Tauberian theorems for Abel limitability method

This paper addresses conditions for the Abel method of limitability to imply convergence and subsequential convergence.      

On geometric interpretations of split quaternions

Quaternions are an important tool that provides a convenient and effective mathematical method for representing reflections and rotations in three-dimensional space. A unit timelike split quaternion represents a rotation in the Lorentzian space. In this paper, we give some geometric interpretations of split quaternions for lines and planes in the Minkowski 3-space with the help of mutual pseudo orthogonal planes. We classified mutual planes with respect to the casual character of the normals of the plane as follows; if the normal is timelike, then the mutual plane is isomorphic to the complex plane; if the normal is spacelike, then the plane is isomorphic to the hyperbolic number plane (Lorentzian plane); if the normal is lightlike, then the plane is isomorphic to the dual number plane (Galilean plane).     

New olefinic centred binuclear clamshell type phthalocyanines: Design, synthesis, structural characterisation, the stability and the change in the electron cloud at olefine-based symmetrical diphthalonitrile fragment by the combined application of UV–Vis electronic structure and theoretical methods

The olefinic centred Schiff base (3) was obtained from the condensation of substituted dialdehyde (1) with 2-amino-4-methylphenol (2) in a 1:2 ratio. The diphthalonitrile derivative (5) was prepared by the reaction of 4-nitrophthalonitrile (4) and compound (3) in dry dimethylformamide/potassium carbonate. The key product (5) was obtained by nucleophilic substitution of an activated nitro group into an aromatic ring. The cyclotetramerization of compound (5) with phthalonitrile (6) in 1:6.15 ratio gave the expected metal-free phthalocyanine of clamshell type (7), and with metal salts of Zn(II), Ni(II), Co(II) and Cu(II) gave metallophthalocyanines of clamshell types (8-11), respectively in dimethylaminoethanol/1,8-diazabycyclo[5.4.0]undec-7-ene system. The products were purified by several techniques such as crystallization and preparative thin layer chromatography. The newly prepared compounds were characterised by a combination of elemental analyses, IR, ¹H/¹³C NMR, MS and UV-Vis spectroscopy.     

Plain ditopological texture spaces

Using the characterization of plain textures in terms of posets given by Mustafa Demirci (M. Demirci, Textures and C-spaces, Fuzzy Sets and Systems 158 (11) (2007) 1237-1245), the authors consider the important class of plain ditopological texture spaces and give several new results.     

Constructing rational and multi‐wave solutions to higher order NEEs via the Exp‐function method

In this paper, we present an application of some known generalizations of the Exp‐function method to the fifth‐order Burgers and to the seventh‐order Korteweg de Vries equations for the first time. The two examples show that the Exp‐function method can be an effective alternative tool for explicitly constructing rational and multi‐wave solutions with arbitrary parameters to higher order nonlinear evolution equations. Being straightforward and concise, as pointed out previously, this procedure does not require the bilinear representation of the equation considered. Copyright © 2011 John Wiley & Sons, Ltd.     

Surface wave scattering by a material filled rectangular impedance groove in a reactive plane

The problem of TE-polarized surface wave scattering from a rectangular impedance groove located on an infinite reactive plane which is filled with dielectric material is considered for a rather general case where the impedances of the horizontal and vertical sides of the groove have different values. The multiple interactions up to the second order between the edges of the groove are obtained to yield diffracted field. The diffraction problem is first reduced into a modified Wiener-Hopf equation and then solved approximately. The solution contains branch-cut integrals and two infinite sets of constants satisfying two infinite systems of linear algebraic equations. The approximate analytical evaluations of the corresponding integrals as well as the numerical solutions of the linear algebraic equation systems are obtained for various values of the parameters such as the surface reactance of the guiding plane, the vertical and horizontal wall impedances of the groove, the permittivity of the material loading, the width and the height of the groove which permit one to study the effect of these parameters on the diffraction phenomenon.     

Benzimidazolin‐2‐ylidene–palladium‐catalysed coupling reactions of aryl halides

The in situ prepared three‐component system Pd(OAc)₂–1,3‐dialkylbenzimidazolium chlorides ( 2a – f ) and Cs₂CO₃ catalyses, quantitatively, the Suzuki cross‐coupling of deactivated aryl chlorides and Heck coupling reactions of aryl bromide and iodide substrates. The 1,3‐dialkylbenzimidazolium salts ( 2a – f ) were characterized by conventional spectroscopic methods and elemental analysis. Copyright © 2005 John Wiley & Sons, Ltd.