A fast polygon inflation algorithm to compute the area of feasible solutions for three‐component systems. I: concepts and applications

The multicomponent factorization of multivariate data often results in nonunique solutions. The so‐called rotational ambiguity paraphrases the existence of multiple solutions that can be represented by the area of feasible solutions (AFS). The AFS is a bounded set that may consist of isolated subsets. The numerical computation of the AFS is well understood for two‐component systems and is an expensive numerical process for three‐component systems. In this paper, a new fast and accurate algorithm is suggested that is based on the inflation of polygons. Starting with an initial triangle located in a topologically connected subset of the AFS, an automatic extrusion algorithm is used to form a sequence of growing polygons that approximate the AFS from the interior. The polygon inflation algorithm can be generalized to systems with more than three components. The efficiency of this algorithm is demonstrated for a model problem including noise and a multicomponent chemical reaction system. Further, the method is compared with the recent triangle‐boundary‐enclosing scheme of Golshan, Abdollahi, and Maeder (Anal. Chem. 2011, 83, 836–841). Copyright © 2013 John Wiley & Sons, Ltd.     

Testing the performance of pure spectrum resolution from Raman hyperspectral images of differently manufactured pharmaceutical tablets

Chemical imaging is a rapidly emerging analytical method in pharmaceutical technology. Due to the numerous chemometric solutions available, characterization of pharmaceutical samples with unknown components present has also become possible. This study compares the performance of current state-of-the-art curve resolution methods (multivariate curve resolution-alternating least squares, positive matrix factorization, simplex identification via split augmented Lagrangian and self-modelling mixture analysis) in the estimation of pure component spectra from Raman maps of differently manufactured pharmaceutical tablets. The batches of different technologies differ in the homogeneity level of the active ingredient, thus, the curve resolution methods are tested under different conditions. An empirical approach is shown to determine the number of components present in a sample. The chemometric algorithms are compared regarding the number of detected components, the quality of the resolved spectra and the accuracy of scores (spectral concentrations) compared to those calculated with classical least squares, using the true pure component (reference) spectra. It is demonstrated that using appropriate multivariate methods, Raman chemical imaging can be a useful tool in the non-invasive characterization of unknown (e.g. illegal or counterfeit) pharmaceutical products.     

Multi‐component analysis: blind extraction of pure components mass spectra using sparse component analysis

The paper presents sparse component analysis (SCA)‐based blind decomposition of the mixtures of mass spectra into pure components, wherein the number of mixtures is less than number of pure components. Standard solutions of the related blind source separation (BSS) problem that are published in the open literature require the number of mixtures to be greater than or equal to the unknown number of pure components. Specifically, we have demonstrated experimentally the capability of the SCA to blindly extract five pure components mass spectra from two mixtures only. Two approaches to SCA are tested: the first one based on ?₁ norm minimization implemented through linear programming and the second one implemented through multilayer hierarchical alternating least square nonnegative matrix factorization with sparseness constraints imposed on pure components spectra. In contrast to many existing blind decomposition methods no a priori information about the number of pure components is required. It is estimated from the mixtures using robust data clustering algorithm together with pure components concentration matrix. Proposed methodology can be implemented as a part of software packages used for the analysis of mass spectra and identification of chemical compounds. Copyright © 2009 John Wiley & Sons, Ltd.     

On an SVD‐free approach to the complementarity and coupling theory A note on the elimination of unknowns in sums of dyadic products

The partial knowledge of the factors in a multivariate curve resolution problem can simplify the factorization problem. The complementarity and coupling theory as published in 2012 provides precise mathematical conditions for certain unknown parts of the factors. These constraints are based on a singular value decomposition of the data matrix; they have the form of linear or affine linear spaces which contain the unknown parts of the pure component factors. This paper presents a new and simple singular value decomposition‐free form of the complementarity and coupling theory. The derivation of these theorems is based on elementary arguments of linear algebra. The new mathematical form of the theory allows its easy and straightforward applicability. Copyright © 2015 John Wiley & Sons, Ltd.     

Detection of malathion in food peels by surface-enhanced Raman imaging spectroscopy and multivariate curve resolution

An analytical methodology was developed for detection of malathion in the peels of tomatoes and Damson plums by surface-enhanced Raman imaging spectroscopy and multivariate curve resolution. To recover the pure spectra and the distribution mapping of the analyzed surfaces, non-negative matrix factorization (NMF), multivariate curve calibration methods with alternating least squares (MCR-ALS) and MCR with weighted alternating least square (MCR-WALS) were utilized. Error covariance matrices were estimated to evaluate the structure of the error over all the data. For the tomato data, NMF-ALS and MCR-ALS presented excellent spectral recovery even in the absence of initial knowledge of the pesticide spectrum. For the Damson plum data, owing to heteroscedastic noise, MCR-WALS produced better results. This methodology enabled detection below to the maximum residue limit permitted for this pesticide. This approach can be implemented for in situ monitoring because it is fast and does not require extensive manipulation of samples, making its use feasible for other fruits and pesticides as well.     

Existence, uniqueness, and calculation of the physically acceptable solution of a particular case of the Sherman equations

When a chemical sample made of N elements is analyzed by using sequential selective excitation by monochromatic X-ray beams and selective measurement of the characteristic X-rays, the production of secondary fluorescence does not interfere with the measurements. This experimental situation leads to a particular simple case of the Sherman equations which can be written in this instance as linear equations. The linear equations thus obtained are shown to be very similar to the equations appearing in the classical models of Beattie and Brissey and of Lachance and Traill. The linear algebra proves the existence of N different sets of solutions, but the Perron Frobenius theorem ensures that there is one and only one physically feasible solution, and also leads to the method for obtaining it. This equation solution method can be extended to the equations appearing when standard samples of pure elements are also measured.The propagation of the errors in the measurements to the errors in the sample concentrations has been calculated and simulated, and the results have shown that the solution is well conditioned.     

水＋甲酰胺＋环己烷＋二乙二醇丁醚拟三元系的三临界现象

用改进的体积法研究了拟三元体系水＋甲酰胺＋环己烷＋二乙二醇丁醚的三临界现象. 以水+甲酰胺作为拟“纯组分”，通过控制甲酰胺在拟“纯组分”中的质量分数w，逐渐向三临界点逼近，当w从0增加到0.206时，三相区范围ΔT从1.500 ℃减小到0.228 ℃. 由实验数据得到体系的临界质量分数w=0.286±0.003，临界温度为（54.8±0.7） ℃. 实验发现平均温度下三相区三角形的水-油边与三角坐标中的水-油边几乎平行，而且三相区三角形水-油边的边长及其对应的高与ΔT的指数关系符合三临界的经典理论.     

Reduction of the rotational ambiguity of curve resolution techniques under partial knowledge of the factors. Complementarity and coupling theorems

Multivariate curve resolution techniques allow to uncover from a series of spectra (of a chemical reaction system) the underlying spectra of the pure components and the associated concentration profiles along the time axis. Usually, a range of feasible solutions exists because of the so‐called rotational ambiguity. Any additional information on the system should be utilized to reduce this ambiguity. Sometimes the pure component spectra of certain reactants or products are known, or the concentration profiles of the same or other species are available. This valuable information should be used in the computational procedure of a multivariate curve resolution technique. The aim of this paper is to show how such supplemental information on the components can be exploited. The knowledge of spectra leads to linear restrictions on the concentration profiles of the complementary species and vice versa. Further, affine–linear restrictions can be applied to pairs of a concentration profile and the associated spectrum of a species. These (affine) linear constraints can also be combined with the usual non‐negativity restrictions. These arguments can reduce the rotational ambiguity considerably. In special cases, it is possible to determine the unknown concentration profile or the spectrum of a species only from these constraints. Copyright © 2012 John Wiley & Sons, Ltd.     

基于峭度的重叠峰解析新方法

峭度是表征曲线陡峭程度的物理量。本实验提出了基于峭度的组分分析(component analysis based on kurtosis,CABK)方法来解析重叠峰,从中分离出纯组分信息。这种新方法的优点在于半盲源分离。即只要判断出重叠峰所含组分的数目,就可从混合谱中分离出各组分的纯谱信息。将它用来解析模拟两组分重叠峰体系和酒样的GC-MS混合谱,得到了令人满意的结果。     

Hydrogen-bond configuration in hydrogen-bond solutions. 1. Alcohol/alcohol solutions

A theory based on the hydrogen-bond configuration is proposed and applied to alcohol/alcohol binary solutions. The theory leads explicit expressions for the mixing Gibbs energy and reproduces the experiments on the crystal–liquid phase-diagrams of pure crystals and co-crystals and mixing heats with the parameters common to these experiments. The mixing entropy arises from the increase in the hydrogen-bonding availability of proton donors to approach hydrogen-bond-free proton acceptors. The mixing heat arises from a balance between the contribution from maintaining the original associations in pure liquids and the contribution from a construction of hydrogen bonds freely to hydrogen-bond-free acceptors. When hydrogen-bond associations between component-1 and component-2 are distinguished statistically from associations in each pure component, we call the solution as a cooperative solution that has at least one stoichiometric cooperative concentration point. Some shorter alcohol/alcohol solutions and some aromatic alcohol/aromatic alcohol solutions, however, have no cooperative point and we call those solutions as the ideal hydrogen-bond solutions of which properties are mainly governed by the ideal-gas-like mixing hydrogen-bond entropy. The hydrogen-bond energies of various combinations of the proton acceptor and the proton donor have been estimated consistently from the fittings of the theory, the shifts by hydrogen bonding of the OH stretching in the Raman or IR spectroscopy, and the sublimation energy of crystals. The present theory reveals the characteristics of hydrogen-bond solutions and gives some predictions.     

Radiation of multicomponent aqueous salt solutions in the millimeter-wave spectral region

The low-intensity radiation and radiobrightness contrasts of multicomponent aqueous solutions of alkali metal chlorides have been studied at a frequency of 61.2 GHz. An additive character of changes in radiobrightness contrasts relative to initial binary systems (component ratio, 1: 1) is shown. The concentration dependence for solutions of potassium and lithium chlorides shows that initial different-sign radiation effects are eliminated. The radiation of mixed solutions proves to be close to the radiation of pure water (up to 4 m solutions).     

Surface-tension-driven instabilities of a pure liquid layer evaporating into an inert gas

A theoretical model of the evaporation of a pure liquid layer is developed. We focus on the influence of an inert gaseous component, in addition to vapor, on surface-tension-driven Bénard instabilities. It is assumed that the gas phase is perfectly mixed at some distance from the liquid-gas interface (given composition, pressure, and temperature). If this distance is not much larger than the liquid layer thickness, it is shown that a reduction of the full two-layer problem to a one-layer problem is possible, provided the evaporation rate is not too large. An analytical expression is given for the corresponding dimensionless heat transfer coefficient (a generalized, wavenumber-dependent Biot number) at the evaporating interface. The approach is validated through a comparison with a direct numerical resolution of the full two-layer problem.     

基于非负分解方法的质谱成像数据特征提取

质谱成像技术能够在同一个实验里无需标记手段而获得样品表面的分子信息及其分布信息,是当前质谱分析的热点.其分析所得数据量大且复杂,使其特征难以提取.多元统计分析方法,特别是主成分分析法已应用于质谱成像数据的压缩和特征提取.然而由于主成分分析常产生负的数据结果,其意义难以解释且不易分解为单一的特征.本研究开发出一种基于非负分解的质谱成像数据提取方法,能够提取单一的分子特征及其在样品上的分布特征,并将多个单一的特征分布通过红、绿、蓝三色叠加显示,获得轮廓直观的综合特征分布.应用本方法对小鼠脑组织切片质谱成像数据进行分析,可直观分解出灰质区域、白质区域和背景区域,相对主成分分析方法更直观且易于解释.应用本方法对在同一个样品靶上的人膀胱癌变组织和其相邻非癌变组织切片质谱成像数据进行分析,癌变与非癌变组织间差异清晰直观.本研究设计的质谱成像软件可由http://www.msimaging.net获取.     

Identification of essential components of Houttuynia cordata by gas chromatography/mass spectrometry and the integrated chemometric approach

Starting with Biller-Biemann's work [J.E. Biller, K. Biemann, Anal. Lett. 7 (1974) 515], various kinds of approaches have been proposed to extract GC/MS data to obtain pure components responses. In this paper, an integrated chemometric approach is proposed, which combine four sequential steps, data pretreatment, component perception, resolution and component identification, and then the proposed approach is manipulated to analyze the essential oils of a herbal medicine named Houttuynia cordata (HC). On the basis of the selective information obtained from both chromatograms and mass spectra, the proposed integrated chemometric approach can resolve the two-way GC/MS responses matrix into pure chromatograms and mass spectra without any model assumption on the peak shape. The resolution results obtained from HC samples demonstrate the performance of the proposed approach and indicate that it may be a promising one for analyzing complex chromatograms.     

Remarks on the formulations of the adiabatic theorem

By comparing the adiabatic limit of the exact solutions of the time-dependent Schrödinger equation for spin in rotating magnetic field and for harmonic oscillator with time-dependent frequency with the solutions obtained using the quantum adiabatic theorem we have demonstrated the complete agreement of the two sets of solutions and the importance of phase fixing condition for this agreement. We argue that the notions like “familiar dynamical phase” of the “usual quantum adiabatic theorem” and “an additional phase” of “geometrical origin” have been based on the unjustified neglection of the mentioned condition by applying the quantum adiabatic theorem. There is nothing to add to the quantum adiabatic theorem in which time-dependent eigenbasis satisfies phase fixing condition.     

Inhomogeneous multiscale dynamics in harmonic lattices

We use projection operators to address the coarse-grained multiscale problem in harmonic systems. Stochastic equations of motion for the coarse-grained variables, with an inhomogeneous level of coarse graining in both time and space, are presented. In contrast to previous approaches that typically start with thermodynamic averages, the key element of our approach is the use of a projection matrix chosen both for its physical appeal in analogy to mechanical stability theory and for its algebraic properties. We show that thermodynamic equilibrium can be recovered and obtain the fluctuation dissipation theorem a posteriori. All system-specific information can be computed from a series of feasible molecular dynamics simulations. We recover previous results in the literature and show how this approach can be used to extend the quasicontinuum approach and comment on implications for dissipative particle dynamics type of methods. Contrary to what is assumed in the latter models, the stochastic process of all coarse-grained variables is not necessarily Markovian, even though the variables are slow. Our approach is applicable to any system in which the coarse-grained regions are linear. As an example, we apply it to the dynamics of a single mesoscopic particle in the infinite one-dimensional harmonic chain.     

A structure-preserving computational method in the simulation of the dynamics of cancer growth with radiotherapy

In this work, we consider a two-dimensional mathematical model that describes the growth dynamics of cancer when radiotherapy is considered. The mathematical model for the local density of the tumor is described by a parabolic partial differential equation with variable diffusion coefficient. The nonlinear reaction term considers both the logistic law of proliferation of tumor cells and the effect of a treatment against cancer. Suitable initial-boundary conditions are imposed on a bounded spatial domain, and a theorem on the existence and the uniqueness of solutions for the initial-boundary-value problem is proved. Motivated by this result, we design a finite-difference methodology to approximate the solutions of our mathematical model. The scheme is a linear method that is capable of preserving the positivity and the boundedness of the approximations. Some simulations are presented in order to illustrate the performance of the method. Among other conclusions, the numerical results show that the method is able to preserve the analytical features of the relevant solutions of the model.     

On the low-temperature limit of the mean square amplitude matrix

In a previous paper Cauchy's residue theorem has been shown to furnish an expedient tool for dealing with the mean square amplitude matrix Σ. Here the application of a D-shaped contour is found to provide a convenient point of departure in deriving a non-spectral computational approach to Σ in the lowtemperature limit, the basic result being an integral representation involving the values of a generalized compliance matrix at pure imaginary frequency values. Alternatively, this result is also obtainable from Poisson- or Euler-transforms of a previously derived series relating to an O-shaped circular path of integration. In addition, bounds on the thermal contribution are included.     

Component recognition with three‐dimensional fluorescence spectra based on non‐negative matrix factorization

Non‐negative matrix factorization (NMF) is a widely used approach in signal processing. In this work, we apply it to the component recognition of mixtures with multicomponent three‐dimensional fluorescence spectra. Compared with the popular PARAFAC for component recognition, NMF has the following advantages: on one hand, the decomposed spectra are three dimensional, and thus, more information can be obtained, which is beneficial for component recognition; on the other hand, the decomposed spectra are non‐negative and thus have a certain physical significance. More importantly, we propose a type of integrated similarity indices for the three‐dimensional fluorescence spectra, which, by construction, is good at component recognition from overlapping fluorescence spectra. Experiment results demonstrate that NMF combined with integrated similarity index provides an effective method for component recognition of multicomponent three‐dimensional overlapping fluorescence spectra. Copyright © 2011 John Wiley & Sons, Ltd.     

Calorimetry of model biomolecules in aqueous solution

A survey of the thermodynamic properties of aqueous solutions of model compounds is presented. The review is focused on the analysis of the thermal data obtained from dilution and mixing experiments monitoring weak interactions occurring between solvated solute species. The treatment of experimental results in the form of a virial-like expansion and the factorization analysis known as “group-contribution approach” is discussed in order to gain insight into the phenomenological process at molecular level.Examples are chosen in three dominating classes of biomolecular interactions: purine-like, peptide and carbohydrate molecules.