Study in Fractional Differential Equations by Means of Topological Degree Methods

In this article, we study some class of fractional differential equations involving the Caputo fractional derivative. By using a fixed point theorem on topological degree for condensing maps via a priori estimate method, some sufficient conditions for the existence of solutions are presented. Uniqueness and data dependence results of solutions are also deduced.     

Significance of data treatment and experimental setup on the determination of copper complexing parameters by anodic stripping voltammetry

Different procedures of voltammetric peak intensities determination, as well as various experimental setups were systematically tested on simulated and real experimental data in order to identify critical points in the determination of copper complexation parameters (ligand concentration and conditional stability constant) by anodic stripping voltammetry (ASV). Varieties of titration data sets (Cu_measuredvs. Cu_total) were fitted by models encompassing discrete sites distribution of one-class and two-class of binding ligands (by PROSECE software). Examination of different procedures for peak intensities determination applied on voltammograms with known preset values revealed that tangent fit (TF) routine should be avoided, as for both simulated and experimental titration data it produced an additional class of strong ligand (actually not present). Peak intensities determination by fitting of the whole voltammogram was found to be the most appropriate, as it provided most reliable complexation parameters.Tests performed on real seawater samples under different experimental conditions revealed that in addition to importance of proper peak intensities determination, an accumulation time (control of the sensitivity) and an equilibration time needed for complete complexation of added copper during titration (control of complexation kinetics) are the keypoints to obtain reliable results free of artefacts.The consequence of overestimation and underestimation of complexing parameters is supported and illustrated by the example of free copper concentrations (the most bioavailable/toxic specie) calculated for all studied cases. Errors up to 80% of underestimation of free copper concentration and almost two orders of magnitude overestimation of conditional stability constant were registered for the simulated case with two ligands.     

Density,viscosity, vapour–liquid equilibrium,excess molar enthalpy,and their correlations of the binary system [1-pentanol + R-(+)-limonene] over the complete concentration range,at different temperatures

Density and viscosity measurements in the T = (293.15–373.15) K range of pure 1-pentanol, R-(+)-limonene, as well as of the binary system {x₁ 1-pentanol + (1 − x₁) limonene} over the whole concentration range were made. The experimental results were fitted to empirical equations, which permit the calculation of these properties in the studied temperature range. Calculated values are in agreement with the experimental ones. Data of the binary mixtures were further used to calculate the excess molar volume and viscosity deviations. Excess enthalpy at 303 K and vapour–liquid equilibrium measurements in the T = (328.15–343.15) K range were also obtained for the binary system. These last experimental results were used to calculate activity coefficients and the excess molar Gibbs energy. This binary system exhibits a maximum pressure azeotrope. Excess or deviation properties were fitted to the Redlich–Kister polynomial relation to obtain their coefficients and standard deviations. Vapour pressure of 1-pentanol over the P = (2.3–95.1) kPa range were also measured. Furthermore, functional relationships between the total pressure and the mole fraction of 1-pentanol with the temperature of the azeotropic point were also deduced. These equations are useful to calculate the azeotropic point coordinates in the temperature and pressure ranges studied in this work.     

(Liquid + liquid) equilibria of the quaternary system methanol + isooctane + cyclohexane + benzene at T = 303.15 K

Tie line data of the ternary system {methanol + isooctane + cyclohexane} were obtained at T = 303.15 K. A quaternary system containing these three compounds and benzene was also studied at the same temperature, while data for {methanol + benzene + cyclohexane} and {methanol + benzene + isooctane} were taken from literature. In order to obtain the binodal surface of the quaternary system, four quaternary sectional planes with several cyclohexane/isooctane ratios were studied. The distribution of benzene between both phases was also analysed. Ternary experimental results were correlated with the UNIQUAC and NRTL equations and compared with predictions using the UNIFAC group contribution method.     

Scaled unscented transform Gaussian sum filter: Theory and application

In this work we consider the state estimation problem in nonlinear/non-Gaussian systems. We introduce a framework, called the scaled unscented transform Gaussian sum filter (SUT-GSF), which combines two ideas: the scaled unscented Kalman filter (SUKF) based on the concept of scaled unscented transform (SUT) (Julier and Uhlmann (2004) [16]), and the Gaussian mixture model (GMM). The SUT is used to approximate the mean and covariance of a Gaussian random variable which is transformed by a nonlinear function, while the GMM is adopted to approximate the probability density function (pdf) of a random variable through a set of Gaussian distributions. With these two tools, a framework can be set up to assimilate nonlinear systems in a recursive way. Within this framework, one can treat a nonlinear stochastic system as a mixture model of a set of sub-systems, each of which takes the form of a nonlinear system driven by a known Gaussian random process. Then, for each sub-system, one applies the SUKF to estimate the mean and covariance of the underlying Gaussian random variable transformed by the nonlinear governing equations of the sub-system. Incorporating the estimations of the sub-systems into the GMM gives an explicit (approximate) form of the pdf, which can be regarded as a “complete” solution to the state estimation problem, as all of the statistical information of interest can be obtained from the explicit form of the pdf (Arulampalam et al. (2002) [7]).In applications, a potential problem of a Gaussian sum filter is that the number of Gaussian distributions may increase very rapidly. To this end, we also propose an auxiliary algorithm to conduct pdf re-approximation so that the number of Gaussian distributions can be reduced. With the auxiliary algorithm, in principle the SUT-GSF can achieve almost the same computational speed as the SUKF if the SUT-GSF is implemented in parallel.As an example, we will use the SUT-GSF to assimilate a 40-dimensional system due to Lorenz and Emanuel (1998) [27]. We will present the details of implementing the SUT-GSF and examine the effects of filter parameters on the performance of the SUT-GSF.     

Phase retrieval in optical fiber modal interference for structural health monitoring

A method based on data dependent system (DDS) for extraction of phase in fiber modal interference is presented. The interference patterns of LP₀₁ & LP₁₁, LP₀₁ & LP₀₂ and LP₀₆ &LP₀₇ within the fiber have been recorded under different launching conditions. The patterns were characterized by means of autoregressive model and the self coherence functions of the corresponding interferogram were determined. It would provide the phase distribution of the pattern and the modulation of group delay due to the measurand. An application has been made for measuring strain in a simply supported beam under different loading conditions. Results are presented for the applied strain in the range of 270-1500 μ strain.     

Combining orthogonal signal correction and wavelet packet transform with partial least squares to analyze overlapping spectra of three kinds of metal ions

A novel method named OSC-WPT-PLS approach based on partial least squares (PLS) regression with orthogonal signal correction (OSC) and wavelet packet transform (WPT) as pre-processed tools was proposed for the simultaneous spectrophotometric determination of Al(III), Mn(II) and Co(II). This method combines the ideas of OSC and WPT with PLS regression for enhancing the ability of extracting characteristic information and the quality of regression. OSC is used to remove information in the response matrix D by subtracting the structured noise that is orthogonal to the concentration matrix C. Wavelet packet transform was applied to perform data compression, to extract relevant information, and to eliminate noise and collinearity. PLS was applied for multivariate calibration and noise reduction by eliminating the less important latent variables. In this case, using trials, the kind of wavelet function, the decomposition level, the number of OSC components and the number of PLS factors for the OSC-WPT-PLS method were selected as Daubechies 4, 3, 2 and 3, respectively. A program (POSCWPTPLS) was designed to perform the simultaneous spectrophotometric determination of Al(III), Mn(II) and Co(II). The relative standard errors of prediction (RSEP) obtained for total elements using OSC-WPT-PLS, WPT-PLS and PLS were compared. Experimental results demonstrated that the OSC-WPT-PLS method had the best performance among the three methods and was successful even when there was severe overlap of spectra.     

Capture of asymmetric top dipolar molecules by ions: Rate constants for capture of H2O, HDO, and D2O by arbitrary ions

The capture of rotationally state-selected and unselected asymmetric top polar molecules by ions is investigated. Analytical expressions (for all rotational states up to j = 2) of capture rate constants in the perturbed-rotor second-order limit are derived for application to low temperature conditions. Approximate analytical representations over wider temperature ranges are also given for rotationally unselected molecules. The capture of H₂O, D₂O, and HDO by arbitrary ions is chosen for demonstration of the approach. Capture rate constants for the about 60 reactions of H₂O with ions listed in the UMIST 2006 data base for astrochemistry are calculated, compared with experimental data, and represented in the format k_cap(T) ≈ c₁ + c₂(T/300 K)^−1/2. The parameters c₁ and c₂ can be predicted in a very simple way. The approach allows one to identify capture-controlled mechanisms and/or to trace experimental artifacts. The approach applies equally well to the capture of symmetric top and linear dipole molecules by arbitrary ions.     

Developing an efficient knowledge discovering model for mining fuzzy multi-level sequential patterns in sequence databases

Sequential pattern mining from sequence databases has been recognized as an important data mining problem with various applications. Items in a sequence database can be organized into a concept hierarchy according to taxonomy. Based on the hierarchy, sequential patterns can be found not only at the leaf nodes (individual items) of the hierarchy, but also at higher levels of the hierarchy; this is called multiple-level sequential pattern mining. In previous research, taxonomies based on crisp relationships between any two disjointed levels, however, cannot handle the uncertainties and fuzziness in real life. For example, Tomatoes could be classified into the Fruit category, but could be also regarded as the Vegetable category. To deal with the fuzzy nature of taxonomy, Chen and Huang developed a novel knowledge discovering model to mine fuzzy multi-level sequential patterns, where the relationships from one level to another can be represented by a value between 0 and 1. In their work, a generalized sequential patterns (GSP)-like algorithm was developed to find fuzzy multi-level sequential patterns. This algorithm, however, faces a difficult problem since the mining process may have to generate and examine a huge set of combinatorial subsequences and requires multiple scans of the database. In this paper, we propose a new efficient algorithm to mine this type of pattern based on the divide-and-conquer strategy. In addition, another efficient algorithm is developed to discover fuzzy cross-level sequential patterns. Since the proposed algorithm greatly reduces the candidate subsequence generation efforts, the performance is improved significantly. Experiments show that the proposed algorithm is much more efficient and scalable than the previous one. In mining real-life databases, our works enhance the model's practicability and could promote more applications in business.