scholarly journals Density Function of the FK4 Systematic Errors

1990 ◽  
Vol 141 ◽  
pp. 94-94
Author(s):  
V. S. Gubanov ◽  
N. I. Solina
Keyword(s):  
Density Function ◽  
Spherical Harmonics ◽  
Unit Sphere ◽  
Systematic Errors ◽  
Layer Potential ◽  
Partial Derivatives ◽  
Derivatives Of

A notion of density function of systematic errors of astrometric catalogues distributed on the unit sphere as a simple layer is introduced. Components of the catalogue errors in any direction are determined as partial derivatives of layer potential in the same direction. For example, the density function of FK4 errors is computed as an expansion of spherical harmonics.

Solid waste shape description and generation based on spherical harmonics and probability density function

2021 ◽  
pp. 0734242X2110450
Author(s):  
Yifeng Li ◽  
Xunpeng Qin ◽  
Zhenyuan Zhang ◽  
Huanyu Dong
Keyword(s):  
Probability Density Function ◽  
Solid Waste ◽  
Probability Density ◽  
Density Function ◽  
Spherical Harmonics ◽  
Morphological Data ◽  
Control Factor ◽  
Shape Description ◽  
Separation Processes ◽  
Waste Particles

Transport and separation processes of solid waste can only be modelled successfully with discrete element methods in case the shape of the particles can be described accurately. The existing techniques for morphological data acquisition, such as computed tomography, laser scanning technique, optical interferometer, stereo photography and structured light technique, are laborious and require a large amount of realistic solid waste samples. Therefore, there is a pressing need for an alternative method to describe the shape of solid waste particles and to generate multiple variations of particles with almost similar shapes. In this paper, a new method to describe solid waste particles is proposed that is frequency-based and uses spherical harmonics (SHs). Additionally, a new shape generation method is introduced that uses the shape description of a single particle to generate an array of related shapes based on a probability density function with a dimensionless control factor η. The newly proposed methods were successfully applied to describe the complex shapes of pieces of metal and plastic scrap. The shapes of these pieces of scrap can be described adequately with 15° of SH expansion and the overall divergence is within 0.1 mm. Five different values for η were tested, which generated shapes with the same distribution as the original particle. Rising levels of η cause the morphological variation of the generated particles to increase. These new methods improve the modelling of transportation and separation processes.

Partial Derivatives of Regular Expressions over Alphabet-Invariant and User-Defined Labels

2019 ◽  
pp. 184-196 ◽  
Author(s):  
Stavros Konstantinidis ◽  
Nelma Moreira ◽  
João Pires ◽  
Rogério Reis
Keyword(s):  
Regular Expressions ◽  
Partial Derivatives ◽  
Derivatives Of

scholarly journals Derivatives of addition theorems for Legendre functions

1995 ◽  
Vol 37 (2) ◽  
pp. 212-234 ◽  
Author(s):  
D.E. Winch ◽  
P.H. Roberts
Keyword(s):  
Spherical Harmonics ◽  
Chebyshev Polynomials ◽  
Legendre Polynomials ◽  
Addition Theorem ◽  
Legendre Functions ◽  
Addition Theorems ◽  
Associated Legendre Functions ◽  
Tensor Spherical Harmonics ◽  
Derivatives Of

AbstractDifferentiation of the well-known addition theorem for Legendre polynomials produces results for sums over order m of products of various derivatives of associated Legendre functions. The same method is applied to the corresponding addition theorems for vector and tensor spherical harmonics. Results are also given for Chebyshev polynomials of the second kind, corresponding to ‘spin-weighted’ associated Legendre functions, as used in studies of distributions of rotations.

Partial Derivatives of the Lambert Problem

2014 ◽  
Author(s):  
Nitin Arora ◽  
Ryan P. Russell ◽  
Nathan J. Strange
Keyword(s):  
Partial Derivatives ◽  
Lambert Problem ◽  
Derivatives Of

Perturbation theory and finite Markov chains

1968 ◽  
Vol 5 (2) ◽  
pp. 401-413 ◽  
Author(s):  
Paul J. Schweitzer
Keyword(s):  
Perturbation Theory ◽  
Markov Chain ◽  
Markov Chains ◽  
Stationary Distribution ◽  
Fundamental Matrix ◽  
Transition Probabilities ◽  
Semi Group ◽  
Partial Derivatives ◽  
Finite Markov Chains ◽  
Derivatives Of

A perturbation formalism is presented which shows how the stationary distribution and fundamental matrix of a Markov chain containing a single irreducible set of states change as the transition probabilities vary. Expressions are given for the partial derivatives of the stationary distribution and fundamental matrix with respect to the transition probabilities. Semi-group properties of the generators of transformations from one Markov chain to another are investigated. It is shown that a perturbation formalism exists in the multiple subchain case if and only if the change in the transition probabilities does not alter the number of, or intermix the various subchains. The formalism is presented when this condition is satisfied.

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