Simple Equations Method (SEsM): Algorithm,Connection with Hirota Method,Inverse Scattering Transform Method,and Several Other Methods

The goal of this article is to discuss the Simple Equations Method (SEsM) for obtaining exact solutions of nonlinear partial differential equations and to show that several well-known methods for obtaining exact solutions of such equations are connected to SEsM. In more detail, we show that the Hirota method is connected to a particular case of SEsM for a specific form of the function from Step 2 of SEsM and for simple equations of the kinds of differential equations for exponential functions. We illustrate this particular case of SEsM by obtaining the three- soliton solution of the Korteweg-de Vries equation, two-soliton solution of the nonlinear Schrödinger equation, and the soliton solution of the Ishimori equation for the spin dynamics of ferromagnetic materials. Then we show that a particular case of SEsM can be used in order to reproduce the methodology of the inverse scattering transform method for the case of the Burgers equation and Korteweg-de Vries equation. This particular case is connected to use of a specific case of Step 2 of SEsM. This step is connected to: (i) representation of the solution of the solved nonlinear partial differential equation as expansion as power series containing powers of a “small” parameter ϵ; (ii) solving the differential equations arising from this representation by means of Fourier series, and (iii) transition from the obtained solution for small values of ϵ to solution for arbitrary finite values of ϵ. Finally, we show that the much-used homogeneous balance method, extended homogeneous balance method, auxiliary equation method, Jacobi elliptic function expansion method, F-expansion method, modified simple equation method, trial function method and first integral method are connected to particular cases of SEsM.     

Local Sensitivity Analysis in Estimation Problems

This article deals with the problem of local sensitivity analysis, that is, how sensitive are the results of a statistical analysis to changes in the data? A general methodology of sensitivity analysis is applied to some statistical problems. The proposed methods are applicable to any statistical problem that can be expressed as an optimization problem or that involves solving a system of equations. As some particular examples, the methodology is applied to the maximum likelihood method, the standard and constrained methods of moments and the standard and constrained probability weighted moments methods. Unlike the standard method of moments, the constrained method of moments ensures that the obtained estimates are always consistent with the data. Closed analytical formulas for the calculation of these local sensitivities are derived. The obtained sensitivities include: (a) the objective function sensitivities to data points and (b) the sensitivities of the estimated parameters to data points. The derived formulas for the sensitivities are based on recent results in the area of mathematical programming. Several examples of parameter estimation problems are used to illustrate the methodology.     

Rapid simulation procedure for fretting wear on the basis of the method of dimensionality reduction

We suggest a numerical procedure for rapid simulation of fretting wear in a contact of two bodies subjected to tangential oscillations with a small amplitude. The incremental wear in each point of contact area is calculated using the Reye–Archard–Khrushchov wear criterion. For applying this criterion, the distributions of pressure and relative displacements of bodies are required. These are calculated using the method of dimensionality reduction (MDR).     

Path-following barrier and penalty methods for linearly constrained problems

In the present paper some barrier and penalty methods (e.g. logarithmic barriers, SUMT, exponential penalties), which define a continuously differentiable primal and dual path, applied to linearly constrained convex problems are studied, in particular, the radius of convergence of Newton’s method depending on the barrier and penalty para-meter is estimated, Unlike using self-concordance properties the convergence bounds are derived by direct estimations of the solutions of the Newton equations. The obtained results establish parameter selection rules which guarantee the overall convergence of the considered barrier and penalty techniques with only a finite number of Newton steps at each parameter level. Moreover, the obtained estimates support scaling method which uses approximate dual multipliers as available in barrier and penalty methods     

Rapid determination of catecholamines in urine samples by nonaqueous microchip electrophoresis with LIF detection

A method was developed for the rapid separation of catecholamines by nonaqueous microchip electrophoresis (NAMCE) with LIF detection, A homemade pump‐free negative pressure sampling device was used for rapid bias‐free sampling in NAMCE, the injection time was 0.5 s and the electrophoresis separation conditions were optimized. Under the optimized conditions, the samples were separated completely in <1 min. The average migration times of the epinephrine (E), dopamine (DA), and norepinephrine (NE) were 34.26, 43.81, and 50.07 s, with an RSD of 1.05, 1.26, and 0.89% (n = 7), respectively. The linearity of the method ranged from 0.0125 to 2.0 mg/L for E and 0.025～4.0 mg/L for DA and NE, with correlation coefficients ranging between 0.9978 and 0.9986. The detection limits of E, DA, and NE were 2.5, 5.0, and 5.0 μg/L, respectively. The recoveries of E, DA, and NE in spiked urine samples were between 86 and 103%, with RSDs of 4.5～6.8% (n = 5). The proposed NAMCE with LIF detection combined with a pump‐free negative pressure sampling device is a simple, inexpensive, energy efficient, miniaturized system that can be successfully applied for the determination of catecholamines in urine samples.     

Development of a sensitive and reliable high performance liquid chromatography method with fluorescence detection for high-throughput analysis of multi-class mycotoxins in Coix seed

As an edible and medicinal plant, Coix seed is readily contaminated by more than one group of mycotoxins resulting in potential risk to human health. A reliable and sensitive method has been developed to determine seven mycotoxins (aflatoxins B₁, B₂, G₁, G₂, zearalenone, α-zearalenol, and β-zearalenol) simultaneously in 10 batches of Coix seed marketed in China. The method is based on a rapid ultrasound-assisted solid–liquid extraction (USLE) using methanol/water (80/20) followed by immunoaffinity column (IAC) clean-up, on-line photochemical derivatization (PCD), and high performance liquid chromatography coupled with fluorescence detection (HPLC-FLD). Careful optimization of extraction, clean-up, separation and detection conditions was accomplished to increase sample throughput and to attain rapid separation and sensitive detection. Method validation was performed by analyzing samples spiked at three different concentrations for the seven mycotoxins. Recoveries were from 73.5% to 107.3%, with relative standard deviations (RSDs) lower than 7.7%. The intra- and inter-day precisions, expressed as RSDs, were lower than 4% for all studied analytes. Limits of detection and quantification ranged from 0.01 to 50.2 μg kg⁻¹, and from 0.04 to 125.5 μg kg⁻¹, respectively, which were below the tolerance levels for mycotoxins set by the European Union. Samples that tested positive were further analyzed by HPLC tandem electrospray ionization mass spectrometry for confirmatory purposes. This is the first application of USLE-IAC-HPLC-PCD-FLD for detecting the occurrence of multi-class mycotoxins in Coix seed.     

Ammonium Determination in Water Samples by Using Opa-Nac Reagent: A Comparative Study with Nessler and Ammonium Selective Electrode Methods

A selective and sensitive method based on the ammonium derivatisation with o -phthaldialdehyde (OPA) and N -acetyl-cysteine (NAC) has been developed for ammonium determination in real water samples. The proposed procedure has been compared with ammonium reference methods such as Nessler reagent method and ammonium selective electrode. All procedures have been chemometrically tested and compared in terms of the main analytical properties. These procedures have been used to determine ammonium in unknown water samples. The OPA-NAC reagent method does not present any systematic error (proportional or constant), while Nessler reagent presents both of them for some samples assayed. The ammonium selective electrode is free of corrigible systematic errors, however presents amine interference. The OPA-NAC ammonium method is able to achieve a detection limit (LOD) of 0.07 mg/L in the sample, with a linear dynamic range up to 1.4 mg/L of ammonium.     

Development and validation of high-performance liquid chromatography method for analysis of β-CCT and its related substances

Compound 2β-carbomethoxy-3β-(4-chlorophenyl)tropane (β-CCT) is a key intermediate for the synthesis of some clinical dopamine transporter (DAT) imaging agents. Potential impurities from synthesis process of β-CCT and degradation during storage might have detrimental effect on the final imaging agents. Thus, it is necessary to guarantee the quality of β-CCT. In this study, a rapid, sensitive and accurate high-performance liquid chromatography (HPLC) method was developed and validated for the analysis of β-CCT and its related substances. The chromatographic separation was achieved on a reverse-phase phenomenex? Gemini C18 column with an isocratic mobile phase consisted of methanol, water and TFA (30:70:0.1 v/v/v). The flow rate was 1.0 mL/min at 30 °C and samples were monitored at 220 nm. The method was validated concerning system suitability, linearity, accuracy, precision, specificity, robustness and stability. The limit of detection (LOD) and the limit of quantification (LOQ) of β-CCT were 0.5 and 1.5 μg/mL, respectively. The linearity range of β-CCT was 1.5–450 μg/mL with a good linear correlation coefficient (R² = 0.9999) between the peak response and concentration. Specificity investigation through forced degradation experiments displayed that β-CCT was stable in acidic, thermal and photolytic degradation conditions, but significantly unstable in alkaline and oxidative conditions. With the developed chromatographic method, possible impurity α-CCT from synthetic process and potential degradation products could be well separated from β-CCT. Good recovery and precision were manifested in the assay method. These results indicated that the present method would be suitable for not only the quality assurance of β-CCT in regular production sample assays but also the monitoring and determination of its related substances.     

Development of GC–MS/MS method for environmental monitoring of 49 pesticide residues in food commodities in Al-Rass,Al-Qassim region,Saudi Arabia

The harmful effects of pesticide residues are a threat to our health. Therefore, the current study aimed to validate a simple method for the determination of pesticide residues in commonly consumed fruits and vegetables from Al-Rass, Al-Qassim region, Saudi Arabia. A total of 1430 samples were collected from a local market and then analyzed for monitoring of 49 pesticide residues. A quick, easy, cheap, effective, rugged, and safe (QuEChERS) multi-residue extraction method followed by gas chromatography equipped with triple-quadrupole mass spectrometry (GC–MS/MS) was successfully implemented. This 17-min-run analytical method detects and quantifies pesticide residues with acceptable validation performance parameters in terms of sensitivity, selectivity, linearity, the limit of quantification, accuracy, and precision. The linear range of the calibration curves ranged from 10 to 300 µg/L, all the pesticide LODs ranged from 0.0005 to 0.0024 mg/kg, and the pesticide LOQs ranged from 0.0011 to 0.0047 mg/kg. The recovery values at the three fortification levels ranged from 78 % to 107 %, and the precision values (expressed as RSD%) were less than 20 % for all of the investigated analytes. The results showed that 138 (9.65 %) of the analyzed samples were contaminated with pesticide residues, 40 (2.80 %) of the analyzed samples exceeded the maximum residue limit (MRL) of the European Commission regulations (EC) for pesticides residues, 98 (6.85 %) of the analyzed samples were contaminated with residues below the MRL, and 1292 (90.35 %) of the analyzed samples were pesticide residue-free. Coriander contained the highest percentage (46.88 %) of pesticide residues, particularly tetradifon that representing 18.75 % noncompliance with the MRL, followed by parsley, with 20.59 % pesticide residues (10.29 % non-compliance). Multiple pesticide residues were observed most frequently in tomatoes and dates which were contaminated with buprofezin and ethion respectively.