scholarly journals Example of Bianchi Transformation of Kuen’s Surface

2021 ◽  
pp. 126-128
Author(s):  
М.A. Cheshkova ◽  
A.A. Pavlova
Keyword(s):  
Gaussian Curvature ◽  
Software Package ◽  
Gordon Equation ◽  
Nonlinear Differential Equations ◽  
Mathematical Software ◽  
Negative Gaussian Curvature ◽  
Pseudospherical Surfaces ◽  
Modern Physics ◽  
Beltrami Surface ◽  
Pseudospherical Surface

The work is devoted to the study of the Bianchi transformation for surfaces of constant negative Gaussian curvature. The surfaces of rotation of constant negative Gaussian curvature are the Minding top, the Minding coil, and the pseudosphere (Beltrami surface). Surfaces of constant negative Gaussian curvature also include Kuen’s surface and the Dini’s surface. Studying the surfaces of constant negative Gaussian curvature (pseudospherical surfaces) is of great importance for the interpretation of Lobachevsky planimetry. Geometric characteristics of pseudospherical surfaces are found to be related to the theory of networks, the theory of solitons, nonlinear differential equations, and sin-Gordon equations. The sin-Gordon equation plays an important role in modern physics. Bianchi transformations make it possible to obtain new pseudospherical surfaces from a given pseudospherical surface. The Bianchi transformation for the Kuen’s surface is constructed using a mathematical software package.

Transformation of Bianchi for Minding Top

2020 ◽  
pp. 135-142
Author(s):  
M. A. Cheshkova
Keyword(s):  
Differential Equations ◽  
Gaussian Curvature ◽  
Gordon Equation ◽  
Nonlinear Differential Equations ◽  
Geometric Characteristics ◽  
Negative Gaussian Curvature ◽  
Pseudospherical Surfaces ◽  
Modern Physics ◽  
Beltrami Surface ◽  
Pseudospherical Surface

The work is devoted to the study of the Bianchi transform for surfac­es of revolution of constant negative Gaussian curvature. The surfaces of rotation of constant negative Gaussian curvature are the Minding top, the Minding coil, the pseudosphere (Beltrami surface). The study of surfaces of constant negative Gaussian curvature (pseudospherical surfaces) is of great importance for the interpretation of Lobachevsky planimetry. The connection of the geometric characteristics of pseudospherical surfaces with the theory of networks, with the theory of solitons, with nonlinear differential equations and sin-Gordon equations is established. The sin-Gordon equation plays an important role in modern physics. Bianchi transformations make it possible to obtain new pseudospherical surfaces from a given pseudospherical surface. The Bianchi transform for the Minding top is constructed. Using a mathematical package, Minding's top and its Bianchi transform are constructed.

scholarly journals On the solution of a class of second-order quasi-linear PDEs and the Gauss equation

2001 ◽  
Vol 42 (3) ◽  
pp. 312-323
Author(s):  
A. R. Selvaratnam ◽  
M. Vlieg-Hulstman ◽  
B. van-Brunt ◽  
W. D. Halford
Keyword(s):  
Gaussian Curvature ◽  
Gordon Equation ◽  
Second Order ◽  
Bäcklund Transformations ◽  
Coordinate Systems ◽  
Gauss Equation ◽  
Sine Gordon Equation ◽  
Partial Differential ◽  
Metric Tensor ◽  
Backlund Transformations

AbstractGauss' Theorema Egregium produces a partial differential equation which relates the Gaussian curvature K to components of the metric tensor and its derivatives. Well-known partial differential equations (PDEs) such as the Schrödinger equation and the sine-Gordon equation can be derived from Gauss' equation for specific choices of K and coördinate systems. In this paper we consider a class of Bäcklund Transformations which corresponds to coördinate transformations on surfaces with a given Gaussian curvature. These Bäcklund Transformations lead to the construction of solutions to certain classes of non-linear second order PDEs of hyperbolic type by identifying these PDEs as the Gauss equation in some coördinate system. The possibility of solving the Cauchy Problem has also been explored for these classes of equations.

Negative Gaussian curvature induces significant suppression of thermal conduction in carbon crystals

Nanoscale ◽  
2017 ◽  
Vol 9 (37) ◽  
pp. 14208-14214 ◽  
Author(s):  
Zhongwei Zhang ◽  
Jie Chen ◽  
Baowen Li
Keyword(s):  
Gaussian Curvature ◽  
Thermal Conduction ◽  
Negative Curvature ◽  
Positive Curvature ◽  
Significant Suppression ◽  
Carbon Allotropes ◽  
Zero Curvature ◽  
Negative Gaussian Curvature

From the mathematic category of surface Gaussian curvature, carbon allotropes can be classified into three types: zero curvature, positive curvature, and negative curvature.

scholarly journals Chirality in curved polyaromatic systems

2017 ◽  
Vol 46 (6) ◽  
pp. 1643-1660 ◽  
Author(s):  
Michel Rickhaus ◽  
Marcel Mayor ◽  
Michal Juríček
Keyword(s):  
Gaussian Curvature ◽  
Carbon Allotrope ◽  
Carbon Allotropes ◽  
Negative Gaussian Curvature ◽  
Helical Chirality

Chiral non-planar polyaromatic systems that display zero, positive or negative Gaussian curvature are analysed and their potential to ‘encode’ chirality of larger sp2-carbon allotropes is evaluated. Shown is a hypothetical peanut-shaped carbon allotrope, where helical chirality results from the interplay of various curvature types.

scholarly journals Vibro-Impact System Based on Forced Oscillations of Heavy Mass Particle along a Rough Parabolic Line

2012 ◽  
Vol 2012 ◽  
pp. 1-17 ◽  
Author(s):  
Srdjan Jović ◽  
Vladimir Raičević ◽  
Ljubiša Garić
Keyword(s):  
Software Package ◽  
Nonlinear Differential Equations ◽  
Vertical Plane ◽  
Interpolation Polynomial ◽  
Primary Objective ◽  
Forced Oscillations ◽  
Single Frequency ◽  
Mass Particle ◽  
Impact System ◽  
Wolfram Mathematica

This paper analyses motion trajectory of vibro-impact system based on the oscillator moving along the rough parabolic line in the vertical plane, under the action of external single-frequency force. Nonideality of the bond originates of slidingCoulomb’stype friction force with coefficientμ=tgα0. The oscillator consists of one heavy mass particle whose forced motion is limited by two angular elongation fixed limiters. The differential equation of motion of the analyzed vibro-impact system, which belongs to the group of common second order nonhomogenous nonlinear differential equations, cannot be solved explicitly (in closed form). For its approximate solving, the software package WOLFRAM Mathematica 7 is used. The results are tested by using the software package MATLAB R2008a. The combination of analytical-numerical results for the defined parameters of analyzed vibro-impact system is a base for the motion analysis visualization, which was the primary objective of this analytic research. Upon the phase portrait of the heavy mass particle obtained, the energy of the considered vibro-impact system is analyzed. During the graphical visualization of the energetic changes, one of the steps is the process of the phase trajectory equations determination. For this determination, we have used interpolation process that utilizesLagrangeinterpolation polynomial.

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