Fractional derivatives: probability interpretation and frequency response of rational approximations

The theory of fractional calculus (FC) is a useful mathematical tool in many applied sciences. Nevertheless, only in the last decades researchers were motivated for the adoption of the FC concepts. There are several reasons for this state of affairs, namely the co-existence of different definitions and interpretations, and the necessity of approximation methods for the real time calculation of fractional derivatives (FDs). In a first part, this paper introduces a probabilistic interpretation of the fractional derivative based on the Grünwald-Letnikov definition. In a second part, the calculation of fractional derivatives through Padé fraction approximations is analyzed. It is observed that the probabilistic interpretation and the frequency response of fraction approximations of FDs reveal a clear correlation between both concepts.

Algoritmos genéticos: aplicação na síntese de alguns algoritmos de controlo

Mestrado em Engenharia Electrotécnica e de Computadores. Área de Especialização de Automação e Sistemas.

Optimal approximation of fractional derivatives through discrete-time fractions using genetic algorithms

This study addresses the optimization of rational fraction approximations for the discrete-time calculation of fractional derivatives. The article starts by analyzing the standard techniques based on Taylor series and Padé expansions. In a second phase the paper re-evaluates the problem in an optimization perspective by tacking advantage of the flexibility of the genetic algorithms.

Continued fraction solution for the radiative transfer equation in three dimensions

Starting from the radiative transfer equation, we obtain an analytical solution for both the free propagator along one of the axes and an arbitrary phase function in the Fourier-Laplace domain. We also find the effective absorption parameter, which turns out to be very different from the one provided by the diffusion approximation. We finally present an analytical approximation procedure and obtain a differential equation that accurately reproduces the transport process. We test our approximations by means of simulations that use the Henyey-Greenstein phase function with very satisfactory results.

Optimization of Rational Approximations by Continued Fractions

The paper has been presented at the 12th International Conference on Applications of Computer Algebra, Varna, Bulgaria, June, 2006.

On the Wick product and Wong-Zakai approximations.

This thesis is a compilation of 6 papers that the author has written together with Alberto Lanconelli (chapters 3, 5 and 8) and Hyun-Jung Kim (ch 7). The logic thread that link all these chapters together is the interest to analyze and approximate the solutions of certain stochastic differential equations using the so called Wick product as the basic tool. In the first chapter we present arguably the most important achievement of this thesis; namely the generalization to multiple dimensions of a Wick-Wong-Zakai approximation theorem proposed by Hu and Oksendal. By exploiting the relationship between the Wick product and the Malliavin derivative we propose an original reduction method which allows us to approximate semi-linear systems of stochastic differential equations of the Itô type. Furthermore in chapter 4 we present a non-trivial extension of the aforementioned results to the case in which the system of stochastic differential equations are driven by a multi-dimensional fraction Brownian motion with Hurst parameter bigger than 1/2. In chapter 5 we employ our approach and present a “short time” approximation for the solution of the Zakai equation from non-linear filtering theory and provide an estimation of the speed of convergence. In chapters 6 and 7 we study some properties of the unique mild solution for the Stochastic Heat Equation driven by spatial white noise of the Wick-Skorohod type. In particular by means of our reduction method we obtain an alternative derivation of the Feynman-Kac representation for the solution, we find its optimal Hölder regularity in time and space and present a Feynman-Kac-type closed form for its spatial derivative. Chapter 8 treats a somewhat different topic; in particular we investigate some probabilistic aspects of the unique global strong solution of a two dimensional system of semi-linear stochastic differential equations describing a predator-prey model perturbed by Gaussian noise.

Gastroprotective Effects (in Rodents) Of A Flavonoid Rich Fraction Obtained From Syngonanthus Macrolepsis.

Syngonanthus macrolepis, popularly known in Brazil as 'sempre-vivas', is a plant from the family Eriocaulaceae, it is found in the states of Minas Gerais and Bahia. The species contains a variety of constituents, including flavonoids with gastroprotective effect. In this work, a flavonoid-rich fraction (Sm-FRF) obtained from scapes of S. macrolepis was investigated for preventing gastric ulceration in mice and rats. The activity was evaluated in models of induced gastric ulcer (absolute ethanol, stress, non-steroidal anti-inflammatory drugs and pylorus ligation). The cytoprotective mechanisms of the Sm-FRF in relation to sulfhydryl (SH) groups, nitric oxide (NO) and antioxidant enzymes were also evaluated. The Sm-FRF (100 mg/kg, p.o.) significantly reduced gastric injury in all models, and did not alter gastric juice parameters after pylorus ligation. The results indicate significant gastroprotective activity for the Sm-FRF, which probably involves the participation of both SH groups and the antioxidant system. Both are integral parts of the gastrointestinal mucosa's cytoprotective mechanisms against aggressive factors.

Productive performance and milk protein fraction composition of dairy cows supplemented with sodium monensin

The objective of this work was to evaluate the levels of sodium monensin on lactating cows and their effects on productive performance and milk protein fraction composition. It was used 12 Holstein cows, distributed in four balanced 3 × 3 Latin squares, and fed three diets: one control without monensin, and two diets with monensin at the levels of 24 or 48 mg/kg DM added to the concentrate. Milk production was daily measured throughout the entire experimental period. The samples used for analysis of milk composition were collected on two alternated days from the two daily milking. Non-protein nitrogen, total nitrogen and non-casein nitrogen contents were directly evaluated in the milk, and casein, whey protein and true protein contents were indirectly determined. The use of monensin in the rations reduced dry matter and nutrient intake, especially when diet with 48 mg/kg of dry matter was given. The ration with 24 mg/kg of DM increased milk production, with or without correction, and also fat and lactose yield, and it improved productive efficiency. The levels of monensin in the ratios did not influence contents of milk crude protein, non-protein nitrogen, non-casein nitrogen, true protein, casein, casein/true protein ratio, whey protein, and of all those fractions expressed as percentage of crude protein. The utilization of monensin in the ratio at the dose of 24 mg/kg of DM influences positively the productive performance of lactating cows, and it does not influence the composition of milk protein fractions.

Productive performance and composition of milk protein fraction in dairy cows supplemented with fat sources

The objective of this study was to evaluate the use of fat sources in rations for lactating cows on the productive performance and composition of milk protein fraction. Twelve Holstein cows were used, grouped in three balanced 4 × 4 Latin squares, fed with the following rations: control; refined soybean oil; whole raw soybean; and calcium salts of unsaturated fatty acid (Megalac-E). Dry matter and nutrient intake, and daily milk production were evaluated. The samples used to analyze milk composition were collected in two alternate days and were obtained from two daily milking. Milk composition and total nitrogen, non-protein nitrogen and non-casein nitrogen ratios were analyzed. The casein, serum protein and true protein ratios were obtained by difference. Dry matter and nutrient intakes were lower when cows received the diet containing calcium salts of fatty acids, in relation to the control diet. Among the diets with fat sources, the one with whole raw soybean and calcium salts decreased milk production. There was no effect of fat sources added to the diet on crude protein, non-protein nitrogen, non-casein nitrogen, true protein, casein, casein/milk true protein ratio and serum protein. Similarly, the experimental diets did not influence the protein fractions when expressed in percentage of milk crude protein. The utilization of fat sources in diets changes milk production and composition of lactating cows, but does not influence the composition of milk protein fractions.

Bioassay guided purification of the antimicrobial fraction of a Brazilian propolis from Bahia state

Background: Brazilian propolis type 6 (Atlantic forest, Bahia) is distinct from the other types of propolis especially due to absence of flavonoids and presence of other non-polar, long chain compounds, but presenting good in vitro and in vivo antimicrobial activity. Several authors have suggested that fatty acids found in this propolis might be responsible for its antimicrobial activity; however, so far no evidence concerning this finding has been reported in the literature. The goals of this study were to evaluate the antibacterial activity of the main pure fatty acids in the ethanolic extract and fractions and elucidate the chemical nature of the bioactive compounds isolated from Brazilian propolis type 6. Methods: Brazilian propolis type 6 ethanolic extract (EEP), hexane fraction (H-Fr), major fatty acids, and isolated sub-fractions were analyzed using high performance liquid chromatography (HPLC), high resolution gas chromatography with flame ionization detection (HRGC-FID), and gas chromatography-mass spectrometry (GC-MS). Three sub-fractions of H-Fr were obtained through preparative HPLC. Antimicrobial activity of EEP, H-Fr, sub-fractions, and fatty acids were tested against Staphyloccus aureus ATCC 25923 and Streptococcus mutans Ingbritt 1600 using minimum inhibitory concentration (MIC) and minimum bactericidal concentration (MBC). Results: EEP and H-Fr inhibited the growth of the microorganisms tested; nevertheless, no antimicrobial activity was found for the major fatty acids. The three sub-fractions (1, 2, and 3) were isolated from H-Fr by preparative HPLC and only sub-fraction 1 showed antimicrobial activity. Conclusion: a) The major fatty acids tested were not responsible for the antimicrobial activity of propolis type 6; b) Sub-fraction 1, belonging to the benzophenone class, was responsible for the antimicrobial activity observed in the present study. The identification of the bioactive compound will improve the development of more efficient uses of this natural product.

Assessing influence in survival data with a cure fraction and covariates

Diagnostic methods have been an important tool in regression analysis to detect anomalies, such as departures from error assumptions and the presence of outliers and influential observations with the fitted models. Assuming censored data, we considered a classical analysis and Bayesian analysis assuming no informative priors for the parameters of the model with a cure fraction. A Bayesian approach was considered by using Markov Chain Monte Carlo Methods with Metropolis-Hasting algorithms steps to obtain the posterior summaries of interest. Some influence methods, such as the local influence, total local influence of an individual, local influence on predictions and generalized leverage were derived, analyzed and discussed in survival data with a cure fraction and covariates. The relevance of the approach was illustrated with a real data set, where it is shown that, by removing the most influential observations, the decision about which model best fits the data is changed.

Physical approximations for the nonlinear evolution of perturbations in inhomogeneous dark energy scenarios

The abundance and distribution of collapsed objects such as galaxy clusters will become an important tool to investigate the nature of dark energy and dark matter. Number counts of very massive objects are sensitive not only to the equation of state of dark energy, which parametrizes the smooth component of its pressure, but also to the sound speed of dark energy, which determines the amount of pressure in inhomogeneous and collapsed structures. Since the evolution of these structures must be followed well into the nonlinear regime, and a fully relativistic framework for this regime does not exist yet, we compare two approximate schemes: the widely used spherical collapse model and the pseudo-Newtonian approach. We show that both approximation schemes convey identical equations for the density contrast, when the pressure perturbation of dark energy is parametrized in terms of an effective sound speed. We also make a comparison of these approximate approaches to general relativity in the linearized regime, which lends some support to the approximations.

Local approximations for polarization potentials

We discuss the derivation of an equivalent polarization potential independent of angular momentum l for use in the optical Schrodinger equation that describes the elastic scattering of heavy ions. Three different methods are used for this purpose. Application of our theory to the low energy scattering of light heavy-ion systems at near-barrier energies is made. It is found that the notion of an l-independent polarization potential has some validity but cannot be a good substitute for the l-dependent local equivalent Feshbach polarization potential.

A novel use for sugarcane bagasse hemicellulosic fraction: Xylitol enzymatic production

The excess of sugarcane bagasse (SCB) from the sugar-alcohol industry is considered a by-product with great potential for many bioproducts production. This work had as objective to verify the performance of sugarcane bagasse hemicellulosic hydrolysate (SCBHH) as source of sugars for enzymatic or in vitro xylitol production. For this purpose, xylitol enzymatic production was evaluated using different concentrations of treated SCBHH in the commercial reaction media. The weak acid hydrolysis of SCB provided a hydrolysate with 18 g L(-1) and 6 g L(-1) of xylose and glucose, respectively. Considering the reactions, changes at xylose xylitol conversion efficiency and volumetric productivity in xylitol were not observed for the control experiment and using 20 and 40% v.v (1) of SCBHH in the reaction media. The conversion efficiency achieved 100% in all the experiments tested. The results showed that treated SCBHH is suitable as xylose and glucose source for the enzymatic xylitol production and that this process has potential as an alternative for traditional xylitol production ways. (C) 2011 Published by Elsevier Ltd.

Phase fraction distribution measurement of oil-water flow using a capacitance wire-mesh sensor

In this paper, a novel wire-mesh sensor based on permittivity (capacitance) measurements is applied to generate images of the phase fraction distribution and investigate the flow of viscous oil and water in a horizontal pipe. Phase fraction values were calculated from the raw data delivered by the wire-mesh sensor using different mixture permittivity models. Furthermore, these data were validated against quick-closing valve measurements. Investigated flow patterns were dispersion of oil in water (Do/w) and dispersion of oil in water and water in oil (Do/w&w/o). The Maxwell-Garnett mixing model is better suited for Dw/o and the logarithmic model for Do/w&w/o flow pattern. Images of the time-averaged cross-sectional oil fraction distribution along with axial slice images were used to visualize and disclose some details of the flow.