A principal component analysis based method to discover chemical differences in comprehensive two-dimensional gas chromatography with time-of-flight mass spectrometry (GCxGC-TOFMS) separations of metabolites in plant samples

Comprehensive two-dimensional gas chromatography combined with time-of-flight mass spectrometry (GC × GC-TOFMS) provides high resolution separations of complex samples with a mass spectrum at every point in the separation space. The large volumes of multidimensional data obtained by GC × GC-TOFMS analysis are analyzed using a principal component analysis (PCA) method described herein to quickly and objectively discover differences between complex samples. In this work, we submitted 54 chromatograms to PCA to automatically compare the metabolite profiles of three different species of plants, namely basil (Ocimum basilicum), peppermint (Mentha piperita), and sweet herb stevia (Stevia rebaudiana), where there were 18 chromatograms for each type of plant. The 54 scores of the m/z 73 data set clustered in three groups according to the three types of plants. Principal component 1 (PC 1) separated the stevia cluster from the basil and peppermint clusters, capturing 61.84% of the total variance. Principal component 2 (PC 2) separated the basil cluster from the peppermint cluster, capturing 16.78% of the total variance. The PCA method revealed that relative abundances of amino acids, carboxylic acids, and carbohydrates were responsible for differentiating the three plants. A brief list of the 16 most significant metabolites is reported. After PCA, the 54 scores of the m/z 217 data set clustered in three groups according to the three types of plants, as well, yielding highly loaded variables corresponding with chemical differences between plants that were complementary to the m/z 73 information. The PCA data mining method is applicable to all of the monitored selective mass channels, utilizing all of the collected data, to discover unknown differences in complex sample profiles.     

Bounded variation in the mean

It is shown that the concept of bounded variation in the mean is not a meaningful generalization of ordinary bounded variation. In fact, it is a characterization of functions which differ from functions of bounded variation on a zero set.

     

Chapter 1:general introduction

One of the most active and continuing subjects in matrix theory during the past century is the study of those linear operators on matrices that leave certain properties or relations of subsets invariant. Such questions are usually called "Linear Preserver Problems." The earliest papers in our reference list are [Frobenius, 1897] and [Kantor, 1897]. Since much effort has been devoted to this type of problem, there have been several excellent survey papers, e.g., [Marcus, 1962], [Marcus, 1971], and [Grone, 1976]. Since the time of Grone's thesis, there have been many developments, encouraged by the introduction of new techniques, many from outside matrix theory itself. We hope this monograph will provide a useful update and convenient reference for the reader. In this introduction, we describe some typical questions and results on linear preserver problems, give some motivation for the study of the subject and finally provide some guidance fur the use of the monograph.     

Validity of linear acoustics for prediction of waveforms caused by sonically moving laser beams

The question is raised as to whether the analysis of the generation of sound by a laser beam moving over a water surface at the sound speed c for an interminable time period requires consideration of nonlinear effects. A principal consideration in this regard is whether the linear acoustics theory predicts a pressure waveform that is bounded in the asymptotic limit when the laser irradiation time is arbitrarily large. It is shown that a bounded asymptotic limit exists when the upper boundary condition corresponds (as is more nearly appropriate) to that of a pressure release surface, but not when it corresponds to that of a rigid surface. The asymptotic solution to the appropriate inhomogeneous wave equation is given exactly for the former case, and it is shown that the highest asymptotic amplitudes, given specified laser power and beam radius a, occur in the limit of a very small light absorption coefficient mu. In this limit, the peak amplitude is independent of mu and occurs at a depth of 0.88/mu. An approximate solution for the pressure waveform at intermediate times establishes that the characteristic time for buildup to the asymptotic limit is of the order of 2.5/(c mu 2a). If this time is substantially shorter than the time that a plane-wave pulse with the asymptotic waveform would take to develop a shock wave, then accumulative nonlinear effects are of minor importance.     

On the mechanism of the reduction of 1-phenyl-3-(4-pyridinylmethylene)-2-oxindole with sodium borohydride in methanol

Sodium borohydride reduction of 1-phenyl-3-(4-pyridinylmethylene)-2-oxindole 1 in methanol gives a high yield of the oxindole 2 , a precursor to linopirdine. The reduction is accelerated by methanol and the major by-products during this reduction are the diastereomeric Michael adducts between 1 and 2 and the overreduced products, 9 and 11 . Surprisingly, indole 9 is not derived from further reduction of 2 but is formed concurrently, whereas the ring opened product 11 is the unexpected reduction product derived from 2 .     

Adjoint and defect error bounding and correction for functional estimates

We present two error estimation approaches for bounding or correcting the error in functional estimates such as lift or drag. Adjoint methods quantify the error in a particular output functional that results from residual errors in approximating the solution to the partial differential equation. Defect methods can be used to bound or reduce the error in the entire solution, with corresponding improvements to functional estimates. Both approaches rely on smooth solution reconstructions and may be used separately or in combination to obtain highly accurate solutions with asymptotically sharp error bounds. The adjoint theory is presented for both smooth and shocked problems; numerical experiments confirm fourth-order error estimates for a pressure integral of shocked quasi-1D Euler flow. By employing defect and adjoint methods together and accounting for errors in approximating the geometry, it is possible to obtain functional estimates that exceed the order of accuracy of the discretization process and the reconstruction approach. Superconvergent drag estimates are obtained for subsonic Euler flow over a lifting airfoil.