On Marcinkiewicz-Zygmund Inequalities at Hermite Zeros and Their Airy Function Cousins

2020 ◽  
pp. 119-147
Author(s):  
D. S. Lubinsky
Keyword(s):  
Airy Function

An Invariant Membrane Stress Function for Shells

1953 ◽  
Vol 20 (2) ◽  
pp. 178-182
Author(s):  
H. L. Langhaar
Keyword(s):  
Gaussian Curvature ◽  
Stress Function ◽  
Airy Function ◽  
Membrane Stress ◽  
Constant Gaussian Curvature ◽  
Bending Stresses ◽  
Small Deformations

Abstract Inextensional shells that have no thickness are idealized representations of real shells that have small bending stresses and small deformations. With certain restrictions, the stresses in these shells are derivable from a generalized Airy function. For shells of constant Gaussian curvature, the stress function is unrestricted, but, for other shells, it is expressed as a function of the Gaussian curvature. Although, in this respect, it is less general than Pucher’s stress function, it has the advantage that it may be used with any surface co-ordinates.

scholarly journals The Cauchy Problem for the Generalized Hyperbolic Novikov–Veselov Equation via the Moutard Symmetries

Symmetry ◽  
2020 ◽  
Vol 12 (12) ◽  
pp. 2113
Author(s):  
Alla A. Yurova ◽  
Artyom V. Yurov ◽  
Valerian A. Yurov
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Cauchy Problem ◽  
Exact Solutions ◽  
Airy Function ◽  
The Real ◽  
The Cauchy Problem

We begin by introducing a new procedure for construction of the exact solutions to Cauchy problem of the real-valued (hyperbolic) Novikov–Veselov equation which is based on the Moutard symmetry. The procedure shown therein utilizes the well-known Airy function Ai(ξ) which in turn serves as a solution to the ordinary differential equation d2zdξ2=ξz. In the second part of the article we show that the aforementioned procedure can also work for the n-th order generalizations of the Novikov–Veselov equation, provided that one replaces the Airy function with the appropriate solution of the ordinary differential equation dn−1zdξn−1=ξz.

Remark on algorithm 301: Airy function

1967 ◽  
Vol 10 (7) ◽  
pp. 453
Author(s):  
M. L. V. Pitteway
Keyword(s):  
Airy Function

Electron Transmission Through Modified Schottky Barriers

2001 ◽  
Vol 685 ◽  
Author(s):  
Kevin L. Jensen
Keyword(s):  
Schottky Barrier ◽  
Conduction Band ◽  
Current Density ◽  
Transport Mechanism ◽  
Airy Function ◽  
Numerical Approach ◽  
Schottky Barriers ◽  
Present Treatment ◽  
One Dimensional ◽  
Electron Transmission

AbstractThe effects of a Coulomb-like potential in the Schottky barrier existing between a material-diamond interface is analyzed. The inclusion is intended to mimic the effects of an ionized trap within the barrier, and therefore to account for charge injection into the conduction band of diamond via a Poole-Frenkel transport mechanism. The present treatment is to provide a qualitative account of the increase in current density near the inclusion, which can be substantial. The model is first reduced to an analytically tractable one-dimensional tunneling problem addressable by an Airy Function approach in order to investigate the nature of the effect. A more comprehensive numerical approach is then applied. Finally, statistical arguments are used to estimate emission site densities using the results of the aforementioned analysis.

Hyperasymptotics for integrals with finite endpoints

1992 ◽  
Vol 439 (1906) ◽  
pp. 373-396 ◽  
Keyword(s):  
Asymptotic Expansion ◽  
Complex Plane ◽  
Airy Function ◽  
Error Function ◽  
Steepest Descent ◽  
Similar Form ◽  
Complete Asymptotic Expansion ◽  
Complementary Error Function ◽  
Plane Passing ◽  
Method Of Steepest Descent

Berry & Howls (1991) (hereinafter called BH) refined the method of steepest descent to study exponentially accurate asymptotics of a general class of integrals involving exp {– kf ( z )} along doubly infinite contours in the complex plane passing over saddlepoints of f ( z ). Here we derive analogous results for integrals with integrands of a similar form, but whose local expansions in powers of 1/ k are made about the finite endpoints of semi-infinite contours of integration. We treat endpoints where f ( z ) behaves locally linearly or quadratically. Generically, local endpoint expansions made by the method of steepest descent diverge because of the presence of saddles of f ( z ). We derive ‘resurgence relations’ which express the original integral exactly as a truncated endpoint expansion plus a remainder, involving the global saddle structure of f ( z ) via integrals through certain ‘adjacent’ saddles. The saddles adjacent to the endpoint are determined by a topological rule. If the least term of the endpoint expansion is the N 0 ( k ) th, summing to here calculates the endpoint integral up to an error of approximately exp ( – N 0 ( k )). We develop a scheme, involving iteration of the new resurgence relations with a similar one derived in BH, which can reduce this error down to exp( – 2.386 N 0 ( k )). This ‘hyperasymptotic’ formalism parallels that of BH and incorporates automatically any change in the complete asymptotic expansion as the endpoint moves in the complex plane, provided that it does not coincide with other saddles. We illustrate the analytical and numerical use of endpoint hyperasymptotics by application to the complementary error function erfc( x ) and a constructed ‘incomplete’ Airy function.

MAF solution for bounded potential problems

1998 ◽  
Vol 76 (5) ◽  
pp. 351-359 ◽  
Author(s):  
A K Ghatak ◽  
I C Goyal ◽  
R Jindal ◽  
Y P Varshni
Keyword(s):  
Airy Function ◽  
Numerical Results ◽  
Harmonic Potential ◽  
Bounded Linear ◽  
Quartic Potential ◽  
Bounded Potential ◽  
Potential Problems

We present here the solutions of a bounded linear harmonic potential and abounded quartic potential using the modified Airy function (MAF) method. Resultsobtained by the MAF method have been compared with the analytical (numerical)results and with those obtained by the JWKB method. The comparison showsthat the MAF method gives very accurate results and is, in general, the moreaccurate of the two methods. The MAF method also gives an accurate descriptionof the eigenfunction. A perturbation correction when applied to MAF helps usto get very accurate eigenvalues. The method should be useful in determiningthe eigenvalues and the eigenfunctions of any smoothly varying arbitrarypotential confined by infinite walls.PACS No. 03.65

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