Expansions of the mildly relativistic dispersion function for nearly perpendicular electron cyclotron ray tracing

1999 ◽  
Vol 61 (1) ◽  
pp. 121-128 ◽  
Author(s):  
I. P. SHKAROFSKY
Keyword(s):  
Ray Tracing ◽  
Computer Programs ◽  
Higher Order ◽  
Electron Cyclotron ◽  
Dispersion Function ◽  
Relativistic Dispersion ◽  
Plasma Dispersion ◽  
Derivatives Of ◽  
Cyclotron Harmonic ◽  
Experienced Difficulty

To trace rays very close to the nth electron cyclotron harmonic, we need the mildly relativistic plasma dispersion function and its higher-order derivatives. Expressions for these functions have been obtained as an expansion for nearly perpendicular propagation in a region where computer programs have previously experienced difficulty in accuracy, namely when the magnitude of (c/vt)2 (ω−nωc)/ω is between 1 and 10. In this region, the large-argument expansions are not yet valid, but partial cancellations of terms occur. The expansion is expressed as a sum over derivatives of the ordinary dispersion function Z. New expressions are derived to relate higher-order derivatives of Z to Z itself in this region of concern in terms of a finite series.

Generalized Trubnikov functions for unmagnetized plasmas

1999 ◽  
Vol 62 (2) ◽  
pp. 249-253 ◽  
Author(s):  
D. B. MELROSE
Keyword(s):  
Thermal Plasma ◽  
Thermal Plasmas ◽  
Relativistic Plasma ◽  
Dispersion Function ◽  
Recursion Relations ◽  
Relativistic Dispersion ◽  
Plasma Dispersion ◽  
Class Of Functions ◽  
Explicit Expressions

A class of relativistic dispersion functions for unmagnetized thermal plasmas is defined by generalizing functions first defined by Trubnikov in 1958. Recursion relations are derived that allow one to generate explicit expressions for the class of functions in terms of the relativistic plasma dispersion function T(z, ρ) introduced by Godfrey et al. in 1975. These functions are relevant to the description of the response of a weakly mangetized, highly relativistic, thermal plasma.

scholarly journals Perturbation confusion in forward automatic differentiation of higher-order functions

2019 ◽  
Vol 29 ◽  
Author(s):  
OLEKSANDR MANZYUK ◽  
BARAK A. PEARLMUTTER ◽  
ALEXEY ANDREYEVICH RADUL ◽  
DAVID R. RUSH ◽  
JEFFREY MARK SISKIND
Keyword(s):  
Automatic Differentiation ◽  
Computer Programs ◽  
Higher Order ◽  
The Other ◽  
Order Function ◽  
Accumulation Mode ◽  
The Creation ◽  
One To One ◽  
Derivatives Of ◽  
Higher Order Functions

Abstract Automatic differentiation (AD) is a technique for augmenting computer programs to compute derivatives. The essence of AD in its forward accumulation mode is to attach perturbations to each number, and propagate these through the computation by overloading the arithmetic operators. When derivatives are nested, the distinct derivative calculations, and their associated perturbations, must be distinguished. This is typically accomplished by creating a unique tag for each derivative calculation and tagging the perturbations. We exhibit a subtle bug, present in fielded implementations which support derivatives of higher-order functions, in which perturbations are confused despite the tagging machinery, leading to incorrect results. The essence of the bug is as follows: a unique tag is needed for each derivative calculation, but in existing implementations unique tags are created when taking the derivative of a function at a point. When taking derivatives of higher-order functions, these need not correspond! We exhibit a simple example: a higher-order function f whose derivative at a point x, namely f′(x), is itself a function which calculates a derivative. This situation arises naturally when taking derivatives of curried functions. Two potential solutions are presented, and their deficiencies discussed. One uses eta expansion to delay the creation of fresh tags in order to put them into one-to-one correspondence with derivative calculations. The other wraps outputs of derivative operators with tag substitution machinery. Both solutions seem very difficult to implement without violating the desirable complexity guarantees of forward AD.

New representations of dielectric tensor elements in magnetized plasma

1986 ◽  
Vol 35 (2) ◽  
pp. 319-331 ◽  
Author(s):  
I. P. Shkarofsky
Keyword(s):  
Functional Expression ◽  
Relativistic Plasma ◽  
Magnetized Plasma ◽  
Dispersion Function ◽  
Dielectric Tensor ◽  
Relativistic Case ◽  
Plasma Dispersion ◽  
Derivatives Of

Each of the dielectric tensor elements in a Maxwellian magnetoplasma is expressed in terms of various derivatives of a single functional expression. The relationships for all the elements are given, first for the general case of a relativistic plasma, then for the slightly relativistic case, and finally for the non-relativistic case, when the perpendicular wavenumber is either large or small. We also derive new relations useful for the computation of the slightly relativistic plasma dispersion function.

A C0 three-node triangular element based on preprocessing approach for thick sandwich plates

2017 ◽  
Vol 21 (6) ◽  
pp. 1820-1842
Author(s):  
Wu Zhen ◽  
Ma Rui ◽  
Chen Wanji
Keyword(s):  
Transverse Shear ◽  
Equilibrium Equation ◽  
Three Dimensional ◽  
Higher Order ◽  
Triangular Element ◽  
Shear Stresses ◽  
Sandwich Plates ◽  
Transverse Shear Stress ◽  
Transverse Shear Stresses ◽  
Derivatives Of

This paper will try to overcome two difficulties encountered by the C0 three-node triangular element based on the displacement-based higher-order models. They are (i) transverse shear stresses computed from constitutive equations vanish at the clamped edges, and (ii) it is difficult to accurately produce the transverse shear stresses even using the integration of the three-dimensional equilibrium equation. Invalidation of the equilibrium equation approach ought to attribute to the higher-order derivations of displacement parameters involved in transverse shear stress components after integrating three-dimensional equilibrium equation. Thus, the higher-order derivatives of displacement parameters will be taken out from transverse shear stress field by using the three-field Hu–Washizu variational principle before the finite element procedure is implemented. Therefore, such method is named as the preprocessing method for transverse shear stresses in present work. Because the higher-order derivatives of displacement parameters have been eliminated, a C0 three-node triangular element based on the higher-order zig-zag theory can be presented by using the linear interpolation function. Performance of the proposed element is numerically evaluated by analyzing multilayered sandwich plates with different loading conditions, lamination sequences, material constants and boundary conditions, and it can be found that the present model works well in the finite element framework.

scholarly journals The higher-order derivatives of spectral functions

2007 ◽  
Vol 424 (1) ◽  
pp. 240-281 ◽  
Author(s):  
Hristo S. Sendov
Keyword(s):  
Higher Order ◽  
Spectral Functions ◽  
Derivatives Of

scholarly journals Excitation of electron cyclotron harmonic waves in the inner Saturn magnetosphere within local plasma injections

2010 ◽  
Vol 115 (A12) ◽  
pp. n/a-n/a ◽  
Author(s):  
X. Tao ◽  
R. M. Thorne ◽  
R. B. Horne ◽  
S. Grimald ◽  
C. S. Arridge ◽  
...  
Keyword(s):  
Electron Cyclotron ◽  
Harmonic Waves ◽  
Cyclotron Harmonic
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