Invertible affine transformations on integer coordinate system — general theory inn- dimensional space

1993 ◽  
Vol 24 (7) ◽  
pp. 1-12
Author(s):  
Masahiko Shizawa
Keyword(s):  
General Theory ◽  
Coordinate System ◽  
Dimensional Space ◽  
Affine Transformations ◽  
System A

scholarly journals Zur exakten Behandlung von kollektiven Rotationen Teil A: Theorie / On the Exact Treatment of Collective Rotations Part A: Theory

1973 ◽  
Vol 28 (2) ◽  
pp. 206-215
Author(s):  
Hanns Ruder
Keyword(s):  
Coordinate System ◽  
Perturbation Expansion ◽  
Rotational Energy ◽  
The Body ◽  
System A ◽  
N Body Problem ◽  
Definition Of ◽  
Fixed System ◽  
Body Fixed Coordinate System ◽  
Groundstate Energy

Basic in the treatment of collective rotations is the definition of a body-fixed coordinate system. A kinematical method is derived to obtain the Hamiltonian of a n-body problem for a given definition of the body-fixed system. From this exact Hamiltonian, a consequent perturbation expansion in terms of the total angular momentum leads to two exact expressions: one for the collective rotational energy which has to be added to the groundstate energy in this order of perturbation and a second one for the effective inertia tensor in the groundstate. The discussion of these results leads to two criteria how to define the best body-fixed coordinate system, namely a differential equation and a variational principle. The equivalence of both is shown.

HYPER-SPHERICAL HARMONICS AND ANHARMONICS IN m-DIMENSIONAL SPACE

2008 ◽  
Vol 17 (06) ◽  
pp. 1125-1130
Author(s):  
M. R. SHOJAEI ◽  
A. A. RAJABI ◽  
H. HASANABADI
Keyword(s):  
Quantum Mechanics ◽  
General Theory ◽  
Spherical Harmonics ◽  
Interaction Potential ◽  
Dimensional Space ◽  
Harmonic Polynomials ◽  
Central Importance ◽  
Computational Tools ◽  
Relative Coordinate

In quantum mechanics the hyper-spherical method is one of the most well-established and successful computational tools. The general theory of harmonic polynomials and hyper-spherical harmonics is of central importance in this paper. The interaction potential V is assumed to depend on the hyper-radius ρ only where ρ is the function of the Jacobi relative coordinate x1, x2,…, xn which are functions of the particles' relative positions.

scholarly journals A new result under weak and non-relativistic approximation from general theory of relativity

2021 ◽  
Author(s):  
Abhijit Samanta
Keyword(s):  
Field Equation ◽  
Energy Density ◽  
General Theory ◽  
Force Field ◽  
Coordinate System ◽  
Theory Of Relativity ◽  
General Theory Of Relativity ◽  
Inertial Coordinate System ◽  
Relativistic Approximation ◽  
Moving Particle

Abstract We have derived a metric field equation in the locally inertial coordinate system from Einstein's field equation considering the energy density of the moving particle with the approximations that the force field under which the particle is moving is weak and the velocity of the particle is non-relativistic. We study the motion of different microscopic systems using this metric equation and compared the results with the experimentally measured values and we find that the results are identical.

SOLVING SPATIAL CONSTRAINTS WITH GLOBAL DISTANCE COORDINATE SYSTEM

2006 ◽  
Vol 16 (05n06) ◽  
pp. 533-547 ◽  
Author(s):  
LU YANG
Keyword(s):  
Coordinate System ◽  
Systematic Approach ◽  
Distance Geometry ◽  
Spatial Constraints ◽  
Constraint Equations ◽  
System A ◽  
Short Program ◽  
Complete Set ◽  
Point Line ◽  
Made In

A systematic approach making use of distance geometry to solve spatial constraints is introduced. We demonstrate how to create the constraint equations by means of a relevant distance coordinate system. A short program is made (in Maple) which implements the algorithm producing automatically a complete set of constraint equations for a given point-plane configuration. The point-line-plane configurations are converted into point-plane ones beforehand.

Quaternionic Neural Networks

2009 ◽  
pp. 411-439 ◽  
Author(s):  
Teijiro Isokawa ◽  
Nobuyuki Matsui ◽  
Haruhiko Nishimura
Keyword(s):  
Neural Networks ◽  
Dimensional Space ◽  
Three Dimensional ◽  
Network Models ◽  
Recurrent Network ◽  
Affine Transformations ◽  
Neural Network Models ◽  
Fundamental Properties ◽  
Modern Physics ◽  
The Stability

Quaternions are a class of hypercomplex number systems, a four-dimensional extension of imaginary numbers, which are extensively used in various fields such as modern physics and computer graphics. Although the number of applications of neural networks employing quaternions is comparatively less than that of complex-valued neural networks, it has been increasing recently. In this chapter, the authors describe two types of quaternionic neural network models. One type is a multilayer perceptron based on 3D geometrical affine transformations by quaternions. The operations that can be performed in this network are translation, dilatation, and spatial rotation in three-dimensional space. Several examples are provided in order to demonstrate the utility of this network. The other type is a Hopfield-type recurrent network whose parameters are directly encoded into quaternions. The stability of this network is demonstrated by proving that the energy decreases monotonically with respect to the change in neuron states. The fundamental properties of this network are presented through the network with three neurons.

Transfer Equation in General Curvilinear Coordinates

2011 ◽  
Vol 20 (2) ◽  
Author(s):  
J. Freimanis
Keyword(s):  
Differential Operator ◽  
Radiative Transfer ◽  
Coordinate System ◽  
Transfer Equation ◽  
Radiative Transfer Equation ◽  
Dimensional Space ◽  
Polarization Plane ◽  
Curvilinear Coordinates ◽  
Reference Plane ◽  
Metric Tensor

AbstractThe differential operator of the monochromatic polarized radiative transfer equation is studied in case of statistically homogeneous turbid medium in Euclidean three-dimensional space, with arbitrary curvilinear coordinate system defined in it. An apparent rotation of the polarization plane along the light ray with respect to the chosen polarization reference plane generally takes place, due to purely geometric reasons. Using methods of tensor analysis, analytic expressions for the differential operator of the transfer equation depending on the components of the metric tensor and their derivatives are found. Considerable simplifications take place if the coordinate system is orthogonal. As an example, the differential operator of the vector radiative transfer equation in both elliptical conical coordinate system and oblate spheroidal coordinate system is written down. Nonstandard parameterization of standard elliptical conical coordinate system is proposed.

scholarly journals Do Lawyers Improve the Adversary System? A General Theory of Litigation Advice and Its Regulation

1991 ◽  
Vol 79 (2) ◽  
pp. 313 ◽  
Author(s):  
Stephen McG. Bundy ◽  
Einer Richard Elhauge
Keyword(s):  
General Theory ◽  
System A ◽  
Adversary System

scholarly journals Online Correction of the Mutual Miscalibration of Multimodal VIS–IR Sensors and 3D Data on a UAV Platform for Surveillance Applications

2019 ◽  
Vol 11 (21) ◽  
pp. 2469
Author(s):  
Siekański ◽  
Paśko ◽  
Malowany ◽  
Malesa
Keyword(s):  
Coordinate System ◽  
Critical Infrastructure ◽  
Spatial Coherence ◽  
Phase Correlation ◽  
Geometric Calibration ◽  
System A ◽  
3D Data ◽  
Common Coordinate System ◽  
Ir Sensors ◽  
Imaging Camera

Unmanned aerial vehicles (UAVs) are widely used to protect critical infrastructure objects, and they are most often equipped with one or more RGB cameras and, sometimes, with a thermal imaging camera as well. To obtain as much information as possible from them, they should be combined or fused. This article presents a situation in which data from RGB (visible, VIS) and thermovision (infrared, IR) cameras and 3D data have been combined in a common coordinate system. A specially designed calibration target was developed to enable the geometric calibration of IR and VIS cameras in the same coordinate system. 3D data are compatible with the VIS coordinate system when the structure from motion (SfM) algorithm is used. The main focus of this article is to provide the spatial coherence between these data in the case of relative camera movement, which usually results in a miscalibration of the system. Therefore, a new algorithm for the detection of sensor system miscalibration, based on phase correlation with automatic calibration correction in real time, is introduced.

Sign in / Sign up

Export Citation Format

Share Document