scholarly journals On the Adaptive Quadrature of Fermi-Dirac Functions and their Derivatives

2019 ◽  
Vol 18 (1) ◽  
pp. 1-20
Author(s):  
Mandyam N Anandaram
Keyword(s):  
Numerical Results ◽  
Adaptive Quadrature ◽  
Third Order ◽  
Partial Derivatives ◽  
The Third ◽  
Break Points ◽  
Fixed Order ◽  
Dirac Function ◽  
Computational Research ◽  
Excellent Accuracy

In this paper, using the Python SciPy module “quad”, a fast auto-adaptive quadrature solver based on the pre-compiled QUADPACK Fortran package, computational research is undertaken to accurately integrate the generalised Fermi-Dirac function and all its partial derivatives up to the third order. The numerical results obtained with quad method when combined with optimised break points achieve an excellent accuracy comparable to that obtained by other publications using fixed-order quadratures.

scholarly journals Non-singular spherical harmonic expressions of geomagnetic vector and gradient tensor fields in the local north-oriented reference frame

2015 ◽  
Vol 8 (7) ◽  
pp. 1979-1990 ◽  
Author(s):  
J. Du ◽  
C. Chen ◽  
V. Lesur ◽  
L. Wang
Keyword(s):  
Reference Frame ◽  
Potential Field ◽  
Spherical Harmonic ◽  
Magnetic Potential ◽  
Tensor Fields ◽  
Third Order ◽  
Gradient Tensor ◽  
Partial Derivatives ◽  
The Third ◽  
Derivatives Of

Abstract. General expressions of magnetic vector (MV) and magnetic gradient tensor (MGT) in terms of the first- and second-order derivatives of spherical harmonics at different degrees/orders are relatively complicated and singular at the poles. In this paper, we derived alternative non-singular expressions for the MV, the MGT and also the third-order partial derivatives of the magnetic potential field in the local north-oriented reference frame. Using our newly derived formulae, the magnetic potential, vector and gradient tensor fields and also the third-order partial derivatives of the magnetic potential field at an altitude of 300 km are calculated based on a global lithospheric magnetic field model GRIMM_L120 (GFZ Reference Internal Magnetic Model, version 0.0) with spherical harmonic degrees 16–90. The corresponding results at the poles are discussed and the validity of the derived formulas is verified using the Laplace equation of the magnetic potential field.

On the Generalization of Displacement-Based Laminate Theories

1989 ◽  
Vol 42 (11S) ◽  
pp. S213-S222 ◽  
Author(s):  
J. N. Reddy
Keyword(s):  
Analytical Solution ◽  
Elasticity Theory ◽  
Shear Deformation ◽  
Single Layer ◽  
Numerical Results ◽  
Two Dimensional ◽  
Third Order ◽  
The Third ◽  
Laminate Theory ◽  
Displacement Based

A review and generalization of the displacement-based two-dimensional plate theories is presented. The classical and shear deformation single–layer theories up to the third-order are presented in a single theory through tracers. The layer–wise laminate theory developed by the author is reviewed. Numerical results are presented to illustrate the accuracy of the layer–wise theory by comparison with the analytical solution of the 3–D elasticity theory.

CONFORMAL CONNECTION AND EQUIVALENCE PROBLEM FOR THIRD ORDER ODEs

2002 ◽  
Vol 17 (20) ◽  
pp. 2770-2770 ◽  
Author(s):  
PAWEL NUROWSKI
Keyword(s):  
Explicit Expression ◽  
Equivalence Problem ◽  
Equivalence Classes ◽  
Third Order ◽  
Partial Derivatives ◽  
The Third ◽  
One To One ◽  
Conformal Connection ◽  
Contact Transformations ◽  
Explicit Expressions

The equivalence problem for the third order ODEs solved by E. Cartan1 and S. S. Chern2 is reconsidered. We consider third order ODEs of the form y′′′ = F(x,y,y′,y′′) for which the Wunshman invariant I vanishes. All such ODEs split into equivalence classes with respect to the contact transformations of the variables. As shown by E. T. Newman3 and collaborators such equations are also in one-to-one correspondence with conformal classes of Lorentian three-metrics. We supplement Cartan-Chern-Newman results by providing explicit expressions for all the contact invariants of an ODE with I = 0. The invariants are explicitly written in terms of the function F and its partial derivatives. Explicit expression for the associated Cartan's O(2,3) connection is also given. The curvature of this conformal connection is reinterpreted in terms of the Cotton-York tensor of the Lorentzian three-metric associated with the equation.

Some aspects of nonminimal inflation driven by a superpotential

2016 ◽  
Vol 26 (07) ◽  
pp. 1750058
Author(s):  
Kourosh Nozari ◽  
Narges Rashidi
Keyword(s):  
Spectral Index ◽  
Second Order ◽  
Numerical Results ◽  
Nonminimal Coupling ◽  
Third Order ◽  
The Third ◽  
Scalar Spectral Index ◽  
Non Gaussian

We study a nonminimal inflation which is driven by a superpotential. By adopting the Arnowitt–Deser–Misner formalism, we explore the primordial perturbations and its non-Gaussianity in this framework. By expanding the action up to the second-order in perturbations, we seek the scalar spectral index, its running and the tensor-to-scalar ratio. In this regard, we find the ranges of the nonminimal coupling and superpotential parameters which lead to the observationally viable perturbations parameters. The non-Gaussian feature in both the equilateral and orthogonal configurations in this setup, is studied by exploring the third-order action. We show that in some ranges of the nonminimal and superpotential parameters, the model predicts large non-Gaussianity. By comparing the numerical results with Planck2015 data, we test the viability of the model and find some constraints on the model’s parameters space.

Probe size and 5th-order aberration

1987 ◽  
Vol 45 ◽  
pp. 134-135 ◽  
Author(s):  
Zhifeng Shao
Keyword(s):  
Electron Probe ◽  
Spherical Aberration ◽  
Wave Length ◽  
Cylindrical Symmetry ◽  
Electron Wave ◽  
Third Order ◽  
The Third ◽  
Probe Radius ◽  
Small Probe ◽  
Probe Size

A small electron probe has many applications in many fields and in the case of the STEM, the probe size essentially determines the ultimate resolution. However, there are many difficulties in obtaining a very small probe.Spherical aberration is one of them and all existing probe forming systems have non-zero spherical aberration. The ultimate probe radius is given byδ = 0.43Csl/4ƛ3/4where ƛ is the electron wave length and it is apparent that δ decreases only slowly with decreasing Cs. Scherzer pointed out that the third order aberration coefficient always has the same sign regardless of the field distribution, provided only that the fields have cylindrical symmetry, are independent of time and no space charge is present. To overcome this problem, he proposed a corrector consisting of octupoles and quadrupoles.

Children’s Recall of Approximations to English

1973 ◽  
Vol 16 (2) ◽  
pp. 201-212 ◽  
Author(s):  
Elizabeth Carrow ◽  
Michael Mauldin
Keyword(s):  
Language Development ◽  
Fourth Order ◽  
Third Order ◽  
Memory Processes ◽  
The Third ◽  
General Index ◽  
General Linguistic

As a general index of language development, the recall of first through fourth order approximations to English was examined in four, five, six, and seven year olds and adults. Data suggested that recall improved with age, and increases in approximation to English were accompanied by increases in recall for six and seven year olds and adults. Recall improved for four and five year olds through the third order but declined at the fourth. The latter finding was attributed to deficits in semantic structures and memory processes in four and five year olds. The former finding was interpreted as an index of the development of general linguistic processes.

Sign in / Sign up

Export Citation Format

Share Document