Third-order ladder operators, generalized Okamoto and exceptional orthogonal polynomials

We extend and generalize the construction of Sturm-Liouville problems for a family of Hamiltonians constrained to fulfill a third-order shape-invariance condition and focusing on the "-2x/3" hierarchy of solutions to the fourth Painlev\'e transcendent. Such a construction has been previously addressed in the literature for some particular cases but we realize it here in the most general case. The corresponding potential in the Hamiltonian operator is a rationally extended oscillator defined in terms of the conventional Okamoto polynomials, from which we identify three different zero-modes constructed in terms of the generalized Okamoto polynomials. The third-order ladder operators of the system reveal that the complete set of eigenfunctions is decomposed as a union of three disjoint sequences of solutions, generated from a set of three-term recurrence relations. We also identify a link between the eigenfunctions of the Hamiltonian operator and a special family of exceptional Hermite polynomial.

A reciprocal relation for Hermite polynomials

For $x\in \mathbb{R}$, the ordinary Hermite polynomial $H_k(x)$ can be written\begin{eqnarray*}\displaystyleH_k(x)= \mathbb{E} \left[ (x + {\rm i} N)^k \right] =\sum_{j=0}^k {k\choose j} x^{k-j} {\rm i}^j \mathbb{E} \left[ N^j \right],\end{eqnarray*}where ${\rm i} = \sqrt{-1}$ and $N$ is a unit normal random variable. We prove the reciprocal relation\begin{eqnarray*}\displaystylex^k=\sum_{j=0}^k {k\choose j} H_{k-j}(x)\ \mathbb{E} \left[ N^j \right].\end{eqnarray*}A similar result is given for the multivariate Hermite polynomial.

Probabilistic small-signal stability analysis of power systems based on Hermite polynomial approximation

This paper proposes a probabilistic small-signal stability analysis method based on the polynomial approximation approach. Since the correct determination of unknown coefficients has a direct effect on the accuracy of the polynomial approximation method, this paper presents a method for determining these coefficients, with high coverage on the probabilistic input domain of the problem. In this method, by increasing the number of random input variables, the proposed method can continue to maintain its efficiency. After determining the unknown coefficients, the load flow results and the system state matrix are determined for random changes of all loads based on the Hermite polynomial approximation. Then, the small-signal stability of the system is probabilistically evaluated based on stochastic analysis of the system eigenvalues. The consistency and validity of the proposed method are demonstrated based on the simulation studies in the MATLAB® software environment. In the simulation studies, the performance of the proposed method is examined by comparison with Point Estimation, Cumulant, Monte Carlo, and Chebyshev polynomial-based methods, for IEEE 14-bus and IEEE 39-bus test systems. Article highlights Probabilistic small-signal stability analysis of power systems Modeling of governing equations of power system based on Hermite polynomial approximation Forming the Collocation matrix based on a robust method

Transformation of a high-order edge dislocation to optical vortices (spiral dislocations)

We theoretically show that an astigmatic transformation of an nth-order edge dislocation (a zero-intensity straight line) produces n optical elliptical vortices (spiral dislocations) with unit topological charge at the double focal distance from the cylindrical lens, located on a straight line perpendicular to the edge dislocation, at points whose coordinates are the roots of an nth-order Hermite polynomial. The orbital angular momentum of the edge dislocation is proportional to the order n.

A generalized Fourier–Hermite method for the Vlasov–Poisson system

A generalized Fourier–Hermite semi-discretization for the Vlasov–Poisson equation is introduced. The formulation of the method includes as special cases the symmetrically-weighted and asymmetrically-weighted Fourier–Hermite methods from the literature. The numerical scheme is formulated as a weighted Galerkin method with two separate scaling parameters for the Hermite polynomial and the exponential part of the new basis functions. Exact formulas for the error in mass, momentum, and energy conservation of the method depending on the parameters are devised and $$L^2$$ L 2 stability is discussed. The numerical experiments show that an optimal choice of the additional parameter in the generalized method can yield improved accuracy compared to the existing methods, but also reveal the distinct stability properties of the symmetrically-weighted method.

Hermite polynomial normal transformation for structural reliability analysis

Purpose Normal transformation is often required in structural reliability analysis to convert the non-normal random variables into independent standard normal variables. The existing normal transformation techniques, for example, Rosenblatt transformation and Nataf transformation, usually require the joint probability density function (PDF) and/or marginal PDFs of non-normal random variables. In practical problems, however, the joint PDF and marginal PDFs are often unknown due to the lack of data while the statistical information is much easier to be expressed in terms of statistical moments and correlation coefficients. This study aims to address this issue, by presenting an alternative normal transformation method that does not require PDFs of the input random variables. Design/methodology/approach The new approach, namely, the Hermite polynomial normal transformation, expresses the normal transformation function in terms of Hermite polynomials and it works with both uncorrelated and correlated random variables. Its application in structural reliability analysis using different methods is thoroughly investigated via a number of carefully designed comparison studies. Findings Comprehensive comparisons are conducted to examine the performance of the proposed Hermite polynomial normal transformation scheme. The results show that the presented approach has comparable accuracy to previous methods and can be obtained in closed-form. Moreover, the new scheme only requires the first four statistical moments and/or the correlation coefficients between random variables, which greatly widen the applicability of normal transformations in practical problems. Originality/value This study interprets the classical polynomial normal transformation method in terms of Hermite polynomials, namely, Hermite polynomial normal transformation, to convert uncorrelated/correlated random variables into standard normal random variables. The new scheme only requires the first four statistical moments to operate, making it particularly suitable for problems that are constraint by limited data. Besides, the extension to correlated cases can easily be achieved with the introducing of the Hermite polynomials. Compared to existing methods, the new scheme is cheap to compute and delivers comparable accuracy.

Space-Time Analysis of Precipitation Reanalysis Data for the Island of Crete using Gaussian Anamorphosis with Hermite Polynomials

<p>Societies seek to ensure sustainable development in the face of climate change, population increase, and increased demands for natural resources. Understanding, modeling, and forecasting the spatiotemporal patterns of precipitation are central to this effort [1-3]. Spatiotemporal models of precipitation with global validity are not available. This is due to the non-Gaussian distribution of precipitation as well as its intermittent nature and strong dependence on the geographic location and the space-time scales analyzed.  Herein we investigate the spatiotemporal patterns of precipitation on a Mediterranean island using geostatistical methods. </p><p>We use ERA5 reanalysis precipitation products from the Copernicus Climate Change Service [4].  The dataset includes 31980 values of monthly precipitation height (mm) for a period of 492 consecutive months (January 1979 to December 2019) at the nodes of a 5 × 13 spatial grid that covers the island of Crete (Greece). This results in an average spatial resolution of approximately 0.28 degrees (corresponding to an approximate grid cell size of 31 km).  </p><p>We construct a spatial model of monthly precipitation using Gaussian anamorphosis (GA). GA employs nonlinear transformations to normalize the probability distribution of the data. It is extensively used in various environmental applications [5-6].  The methodology that we follow involves (i) normalizing the precipitation data per month using GA with Hermite polynomials, (ii) estimating spatial correlations and fitting them to the Spartan variogram family [6], (iii) ordinary kriging (OK) of the normalized data in order to generate precipitation estimates on a denser map grid, and (iv) application of the inverse GA transform to generate monthly precipitation maps. We also use cross-validation analysis to determine the kriging interpolation performance, first using the untransformed precipitation data and then the Hermite-polynomial GA approach outlined above. We find that Hermite-polynomial GA significantly improves the cross-validation measures.</p><p> </p><p>Keywords: Gaussian anamorphosis, Hermite polynomials, Mediterranean island, non-Gaussian, ordinary kriging, Spartan variogram</p><p> </p><p><strong>References</strong></p><p>1. D. Allard, and M. Bourotte, 2015. Disaggregating daily precipitations into hourly values with a transformed censored latent Gaussian process. Stochastic Environ. Res. Risk Assess, <strong>29</strong>(2), pp. 453– 462. https://doi.org/10.1007/s00477-014-0913-4.</p><p>2. A. Baxevani, and J. Lennartsson, 2015. A spatiotemporal precipitation generator based on a censored latent Gaussian field, Water Resources Research, <strong>51</strong>(6), 4338–4358. https://doi.org/10.1002/2014WR016455.</p><p>3. C. Lussana, T. N. Nipen, I. A. Seierstad, and C. A. Elo, 2020. Ensemble-based statistical interpolation with Gaussian anamorphosis for the spatial analysis of precipitation. Nonlinear Processes in Geophysics, 1–43. https://doi.org/10.5194/npg-2020-20.</p><p>4. C3S, C. C. C. S., 2018. ERA5: Fifth generation of ECMWF atmospheric reanalyses of the global climate. Data retrieved from: https://cds.climate.copernicus.eu/cdsapp#!/home.</p><p>5. N. Cressie, 1993. Spatial Statistics. John Wiley and Sons, New York.</p><p>6. D. T. Hristopulos, 2020. Random Fields for Spatial Data Modeling. Springer Netherlands, http://dx.doi.org/10.1007/978-94-024-1918-4.</p>

The sum of two independent polynomially-modified hyperbolic secant random variables with application in computational finance

In this paper, by reshaping the hyperbolic secant distribution using Hermite polynomial, we devise a polynomially-modified hyperbolic secant distribution which is more flexible than secant distribution to capture the skewness, heavy-tailedness and kurtosis of data. As a portfolio possibly consists of multiple assets, the distribution of the sum of independent polynomially-modified hyperbolic secant random variables is derived. In exceptional cases, we evaluate risk measures such as value at risk and expected shortfall (ES) for the sum of two independent polynomially-modified hyperbolic secant random variables. Finally, using real datasets from four international computers stocks, such as Adobe Systems, Microsoft, Nvidia and Symantec Corporations, the effectiveness of the proposed model is shown by the goodness of Gram–Charlier-like expansion of hyperbolic secant law, for performance of value at risk and ES estimation, both in and out of the sample period.