Poisson Series Research Articles

Using Poisson Series in the Analytic Theory of Irregularly Degenerate Elliptic Differential Operators

Using Poisson Series in the Analytic Theory of Irregularly Degenerate Elliptic Differential Operators

An improved C++ Poisson series processor with its applications

One of major problems in celestial mechanics is the management of the long developments in Fourier or Poisson series used to describe the perturbed motion in the planetary system. In this work we will develop a software package suitable for managing these objects. This package includes the ordinary arithmetic operations with Poisson series—such as sum or product—as well as the most commonly used functions sin , cos , or exp , among others. Derivation or integration procedures with respect to time of these objects have been implemented and inversion or serialization procedures are also attainable to obtain such series. All the programming has been accomplished overloading the arithmetic and functional operators natural to C++ with the intention of allowing the programmer to work in a more friendly way and treating the series as if they were mere numbers. In this work we extend the processor in order to obtain the solution to perturbed problems which solution is in the form of a Poisson series. These algorithms have been written in C++.

Hurricane trend detection

Because a change in the frequency (number/year) of hurricanes could be a result of climate change, we analyzed the historical record of Atlantic basin and US landfalling hurricanes, as well as US continental accumulated cyclone energy to evaluate issues related to trend detection. Hurricane and major hurricane landfall counts exhibited no significant overall trend over 167 years of available data, nor did accumulated cyclone energy over the continental USA over 119 years of available data, although shorter-term trends were evident in all three datasets. Given the χ2 distribution evinced by hurricane and major hurricane counts, we generated synthetic series to test the effect of segment length, demonstrating that shorter series were increasingly likely to exhibit spurious trends. Compared to synthetic data with the same mean, the historical all-storm data were more likely to exhibit short-term trends, providing some evidence for long-term persistence at timescales below 10 years. Because this might be due to known climate modes, we examined the relationship between the Atlantic multidecadal oscillation (AMO) and hurricane frequency in light of these short-term excursions. We found that while ratios of hurricane counts with AMO phase matched expectations, statistical tests were less clear due to noise. Over a period of 167 years, we found that an upward trend of roughly 0.7/century is sufficient to be detectable with 80% confidence over the range from 1 to 21 storms/year. Storm energy data 1900–2018 over land were also analyzed. The trend was again zero. The pattern of spurious trends for short segments was again found. Results for AMO periods were similar to count data. Atlantic basin all storms and major storms (1950–2018) did not exhibit any trend over the whole period or after 1990. Major storms 1950–1989 exhibited a significant downward trend. All-storm basin scale storms exhibited short-term trends matching those expected from a Poisson process. A new test for Poisson series was developed based on the 95% distribution of slopes for simulated data across a range of series lengths. Because short data series are inherently likely to yield spurious trends, care is needed when interpreting hurricane trend data.

The Implementation of Hori–Deprit Method to the Construction Averaged Planetary Motion Theory by Means of Computer Algebra System Piranha

This article is related to the problem of the construction of planetary motion theory. We have expanded the Hamiltonian of the four-planetary problem into the Poisson series in osculating elements of the second Poincare system. The series expansion is constructed up to the third degree of the small parameter. The averaging procedure of the Hamiltonian is performed by the Hori–Deprit method. It allows to eliminate short-periodic perturbations and sufficiently increase time step of the integration of the equations of motion. This method is based on Lie transformation theory. The equations of motion in averaged elements are constructed as the Poisson brackets of the averaged Hamiltonian and corresponding orbital element. The transformation between averaged and osculating elements is given by the change-variable functions, which are obtained in the second approximation of the Hori–Deprit method. We used computer algebra system Piranha for the implementation of the Hori–Deprit method. Piranha is an echeloned Poisson series processor authored by F. Biscani. The properties of the obtained series are discussed. The numerical integration of the equations of motion is performed by Everhart method for the Solar system’s giant planets.

Population parameters of two water scavenger beetles:Derallus angustusSharp, 1882 andEnochrus vulgaris(Steinheil, 1869) (Coleoptera: Hydrophilidae) in permanent ponds: spatial distribution and microhabitat preference

The spatial distribution, temporal variation, and microhabitat preference of two populations of aquatic Coleoptera were analysed in permanent ponds located in the province of Corrientes, Argentina. Samples were collected every 15 days, between October 2010 and October 2011. The dispersion index and distribution models (Poisson Series, positive binomial and negative binomial) were used to evaluate the spatial distribution. For the evaluation of the microhabitat preference we used non-parametric Kruskal–Wallis test and then the Dunn test with the Bonferroni’s correction. In the test of the use of different microhabitats, we used a K proportion test and, when significant, a two proportion Z test was applied between pairs of microhabitats. The species analysed were the hydrophilids Derallus angustus Sharp and Enochrus vulgaris (Steinheil). The results indicate that the spatial distribution of individuals was mostly related to the homogeneity or heterogeneity of the ecosystem. However, when species were analys...

Modelling of nonlinear filtering Poisson time series

In this article, algorithms of non-linear filtering of Poisson time series are tested using statistical modelling. The objective is to find a representation of a time series as a wavelet series with a small number of non-linear coefficients, which allows distinguishing statistically significant details. There are well-known efficient algorithms of non-linear wavelet filtering for the case when the values of a time series have a normal distribution. However, if the distribution is not normal, good results can be expected using the maximum likelihood estimations. The filtration is studied according to the criterion of maximum likelihood by the example of Poisson time series. For direct optimisation of the likelihood function, different stochastic (genetic algorithms, annealing method) and deterministic optimization algorithms are used. Testing of the algorithm using both simulated series and empirical data (series of rare words frequencies according to the Google Books Ngram data were used) showed that filtering based on the criterion of maximum likelihood has a great advantage over well-known algorithms for the case of Poisson series. Also, the most perspective methods of optimisation were selected for this problem.

Modified poisson series to implement the method of multiple scales

The method of multiple scales is a global perturbation technique that has resulted to be very useful in perturbed ordinary differential equations characterized by disparate time scales. The general principle behind the method is that the solution to the differential equation is uniformly expanded in terms of two or more independent variables, referred to as time scales. In this article, we present a mathematical object based on a Poisson series to apply the method of multiple scales via specific symbolic computation. Copyright © 2016 John Wiley & Sons, Ltd.

Expansion of the Hamiltonian of the planetary problem into the Poisson series in elements of the second Poincare system

The Hamiltonian of the N-planetary problem is written in the Jacobi coordinates using the second system of Poincare elements. The Hamiltonian is expanded into the Poisson series for the four-planet system. The computer algebra system Piranha is used for analytical transformations. Obtained expansions provide the Hamiltonian expression accuracy up to the third degree of the small parameter for giant planets of the Solar System and up to the second degree of the small parameter for extrasolar planetary systems. The ratio of sums of masses of the planets to the star mass can be selected as a small parameter.

Fully Bayesian inference for α-stable distributions using a Poisson series representation

In this paper we develop an approach to Bayesian Monte Carlo inference for skewed α-stable distributions. Based on a series representation of the stable law in terms of infinite summations of random Poisson process arrival times, our framework leads to a simple representation in terms of conditionally Gaussian distributions for certain latent variables. Inference can therefore be carried out straightforwardly using techniques such as auxiliary variables versions of Markov chain Monte Carlo (MCMC) methods. The Poisson series representation (PSR) is further extended to practical application by introducing an approximation of the series residual terms based on exact moment calculations. Simulations illustrate the proposed framework applied to skewed α-stable simulated and real-world data, successfully estimating the distribution parameter values and being consistent with other (non-Bayesian) approaches. The methods are highly suitable for incorporation into hierarchical Bayesian models, and in this case the conditionally Gaussian structure of our model will lead to very efficient computations compared to other approaches.

An improved algorithm to develop semi-analytical planetary theories using Sundman generalized variables

One of the main problems in celestial mechanics is the construction of analytical theories of planetary motion. The most common solution of this problem is arranged by means of Poisson series developments. These developments depend on the selection of the anomaly to be used as temporal variable. In this paper we develop an improved algorithm in order to use arbitrary anomalies included in the family of the generalized Sundman anomalies as temporal variables.

Generation of a secular system in the analytical theory for the motion of the moon

In order to generate an analytical theory of the motion of the Moon by considering planetary perturbations, a procedure of general planetary theory (GPT) is used. In this case, the Moon is considered as an addition planet to the eight principal planets. Therefore, according to the GPT procedure, the theory of the Moon’s orbital motion can be presented in the form of series with respect to the evolution of eccentric and oblique variables with quasi-periodic coefficients, which are the functions of mean longitudes for principal planets and the Moon. The relationship between evolution variables and the time is determined by a trigonometric solution for the independent secular system that describes the secular motion of a perigee and the Moon node by considering secular planetary inequalities. Principal planetary coordinates required for generating the theory of the motion of the Moon includes only Keplerian terms, the intermediate orbit, and the linear theory with respect to eccentricities and inclinations in the first order relative to the masses. All analytical calculations are performed by means of the specialized echeloned Poisson Series Processor EPSP.

Analytical derivation of higher-order terms of Molière’s series and accuracy of Molière’s angular distribution of fast charged particles

Molière’s series functions of higher orders describing angular distributions of charged particle under the multiple scattering process are solved by exactly evaluating Cauchy integral with poles within the contour of integration. The functions of Molière’s series giving higher-order terms are evaluated accurately by Poisson series expansion, both for spatial and projected angular distributions. Molière’s series for the integrated angular distributions are also derived. Accuracy of the Molière’s series expansion of higher orders is examined by comparing the reconstructed angular distributions with those derived exactly through the numerical Hankel transforms.

Ecological morphotaxometry of trematodes of garfish (Teleostomi: Belonidae) from Gangetic riverine ecosystem in India. II. Correlation of seasonality and host biology with distribution pattern of Cephalogonimus yamunii n.sp.

The seasonality influence on the biology and distribution dynamics of Cephalogonimus yamunii n.sp. in Xenentodon cancilla (Teleostomi: Belonidae) was reflected during 2years' investigations. The highest infection prevalence appeared to be in late summer. Size, weight, and sex-biased population dynamics of trematode was well marked in Gangetic riverine ecosystem. The biostatistical non-parametric Mann-Whitney χ (2) test values were significant for dissolved oxygen and magnesium. The distribution of infection data on Poisson series ruled out overdispersion because good agreement was observed (P<0.01).

PRICING AND SEMIMARTINGALE REPRESENTATIONS OF VULNERABLE CONTINGENT CLAIMS IN REGIME‐SWITCHING MARKETS

Using a suitable change of probability measure, we obtain a Poisson series representation for the arbitrage‐free price process of vulnerable contingent claims in a regime‐switching market driven by an underlying continuous‐time Markov process. As a result of this representation, along with a short‐time asymptotic expansion of the claim's price process, we develop an efficient novel method for pricing claims whose payoffs may depend on the full path of the underlying Markov chain. The proposed approach is applied to price not only simple European claims such as defaultable bonds, but also a new type of path‐dependent claims that we term self‐decomposable, as well as the important class of vulnerable call and put options on a stock. We provide a detailed error analysis and illustrate the accuracy and computational complexity of our method on several market traded instruments, such as defaultable bond prices, barrier options, and vulnerable call options. Using again our Poisson series representation, we show differentiability in time of the predefault price function of European vulnerable claims, which enables us to rigorously deduce Feynman‐Kač representations for the predefault pricing function and new semimartingale representations for the price process of the vulnerable claim under both risk‐neutral and objective probability measures.

Periodicity characterization of orbital prediction error and Poisson series fitting

Publicly available Two-Line Element Sets (TLE) contains no associated error or accuracy information. The historical-data-based method is a feasible choice for those objects only TLE data are available. Most of current TLE error analysis methods use polynomial fitting which cannot represent the periodic characteristics. This paper has presented a methodology for periodicity characterization and Poisson series fitting for orbital prediction error based on historical orbital data. As error-fitting function, the Poisson series can describe variation of error with respect to propagation duration and on-orbit position of objects. The Poisson coefficient matrices of each error components are fitted using least squares method. Effects of polynomial terms, trigonometric terms, and mixed terms of Poisson series are discussed. Substituting time difference and mean anomaly into the Poisson series one can obtain the error information at specific time. Four satellites (Cosmos-2251, GPS-62, SLOSHSAT, TelStar-10) from four orbital type (LEO, MEO, HEO, GEO, respectively) were selected as examples to demonstrate and validate the method. The results indicated that the periodic characteristics exist in all three components of four objects, especially HEO and MEO. The periodicity characterization and Poisson series fitting could improve accuracy of the orbit covariance information. The Poisson series is a common form for describing orbital prediction error, the commonly used polynomial fitting is a special case of the Poisson series fitting. The Poisson coefficient matrices can be obtained before close approach analysis. This method does not require any knowledge about how the state vectors are generated, so it can handle not only TLE data but also other orbit models and elements.

A note on the use of the generalized Sundman transformations as temporal variables in celestial mechanics

The main aim of this paper is to build up a set of semi-analytical integrators based on the use of a class of anomalies defined by means of a generalized Sundman transformation. The integrators are based on the developments of the mean anomaly and the vector radius according to the new anomaly. To manipulate these developments, a Poisson series processor called poison.h has been used. This processor has been written as a C++class and it contains a set of methods to manage the most common arithmetic and functional operations with these objects.

Refined equations in dynamic problems for glued beams

Two methods for solving problems of the plane dynamic deformation of two beams glued together are proposed. The beams are made of isotropic homogeneous materials and have rectangular cross section. One method is differential (based on equations of motion), and the other method is integral (based on the virtual-displacement principle (VDP)). Some hypotheses and equations of linear elasticity are used to approximate the displacement vectors of the beams and adhesive interlayer (AI). The approximations satisfy the kinematic interface conditions. The correctness and existence of the solutions of the initial–boundary-value problems are justified. Methods to solve them are proposed. The VDPs for the beams and AI are represented in general (explicit) form. It is shown how the practical convergence of the Cauchy–Poisson series used can be improved

The Conservation of Area Integrals in Averaging Transformations

It is shown for the two-planetary version of the weakly perturbed two-body problem that, in a system defined by a finite part of a Poisson expansion of the averaged Hamiltonian, only one of the three components of the area vector is conserved, corresponding to the longitudes measuring plane. The variability of the other two components is demonstrated in two ways. The first is based on calculating the Poisson bracket of the averaged Hamiltonian and the components of the area vector written in closed form. In the second, an echeloned Poisson series processor (EPSP) is used when calculating the Poisson bracket. The averaged Hamiltonian is taken with accuracy to second order in the small parameter of the problem, and the components of the area vector are expanded in a Poisson series.

Accurate analytical representation of Pluto modern ephemeris

An accurate development of the latest JPL’s numerical ephemeris of Pluto, DE421, to compact analytical series is done. Rectangular barycentric ICRF coordinates of Pluto from DE421 are approximated by compact Fourier series with a maximum error of 1.3 km over 1900–2050 (the entire time interval covered by the ephemeris). To calculate Pluto positions relative to the Sun, a development of rectangular heliocentric ICRF coordinates of the Solar System barycenter to Poisson series is additionally made. As a result, DE421 Pluto heliocentric positions by the new analytical series are represented to an accuracy of better than 5 km over 1900–2050.

Spatial Patterns of Parasitism of the Solitary ParasitoidPseudapanteles dignus(Hymenoptera: Braconidae) onTuta absoluta(Lepidoptera: Gelechiidae)

We examined the interaction between the tomato leafminer, Tuta absoluta (Meyrick) (Lepidoptera: Gelechiidae), a key pest of tomato crops in South America, and its main solitary larval parasitoid, Pseudapanteles dignus (Muesebeck) (Hymenoptera: Braconidae). The pattern of parasitism of T. absoluta by the parasitoid was studied at three scales on tomato crops: plant, leaf, and leaflet. Host density, spatial distributions of both host and parasitoid, percentages of parasitism, variation in the probability and risk of parasitism in relation to host density, and the spatial density dependence were assessed in a horticultural region in Argentina. The spatial distribution of T. absoluta was clumped at all sites and scales, whereas that of P. dignus was much more variable, fitting to negative, positive binomial distributions and to Poisson series. Percentages of parasitism were as follows: site 1, 17.06%; site 2, 27.53%; site 3, 26.47%; site 4, 45.95%. Parasitoid aggregation in relation to host density was found at leaf and leaflet scales. However, the proportion of parasitized hosts was independent of host density. The variability of parasitism rates exhibited at the three spatial scales seems to result in partial refuges for the host, which might contribute to the persistence of the interaction between host and parasitoid. We discuss our field observations in relation to ecological theory and its potential application to the biological control of T. absoluta on tomato.