A local interpolation scheme using no derivatives in potential sampling: application to O(1D) + H2 system

We recently proposed a local interpolation scheme, in which interpolant moving least squares (IMLS) and Shepard interpolation are employed to describe potential energy surfaces. This IMLS/Shepard scheme is used to interpolate quantum chemical potential energy surfaces for which analytical derivatives are not available. In this study, we apply the scheme to the highly exothermic O((1)D) + H(2) --> H + OH reaction and compare it with results based on Shepard interpolation using second-order Taylor expansions. An analytical surface is used to define the potential function so that errors in the interpolation function may accurately be determined. We find that the present scheme reproduces the correct reactive cross-sections more accurately than the Shepard scheme, and with rms errors for energy and gradients that are significantly smaller than those from Shepard interpolation. This occurs even though the present scheme does not utilize derivative and Hessian information, whereas the Shepard interpolation does. The Bayesian approach proposed by Bettens and Collins does not improve the IMLS/Shepard results significantly, although it does the Shepard-only approach. The accuracy of the IMLS/Shepard scheme is surprising, but can be explained by the more global nature of the interpolation.     

Fast and high precision algorithms for optimization in large-scale genomic problems

There are several very difficult problems related to genetic or genomic analysis that belong to the field of discrete optimization in a set of all possible orders. With n elements (points, markers, clones, sequences, etc.), the number of all possible orders is n!/2 and only one of these is considered to be the true order. A classical formulation of a similar mathematical problem is the well-known traveling salesperson problem model (TSP). Genetic analogues of this problem include: ordering in multilocus genetic mapping, evolutionary tree reconstruction, building physical maps (contig assembling for overlapping clones and radiation hybrid mapping), and others. A novel, fast and reliable hybrid algorithm based on evolution strategy and guided local search discrete optimization was developed for TSP formulation of the multilocus mapping problems. High performance and high precision of the employed algorithm named guided evolution strategy (GES) allows verification of the obtained multilocus orders based on different computing-intensive approaches (e.g., bootstrap or jackknife) for detection and removing unreliable marker loci, hence, stabilizing the resulting paths. The efficiency of the proposed algorithm is demonstrated on standard TSP problems and on simulated data of multilocus genetic maps up to 1000 points per linkage group.     

Calculation of retention indices and peak widths in temperature programmed gas-liquid chromatography

Summary Isothermal chromatographic measurements lead directly to H _v ^o and A (entropy term) of solutes, and three constants of an empirical relationship between peak width and column temperature. From the thermodynamic parameters H _v ^o and A retention temperatures have been computed by means of a theoretical model including temperature dependence of carrier gas viscosity, and subsequently retention times; programmed retention indices have been determined by linear and polynomial interpolations. By substituting the value of the calculated retention temperature in the above-mentioned relationship, peak width at half-height for a linear temperature may be estimated. Predicted retentions correlate with observed data, with a P-value 0.01; simulation accuracy is generally 6–10% for peak widths.Retention indices of some organochlorine species, separated on an OV-101 capillary column, may differ by as much as 26 units depending on the method of calculation. Polynomial-calculated indices are more consistent with the retention index scheme, and have smaller standard deviations and better constancy at different heating rates.     

Optimisation of data handling and detection conditions for automated chromatographic assay software

Summary The principle of automated chromatographic peak detection and analysis software is summarized, and critical steps are systematically studied. As the only parameter to be entered is the acquisition frequency, evaluation of its effect on software performance is discussed. In the case of relatively noisy chromatograms, it is shown experimentally that numerous points per peak have to be taken, leading to quite fast computer acquisition procedures. The use of discrete Fourier transform filtration techniques can modify peak shapes and a comparative study evaluates the relative errors induced in the shapes and characteristics of the chromatographic profiles. Optimisation of filtering conditions is achieved and it is shown that for a filter position only 2% of the Nyquist frequency no deformation occurs in the chromatographic profile. Detection of the start and finish of chromatographic peaks is optimized according to a simple four step iterative procedure. In the case of simulations, the difference between the values used to simulate peaks and those calculated by the software are less than 1%.     

处理无模型体系优化问题的一种新方法

本文提出一种处理无模型体系优化问题的方法，按正交方向进行一维搜索，辅之于样条插值方法最优点。同时，通过将Ｎ维优化问题转化为Ｎ维空间超平面上的优化问题的途径，使试验点的数目大为减小，从计算机模拟和原子吸收条件优化结果看，可获得满意结果。     

Some further considerations on deriving matrix elements of non-periodic potentials in the basis of the non-degenerate harmonic oscillator wavefunctions

Using the properties of the generating function of the Hermite polynomials, we have calculated the matrix elements for the Gaussian-type potential V_G(x)=A exp{−B(x−C)²} and for the Morse-type potential V_M(x)=D_e[1−exp(−ax)]² in the basis of the non-degenerate harmonic oscillator wavefunctions. The final forms of these matrix elements are very simple to use and hence suitable for the numerical resolution of the Schrödinger equation for multiple-well potentials or anharmonic oscillators.     

New generating function formulae of even- and odd-Hermite polynomials obtained and applied in the context of quantum optics

By combining the operator Hermite polynomial method and the technique of integration within an ordered product of operators, for the first time we derive the generating function of even- and odd-Hermite polynomials which will be useful in constructing new optical field states. We then show that the squeezed state and photon-added squeezed state can be expressed by even- and odd-Hermite polynomials.     

Inequalities for Wavelet Frames

One of the fundamental problems in the study of wavelet frames is to find conditions on the wavelet function and the dilation and translation parameters so that the corresponding wavelet system forms a frame. In this article, we obtain some inequalities for a discrete wavelet system to be a frame. Our result improves known ones by Chui, Shi, and Chen.     

连续可导函数类的最优拉格朗日插值

在构造拉格朗日插值算法时,插值结点的选择是十分重要的.给定一个足够光滑的函数,如果结点选择的不好,当插值结点个数趋于无穷时,插值函数不收敛于函数本身.例如龙格现象:对于龙格函数f(x)=1/1+25x^2,如果拉格朗日插值的结点取[-1,1]上的等距结点,那么逼近的误差会随着结点个数增多而趋于无穷大⑴,由此可知插值结点的选择尤为重要.