A <Emphasis Type="Italic">QR</Emphasis>–method for computing the singular values via semiseparable matrices

Summary. The standard procedure to compute the singular value decomposition of a dense matrix, first reduces it into a bidiagonal one by means of orthogonal transformations. Once the bidiagonal matrix has been computed, the QR–method is applied to reduce the latter matrix into a diagonal one. In this paper we propose a new method for computing the singular value decomposition of a real matrix. In a first phase, an algorithm for reducing the matrix A into an upper triangular semiseparable matrix by means of orthogonal transformations is described. A remarkable feature of this phase is that, depending on the distribution of the singular values, after few steps of the reduction, the largest singular values are already computed with a precision depending on the gaps between the singular values. An implicit QR–method for upper triangular semiseparable matrices is derived and applied to the latter matrix for computing its singular values. The numerical tests show that the proposed method can compete with the standard method (using an intermediate bidiagonal matrix) for computing the singular values of a matrix.Mathematics Subject Classification (2000): 65F15, 15A18The research of the first two authors was partially supported by the Research Council K.U.Leuven, project OT/00/16 (SLAP: Structured Linear Algebra Package), by the Fund for Scientific Research–Flanders (Belgium), projects G.0078.01 (SMA: Structured Matrices and their Applications), G.0176.02 (ANCILA: Asymptotic aNalysis of the Convergence behavior of Iterative methods in numerical Linear Algebra), G.0184.02 (CORFU: Constructive study of Orthogonal Functions) and G.0455.0 (RHPH: Riemann-Hilbert problems, random matrices and Padé-Hermite approximation), and by the Belgian Programme on Interuniversity Poles of Attraction, initiated by the Belgian State, Prime Ministers Office for Science, Technology and Culture, project IUAP V-22 (Dynamical Systems and Control: Computation, Identification & Modelling). The work of the third author was partially supported by MIUR, grant number 2002014121. The scientific responsibility rests with the authors.Acknowledgments.We thank the referees for their suggestions which increased the readability of the paper.     

Symmetry in a free boundary problem for degenerate parabolic equations on unbounded domains

We use the method of Alexandroff-Serrin to establish the spherical symmetry of the ground domain and of the weak solution to a free boundary problem for a class of quasi-linear parabolic equations in an unbounded cylinder , where , with a simply connected bounded domain. The equations considered are of the type , with modeled on . We consider a solution satisfying the boundary conditions: for , and , as . We show that the overdetermined co-normal condition for , with 0$"> for at least one value , forces the spherical symmetry of the ground domain and of the solution.

     

3-Sasakian manifolds, 3-cosymplectic manifolds and Darboux theorem

We present a compared analysis of some properties of 3-Sasakian and 3-cosymplectic manifolds. We construct a canonical connection on an almost 3-contact metric manifold which generalises the Tanaka–Webster connection of a contact metric manifold and we use this connection to show that a 3-Sasakian manifold does not admit any Darboux-like coordinate system. Moreover, we prove that any 3-cosymplectic manifold is Ricci-flat and admits a Darboux coordinate system if and only if it is flat.     

Synthesis and Antiproliferative Effect of Halogenated Coumarin Derivatives

A series of 6- and 6,8-halocoumarin derivatives have been investigated as potential antiproliferative compounds against a panel of tumor and normal cell lines. Cytotoxic effects were determined by the MTT method. To investigate the potential molecular mechanism involved in the cytotoxic effect, apoptosis assay, cell cycle analysis, reactive oxygen species (ROS), and reduced glutathione analysis were performed. Among the screened compounds, coumarins 6,8-dibromo-2-oxo-2H-chromene-3-carbonitrile 2h and 6,8-diiodo-2-oxo-2H-chromene-3-carbonitrile 2k exhibited the most antiproliferative effect in thyroid cancer-derived cells TPC-1. The apoptosis assay showed that both 2h and 2k induced apoptosis in TPC-1 thyroid cancer cells. According to these experiments, both coumarins induced a slight increase in TPC-1 cells in the G2/M phase and a decrease in the S phase. A significant increase in ROS levels was observed in TPC-1 treated with diiodocoumarin 2k, while the dibromocoumarin 2h induced a decrease in ROS in a dose and time-dependent manner.     

Polymeric photoinitiators based on side-chain benzoin methyl ether and tertiary amine moieties for fast UV curable coatings

The synthesis is reported of a new series of polymeric photoinitiators obtained by copolymerization of a-methylolbenzoin methyl ether acrylate (MBA) with different N,N-dialkylamino alkyl acrylates. The copolymers have been fully characterized and employed in the photoinitiated polymerization and crosslinking reactions of a standard acrylic formulation for clear curable coatings. The photoinitiation activity of the above systems has been measured by using microwave dielectrometry and differential photocalorimetry, and compared with that found for the corresponding low-molecular-weight models. The results clearly indicate that the copolymeric systems display a remarkable decrease of the induction period and an improved overall activity in the UV curing of the acrylic coatings. An interpretation of the experimental findings is also reported.     

6-Substituted-5H-pyrrolo[2,3-b]pyrazines via palladium-catalyzed heteroannulation from N-(3-chloropyrazin-2-yl)-methanesulfonamide and alkynes

We herein report the efficient and convenient synthesis of 6-substituted-5H-pyrrolo[2,3-b]pyrazines. The reaction is a palladium-catalyzed heteroannulation process followed by deprotection to yield the desired pyrrolo[2,3-b]pyrazine substrates. The reaction starts with readily accessible N-(3-chloropyrazin-2-yl)-methanesulfonamide and commercially available terminal alkynes and works with aryl- and alkylalkynes.     

The consistency of the postulates of special relativity

In a recent article in this journal, Kingsley has tried to show that the postulates of special relativity contradict each other. Here we show that the arguments of Kingsley are invalid because of an erroneous appeal to symmetry in a non-symmetric situation. The consistency of the postulates of special relativity and the relativistic kinematics deduced from them is restated.