J. A. Reyes

Numeric multistep variable methods for perturbed linear system integration

Abstract A new special-purpose method of numerical integration for a perturbed linear system of ordinary differential equations is presented, based on Scheifele’s G -functions method. The new method converts Scheifele’s method in a multistep scheme that conserves the capability of integrating without truncation errors the unperturbed linear system of ODE’s. The new numerical method presents a good property contrasted with other VSVO multistep methods using G -functions; it adds a simple algebraic procedure for the computation of the coefficients irrespective of what the order is, obtained through recurrence formulas. Some numerical examples are included showing the good behaviour of the new code when it is applied to the class of problems to which it is addressed.

Accurate Numerical Integration of Perturbed Oscillatory Systems in Two Frequencies

Highly accurate long-term numerical integration of nearly oscillatory systems of ordinary differential equations (ODEs) is a common problem in astrodynamics. Scheifele’s algorithm is one of the excellent integrators developed in the past years to take advantage of special transformations of variables such as the K-S set. It is based on using expansions in series of the so-called G-functions, and generalizes the Taylor series integrators but with the remarkable property of integrating without truncation error oscillations in one basic known frequency. A generalization of Scheifele’s method capable of integrating exactly harmonic oscillations in two known frequencies is developed here, after introducing a two parametric family of analytical ϕ-functions. Moreover, the local error contains the perturbation parameter as a factor when the algorithm is applied to perturbed problems. The good behavior and the long-term accuracy of the new method are shown through several examples, including systems with low- and high-frequency constituents and a perturbed satellite orbit. The new methods provide significantly higher accuracy and efficiency than a selection of well-reputed general-purpose integrators and even recent symplectic or symmetric integrators, whose good behavior in the long-term integration of the Kepler problem and the other oscillatory systems is well stated in recent literature.

Multistep numerical methods for the integration of oscillatory problems in several frequencies

The perturbed harmonic oscillator appear frequently in the mathematical modelling of many problems in physics and engineering. The harmonic oscillator has a special purpose in Astrodynamics, because the Kunstaanheimo-Stiefel (KS) and Burdet-Ferrandiz (BF) transformations reduce the Kepler problem to harmonic oscillators. A new multi-step methods of numerical integration are introduced that generalize SMF ones. They are defined for arbitrary order and have similar properties to the former methods. Modified methods allowing step variations, whose coefficients are computed from relations of recurrence, are derived, what considerably improve the implementation of the algorithms. These methods are based in a sequence of analytical @f-functions dependent on two parameters @a and @b that generalizes the Scheifeles G-functions and that, under wide hypothesis, allow us obtain the solution of harmonic oscillator. In this paper a new methodology is generated to solve the problem which the @f-functions series create regarding the calculus of recurrence relations transforming the method into a multistep scheme. The methodology is implemented in a computational algorithm, which lets us solve in a general way the problems of physics and engineering which are modalized by means of the study of the harmonic oscillators. Numerical examples already used by other authors are presented. They show how the new developed methods in this paper may compete in accuracy or efficiency with other well-reputed algorithms.

Lotus glaber Mill: comparison of some morphological-physiological characters between an induced autotetraploid population and diploid cultivars.

Lotus glaber Mill, a diploid forage legume (2n=2x=12), has successfully adapted to the ecological characteristics of the Flooding Pampas region of the Province of Buenos Aires, Argentina. This species is highly valued, basically due to its ability to adapt to soils that do not allow other forage legumes to prosper, as well as for its high nutritional value. A genetic improvement programme of the species resulted in an induced autotetraploid population known as “Leonel” (2n=4x= 24). Generally speaking, induced autotetraploid forms are larger and also more vigorous and productive than their diploid counterparts. Although fewer seeds are produced, they are heavier. This article compares certain morphological-physiological characters of the “Leonel” population and diploid cultivars. The characteristics evaluated were length (CFL), width (CFW), area (CFA), length/width ratio (L/W) of the central foliolae of the first expanded leaf from the stem apex and the vigour of the plants (V) from May to December. The following variables were also studied: days to flowering (BFP), air biomass, expressed as green material (GM) and dry material (DM), weight of a thousand seeds (WS), number of seeds per pod (NS), number of empty seeds per pod (ES) and the number of full seeds per pod (FS). Variance analysis was carried out with a completely random design using two factors, cytotype and month, for the following variables: CFL, CFW, CFA, L/W and V. For the remaining variables, variance analysis was carried out with a completely random design using one factor: cytotype. The Duncan’s Test, using a level of confidence of 5%, was used to compare the mean values. In addition, analysis of the L/W ratio allowed the form of the foliolae to be determined.

A new method for exact integration of some perturbed stiff linear systems of oscillatory type

This article presents a new family of real functions with values within the ring of M(m,R) matrices, @F-functions for perturbed linear systems and a numerical method adapted for integration of this type of problem. This method permits the system solution to be expressed as a series of @F-functions. The coefficients of this series are obtained through recurrences in which the perturbation intervenes. The @F-functions series method has the advantage of being exactly integrated in the perturbed problem. For this purpose an appropriate B matrix is selected and used to construct the operator described in this article, thus annihilating the disturbance terms, transforming the system into a homogenous second-order system, which is exactly integrated with the two first @F-functions. The article ends with a detailed study of four perturbed systems which illustrate how the method is used in stiff problems or in highly oscillatory problems, contrasting its behaviour by studying its accuracy in comparison with other well-known codes.

INFORMATION STORAGE SYSTEM

This paper examines an Information Storage System (ISS), in which a vocabulary and set of experimental data are defined, on the basis of which both linear and linearized (meaning that they are treated in linear fashion) mathematical models are generated. The studies obtained lead to establish certain objective criteria that let us select the best characters to observe and establish a methodology, which allows us to select the mathematical functions that best represent a given system. The studies provide the theoretical basis for obtaining computational algorithms that allow the computational implementation of model selection.

Model information transmission system

Purpose – The purpose of this paper is to study a theory of systems and their models on the basis of studying the information channels between their elements, meaning that it is a continuation of those studies already carried out on the theory of systems and their models applied to environmental systems.Design/methodology/approach – The model information transmission system (MITS) can be defined as the theoretical structure that makes it possible to describe the way information is stored and transmitted in a modelling process, and which will be completed in future studies. The basis for a taxonomy of languages is also presented.Findings – The theoretical structure described will lead to the study of a theory of information from the point of view of the analysis of text systems and models, but analysed within the theoretical structure presented, which includes the study of an information source and an information channel.Originality/value – These theoretical findings will improve the information channels u...

INTERPRETATION OF THE GENERALIZED ZIPF-MANDELBROT LAW PARAMETERS

This article provides an interpretation of the parameters of the generalized Zipf-Mandelbrot law, which the authors have called the generalized range-frequency law which has been developed in 2009. Said law models the distribution of frequencies of words in school textbooks obtained using a Pearson system. The significance of the parameters gathered was obtained by analyzing certain variables. The fact that the statistical law can be applied to texts written using natural language means that it may be applicable to ecological models, these being considered as a particular case of mathematical models written as texts in a formal language, as developed in 1999. After establishing certain hypotheses regarding the significance of the parameters of the generalized law obtained for natural language, this study was designed to analyze whether this law is applicable to models considered as texts written in formal language. If this were so, the law would be a very useful tool for comparing models.

MODELLING OF STATISTICAL LAWS FOR TEXTS USING A PEARSON SYSTEM

In this article, we have obtained a generalization of statistical laws for texts for application to natural languages, in order to determine whether this general law is applicable to model texts in accordance with the definition given in Villacampa et al. (1999a). The studies were carried out in the context of a system whose components generated model texts, through which we confirmed that the laws of Zipf and Mandelbrot are appropriate for explaining the texts. On the basis of a Pearson System, we also obtained a unique function for the distribution of frequencies, with said distribution being a generalization of the above-mentioned laws. For the problem analyzed in this article, the text or text system constitutes a model of the cognitive development of individuals. We can thus infer the cognitive characteristics of different individuals by studying text systems.

Lotus glaber Mill. induced autotetraploid: new forage resource for the Flooding Pampas.

Lotus glaber Mill., a perennial leguminous forage plant endemic to Europe and introduced into Argentina in 1930, has adapted to the Flooding Pampas region’s ecological characteristics, successfully becoming part of the native vegetation. Given its high nutritional value and the fact that it does not cause meteorism, this plant has great potential for increasing the productivity and quality of the Pampas grazing land. In Azul, a Buenos Aires province, an L. glaber genetic improvement programme resulted in an induced autotetraploid population called Leonel, through the use of colchicine. In addition to preserving the species’ valuable characteristics, this population has a series of significant morphological modifications compared to diploid populations. This paper compares the Leonel population in terms of the length and width of its central foliolae (LCF and WCF), its area (A) and the length/width ratio (L/W) of the central foliolae of the first expanded leaf from the apex, during the following seasons: mid-winter (1), late winter (2), and late spring (3) of 2006. Variance analyses, in a completely random sample with one factor, seasons, carried out for each variable, detected significant differences (1%) in all cases. Analysis of the L/W ratio has made it possible to determine the foliolae’s forms. During the winter period, seasons 1 and 2, the foliolae took a narrowly obovate shape, while in the spring they had an oblanceaolate shape. The results obtained lead to the conclusion that there is a high level of variation in length and width, as well as the foliolae’s shape and size.