Clustering via minimum volume ellipsoids

We propose minimum volume ellipsoids (MVE) clustering as an alternative clustering technique to k-means for data clusters with ellipsoidal shapes and explore its value and practicality. MVE clustering allocates data points into clusters in a way that minimizes the geometric mean of the volumes of each cluster’s covering ellipsoids. Motivations for this approach include its scale-invariance, its ability to handle asymmetric and unequal clusters, and our ability to formulate it as a mixed-integer semidefinite programming problem that can be solved to global optimality. We present some preliminary empirical results that illustrate MVE clustering as an appropriate method for clustering data from mixtures of “ellipsoidal” distributions and compare its performance with the k-means clustering algorithm as well as the MCLUST algorithm (which is based on a maximum likelihood EM algorithm) available in the statistical package R. Research of the first author was supported in part by a Discovery Grant from NSERC and a research grant from Faculty of Mathematics, University of Waterloo. Research of the second author was supported in part by a Discovery Grant from NSERC and a PREA from Ontario, Canada.     

The elliptic K3 surfaces with a maximal singular fibre

We give the defining equation of complex elliptic K3 surfaces with a maximal singular fibre. Then we study the reduction modulo p at a particularly interesting prime p. To cite this article: T. Shioda, C. R. Acad. Sci. Paris, Ser. I 337 (2003).     

Transport with a Very Low Density Contrast in Hele–Shaw Cell and Porous Medium: Evolution of the Mixing Zone

The optimal concentration of a blue dye solution with 'tracer' properties, enabling a pollutant to be marked was determined by the use of numerical, theoretical and experimental approaches. Experimental investigations were performed on a transparent Hele–Shaw cell and the concentration distribution was analyzed using an optical technique based on dye light absorption properties. The injected optimal concentration was established thanks to a theoretical and experimental study carried out on the output signal dynamics. Using the same experimental conditions, numerical simulations were performed. The very good agreement between the data (experimental and numerical) clarified that: (i) the choice of the blue dye optimal concentration was valid and (ii) the concentration-dependent density should not be neglected in flow and transport equations even if it concerns a so-called 'tracer'. Following this remark, a theoretical aspect was developed in order to determine the analogous conditions between a Hele–Shaw cell and a porous medium for the variable density transport phenomenon. The structure of the concentration-dependent dispersion tensor used in the numerical code was obtained by homogenizing the Stokes flow of a bi-component mixture. The numerical results show that, as long as the tracer density does not exceed a certain value, it is not necessary to take into account a density contrast in terms of the dispersion tensor. The classical form of the Taylor dispersion tensor can be used successfully.     

Linearly implicit time discretization of non-linear parabolic equations

We give a stability and error analysis of linearly implicitone-step methods for time discretization of non-linear parabolicequations. We derive precise error bounds for Rosenbrock andW-methods, and we explain the error reduction by Richardsonextrapolation of the linearly implicit Euler method which occursin spite of the breakdown of asymptotic expansions. The parabolicequations are studied in a Hilbert space framework that includessemilinear and quasilinear parabolic equations, and also stiffreaction-diffusion equations with reactions at different timescales.     

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Book reviews

     

Zur konstruktion bewerteter dicksonscher fastkörper

Zusammenfassung  After a short introduction to the valuation theory of nearfields, we give criteria for when a valuation of a skewfield F is also a valuation of the Dickson nearfield F^K which is derived from F by the coupling K on F. For the construction of examples, a rational function field F =K(t) is given. The set of all prolongations of a valuation v on K to F is well known. Sufficient conditions are given which guarantee that couplings K on F and elements are in this sense compatible so that w is a valuation of the Dickson nearfield F^K. Examples demonstrate the results.

     

Box-modeling of bone and tooth phosphate oxygen isotope compositions as a function of environmental and physiological parameters*

A time-dependent box model is developed to calculate oxygen isotope compositions of bone phosphate as a function of environmental and physiological parameters. Input and output oxygen fluxes related to body water and bone reservoirs are scaled to the body mass. The oxygen fluxes are evaluated by stoichiometric scaling to the calcium accretion and resorption rates, assuming a pure hydroxylapatite composition for the bone and tooth mineral. The model shows how the diet composition, body mass, ambient relative humidity and temperature may control the oxygen isotope composition of bone phosphate. The model also computes how bones and teeth record short-term variations in relative humidity, air temperature and δ¹⁸O of drinking water, depending on body mass. The documented diversity of oxygen isotope fractionation equations for vertebrates is accounted for by our model when for each specimen the physiological and diet parameters are adjusted in the living range of environmental conditions.     

On the Behavior of the Solutions for Certain Linear Autonomous Difference Equations

Certain linear autonomous delay as well as neutral delay difference equations are considered. A class of linear autonomous delay difference equations with continuous variable is also considered. Some results on the behavior of the solutions are established via two distinct positive roots of the corresponding characteristic equation.     

CuSO4/sodium ascorbate catalysed synthesis of benzosuberone and 1,2,3-triazole conjugates: Design,synthesis and in vitro anti-proliferative activity

A novel series of conjugates of benzosuberone and 1,2,3-triazole i.e. 3-(4-phenyl-1H-1,2,3-triazol-1-yl)propyl-9-chloro-2,3-dimethyl-6,7-dihydro-5H-benzo[7]annulene-8-carboxylic acids (8a-j) were synthesized in good to excellent yields catalysed by CuSO₄ under milder reaction conditions and evaluated for their anti-proliferative activity. The structural elucidation of the prepared compounds was carried out using IR, ¹H NMR, ¹³C NMR and Mass spectral analysis. The newly synthesized derivatives (8a-j) were evaluated for their anti-proliferative activity against four human cell lines and the novel derivatives showed moderate to excellent activity. The obtained results suggest that these compounds can be considered as new hits for anti-proliferative drug development programme and further SAR studies can help obtain better anticancer agents.