Determining the independent parameters in stationary kinetic equations for linear mechanisms

Coloring methods in graph theory are used in estimating the number of parameters to be determined in a kinetic model corresponding to a linear mechanism. The estimates are based on determining the numbers of differently colored frameworks. Examples are discussed of isomerization and an enzymatic reaction catalyzed by ammonoacyl-MRNA synthetase.Translated from Teoreticheskaya i Éksperimental'naya Khimiya, No. 2, pp. 187–191, March–April, 1987.     

pKa prediction for acidic phosphorus‐containing compounds using multiple linear regression with computational descriptors

Ninety‐six acidic phosphorus‐containing molecules with pK_a 1.88 to 6.26 were collected and divided into training and test sets by random sampling. Structural parameters were obtained by density functional theory calculation of the molecules. The relationship between the experimental pK_a values and structural parameters was obtained by multiple linear regression fitting for the training set, and tested with the test set; the R² values were 0.974 and 0.966 for the training and test sets, respectively. This regression equation, which quantitatively describes the influence of structural parameters on pK_a, and can be used to predict pK_a values of similar structures, is significant for the design of new acidic phosphorus‐containing extractants. © 2016 Wiley Periodicals, Inc.     

A simple program for weighted linear regression applied to activation analysis data processing

A simple ALGOL program for activation analysis data handling is presented The program, although designed in principle for processing single-channel spectrometry data, may also be used for multichannel spectrometry, on condition that the peak area is computed separately. The calculations of instrumental error and standard deviation are carried out. The outliers are tested, and the regression line diagram with the related observations are plotted by the program.     

Numerical properties of equations involving high-order derivatives of pressure with respect to volume

This paper presents some unexpected features related to the solution of equations containing a high-order derivative of pressure with respect to volume equated to zero. For pure components, such equations define, in the pressure-temperature plane, nodal curves similar in shape to mixture spinodal curves. The analysis was made for a general form of two-parameter cubic equations of state and various numerical aspects for the Redlich-Kwong equation of state are exemplified.     

The convergence properties of direct energy minimisation with respect to linear coefficients in the MC-LCAO-MO-SCF approach

It is shown that in the LCAO-MO-MC-SCF problem, if the molecular orbital orthonormality constraints are introduced in the manner suggested by Kari and Sutcliffe or indeed by any similar method then the Hessian of the problem with respect to the linear coefficients is singular. The nature of this singularity is analysed and it is shown that in general it is possible to remove it in a level-shifting-like scheme, but that only in certain special cases is this procedure likely to be quickly convergent.     

The convergence properties of direct methods of energy minimization with respect to linear coefficients in the LCAO-MO-SCF approach

It is shown that in the LCAO-MO-SCF problem, if the molecular orbital orthonormality constraints are introduced in the manner first suggested by Fletcher, then the Hessian of the problem is singular. It is suggested that this singularity may well account for the slow convergence observed using direct energy minimization methods to solve the SCF problem. Ways of avoiding the consequences of this singularity are discussed.     

Investigation of the homogeneity of solids with a linear regression model

Testing of the hemogeneity of solids is usually based on the comparison of the variance of measurements at several sample points with the variance of repeated measurements. In the case of destructive analytical methods, empirical mathematical models are needed because repetitions at the same subsample are impossible. Successful application of such models depends on the adequacy of the mathematical model and the type of distribution observed. Gradients and textures with low or medium local frequency require other tests than stochastic distributions with high local frequency. A simple regression model base on a 2² factorial allows conclusions about the type of distribution without great mathematical effort.     

The application of multiple linear regression to the measurement of the median particle size of drugs and pharmaceutical excipients by near-infrared spectroscopy.

A number of powdered drugs and pharmaceutical excipients were used to demonstrate the ability of near-infrared spectroscopy to measure median particle size (d50). Sieved fractions and bulk samples of aspirin, anhydrous caffeine, paracetamol, lactose monohydrate and microcrystalline cellulose were particle sized by forward angle laser light scattering (FALLS) and scanned by fibre-optic probe FT-NIR spectroscopy. Two-wavenumber multiple linear regression (MLR) calibrations were produced using: NIR reflectance; absorbance and Kubelka-Munk function data with each of median particle size, reciprocal median particle size and the logarithm of median particle size. Best calibrations were obtained using reflectance data versus the logarithm of median particle size (NIR predicted lnd50 versus ln(FALLS d50) for microcrystalline cellulose and lactose monohydrate sieve fraction calibrations: r = 0.99 in each case). Working calibrations for lactose monohydrate (median particle size range: 19.2-183 microns) and microcrystalline cellulose (median particle size range: 24-406 microns) were set-up using combinations of machine sieve-fractions and bulk samples. This approach was found to produce more robust calibrations than just the use of sieved fractions. The method has been compared with single wavenumber quadratic least squares regression using reflectance and mean-corrected reflectance data with median particle size. Correlation between NIR predicted and FALLS values was significantly better using the MLR method.     

Comparison of robust regression methods based on least-median and adaptive kalman filtering approaches applied to linear calibration data

Five algorithms for data analysis are evaluated for their abilities to discriminate against outliers in small data sets (4–10 points). These methods included least-squares regression, the least absolute -deviation method, the least median of squares method, and two techniques based on an adaptive Kalman filter. For data sets consisting of 4–9 points with one outlier, the average errors in the estimation of the slope were found to be 18.9 % by least-squares, 17.7% by the least absolute deviation method, 0.5% by the least median of squares algorithm, 9.1% by an adaptive Kalman filter algorithm, and 0.9% by a zero-lag adaptive Kalman filter algorithm. Based on these results, the conclusion is that the zero-lag adaptive Kalman filter and the least median of squares approaches are best suited for the detection of outliers in small calibration data sets.     

Instance-based regression with missing data applied to a photocatalytic oxidation process

In this paper, a modified nearest-neighbor regression method (kNN) is proposed to model a process with incomplete information of the measurements. This technique is based on the variation of the coefficients used to weight the distances of the instances. The case study selected for testing this algorithm was the photocatalytic degradation of Reactive Red 184 (RR184), a dye belonging to the group of azo compounds, which is widely used in manufacturing paint paper, leather and fabrics. The process is conducted with TiO₂ as catalyst (an inexpensive semiconductor material, completely inert chemically and biologically), in the presence of H₂O₂ (with the role of increasing the rate of photo-oxidation), at different pH values. The final concentration of RR184 is predicted accurately with the modified kNN regression method developed in this article. A comparison with other machine learning methods (sequential minimal optimization regression, decision table, reduced error pruning tree, M5 pruned model tree) proves the superiority and efficiency of the proposed algorithm, not only for its results, but for its simplicity and flexibility in manipulating incomplete experimental data.      

Enamel dissolution as a function of solution degree of saturation with respect to hydroxyapatite: a nanoindentation study

The objective of this study was to investigate human enamel dissolution as a function of degree of saturation (DS) of the surrounding solution with respect to hydroxyapatite. Nanoindentation was used to compare changes in enamel nanomechanical properties due to dissolution by two solutions. Citric acid solution (DS=0.000, pH 3.30) and citric acid solution containing calcium (299 mg/l) and phosphate (54.0 mg/l) (DS=0.032, pH 3.30) were compared with a control mineral water (DS=0.673, pH 7.48). Exposure times were 0, 120, 300, 600, 900, and 1200 s. Compared to untreated enamel, there was a statistically significant change in enamel hardness after 120 s exposure to both citric acid solutions, and in elastic modulus after 300 s exposure. The rate of change of both variables decreased with exposure time. This suggests that dissolution rate is diffusion-limited under these conditions, in agreement with previous studies. There was no statistically significant difference between the hardness or elastic modulus of enamel exposed to the two citric acid solutions at any time. This may be due to a change in solution composition during contact with the enamel.     

On the derivative of the associated Legendre function of the first kind of integer degree with respect to its order (with applications to the construction of the associated Legendre function of the second kind of integer degree and order)

The derivative of the associated Legendre function of the first kind of integer degree with respect to its order, , is studied. After deriving and investigating general formulas for μ arbitrary complex, a detailed discussion of , where m is a non-negative integer, is carried out. The results are applied to obtain several explicit expressions for the associated Legendre function of the second kind of integer degree and order, . In particular, we arrive at formulas which generalize to the case of (0 ≤ m ≤ n) the well-known Christoffel’s representation of the Legendre function of the second kind, Q _n (z). The derivatives and , all with m > n, are also evaluated.     

On the derivative of the associated Legendre function of the first kind of integer order with respect to its degree (with applications to the construction of the associated Legendre function of the second kind of integer degree and order)

In our recent works (R. Szmytkowski, J. Phys. A 39:15147, 2006; corrigendum: 40:7819, 2007; addendum: 40:14887, 2007), we have investigated the derivative of the Legendre function of the first kind, P _ν(z), with respect to its degree ν. In the present work, we extend these studies and construct several representations of the derivative of the associated Legendre function of the first kind, \({P_{\nu}^{\pm m}(z)}\), with respect to the degree ν, for \({m \in \mathbb{N}}\). At first, we establish several contour-integral representations of \({\partial P_{\nu}^{\pm m}(z)/\partial\nu}\). They are then used to derive Rodrigues-type formulas for \({[\partial P_{\nu}^{\pm m}(z)/\partial\nu]_{\nu=n}}\) with \({n \in \mathbb{N}}\). Next, some closed-form expressions for \({[\partial P_{\nu}^{\pm m}(z)/\partial\nu]_{\nu=n}}\) are obtained. These results are applied to find several representations, both explicit and of the Rodrigues type, for the associated Legendre function of the second kind of integer degree and order, \({Q_{n}^{\pm m}(z)}\); the explicit representations are suitable for use for numerical purposes in various regions of the complex z-plane. Finally, the derivatives \({[\partial^{2}P_{\nu}^{m}(z)/\partial\nu^{2}]_{\nu=n}, [\partial Q_{\nu}^{m}(z)/\partial\nu]_{\nu=n}}\) and \({[\partial Q_{\nu}^{m}(z)/\partial\nu]_{\nu=-n-1}}\), all with m > n, are evaluated in terms of \({[\partial P_{\nu}^{-m}(\pm z)/\partial\nu]_{\nu=n}}\). The present paper is a complementary to a recent one (R. Szmytkowski, J. Math. Chem 46:231, 2009), in which the derivative \({\partial P_{n}^{\mu}(z)/\partial\mu}\) has been investigated.     

A theoretical model for evaluation of configurational entropy of mixing with respect to shape and size of particles

A general method of evaluation of configurational entropy of a liquid mixture is presented. It is based on a generalized lattice model with no restrictions due to particle shape being introduced. A general formula for the entropy is derived. Achieved results open a way for a rigorous analysis of particle shape effect on mixing process. As an example, a new formula for the entropy of mixing of hard spheres in continuous space is derived which may respect a physical bound for packing ratio. A systematic approach to improve the model accuracy is proposed. The resultant alternative models are discussed in details. A comparison with literature data and the Mansoori-Carnahan-Starling formula is presented. Very good agreement is shown.     

Errors in variables regression with value‐censored data

Samples with analyte concentrations outside a method's dynamic range are a reality of clinical chemistry and are particularly of interest in method comparison studies. The most obvious remedy—to ignore any such values—introduces bias and loses the information that censored data might add to the analysis. Extending conventional errors‐in‐variables methods to incorporate value‐censored data recovers this information. The formulation presented uses a variance model more flexible than either the constant variance or the constant coefficient of variation models. Copyright © 2016 John Wiley & Sons, Ltd.