM) to identify the landslide susceptible zone in the study area. We assessed the landslides distribution in terms of density, number, and area within 85 classes of 13 causal factors including slope, aspect, elevation, formation, land cover, distance to road and river, soil type, total curvature, seismic intensity, topographic wetness index, distance to fault, and flow accumulation. The earthquake-induced landslide is clustered in Northern region of the study area, which is dominated by steep rocky slope, forested land, and low human density. Among the causal factors, 'slope' showed positive correlation for landslide occurrence. Increase in slope in the study area also escalates the landslide distribution, with highest density at 43%, landslide number at 4.34/km2, and landslide area abundance at 2.97% in a slope class (> 50°). We used logistic regression (LR) for LSM integrating with geographic information system. LR analysis depicts that land cover is the best predictor followed by slope and distance to fault with higher positive coefficient values. LSM was validated by assessing the correctly classified landslides under susceptibility categories using area under curve (AUC) and seed cell area index (SCAI). The LSM approach showed good accuracy with respective AUC values for success rate and prediction rate of 0.843 and 0.832. Similarly, the decreasing SCAI value from very low to very high susceptibility categories advise satisfactory accuracy of the LSM approach.

COMPUTATIONAL FLUID DYNAMICS SIMULATIONS OF TAYLOR BUBBLE RISE IN FLOWING LIQUIDS

Systematic analysis of the effect of gravitational, interfacial, viscous and inertia forces acting on a Taylor bubble rising in flowing liquids characterised by the dimensionless Froude (Uc), inverse viscosity (Nf ) and Eötvös numbers (Eo) is carried out using computational fluid dynamic finite element method. Particular attention is paid to cocurrent (i.e upward) liquid flow and the influence of the characterising dimensionless parameters on the bubble rise velocity and morphology analysed for Nf, Eo and Uc ranging between [40, 100], [20, 300] and [−0.20, 0.20], respectively. Analysis of the results of the numerical simulations showed that the existing theoretical model for the prediction of Taylor bubble rise velocity in upward flowing liquids could be modified to accurately predict the rise velocity in liquids with high viscous and surface tension effects. Furthermore, the mechanism governing the change in morphology of the bubble in flowing liquids was shown to be the interplay between the viscous stress and total curvature stress at the interface. Keywords: Taylor bubble, finite element, slug flow, CFD, rise velocity

A Three-Dimensional-reconstruction-based study on the ocular volume of Chinese children with high myopia

Background Highly myopic eyes differ in morphology from emmetropic eyes, and the correct estimation of the vitreous volume is difficult. To explore an effective method to estimate ocular volume using refractive factors in children. Methods This is a retrospective study of children with high myopia who visited the Shenzhen Shekou People’s Hospital (July-December 2018) before undergoing posterior scleral reinforcement surgery. Data on refractive factors and ocular 3D reconstruction imaging based on high-end CT were collected for linear correlation and linear regression analyses. Results Ten patients (20 eyes) were included. There are nine males and one female. They were 4 to 12 years of age. The spherical equivalent ranges from + 0.25 to -20.00 D. The cylindrical equivalent ranges from − 0.50 to -6.25 D. The AL(axial length, AL) ranges from 21.78 to 33.90 mm. The corneal curvature (mean) ranges from 42.44 to 46.75. The 3D reconstruction of the CT images shows that the ocular volume ranges from 4.591 to 10.988 ml. The ocular volume of the 20 eyes decreases with the increase of diopter and total curvature, both presenting a linear trend, with the Pearson correlation coefficients being − 0.776 (P < 0.001) and − 0.633 (P = 0.003), respectively. The ocular volume of the 20 eyes increases with the increasing AL, also presenting a linear trend, with the Pearson correlation coefficient being 0.939 (P < 0.001). Conclusions In children, the ocular volume is negatively and linearly correlated with the diopter and curvature, and positively and linearly correlated with the AL.

Geomorphometric analysis of Olkhon island region’s relief

The use of geomorphometric analysis to obtain relevant data and their interpretation in geomorphological and geographical studies is becoming more widespread in recent times. In this study, such an analysis is used for the Priolkhonsky plateau (Western Baikal region), whose territory has recently been subjected to intense recreational impact. Under such conditions, it becomes relevant to determine the erosion-accumulative potential of the territory, its subsequent regionalization, as well as the possibility of using the obtained results in landscape-recreational zoning. The surface curvature maps were made using a digital elevation model based on ALOS data with a resolution of 30 m. Using geoinformation analysis, slope steepness, horizontal, vertical, and general curvature maps were obtained, and statistical coefficients of these indicators for the study area were determined. A geomorphological interpretation of the areas spatial distribution with different values of curvature was carried out. The analysis of the territory geomorphometric parameters showed that denudation processes prevail within the study area. It was also determined that the territory under favorable climatic conditions could have a significant potential for the development of erosion-accumulative processes. Areas of accumulation associated with the development of linear erosion forms, gullies along unpaved roads, were identified. Areas with the maximum values of curvature in the summit level of the relief have a high potential for the development of landslide processes. In the middle level of relief with positive values of the total curvature, prevail low-intensity processes (downwash, creeping).

Curvature Potential Unveiled Topological Defect Attractors

We consider the theoretical and positional assembling of topological defects (TDs) in effectively two-dimensional nematic liquid crystal films. We use a phenomenological Helfrich–Landau–de Gennes-type mesoscopic model in which geometric shapes and nematic orientational order are expressed in terms of a curvature tensor field and a nematic tensor order parameter field. Extrinsic, intrinsic, and total curvature potentials are introduced using the parallel transport concept. These potentials reveal curvature seeded TD attractors. To test ground configurations, we used axially symmetric nematic films exhibiting spherical topology.

Metrics on a Surface with Bounded Total Curvature

Let $g=e^{2u}g_{euc}$ be a conformal metric defined on the unit disk of ${{\mathbb{C}}}$. We give an estimate of $\|\nabla u\|_{L^{2,\infty }(D_{\frac{1}{2}})}$ when $\|K(g)\|_{L^1}$ is small and $\frac{\mu (B_r^g(z),g)}{\pi r^2}&lt;\Lambda $ for any $r$ and $z\in D_{\frac{3}{4}}$. Then we use this estimate to study the Gromov–Hausdorff convergence of a conformal metric sequence with bounded $\|K\|_{L^1}$ and give some applications.