Crystallographic and conformational analysis of two novel trans-azo benzene compounds

The molecular and crystal structure of (E)-2-Acetyl-4-(2-bromophenyldiazenyl)phenol (1) and (E)-2-Methyl-4-(o-tolyldiazenyl)phenol (2) were characterized and determined by single crystal X-ray diffraction method besides spectroscopic means. The periodic organization of 1 is stabilized by C–H···O type weak H-bond and Br···O type weak halogen bonding and thus, a two dimensional puckered network is established almost parallel to () the plane. Molecules of 2 are linked into C(7) chains generated by translation along the [1 0 1] direction with the aid of O–H···N type H-bonds, and these chains are strengthened by C–H···π interactions involving o-tolylphenol ring. Quantum chemical studies at B3LYP/6-311 ++G(d,p) level reveal that potential barrier of the compounds around Ar–N torsions is of double minimum character unless it is defected by the presence of o-substituent groups in the vicinity of the azo bridge. The results from crystallographic and quantum chemical studies suggest that azo benzene compounds may adapt non-planar geometry apart from the most stable planar conformation, which is located on the secondary minima of double potential barrier regarding rotational motion around Ar–N bonds.     

The Taylor series method of order p and Adams–Bashforth method on time scales

A recent study on the Taylor series method of second order and the trapezoidal rule for dynamic equations on time scales has been continued by introducing a derivation of the Taylor series method of arbitrary order $p$ on time scales. The error and convergence analysis of the method is also obtained. The 2-step Adams–Bashforth method for dynamic equations on time scales is concluded and applied to examples of initial value problems for nonlinear dynamic equations. Numerical results are presented and discussed.     

On the Computation of the Macroscopic Tangent for Multiscale Volumetric Homogenization Problems

For the computation of the macroscopic tangent that is required in multiscale volumetric homogenization techniques, two methods are summarized. First, a condensation approach is followed where the linearity of variational terms associated with the penalty enforcement of boundary conditions is explored in order to extract a macroscopic tangent at any stage of the Newton–Raphson iterations of a microstructural testing procedure. As an alternative approach, a second method is explored that requires only the knowledge of infinitesimally close macroscopic deformation states. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)