Plane Convex
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2021 ◽  
Vol 31 (5) ◽  
Author(s):  
Jonas Allemann ◽  
Norbert Hungerbühler ◽  
Micha Wasem

AbstractWe obtain a formula for the number of horizontal equilibria of a planar convex body K with respect to a center of mass O in terms of the winding number of the evolute of $$\partial K$$ ∂ K with respect to O. The formula extends to the case where O lies on the evolute of $$\partial K$$ ∂ K and a suitably modified version holds true for non-horizontal equilibria.


Nanomaterials ◽  
2021 ◽  
Vol 11 (7) ◽  
pp. 1838
Author(s):  
Lucien Saviot

An extension of the Rayleigh–Ritz variational method to objects with superquadric and superellipsoid shapes and cylinders with cross-sections delimited by a superellipse is presented. It enables the quick calculation of the frequencies and displacements for shapes commonly observed in nano-objects. Original smooth shape variations between objects with plane, convex, and concave faces are presented. The validity of frequently used isotropic approximations for experimentally relevant vibrations is discussed. This extension is expected to facilitate the assignment of features observed with vibrational spectroscopies, in particular in the case of single-nanoparticle measurements.


Author(s):  
Ferenc Fodor

AbstractEggleston (Approximation to plane convex curves. I. Dowker-type theorems. Proc. Lond. Math. Soc. 7, 351–377 (1957)) proved that in the Euclidean plane the best approximating convex n-gon to a convex disc K is always inscribed in K if we measure the distance by perimeter deviation. We prove that the analogue of Eggleston’s statement holds in the hyperbolic plane, and we give an example showing that it fails on the sphere.


2020 ◽  
Vol 63 (4) ◽  
pp. 888-917
Author(s):  
János Pach ◽  
Bruce Reed ◽  
Yelena Yuditsky

2020 ◽  
Vol 299 ◽  
pp. 1099-1103
Author(s):  
Dmitry Kruchinin ◽  
Yulia Aleshina ◽  
Elena Farafontova

The problematic issues, associated with deformations of the outer surfaces of optical parts made of quartz glass, due to the process of cementing, are considered. The effect of the heat treatment temperature of optical assemblies, consisting of parts made of quartz glass and cementing with epoxy cements OK-72FT5 and OK-72FT15, on the deformation of external surfaces was experimentally established. The object of the study was plane-convex and plane-concave lenses made of quartz glass. It has been stated, that on cementing with optical epoxy cements OK72FT5 and OK72FT15 the external quartz glass surface deformations depend primarily on the heat treatment temperature. When cementing parts are made of quartz glass, increasing the polymerization temperature improves the performance characteristics of products and reduces changes in their optical characteristics.


2020 ◽  
Vol 299 ◽  
pp. 311-315
Author(s):  
Dmitry Kruchinin ◽  
Liana Garayshina ◽  
Elena Farafontova

The problematic issues connected with the optical details deformations that arise during the temperature tests of cemented assemblies are considered. Climatic tests of optical devices after assembly result in a change of technical product characteristics. The research problem is to define how N and ∆N of surfaces of cemented optical assemblies are changing after the temperature testing. The research objects are plane convex lenses fabricated of LK7 (LZOS) glass brand and plane concave lenses - of TF7 (LZOS). Before cementing, the bounded surfaces cleaning by the hydromechanical method in the clean room has been purged. The temperature testing of samples is held at 22, 50 and 65 °C. The effect of temperature on a surface deformation of cemented optical assemblies is experimentally established. The comparison of effect of optical cements OK72FT5 and OK72FT15 on the optical surfaces deformation is carried out. In this paper we have compared the results of shape accuracy of the external surfaces of optical assemblies and the flat surfaces. Interferograms of lenses after heat treatment of optical assemblies are presented. It is determined that the cement polymerization temperature equals or higher than the test temperature.


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