Numerical analysis of singularly perturbed delay differential turning point problem

In this paper, we describe a numerical method based on fitted operator finite difference scheme for the boundary value problems for singularly perturbed delay differential equations with turning point and mixed shifts. Similar boundary value problems are encountered while simulating several real life processes for instance, first exit time problem in the modelling of neuronal variability. A rigorous analysis is carried out to obtain priori estimates on the solution and its derivatives for the considered problem. In the development of numerical methods for constructing an approximation to the solution of the problem, a special type of mesh is generated to tackle the delay term along with the turning point. Then, to develop robust numerical scheme and deal with the singularity because of the small parameter multiplying the highest order derivative term, an exponential fitting parameter is used. Several numerical examples are presented to support the theory developed in the paper.     

Rhodium complexes bearing tetradentate diamine-bis(phenolate) ligands

Using tetradentate, dianionic ligands, several new rhodium complexes have been prepared. Some of these diamine-bis(phenolate) compounds, are active for C-H activation of benzene. These complexes are air and thermally stable. All four complexes were characterized by X-ray diffraction.     

Interconversion of host–guest components in supramolecular assemblies of polycarboxylic acids and reduced Schiff bases

Eight supramolecular assemblies of benzene-1,3,5-tricarboxylic acid (H₃BTC) and benzene-1,2,4,5-tetracarboxylic acid (H₄BTtC) with reduced Schiff base of flexible backbone having phenolic and pyridyl groups, i.e., 1,2-bis(2-hydroxybenzylamino)ethane, 1,3-bis(2-hydroxybenzylamino)propane, 1,4-bis(2-hydroxybenzylamino)butane and 1,4-bis(4-pyridinylmethylamino)butane have been constructed by proton transfer reaction. H₃BTC forms host–guest type assemblies with amines having phenolic functionality, while a layered structure was obtained with pyridyl functionalized amine. H₄BTtC also formed host–guest assemblies with the diamines where reduced Schiff base acts as host and acid moiety acts as the guest. Different conformations of the diamines were observed in these assemblies. Theoretical studies were performed to analyze the effect of varied chain lengths of diamines on hydrogen bond interaction energy of the adducts.     

A geometric characterization of arithmetic varieties

A result of Belyi can be stated as follows. Every curve defined over a number field can be expressed as a cover of the projective line with branch locus contained in a rigid divisor. We define the notion of geometrically rigid divisors in surfaces and then show that every surface defined over a number field can be expressed as a cover of the projective plane with branch locus contained in a geometrically rigid divisor in the plane. The main result is the characterization of arithmetically defined divisors in the plane as geometrically rigid divisors in the plane.