The usefulness of the deconvolution of chromatograms into orthogonal polynomials for characterizing the quality of separation

Simulated chromatograms are deconvoluted into series of orthogonal polynomials. The number of terms necessary to achieve a preset goodness of fit, the reproducibility of coefficients in the presence of noise and the effects of signal-to-noise ratio, peak-height ratio and peak asymmetry on the value of coefficients from the deconvolution are investigated. A convenient method for the deconvolution of real chromatograms is described in order to characterize the quality of separation in the chromatogram. A known quality criterion is modified by using coefficients from the deconvolution in order to enhance its performance. Performance is tested on a simulated separation of five sulfa drugs on a reversed- phase column.     

A class of orthogonal polynomials suggested by a trigonometric Hamiltonian: Antisymmetric states

This is the second in a series of papers dealing with the sets of orthogonal polynomials generated by a trigonometric Hamiltonian. In the first of this series, a subclass of the Jacobi polynomials denoted by and referred to as the -polynomial of the first kind, which arises in the investigation of the symmetric state eigenfunctions of the Hamiltonian under consideration, was examined. Another subclass of the Jacobi polynomials denoted by is introduced here representing the antisymmetric states, and is called in accordance the -polynomial of the second kind. Moreover, by the derivation of the ultraspherical polynomial wavefunctions, interrelations between the -polynomials of the first and second kinds as well as the other orthogonal polynomial systems are also emphasized.AMS subject classification: 33C45, 81Q05, 33C05     

Asymptotics (p\rightarrow \infty ) of L_p-norms of hypergeometric orthogonal polynomials

The determination of the weighted $L_p$ norms of the real orthogonal polynomials of hypergeometric type $\left\{ y_n(x)\right\} $ is not only a very important problem per se in the theory of special functions, but also because of their recent entropic characterization and applications in quantum chemistry, quantum physics and information theory. Indeed, they essentially describe the $p$ th-order Rényi and Tsallis entropies of the numerous quantum systems whose wavefunctions are controlled by these polynomials. Moreover, for different values of $p$ , up to a constant factor, these norms characterize various fundamental and experimentally accessible quantities of many-electron systems. As well, the $L_p$ norms have been used to develop and interpret all energy components in the density-functional theory of the ground-state of atoms and molecules. The asymptotics of these quantities when $n \rightarrow \infty $ and $p>0$ have been recently calculated for Hermite polynomials, although not yet for Laguerre and Jacobi polynomials. Here, we determine the asymptotics ( $p\rightarrow \infty $ , $n$ fixed) of the weighted $L_p$ norms for general orthogonal polynomials in terms of the weight function and the coefficients of the second-order hypergeometric differential equation that they satisfy, and we apply it to the three classical families of real orthogonal polynomials. Moreover we analyse and discuss the monotonicity of this asymptotics, and we carry out a detailed numerical study of it.     

On counting theorems for self-avoiding walks

Certain differences are pointed out between two recently published counting theorems for self-avoiding walks on a lattice.     

Computational algorithms for matching polynomials of graphs from the characteristic polynomials of edge-weighted graphs

Computational algorithms are described which provide for constructing the set of associated edge-weighted directed graphs such that the average of the characteristic polynomials of the edge-weighted graphs gives the matching polynomial of the parent graph. The weights were chosen to be unities or purely imaginary numbers so that the adjacency matrix is hermitian. The computer code developed earlier by one of the authors (K.B.) is generalized for complex hermitian matrices. Applications to bridged and spirographs, some lattices and all polycyclic graphs containing up to four cycles are considered.     

The use of chebyshev polynomials orthogonal on a finite arbitrary system of points for interpolating changes in nematic-isotropic liquid phase transition temperatures in homologous series

Chebyshev polynomials Ψ _q (x) orthogonal on a finite arbitrary system of points x _i (i = 1−N) are used to interpolate changes in nematic-isotropic liquid phase transition temperatures t _c(x) in homologous series of liquid crystals (x = 1/n, where n is the number of the homologue). The expansion of the t _c(x) function into a series in Ψ _q (x) polynomials was found to be very effective. Already at q ≤ 3, this series describes the known types of the t _c(x) dependences with high accuracy and very small root-mean-square deviations for mesogenic molecules of various chemical structures and dimensions. The dependence of the limiting t _l = t _c(0) value on the form of X-shaped molecules and linear dimensions of N-mers with N rigid aromatic fragments linked with each other by flexible spacer chains was studied.

     

Chirality polynomials

Chirality algebra uses ideas from permutation group theory and group representation theory to derive chirality polynomials having appropriate transformation properties for estimation of the magnitude and sign of a given pseudoscalar property (e.g. optical rotation, circular dichroism) for a given skeleton using parameters which depend only upon the ligands located at the specific sites of the skeleton, the particular skeleton, and the particular pseudoscalar property. For all but the simplest skeletons, a qualitatively complete chirality polynomial describing all chirality phenomena associated with the skeleton contains more than one component and thus requires more than one set of ligand parameters. Qualitatively complete chirality polynomials are reviewed for the most important transitive skeletons (i.e. skeletons in which all sites are equivalent), including the polarized triangle, tetrahedron, disphenoid (allene), polarized square, polarized rectangle, polarized pentagon, octahedron, trigonal prism (cyclopropane), and polarized heptagon skeletons.     

Reverse orthogonal strategy for oligosaccharide synthesis

Herein, we report the invention of a novel expeditious concept for oligosaccharide synthesis. Unlike the classic orthogonal strategy based on leaving groups, the reverse approach is based on orthogonal protecting groups, herein p-methoxybenzyl and 4-pentenoyl, which allows for efficient oligosaccharide assembly in the reverse direction.     

Expansion and completeness theorems for operator manifolds

Some expansion and completeness theorems for operator manifolds, which are currently being employed in propagator theory, are derived. It is shown that excitation or ionization operators satisfying the conditions Q|0〉 = |Λ〉 and Q_Λ|0〉 = 0 for general excited states |Λ〉 and reference state |0〉 may be expanded uniquely in particular sets of basis operators. These results are then used to discuss rigorous expressions for fermion propagators.     

Automated orthogonal control system for electrospray ionization

Low-flow electrospray ionization is typically a purely electrostatic method, used without supporting sheath-gas nebulization. Complex spray morphology results from a large number of possible spray emission modes. Spray morphology may assume the optimal Taylor cone-jet spray mode under equilibrium conditions. When coupling to nanobore gradient elution chromatography, however, stability of the Taylor cone-jet spray mode is compromised by the gradient of mobile phase physiochemical properties. The common spray modes for aqueous/organic mobile phases were characterized using orthogonal (strobed illumination) transmitted light and (continuous illumination) scattered light imaging. Correlation of image sets from these complementary illumination methods provides the basis for spray mode identification using qualitative and quantitative image analysis. An automated feedback-controlled electrospray source was developed on a computer capable of controlling electrospray potential using an image-processing based algorithm for spray mode identification. The implementation of the feedback loop results in a system that is both self-starting and self-tuning for a specific spray mode or modes. Thus, changes in mobile phase composition and/or flow rate are compensated in real-time and the source is maintained in the cone-jet or pulsed cone-jet spray modes.     

On self-reciprocal polynomials

In this paper we establish a sharp result concerning integral mean estimates for self-reciprocal polynomials.     

Moments and characteristic polynomials for square lattice graphs

The method of moments is used to derive closed analytical expressions for the characteristic polynomials of square lattice graphs. We obtain exact analytical formulae for three-dimensional cubic lattices, square lattices, tubular square lattices, and cylindrical square lattices containing any number of vertices.Camille and Henry Dreyfus Teacher-Scholar.     

Hypervirial theorems for some simple quantum chemistry systems

A very general form of the hypervirial theorems, which can be applied to several quantum systems of great interest for the theoretical quantum chemists is presented.