Iterative methods for linear complementarity problems with upperbounds on primary variables

This paper is concerned with iterative methods for the linear complementarity problem (LCP) of findingx and y in Rⁿ such thatc+Dx+y≥0, b-x≥0, andx ^T (c+Dx+y)=y ^T (b-x)=0 whenb>0. This is the LCP (M, q) withM=( _{l 0} ^{D t} ), which is in turn equivalent to a linear variational inequality over a rectangle. This type of problem arises, for example, from quadratic programming (QP) problems with upper and lower bounds ifD is symmetric, and from multicommodity market equilibrium problems with institutional price controls imposed ifD is not necessarily symmetric. Iterative methods for the LCP (M, q) with symmetricM are well developed and studied using QP applications. For nonsymmetric cases withM being either an H-matrix with positive diagonals, or a Z-matrix, there exists a robust iterative method with guaranteed convergence. This paper extends this algorithm so that the LCP (M, q) withM=( _{l 0} ^{D t} ) which is neither symmetric, nor an H-matrix with positive diagonals, nor a Z-matrix, can be processed when onlyD notM satisfies such properties. The case whereD is nonsymmetric is explicitly discussed.     

Synthesis, Crystal Structure and Conformational Analysis of(E)-5-Methoxy-2-((4-methoxyphenylimino)methyl)phenol

The title compound,(E)-5-methoxy-2-((4-methoxyphenylimino)methyl)phenol(C15H15NO3),crystallizes in monoclinic,space group P21/c with a=9.4361(6),b=10.6212(5),c=12.9338(9),β=93.064(5)o,V=1294.41(14)3,Z=4,Dc=1.320 g/cm3,F(000)=544,Rint=0.116,T=296 K,μ=0.09 mm-1,the final R=0.051 and wR=0.148 for 1836 observed reflections with I2σ(I).An extensive two-dimensional network of C-H…O hydrogen bonds and π-ring interactions are responsible for the crystal stabilization.Intermolecular hydrogen bonds and C-H…π interactions produce R22(14),R44(30) and R44(31) rings.In addition to the molecular geometry from X-ray experiment,the molecular geometry of the title compound in the ground state has been calculated using the semi-empirical(AM1 and PM3) and density functional theory method(DFT)(B3LYP) with 6-31G(d) basis set.To determine the conformational flexibility,molecular energy profile of the title compound was obtained by semi-empirical(PM3 and AM1) and DFT/B3LYP calculations with respect to the selected degree of torsional freedom,which varied from -180° to +180° in a step of 10°.     

Syntheses and Structural Characterization of 1-（4-Phenyl-2-thiazolyl）-3-（4-methylphenyl）-5-（2-furyl）-4,5-dihydro-（1H）-pyrazole and 1-（4-（4-Fluoro）phenyl-2-thiazolyl）-3-（4-methyl-phenyl）-5-（2-furyl）-4,5-dihydro-（1H）-pyrazole

The title compounds were synthesized and characterized by IR,1H-NMR,Mass and elementary analysis and single-crystal X-ray diffraction.In 1a,intermolecular C-H…π interactions produce a three-dimensional network.In 1b,intermolecular C-H…O hydrogen bonds generate an R22(22) ring.The hydrogen bonding is supported by C-H…π interactions.