Solution of sparse quasi-square rectangular systems by Gaussian elimination

We present a general method for the linear least-squares solutionof overdetermined and underdetermined systems. The method isparticularly efficient when the coefficient matrix is quasi-square,that is when the number of rows and number of columns is almostthe same. The numerical methods for linear least-squares problemsand minimum-norm solutions do not generally take account ofthis special characteristic. The proposed method is based onLU factorization of the original quasi-square matrix A, assumingthat A has full rank. In the overdetermined case, the LU factorsare used to compute a basis for the null space of A^T. The right-handside vector b is then projected onto this subspace and the least-squaressolution is obtained from the solution of this reduced problem.In the case of underdetermined systems, the desired solutionis again obtained through the solution of a reduced system.The use of this method may lead to important savings in computationaltime for both dense and sparse matrices. It is also shown inthe paper that, even in cases where the matrices are quite small,sparse solvers perform better than dense solvers. Some practicalexamples that illustrate the use of the method are included.     

Application of dried spot cards as a rapid sample treatment method for determining hydroxytyrosol metabolites in human urine samples. Comparison with microelution solid-phase extraction

Two different rapid sample pretreatment strategies, dried spot cards, and microelution solid-phase extraction plates (μSPE), with ultra-high performance liquid chromatography coupled to tandem mass spectrometry (UPLC-MS/MS) have been developed and validated for the determination of hydroxytyrosol and its metabolites in spiked human urine samples. Hydroxytyrosol, hydroxytyrosol-3′-O-glucuronide, hydroxytyrosol-4′-O-glucuronide, hydroxytyrosol-3-O-sulphate, and homovanillic alcohol-4′-O-glucuronide were used as the target compounds. Using the FTA DMPK-A dried urine spot card under optimum conditions, with 5 μL of preconcentrated urine volume and 100 μL of methanol/water (50/50, v/v) as the elution solvent, the extraction recovery (%R) of the compounds studied was higher than 80 %, and the matrix effect (%ME) was less than 8 %. The stability of these cards and punching at the centre or side of the card were also studied, obtaining an excellent stability after 7 days of storage and complete homogeneity across the surface of the dried drop. The different μSPE parameters that affect the efficiency were also studied, and under optimum conditions, the %R and the %ME were higher than 70 % and lower than 17 %, respectively. The linearity range in dried urine spot cards was 2.5-20 μM for all the metabolites, with the exception of hydroxytyrosol-3-O-sulphate and hydroxytyrosol, which were 0.3-70 μM and 2.5-50 μM respectively. With regards to μSPE, the linearity range was 0.5-5 μM for all the studied compounds, except for hydroxytyrosol-3-O-sulphate, which was 0.08-5 μM. The quantification limits (LOQs) were 0.3-2.5 μM and 0.08-0.5 μM in dried spot cards and in μSPE, respectively. The two developed methods were then applied and compared for determining hydroxytyrosol and its metabolites in human 24 h-urine samples after a sustained consumption (21 days) of a phenol-enriched virgin olive oil. The metabolites identified were hydroxytyrosol in its glucuronide and sulphate forms, homovanillic alcohol in its glucuronide and sulphate forms, homovanillic acid sulphate and hydroxytyrosol acetate sulphate.     

Differential non-destructive image current detection in a fourier transform quadrupole ion trap

Dual-detector differential non-destructive Fourier transform detection in a quadrupole ion trap is shown to improve signal intensity and reduce noise compared with spectra recorded using a single detector. A larger area detector in each end-cap electrode is machined to fit its hyperbolic shape and so minimize field imperfections on the z-axis. Argon, acetophenone and bromobenzene spectra were recorded to allow a comparison between single- and dual-detector (differential) modes of detection and to demonstrate the improvement achieved with differential detection. Copyright 1999 John Wiley & Sons, Ltd.     

Pseudo-steady-state solutions for solidification in a wedge

Polubarinova-Kochina's analytical differential equation methodis used to determine the pseudo-steady-state solution to problemsinvolving the freezing (solidification) of wedges of liquidwhich are initially at their fusion temperature. In particular,we consider four distinct problems for wedges which are: freezingwith the same constant boundary temperature, freezing with thesame constant boundary heat fluxes, freezing with distinct constantboundary temperatures and freezing with distinct constant fluxesat the boundaries. For the last two problems, a Heun's differentialequation with an unknown singularity is derived, which in bothcases admits a particularly elegant simple solution for thespecial case when the wedge angle is . The moving boundariesobtained are shown pictorially.