Anharmonic oscillator analysis using modified Airy functions

1992 ◽  
Vol 70 (12) ◽  
pp. 1218-1221 ◽  
Author(s):  
I. C. Goyal ◽  
Sukhdev Roy ◽  
A. K. Ghatak ◽  
R. L. Gallawa
Keyword(s):  
Anharmonic Oscillator ◽  
Airy Function ◽  
Closed Form Expression ◽  
Airy Functions ◽  
Form Expression ◽  
Eighth Order ◽  
First Order ◽  
Numerical Effort ◽  
Modified Airy Functions ◽  
First Order Perturbation

We have applied the modified Airy function (MAF) method to the analysis of an anharmonic oscillator, characterized by the potential V(x) = k x2/2 + a x4, k > 0, a > 0. The MAF method gives an accurate closed-form expression for the wave function as well as very accurate eigenvalues with much less numerical effort than the five-term (eighth-order) WKBJ approximation. The application of the first-order perturbation correction to the MAF eigenvalues makes them even more accurate than those obtained by the five-term WKBJ approximation.

A Note on Nonparametric Estimation of the CTE

2009 ◽  
Vol 39 (2) ◽  
pp. 717-734 ◽  
Author(s):  
Bangwon Ko ◽  
Ralph P. Russo ◽  
Nariankadu D. Shyamalkumar
Keyword(s):  
Order Term ◽  
Random Variable ◽  
Small Samples ◽  
Closed Form Expression ◽  
Form Expression ◽  
Continuous Random Variable ◽  
Underlying Distribution ◽  
First Order ◽  
Bootstrap Approach ◽  
Conditional Tail Expectation

AbstractThe α-level Conditional Tail Expectation (CTE) of a continuous random variable X is defined as its conditional expectation given the event {X > qα} where qα represents its α-level quantile. It is well known that the empirical CTE (the average of the n (1 – α) largest order statistics in a sample of size n) is a negatively biased estimator of the CTE. This bias vanishes as the sample size increases, but in small samples can be significant. In this article it is shown that an unbiased nonparametric estimator of the CTE does not exist. In addition, the asymptotic behavior of the bias of the empirical CTE is studied, and a closed form expression for its first order term is derived. This expression facilitates the study of the behavior of the empirical CTE with respect to the underlying distribution, and suggests an alternative (to the bootstrap) approach to bias correction. The performance of the resulting estimator is assessed via simulation.

scholarly journals Non-boost invariant fluid dynamics

2020 ◽  
Vol 9 (2) ◽  
Author(s):  
Jan de Boer ◽  
Jelle Hartong ◽  
Emil Have ◽  
Niels Obers ◽  
Watse Sybesma
Keyword(s):  
Rotational Symmetry ◽  
Transport Coefficients ◽  
Momentum Tensor ◽  
Closed Form Expression ◽  
Lagrangian Description ◽  
Form Expression ◽  
First Order ◽  
Entropy Current ◽  
In Parenthesis ◽  
Dissipative Transport

We consider uncharged fluids without any boost symmetry on an arbitrary curved background and classify all allowed transport coefficients up to first order in derivatives. We assume rotational symmetry and we use the entropy current formalism. The curved background geometry in the absence of boost symmetry is called absolute or Aristotelian spacetime. We present a closed-form expression for the energy-momentum tensor in Landau frame which splits into three parts: a dissipative (10), a hydrostatic non-dissipative (2) and a non-hydrostatic non-dissipative part (4), where in parenthesis we have indicated the number of allowed transport coefficients. The non-hydrostatic non-dissipative transport coefficients can be thought of as the generalization of coefficients that would vanish if we were to restrict to linearized perturbations and impose the Onsager relations. For the two hydrostatic and the four non-hydrostatic non-dissipative transport coefficients we present a Lagrangian description. Finally when we impose scale invariance, thus restricting to Lifshitz fluids, we find 7 dissipative, 1 hydrostatic and 2 non-hydrostatic non-dissipative transport coefficients.

Derivation of tunneling probabilities for arbitrarily graded potential barriers using modified Airy functions

2010 ◽  
Vol 42 (2) ◽  
pp. 129-141
Author(s):  
Kyu-Tae Lee ◽  
Eun Joo Jung ◽  
Chul Han Kim ◽  
Chang-Min Kim
Keyword(s):  
Airy Functions ◽  
Potential Barriers ◽  
Modified Airy Functions

scholarly journals Synthesizing of Planar and Conformal Double-Sided Parallel-Cross Dipoles FSS Based on Closed-Form Expression

IEEE Access ◽  
2021 ◽  
pp. 1-1
Author(s):  
Yassine Zouaoui ◽  
Larbi Talbi ◽  
Khelifa Hettak ◽  
Naresh K. Darimireddy
Keyword(s):  
Closed Form ◽  
Closed Form Expression ◽  
Form Expression

Distribution of Consensus in a Broadcasting-based Consensus-forming Algorithm

2021 ◽  
Vol 48 (3) ◽  
pp. 91-96
Author(s):  
Shigeo Shioda
Keyword(s):  
Distribution Function ◽  
Probability Density Function ◽  
Closed Form ◽  
Probability Density ◽  
Infinite Number ◽  
Stable Distribution ◽  
Random Variable ◽  
Cauchy Distribution ◽  
Closed Form Expression ◽  
Form Expression

The consensus achieved in the consensus-forming algorithm is not generally a constant but rather a random variable, even if the initial opinions are the same. In the present paper, we investigate the statistical properties of the consensus in a broadcasting-based consensus-forming algorithm. We focus on two extreme cases: consensus forming by two agents and consensus forming by an infinite number of agents. In the two-agent case, we derive several properties of the distribution function of the consensus. In the infinite-numberof- agents case, we show that if the initial opinions follow a stable distribution, then the consensus also follows a stable distribution. In addition, we derive a closed-form expression of the probability density function of the consensus when the initial opinions follow a Gaussian distribution, a Cauchy distribution, or a L´evy distribution.

scholarly journals Colored HOMFLY-PT for hybrid weaving knot $$ {\hat{\mathrm{W}}}_3 $$(m, n)

2021 ◽  
Vol 2021 (6) ◽  
Author(s):  
Vivek Kumar Singh ◽  
Rama Mishra ◽  
P. Ramadevi
Keyword(s):  
Closed Form ◽  
Topological Strings ◽  
Chebyshev Polynomials ◽  
Fibonacci Numbers ◽  
Infinite Family ◽  
Closed Form Expression ◽  
Two Dimensional ◽  
Laurent Polynomials ◽  
Form Expression ◽  
Torus Knots

Abstract Weaving knots W(p, n) of type (p, n) denote an infinite family of hyperbolic knots which have not been addressed by the knot theorists as yet. Unlike the well known (p, n) torus knots, we do not have a closed-form expression for HOMFLY-PT and the colored HOMFLY-PT for W(p, n). In this paper, we confine to a hybrid generalization of W(3, n) which we denote as $$ {\hat{W}}_3 $$ W ̂ 3 (m, n) and obtain closed form expression for HOMFLY-PT using the Reshitikhin and Turaev method involving $$ \mathrm{\mathcal{R}} $$ ℛ -matrices. Further, we also compute [r]-colored HOMFLY-PT for W(3, n). Surprisingly, we observe that trace of the product of two dimensional $$ \hat{\mathrm{\mathcal{R}}} $$ ℛ ̂ -matrices can be written in terms of infinite family of Laurent polynomials $$ {\mathcal{V}}_{n,t}\left[q\right] $$ V n , t q whose absolute coefficients has interesting relation to the Fibonacci numbers $$ {\mathrm{\mathcal{F}}}_n $$ ℱ n . We also computed reformulated invariants and the BPS integers in the context of topological strings. From our analysis, we propose that certain refined BPS integers for weaving knot W(3, n) can be explicitly derived from the coefficients of Chebyshev polynomials of first kind.

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