B. Cylindrical Coordinates and Spherical Coordinates

2018 ◽  
pp. 309-320
Keyword(s):  
Spherical Coordinates ◽  
Cylindrical Coordinates

Parity inversion property of the double ring-shaped oscillator in cylindrical coordinates

2015 ◽  
Vol 30 (39) ◽  
pp. 1550200 ◽  
Author(s):  
Dong-Sheng Sun ◽  
Fa-Lin Lu ◽  
Yuan You ◽  
Chang-Yuan Chen ◽  
Shi-Hai Dong
Keyword(s):  
Functional Analysis ◽  
Singular Point ◽  
Harmonic Oscillator ◽  
Two Dimensional ◽  
Spherical Coordinates ◽  
Analysis Method ◽  
Cylindrical Coordinates ◽  
One Dimensional ◽  
Double Ring ◽  
The One

Using the functional analysis method, we present the exact solutions of the double ring-shaped oscillator (DRSO) potential with certain parity in the cylindrical coordinates. Such a quantum system is separated to two differential equations, i.e. a one-dimensional harmonic oscillator plus an inverse square term and a two-dimensional harmonic oscillator plus an inverse square term. The key point is how to find the adapted symmetrical solutions of the one-dimensional harmonic oscillator plus an inverse square term at the singular point [Formula: see text]. The obtained results are compared with those in the spherical coordinates. We also explore intimate connections [Formula: see text] and [Formula: see text] by substituting [Formula: see text] and [Formula: see text].

scholarly journals Numerical Simulation of 1D Unsteady Heat Conduction-Convection in Spherical and Cylindrical Coordinates by Fourth-Order FDM

2018 ◽  
Vol 8 (1) ◽  
pp. 2389-2392
Author(s):  
E. C. Romao ◽  
L. H. P. De Assis
Keyword(s):  
Numerical Simulation ◽  
Heat Conduction ◽  
Finite Difference ◽  
Finite Difference Method ◽  
Fourth Order ◽  
Spherical Coordinates ◽  
Cylindrical Coordinates ◽  
One Dimensional ◽  
Unsteady Heat Conduction ◽  
The One

This paper aims to apply the Fourth Order Finite Difference Method (FDM) to solve the one-dimensional unsteady conduction-convection equation with energy generation (or sink) in cylindrical and spherical coordinates. Two applications were compared through exact solutions to demonstrate the accuracy of the proposed formulation.

SAMPL7 TrimerTrip Host-Guest Binding Poses and Binding Affinities from Spherical-Coordinates-Biased Simulations

2020 ◽  
Author(s):  
Zhaoxi Sun
Keyword(s):  
Free Energy ◽  
Computational Modelling ◽  
Free Energy Calculations ◽  
Energy Estimates ◽  
Binding Affinities ◽  
Spherical Coordinates ◽  
Collective Variables ◽  
Guest Binding ◽  
Starting Point ◽  
Sampl Challenge

Host-guest binding remains a major challenge in modern computational modelling. The newest 7<sup>th</sup> statistical assessment of the modeling of proteins and ligands (SAMPL) challenge contains a new series of host-guest systems. The TrimerTrip host binds to 16 structurally diverse guests. Previously, we have successfully employed the spherical coordinates as the collective variables coupled with the enhanced sampling technique metadynamics to enhance the sampling of the binding/unbinding event, search for possible binding poses and predict the binding affinities in all three host-guest binding cases of the 6<sup>th</sup> SAMPL challenge. In this work, we employed the same protocol to investigate the TrimerTrip host in the SAMPL7 challenge. As no binding pose is provided by the SAMPL7 host, our simulations initiate from randomly selected configurations and are proceeded long enough to obtain converged free energy estimates and search for possible binding poses. The predicted binding affinities are in good agreement with the experimental reference, and the obtained binding poses serve as a nice starting point for end-point or alchemical free energy calculations.

The study of the process of evolution of the jet under outflow of water from the supercritical state through a thin nozzle

2016 ◽  
Vol 11 (1) ◽  
pp. 66-71 ◽  
Author(s):  
R.Kh. Bolotnova ◽  
V.A. Korobchinskaya
Keyword(s):  
Stationary Process ◽  
Supercritical State ◽  
System Of Differential Equations ◽  
Jet Formation ◽  
Cylindrical Coordinates ◽  
Initial Stage ◽  
Supersonic Regime ◽  
Water Outflow ◽  
Thin Nozzle ◽  
Supercritical Temperature

The dynamics of the water outflow from the initial supercritical state through a thin nozzle is studied. To describe the initial stage of non-stationary process outflow the system of differential equations of conservation of mass, momentum and energy in a two-dimensional cylindrical coordinates with axial symmetry is used. The spatial distribution of pressure and velocity of jet formation was received. It was established that a supersonic regime of outflow at supercritical temperature of 650 K is formed, which have a qualitative agreement for the velocity compared with the Bernoulli analytical solution and the experimental data.

The experience of employment of the mathematical cartography methods in cosmology

2017 ◽  
Vol 923 (5) ◽  
pp. 7-16
Author(s):  
A.V. Kavrayskiy
Keyword(s):  
Background Radiation ◽  
Three Dimensional ◽  
Spherical Coordinates ◽  
Linear Element ◽  
Microwave Background ◽  
Euclidian Space ◽  
Microwave Background Radiation ◽  
Rectangular Coordinates ◽  
The Universe ◽  
Length Distortion

The experience of mathematical modeling of the 3D-sphere in the 4D-space and projecting it by mathematical cartography methods in the 3D-Euclidian space is presented. The problem is solved by introduction of spherical coordinates for the 3D-sphere and their transformation into the rectangular coordinates, using the mathematical cartography methods. The mathematical relationship for calculating the length distortion mp(s) of the ds linear element when projecting the 3D-sphere from the 4-dimensional Euclidian space into three-dimensional Euclidian space is derived. Numerical examples, containing the modeling of the ds small linear element by spherical coordinates of 3D-sphere, projecting this sphere into the 3D-Euclidian space and length of ds calculating by means of its projection dL and size of distortion mp(s) are solved. Based on the model of the Universe known in cosmology as the 3D-sphere, the hypothesis of connection between distortion mp(s) and the known observed effects Redshift and Microwave Background Radiation is considered.

Certain problems with the application of stochastic diffusion processes for description of chemical engineering phenomena; Diffusion equations in curvilinear coordinates

1991 ◽  
Vol 56 (3) ◽  
pp. 602-618
Author(s):  
Vladimír Kudrna
Keyword(s):  
Heat Transfer ◽  
Mass Transport ◽  
Probability Density ◽  
Chemical Engineering ◽  
Diffusion Processes ◽  
Curvilinear Coordinates ◽  
Diffusion Equations ◽  
Parabolic Partial Differential Equations ◽  
Spherical Coordinates ◽  
Transformation Procedure

Parabolic partial differential equations used in chemical engineering for the description of mass transport and heat transfer and analogous relationship derived in stochastic processes theory are given. A standard transformation procedure is applied, allowing these relations to be generally written in curvilinear coordinates and particular expressions for cylindrical and spherical coordinates to be derived. The relation between the probability density for the position of a discernible particle and the concentration of a set of such particles is discussed.

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