A novel piecewise multivariate function approximation method via universal matrix representation

Multivariance in science and engineering causes problematic situations even for continous and discrete cases. One way to overcome this situation is to decrease the multivariance level of the problem by using a divide—and—conquer based method. In this sense, Enhanced Multivariance Product Representation (EMPR) plays a part in the considered scenario and acts successfully. This method brings up a finite expansion to represent a multivariate function in terms of less-variate functions with the assistance of univariate support functions. This work aims to propose a new EMPR based algorithm which has two new features that improves the determination process of each expansion component through Fluctuation Free Integration method, which is an efficient method in evaluating multiple integrals through a universal matrix representation, and increases the approximation quality through inserting a piecewise structure into the standard EMPR algorithm. This new method is called Fluctuation Free Integration based piecewise EMPR. Some numerical implementations are also given to examine the performance of this proposed method.     

Block tridiagonal matrix enhanced multivariance products representation (BTMEMPR)

The basic aim of this work is to design a new tridiagonal matrix enhanced multivariance products representation (TMEMPR) which uses not Cartesian vectors but matrices as the support entities. What we obtain after the construction of the representation has been a singular value decomposition like structure where the core matrix becomes a block tridiagonal matrix in contrast to the diagonal and tridiagonal matrix structures of the singular value decomposition of matrices and TMEMPR respectively. We have used support matrices in the construction and not directly orthogonality of the constructed support matrices but block orthogonality which means the mutual ortho normality of the columns of produced support matrices. Certain confirmative implementations finalize the paper.     

The influence of the support functions on the quality of enhanced multivariance product representation

This paper presents recently developed Enhanced Multivariance Product Representation (EMPR) method for multivariate functions. EMPR disintegrates a multivariate function to components which are respectively constant, univariate, bivariate and so on in ascending multivariance. Although the EMPR method has the same philosophy with the High Dimensional Model Representation (HDMR) method, it has been proposed to get better quality than HDMR’s with the help of the support functions. For this purpose, we investigate the EMPR truncation qualities with respect to the selection of the support functions. The obtained results and a number of numerical implementations to show the efficiency of the method are also given in this paper.     

Analytical arbitrary ℓ‐wave solutions of the manning–rosen potential in the tridiagonalization program

By working in a complete square integrable basis that carries a tridiagonal matrix representation of the wave operator, the arbitrary ?‐wave solutions of the Schrödinger equation for the Manning–Rosen potential is investigated with an approximation of centrifugal term. The resulting three‐term recursion relation for the expansion coefficients of the wavefunction is presented. The bound‐state wavefunctions are expressed in terms of the Jocobi polynomial, and the discrete spectrum of the bound states is obtained by diagonalization of the recursion relation. © 2011 Wiley Periodicals, Inc. Int J Quantum Chem, 2012     

Exact solution of the modified Pöschl–Teller potential in the tridiagonal representation

The Schrödinger equation with the modified Pöschl‐Teller (MPT) potential is studied by working in a complete square integrable basis that supports a tridiagonal matrix representation of the wave operator. The resulting three‐term recursion relation for the expansion coefficients of the wavefunction is presented, and the wavefunctions are expressed in terms of the Jocobi polynomial. The discrete spectrum of the bound states is obtained by diagonalization of the recursion relation. © 2011 Wiley Periodicals, Inc. Int J Quantum Chem, 2011     

Analytical treatment of the oscillating Yukawa potential

Using a suitable Laguerre basis set that ensures a tridiagonal matrix representation of the reference Hamiltonian, we were able to evaluate in closed form the matrix elements of the generalized Yukawa potential with a complex screening parameter. This enabled us to treat analytically both the cosine and sine-like Yukawa potentials on equal footing and compute their bound states spectrum as the eigenvalues of the associated analytical matrix representing their Hamiltonians. Finally we used a carefully designed complex scaling method to evaluate the resonance energies and compared our results satisfactorily with those obtained in the literature for the cosine-like Yukawa potential.     

Calculation of the probabilities for transitions into the continuous spectrum by using the L2 basis: A new method of solving the wave-function renormalization problem

A new method is presented for calculating the normalization factor for continuous spectrum functions in the L₂ basis. One variant of this method is applicable to operators given in the form of a tridiagonal matrix and does not require explicit calculation of the roots of the characteristic equation. The other variant of the method can be used in the general case, where after the basis is introduced, its length is gradually increased.     

Full S matrix calculation via a single real-symmetric Lanczos recursion: the Lanczos artificial boundary inhomogeneity method

We present an efficient and robust method for the calculation of all S matrix elements (elastic, inelastic, and reactive) over an arbitrary energy range from a single real-symmetric Lanczos recursion. Our new method transforms the fundamental equations associated with Light's artificial boundary inhomogeneity approach from the primary representation (original grid or basis representation of the Hamiltonian or its function) into a single tridiagonal Lanczos representation, thereby affording an iterative version of the original algorithm with greatly superior scaling properties. The method has important advantages over existing iterative quantum dynamical scattering methods: (a) the numerically intensive matrix propagation proceeds with real symmetric algebra, which is inherently more stable than its complex symmetric counterpart; (b) no complex absorbing potential or real damping operator is required, saving much of the exterior grid space which is commonly needed to support these operators and also removing the associated parameter dependence. Test calculations are presented for the collinear H+H(2) reaction, revealing excellent performance characteristics.     

Determination of rate constants from two-way kinetic-spectral data by using rank annihilation factor analysis

Rank annihilation factor analysis (RAFA) is applied to resolve the two-way kinetic-spectral data measured from spectroscopic reactions and acquire rate constants and the absorption spectrum of each component. A two-step first-order consecutive reaction is studied in this paper. When the first step rate constant acts as an optimizing object, and simply combined with the pure spectrum of reactant, the rank of the original data matrix can be reduced by one by annihilating the information of reactant from the original data matrix. The residual standard deviation (R.S.D.) of the residual matrix after bilineaization of the background matrix is regarded as the evaluation function. Owing to the correlation between kinetic functions of species in the reaction, two optimal resolutions, corresponding to the rate constants of the first and second step, respectively, can both be obtained in one computing process. Given the kinetic parameters, the absorption spectrum of each component including the intermediate can be obtained through least square regression. This approach can also be applied to reaction systems where the intermediate or the final product doesnot absorb. The performance of the method has been evaluated by using synthetic data. Electrodegradation of phenol solution and alkaline hydrolysis of dimethyl phthalate were also studied by the present method.     

First-principles electronic transport calculations in finite elongated systems: a divide and conquer approach

We present a first-principles method for the evaluation of the transmittance probability and the coherent conductance through elongated systems composed of a repeating molecular unit and terminated at both ends. Our method is based on a divide and conquer approach in which the Hamiltonian of the elongated system can be represented by a block tridiagonal matrix, and therefore can be readily inverted. This allows us to evaluate the transmittance and the conductance using first-principles electronic structure methods without explicitly performing calculations involving the entire system. A proof of concept model based on a trans-polyacetylene chain bridging two aluminum leads indicates that our divide and conquer approach is able to capture all the features appearing in the transmittance probability curves obtained by a full scale calculation.     

Matrix-free analysis of aflatoxins in pistachio nuts using parallel factor modeling of liquid chromatography diode-array detection data

In the present study a second-order calibration strategy for high performance liquid chromatography with diode-array detection (HPLC-DAD) has been developed using parallel factor analysis (PARAFAC) and has been applied for simultaneous determination of aflatoxins B₁, B₂, G₁ and G₂ in pistachio nuts in the presence of matrix interferences. Sample preparation was based on solvent extraction (SE) followed by solid phase extraction (SPE) on Bond Elut C18 cartridges. Since the sample preparation procedure was not selective to the analytes of interest, exploiting second-order advantage to obtain concentrations of individual analytes in the presence of uncalibrated interfering compounds seemed necessary. Appropriate pre-processing steps have been applied to correct background signals and the effect of retention time shifts. Transferred calibration data set obtained from standardization of solvent based calibration data has been used in prediction step. The results of PARAFAC on a set of spiked and naturally contaminated pistachio nuts indicated that the four aflatoxins could be successfully determined. The method was validated and multivariate analytical figures of merit were calculated. The advantages of the proposed method are using a low-cost SPE step relative to standard method of aflatoxin analysis (immune affinity column assay), a unique and simple isocratic elution program for all samples and a calibration transfer for saving both chemicals and time of analysis. This study show that coupling of SPE-HPLC-DAD with PARAFAC as a powerful second-order calibration method can be considered as an alternative method for resolution and quantification of aflatoxins in the presence of unknown interferences obtained through analysis of highly complex matrix of pistachio samples and cost per analysis can be reduced significantly.     

Quantum solution of coupled harmonic oscillator systems beyond normal coordinates

Normal coordinates can be defined as orthogonal linear combinations of coordinates that remove the second order couplings in coupled harmonic oscillator systems. In this paper we go further and explore the possibility of using linear although non-orthogonal coordinate transformations to get the quantum solution of coupled systems. The idea is to use as non-orthogonal linear coordinates those which allow us to express the second-order Hamiltonian matrix in a block diagonal form. To illustrate the viability of this treatment, we first apply it to a system of two bilinearly coupled harmonic oscillators which admits analytical exact solutions. The method provides in this case, as an extra mathematical result, the analytical expressions for the eigenvalues of a certain type of symmetrical tridiagonal matrices. Second, we carry out a numerical application to the Barbanis coupled oscillators system, which contains a third order coupling term and cannot be solved in closed form. We demonstrate that the non-orthogonal coordinates used, named oblique coordinates, are much more efficient than normal coordinates to determine the energy levels and eigenfunctions of this system variationally.     

The extended negative factor counting method for tridiagonal block matrices with cross links

The NFC (negative factor counting) method ws extended to solve the eigenvalue problem of tridiagonal block matrices with elements corresponding to cross links which may be derived from the quantum-chemical calculation on a native protein molecule. The mathematical proof of the necessary theorem is given in detail.     

On 3-D graphical representation of proteomics maps and their numerical characterization

We consider numerical characterization of proteomics maps by representing a map as a three-dimensional graphical object based on x, y coordinates of the spots and using their relative abundance as the z coordinate. In our representation the protein spots are first ordered based on their relative abundance and labeled accordingly. In the next step a 3-D path is constructed connecting spots having adjacent labels. Finally a matrix is constructed by assigning to each pairs of labels (i, j) matrix element, the numerical value of which is based on the quotients of the Euclidean distance and the distance along the 3-D zigzag between the two points. The approach has been illustrated on a fragment of a proteomics map and compared with 2-D graphical representation of proteomics maps.     

A divide and conquer real-space approach for all-electron molecular electrostatic potentials and interaction energies

A computational scheme to perform accurate numerical calculations of electrostatic potentials and interaction energies for molecular systems has been developed and implemented. Molecular electron and energy densities are divided into overlapping atom-centered atomic contributions and a three-dimensional molecular remainder. The steep nuclear cusps are included in the atom-centered functions making the three-dimensional remainder smooth enough to be accurately represented with a tractable amount of grid points. The one-dimensional radial functions of the atom-centered contributions as well as the three-dimensional remainder are expanded using finite element functions. The electrostatic potential is calculated by integrating the Coulomb potential for each separate density contribution, using our tensorial finite element method for the three-dimensional remainder. We also provide algorithms to compute accurate electron-electron and electron-nuclear interactions numerically using the proposed partitioning. The methods have been tested on all-electron densities of 18 reasonable large molecules containing elements up to Zn. The accuracy of the calculated Coulomb interaction energies is in the range of 10(-3) to 10(-6) E(h) when using an equidistant grid with a step length of 0.05 a(0).     

A method of calculating eigenvectors of real symmetric tridiagonal matrices in a hyperspherical space

A novel direct method of calculating eigenvectors for real symmetric tridiagonal matrices has been proposed. An eigenvector is represented as a unit vector in a hyperspherical space set by hyperpolar angle coordinates. An algorithm has been examined for a series of tridiagonal matrices. Numerical calculations as compared with a conventional Householder algorithm have shown a good agreement within 8-byte double precision for most eigenvectors computed.     

不定物种的多组分体系同时测定：逐步回归法解析紫外光谱矢量数据的研究

本文采用逐步回归法对不定物种的多组分体系的紫外光谱矢量数据进行解析，可对物种及物种含量同时进行定性和定量分析，通过对两个复方药物体系（复方扑热息痛与复方维生素Ｂ）的紫外光谱数据的实际解析，表明本法能有效地完成混合物中物种及物种含量的同时定性和定量测定。     

Mathematical optimization in the determination of the dissociation constants of a tribasic organic acid by 13C NMR spectroscopy

The calculation of dissociation constants from the chemical shifts of (13)C NMR spectra leads to a complicated non-linear equation. Two different mathematical methods for solution of this equation have been chosen--an iterative step method and a matrix pseudo-inversion method. When the iteration method is used the initial guesses for the parameters, the initial value of the step size and the escalation of the iteration must be optimized. For comparison the matrix pseudo-inversion method was used because it gives a unique result. With the optimized step method the results were as accurate or even better than those obtained with the matrix method. Although it takes time to optimize the system, the step method is the more suitable method of solving the problem. The matrix inversion can be done only with a computer with 13 significant digits and exponent capacity larger than +/- 38.     

Establishment of signal-recovery functions for calculation of recovery factor. Application to monitoring of contaminant residues in vegetables by chemiluminescence detection

A new strategy is proposed for verifying if recovery factor is constant and independent of the real analyte content of samples. A signal-recovery function has been developed on the basis of measurement of spiked test samples before and after a pre-treatment step and considering, as starting point, a recent IUPAC recommendation which distinguishes between two terms—recovery factor, R, and apparent recovery, R*. Apparent recovery includes recovery factor and a new recovery term proposed in a previous paper by the authors, named calibration recovery, R ^C. The signal-recovery function is obtained directly from the measured analytical signals instead of from the concentrations, simplifying the calculations. A linear signal-recovery curve indicates that the recovery factor is constant in the analyte concentration range studied experimentally and, in this way, a single recovery factor can be calculated. The usefulness of the proposed method has been shown by quantification of the pesticide carbaryl by two different flow-injection analysis methods with chemiluminescent detection based on the luminol and TCPO systems. Good results were obtained from both methods.     

Transformation of Bloch representation into atomic orbital representation and combined matrix element calculation technique for spatially bounded basis functions in crystals

A method is proposed for transforming the Hamiltonian from Bloch to atomic function representation. For spatially bounded functions, this is a rigorous method based on solution of a certain algebraic system of equations. Unlike the conventional procedure based on integration over the Brillouin zone, the new method requires knowledge of the matrix elements of the Bloch representation only at several points of the Brillouin zone. The number of these points is determined by the trimming radius for the spatially bounded functions and by the lattice constant. The method can be used for calculating matrix elements in a basis of atomic functions and for reducing computations in matrix element calculations of the Bloch representation for procedures using numerical integration.