First-order Derivative Spectrophotometry for the Determination of Vitamin C in Medicament

A novel method for the determination of vitamin C（Vc） is proposed in this article. After the reaction with Folin-Ciocalteau reagent at ambient temperature, Vc solution was scanned at 750--1100 nm, and its first-order derivative spectrum were obtained from the original spectrum. The values of derivative selected at 995 nm were used for determination. It was proved that Vc could quickly react with Folin-Ciocalteau reagent within 5 min and the product was quite stable for a long time. The conditions required for this method is not very complicated, its precision and accuracy are similar to those of the iodometric titration described in Chinese Pharmacopoeia, and the limit of detection is 0.312 μg/mL. The determination of the results of vitamin C tablet, pill, and injection demonstrates that this method has wide pharmaceutical applications.

Estimation of the galanthamine using derivative spectrophotometry in bulk drug and formulation

Two simple, rapid, accurate, precise, reliable and economical spectrophotometric methods have been proposed for the determination of galanthamine hydrobromide (GH) in bulk and pharmaceutical formulation. First method is zero order UV spectrophotometry and second is 1st derivative zero crossing spectrophotometry. The developed methods have shown best results in terms of linearity, accuracy, precision, LOD and LOQ for bulk drugs and marketed formulations. Absorbance was measured at 287 nm for zero order and 277.4 nm for first derivative. It obeyed Lambert-Beer’s law in the range of 30-80 μg mL-1. Both methods have good linearity (r2 = 0.9997) and accuracy found to be 100.5% and 101.2% for both methods respectively.

Studies on the Combined Derivative Spectrophotometry

The combined derivative spectrophotometry developed in this paper is a new method, which can be used to determine multicomporent mixture simultaneously and may improve the sensitivity and accuracy of the measurement remarkably. Two practical tests were carried out to verify the correctness of the theory, and the results are satisfactory.

Application of derivative ratio spectrophotometry for determination of β-carotene and astaxanthin from Phaffia rhodozyma extract

A derivative ratio spectrophotometric method was used for the simultaneous determination of β-carotene and astaxanthin produced from Phaffia rhodozyma. Absorbencies of a series of the standard carotenoids in the range of 441 nm to 490 nm demonstrated that their absorptive spectra accorded with Beer’s law and that the additivity when the concentrations of β-carotene and astaxanthin and their mixture were within the range of 0 to 5 μg/ml, 0 to 6 μg/ml, and 0 to 6 μg/ml, respectively. When the wavelength interval (?λ) at 2 nm was selected to calculate the first derivative ratio spectra values, the first derivative amplitudes at 461 nm and 466 nm were suitable for quantitatively determining β-carotene and astaxanthin, respectively. Effect of divisor on derivative ratio spectra could be neglected; any concentration used as divisor in range of 1.0 to 4.0 μg/ml is ideal for calculating the derivative ratio spectra values of the two carotenoids. Calibration graphs were established for β-carotene within 0?6.0 μg/ml and for astaxanthin within 0?5.0 μg/ml with their corresponding regressive equations in: y=?0.0082x?0.0002 and y=0.0146x?0.0006, respectively. R-square values in excess of 0.999 indicated the good linearity of the calibration graphs. Sample recovery rates were found satisfactory (>99%) with relative standard deviations (RSD) of less than 5%. This method was suc- cessfully applied to simultaneous determination of β-carotene and astaxanthin in the laboratory-prepared mixtures and the extract from the Phaffia rhodozyma culture.

Spectrophotometric Determination of Water-Soluble Hexavalent Chromium and Determination of Total Hexavalent Chromium Content of Portland Cement in the Presence of Iron (III) and Titanium (IV) Using Derivative Ratio Spectrophotometry

A rapid, reliable and accurate method for the determination of hexavalent chromium in Portland cement is developed. The proposed method includes direct spectrophotometric determination of Cr (VI) in Portland cement with 1, 2, 5, 8 Tetrahydroxyanthraquinone, (Quinalizarin, QINZ) at pH 1.5. The European Directive (2003/53/EC) limits the use of cements so that it contains no more than 2 mg.Kg-1 of water-soluble Cr (VI). The absorbance at 565 nm due to Cr (VI)-QINZ complex is recommended for the determination of water-soluble Cr (VI) in Portland cement. The quantification of Cr (VI) released from cement when mixed with water is performed according to TRGS 613 (Technical Rules of Hazardous Substances). The validity of the method is thoroughly examined and the proposed method gives satisfactory results. A derivative spectrophotometric method has been developed for the determination of total Cr (VI) in Portland cement in the presence of Fe (III) and Ti (IV). The hexavalent chromium complex formed at pH 1.5 allows precise and accurate determination of chromium (VI) over the concentration range 0.05 to 3.0 mg.L-1of chromium (VI). The validity of the method was examined by analyzing several Standard Reference Material (SRM) Portland cement samples. The MDL (at 95% confidence level) was found to be 25 ng/mL for chromium (VI) in National Institute of Standards and Technology (NIST) cement samples using the proposed method.

DIRECT DETERMINATION OF NEODYMIUM AND ERBIUM IN RARE EARTH MIXTURE BY DERIVATIVE SPECTROPHOTOMETRY

The absorption spectra of 4f electron transitions of the complexes of neodymium and erbium with 8-hydroxyquinoline-5-sulphonic acid in the presence of diethylamine and ethanol have been measured by normal and third-derivative spectrophotometry. Their molar absorptivities are 70.7 l.mol^(-1).cm^(-1) for Nd and 62.5 l.mol^(-1).cm^(-1) for Er. They are 7.6 times and 14.9 times greater than those of corresponding chlorides, respectively. Use of the third-derivative spectra both eliminates the interference of Ce(Ⅳ) and increases the sensitivity for Nd and Er. Beer,s Law was obeyed from 0-10 ug/ml of Nd and Er. The method has been applied to the determination of neodymium and erbium in rare earth mixtures.

Serum paraquat concentration detected by spectrophotometry in patients with paraquat poisoning

BACKGROUND：Paraquat （PQ） is a world-wide used herbicide and also a type of common poison for suicide and accidental poisoning. Numerous studies have proved that the concentration of serum PQ plays an important role in prognosis. Spectrophotometry, including common spectrophotometry and second-derivative spectrophotometry, is commonly used for PQ detection in primary hospitals. So far, lack of systematic research on the reliability of the method and the correlation between clinical features of patients with PQ poisoning and the test results has restricted the clinical use of spectrophotometry. This study aimed to evaluate the reliability and value of spectrophotometry in detecting the concentration of serum PQ. METHODS：The wavelengths for detecting the concentration of serum PQ by common and second-derivative spectrophotometry were determined. Second-derivative spectrophotometry was applied to detect the concentration of serum PQ. The linear range and precision for detection of PQ concentration by this method were confirmed. The concentration of serum PQ shown by second- derivative spectrophotometry and HPLC were compared in 8 patients with PQ poisoning. Altogether 21 patients with acute poisoning 4 hours after PQ ingestion treated in the period of October 2008 to September 2010 were retrospectively reviewed. The patients were divided into higher and lower than 1.8 μg/mL groups based on their concentrations of serum PQ measured by second-derivative spectrophotometry on admission. The severity of clinical manifestations between the two groups were analyzed with Student＇s t test or Fisher＇s exact test. RESULTS：The absorption peak of 257 nm could not be found when common spectrophotometry was used to detect the PQ concentration in serum. The calibration curve in the 0.4-8.0 μg/mL range for PQ concentration shown by second-derivative spectrophotometry obeyed Beer＇s law with r=0.996. The average recovery rates of PQ were within a range of 95.0% to 99.5%, relative standard deviation （RSD） was within 1.35% to 5.41% （n=6）, and the lower detection limit was 0.05 μg/mL. The PQ concentrations in serum of 8 patients with PQ poisoning shown by second-derivative spectrophotometry were consistent with the quantitative determinations by HPLC （r=0.995, P〈0.0001）. The survival rate was 22.2% in patients whose PQ concentration in serum was more than 1.8 μg/mL, and the incidences of acidosis, oliguria and pneumomediastinum in these patients were 55.6%, 55.6% and 77.8%, respectively. These clinical manifestations were different significantly from those of the patients whose PQ concentration in serum was less than 1.8 pg/mL （P〈0.05）. CONCLUSIONS： For common spectrophotometry, the wavelength at 257 nm was not suitable for detecting serum PQ as no absorbance was shown. Second-derivative spectrophotometry was reliable for detecting serum paraquat concentration. Serum PQ concentration detected by second- derivative spectrophotometry could be used to predict the severity of clinical manifestations of patients with PQ poisoning, and PQ content higher than 1.8 tJg/mL 4 hours after ingestion could be an important predictive factor for poor prognosis.

Numerical Treatments for Crossover Cancer Model of Hybrid Variable-Order Fractional Derivatives

In this paper,two crossover hybrid variable-order derivatives of the cancer model are developed.Grünwald-Letnikov approximation is used to approximate the hybrid fractional and variable-order fractional operators.The existence,uniqueness,and stability of the proposed model are discussed.Adams Bashfourth’s fifth-step method with a hybrid variable-order fractional operator is developed to study the proposed models.Comparative studies with generalized fifth-order Runge-Kutta method are given.Numerical examples and comparative studies to verify the applicability of the used methods and to demonstrate the simplicity of these approximations are presented.We have showcased the efficiency of the proposed method and garnered robust empirical support for our theoretical findings.

A Dual Wavelength Differential First Derivative Spectrophotometric Method for Identification and Determination of Carbon Monoxide Generated During the Microsomal Metabolism of Xenobiotics in vitro

A dual wavelength differential first derivative spectrophotometric method has been developed to standardize the concentration of a saturated aqueous solution of carbon monoxide (CO) as the standard and to identify and to determine CO formed during the microsomal metabolism of xenobiotics in vitro. The method can significantly eliminate the background interference in the assay media and increase the quantitative accuracy and the sensitivity. There is a good linear relationship between CO concentration in the range of 2～10 μmol·L 1 CO and the distance D between the first derivative peak at 415 nm amd valley at 426 nm with r=0.9999(n=5),the regression equation being C (mmol·L 1 )=17.6D 0.4, the detection limit lower than 0.1 μmol·L 1 CO. The average recoveries of CO from the assay system and the sample were 102.1%, RSD=2.9% (n=7) and 79.7%, RSD=6.8% (n=12),respectively. The RSD of within day was 4.4%(n=18),and the RSD of day to day was 6.1%(n=16). By this method, four trihaloanilines and one trihalobenzene were tested, the results showed that only 2,4,5 trifluoroaniline could be converted to CO by the incubation with rat hepatic microsomes, NADPH and oxygen, the ability of phenobarbital or dexamethasone to induce rat hepatic microsomes to catalyze CO formation was 3 or 8 times higher than that of the control.

Terahertz spectrum clustering of traditional Chinese medicine based on first derivative characteristics

In order to deal with the unclear absorption peak caused by the absorption peak overlap of traditional Chinese medicine(TCM)and other mixtures,a method of three unsupervised clustering algorithms as K-means,K-medoids and Fuzzy C-means(FCM)combined with the first derivative characteristics of terahertz absorption spectrum,is proposed to perform the terahertz spectra clustering of Sanchi and other three kinds of TCM compared with their easily-confused products(ECPs).These three unsupervised clustering methods complement the scope of the supervised learning classification method.The first derivative of the spectrum could amplify the difference in the absorption coefficient with different substances,so that the obvious absorption peak can be revealed.Experiments shows that these three clustering algorithms can achieve good results by combining the origin absorption coefficient with its first-order derivative as the characteristic data,and among which K-means does the best with the accuracy of95.32%.Compared with pure absorption coefficient data clustering,the accuracy in this study has been significantly improved,especially for the non-absorption-peak TCM classification.And the accuracy of K-means algorithm is improved by5.38%.Besides,clustering algorithms in this study have strong anti-interference ability to the error data.

A Method to Improve First Order Approximation of Smoothed Particle Hydrodynamics

Smoothed particle hydrodynamics (SPH) is a useful meshless method.The first and second orders are the most popular derivatives of the field function in the mechanical governing equations.New methods were proposed to improve accuracy of SPH approximation by the lemma proved.The lemma describes the relationship of functions and their SPH approximation.Finally,the error comparison of SPH method with or without our improvement was carried out.

Application of the Ion Pair of 2-(5-Bromo-2-pyridylazo)-5-diethylaminophenol and Ammonium Tetraphenylborate for Preconcentration of Trace Vanadium and Determination by Third Derivative Spectrophotometry

Vanadium is quantitatively retained on 2-(5-bromo-2-pyridylazo)-5-diethylaminophenol (5-Br-PADAP)-ammonium tetraphenylborate with microcrystalline naphthalene or by a column method in the pH range of 2.1–4.5 from a large volume of aqueous solutions of various samples. After filtration, the solid mass consisting of the vanadium complex and naphthalene was dissolved with 5 mL of dimethylformamide and the metal was determined by the third derivative spectrophotometry. Vanadium complex can alternatively be quantitatively adsorbed on ammonium tetraphenylborate-naphthalene adsorbent packed in a column and determined similarly. About 0.05 μg of vanadium can be concentrated in a column from 250 mL of aqueous sample, where its concentration is as low as 0.2 ng/mL. The interference of a large number of anions and cations has been studied and the optimized conditions developed were utilized for the determination of trace amount of vanadium in various samples.

On the Variational Problems of the Functionals with Derivatives of Higher Orders and Undetermined Boundary

According to the necessary condition of the functional taking the extremum, that is its first variation is equal to zero, the variational problems of the functionals for the undetermined boundary in the calculus of variations are researched, the functionals depend on single argument, arbitrary unknown functions and their derivatives of higher orders. A new view point is posed and demonstrated, i.e. when the first variation of the functional is equal to zero, all the variational terms are not independent to each other, and at least one of them is equal to zero. Some theorems and corollaries of the variational problems of the functionals are obtained.

The Normal Meromorphic Functions Family Concerning Higher Order Derivative and Shared Set by One-Way

Let F be a meromorphic functions family on the unit disc Δ, If for every (the zeros of f is a multiplicity of at least k) and if then and ( ), then F is normal on Δ.

Extremum of second-order directional derivatives

In this paper, the extremum of second-order directional derivatives, i.e. the gradient of first-order derivatives is discussed. Given second-order directional derivatives in three nonparallel directions, or given second-order directional derivatives and mixed directional derivatives in two nonparallel directions, the formulae for the extremum of second-order directional derivatives are derived, and the directions corresponding to maximum and minimum are perpendicular to each other.

Catalytic Kinetic Determination of Micro Amounts of Oxalic Acid by Second-Order Derivative Oscillopolarography

It was found that micro amounts of oxalate showed a very strong catalytic effect on the slow reaction between K 2Cr 2O 7 and Orange Ⅳ in a diluted sulfuric acid medium in a water bath at 70 ℃ . Orange Ⅳ exhibited a sensitive second order derivative polarographic wave at -0 50 V( vs . SCE). This provides the basis for a sensitive and selective catalytic kinetic method for oxalate determination with second order derivative oscillopolarography. The effects of sulphuric acid, K 2Cr 2O 7, and orange Ⅳ concentrations, reaction temperature and reaction time were investigated. A calibration curve of oxalate in the range of 0 1-2 0 μg/mL was obtained by the fixed time procedure. The detection limit was 0 03 μg/ mL. The possible interference from co existing substances or ions was examined. The new method has a high sensitivity and a good selectivity compared to other existing methods for oxalic acid determination. It has been applied to the determination of micro amounts of oxalate in real urine samples with satisfactory results.

Peano Derivatives and Singular Integrals of Arbitrary Order

The Peano derivatives are introduced for functions along an arc in the complex plane. Singular integrals of arbitrary order with singularities at its end-points are defined so that a unified theory for such integrals and Cauchy principal value integrals is established.

Fractional Order Proportional Integral Derivative Controller Design and Simulation for Bioengineering Systems

This paper deals with the study of fractional order system tuning method based on Factional Order Proportional Integral Derivative( FOPID) controller in allusion to the nonlinear characteristics and fractional order mathematical model of bioengineering systems. The main contents include the design of FOPID controller and the simulation for bioengineering systems. The simulation results show that the tuning method of fractional order system based on the FOPID controller outperforms the fractional order system based on Fractional Order Proportional Integral( FOPI) controller. As it can enhance control character and improve the robustness of the system.

Pfaff-Birkhoff Variational Problem and Noether Symmetry in Terms of RiemannLiouville Fractional Derivatives of Variable Order

The Pfaff-Birkhoff variational problem and its Noether symmetry are studied in terms of Riemann-Liouville fractional derivatives of variable order. Based on the combination of variational principle and fractional calculus of variable order,the Pfaff-Birkhoff variational principle with Riemann-Liouville fractional derivatives of variable order is proposed, and the fractional Birkhoff's equations of variable order are derived. Then,the Noether 's theorem for the fractional Birkhoffian system of variable order is given. At last,an example is expressed to illustrate the application of the results.

Fractal Dimension of Graph of Weierstrass Function and Its Derivative of the Fractional Order

In this paper,we obtain the fractal dimension of the graph of the Weierstrass function, its derivative of the fractional order and the relation between the dimension and the order of the fractional derivative.