Component reconstruction in the primary space of spectra and concentrations. Alternating regression and related direct methods

A set of methods that extract the spectral components in a chromatographic run is considered. The methods do not need libraries of previously known spectra or retention times. The methods have been developed for two-dimensional spectra but they can also be used for chromatographic analyses with a single-channel detector. The methods are direct; they do not use principal components as the starting point. Alternating regression (AR) remains in the primary space of spectra and concentrations during the calculations. Random numbers are used as the starting spectra. Regression is used to solve first for the concentrations, then for the spectra. The method uses two kinds of constraints: all spectra and concentrations are forced to be positive; and all concentration profiles are forced to a unimodal shape with a single local maximum. It is assumed that all observations are a linear sum of components. Compact alternating regression (CAR) is a new variant of the basic AR. The idea is to replace multiplication of a large matrix by two multiplications of smaller matrices. This typically speeds up the iterations by a factor of ten. AR and CAR have been successfully used with combined techniques such as gas chromatography—mass spectrometry and liquid chromatography with UV—visible detection. The reliability of the solution is checked by repeatedly injecting noise and performing the analysis several times. This produces estimates of confidence intervals. AR and CAR have recently been extended to handle single-dimensional signals. Examples are single-channel detectors such as the flame ionization detector in gas or liquid chromatography with a fixed-wavelength UV detector. A batch of samples is used as the observation matrix. As a result, one obtains both the concentrations and the elution shapes of individual chromatographic peaks.     

First principle calculations study of AlN surface terminal structure evolution under different conditions

The surface structure and properties of aluminum nitride (AlN) play an important role in many applications. Using the first principle calculations method, we analyzed the surface terminal structure of AlN and its evolution under different conditions by determining the surface energy, adsorption energy, and evaporation energy of the Al and N terminals on the AlN(0001) surface. Our results show that the reason why the N terminal is less stable than the Al terminal is not only because of its high surface energy but also because its adsorption performance is extremely sensitive to the adsorption position. The surface N atoms combine to form N₂ molecules that escape during the evaporation process at high temperature. After surface N atoms escape, the AlN surface structure reconstitutes to form a hexagonal closest packing (HCP)–like structure, and the energy barrier for the reconstructing process is 3.2 eV. This shows that the structure and form of the AlN(0001) terminals depend on the environmental conditions.     

Fourier transform Raman spectral measurements of powdered quaternary mixtures of organic compounds: Exceptional pure component spectral reconstruction using band-target entropy minimization (BTEM)

Fourier transform Raman spectra of eight mixtures of four organic solids, namely dicyandiamide, melamine, acetamide and urea were measured. Matrices formed from these spectra were first subjected to singular value decomposition to obtain the right singular vectors. The right singular vectors were then subjected to blind source separation using band-target entropy minimization (BTEM), thus no a priori information (i.e. involving the nature of the components present, their spectra, nor their concentrations) was included in the analysis. The recovered pure component spectra are of exceptionally high quality and are consistent with pure reference spectra. Various empirical and statistical tests, such as the Euclidean norm and target transform factor analysis, were employed to assess the quality of the recovered spectra. The present results indicate the applicability of combined Raman and BTEM analysis for solid mixtures.     

随机变量序列加权和的强收敛性

本文讨论了一般随机变量序列加权和的强收敛性．作为推论，得到一类鞅差序列加权和的收敛定理和若干经典的独立随机变量序列的强大数定律；已有的若干结论是本文结果的特例．     

同分布NA列Jamison型加权和强稳定的充要条件

本文将Ｊａｍｉｓｏｎ等（１９６５）关于ｉｉｄ列加权和强稳定的充分性结果推广到ＮＡ列的场合并且得到了ＮＡ列Ｊａｍｉｓｏｎ型加权和强稳定的必要性结果。     

Improved estimation of duality gap in binary quadratic programming using a weighted distance measure

We present in this paper an improved estimation of duality gap between binary quadratic program and its Lagrangian dual. More specifically, we obtain this improved estimation using a weighted distance measure between the binary set and certain affine subspace. We show that the optimal weights can be computed by solving a semidefinite programming problem. We further establish a necessary and sufficient condition under which the weighted distance measure gives a strictly tighter estimation of the duality gap than the existing estimations.     

The polynomial reconstruction problem: The first 50 years

The problem of reconstructing the characteristic polynomial of a graph of order at least 3 from the collection of characteristic polynomials of its vertex-deleted subgraphs was posed by Cvetkovi? in 1973 as a spectral counter part to the well-known Ulam's reconstruction conjecture. Over the last 50 years, this problem has received notable attention, many positive results have been obtained, but in the general case the problem is still unresolved. In particular, no counter example is found in literature. In this expository paper we survey classical and some more recent results concerning the polynomial reconstruction problem, discuss some related problems, variations and generalizations.     

Higher-order time integration with a local Lax–Wendroff procedure for a central scheme on an overlapping grid

We have implemented a high-order Lax–Wendroff type time integration for a central scheme on an overlapping grid for conservation law problems. Using a local iterative approach presented by Dumbser et al. (JCP, 2008) [12], we extend a local high-order spatial reconstruction on each cell to a local higher-order space–time polynomial on the cell. We rewrite the central scheme in a fully discrete form to avoid volume integration in the space–time domain. The fluxes at cell interfaces are calculated directly via integrating a higher-order space–time reconstruction of the flux. We compare this approach with the corresponding multi-stage Runge–Kutta time integration (RK). Numerical results show that the new time integration is more cost-effective.     

A simultaneous canonical form of a pair of matrices and applications involving the weighted Moore–Penrose inverse

In this paper, a simultaneous canonical form of a pair of rectangular complex matrices is developed. Using this new tool we give a necessary and sufficient condition to assure that the reverse order law is valid for the weighted Moore–Penrose inverse. Additionally, we characterize matrices ordered by the weighted star partial order and adjacent matrices as applications.     

Equilibrium analysis for a mass-conserving model in presence of cavitation

We study the existence of equilibrium positions for the load problem in Lubrication Theory. The problem consists of two surfaces in relative motion separated by a small distance filled by a lubricant. The system is described by the modified Reynolds equation (Elrod–Adams model) which describes the behavior of the lubricant and an extra integral equation given the balance of forces. The balance of forces allows to obtain the unknown position of the surfaces, defined with one degree of freedom.