A facile protocol for the preparation of 5-alkylidene and 5-imino substituted hydantoins from N,N′-disubstituted parabanic acids

A convenient procedure for the preparation of various substituted (thio)hydantoins is described. The method is based on Wittig and aza-Wittig reactions of parabanic acids with phosphonium ylides. The reactions occurred both regio- and stereo-selectively.     

Continuous production of (S)-1-phenylethanol by immobilized cells of Rhodotorula glutinis with a specially designed process

A specially designed process for the continuous production of a chiral alcohol by immobilized Rhodotorula glutinis was prepared and is reported. The performance of the process with immobilized cells was also investigated. The run time of the immobilized cells, the flow rates of the substrate, and tris buffer containing 4% glucose in the process were optimized. The immobilized R. glutinis biocatalyst could be used for 15 days with maximum reaction activity. Under the optimized conditions, the continuous production process was operated for 30 days and resulted in 10.8 mL (S)-1-phenylethanol [(S)-1-PE]. The process has been demonstrated on a multigram scale in 75% overall yield with a purity of >99%.     

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).     

On Tauberian theorems for (A)(C, α) summability method

In this paper we introduce some Tauberian conditions for the (A)(C, α) summability method. These results extend and generalize some of the classical Tauberian theorems for the Abel summability method.     

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.     

On Korovkin Type Theorem in the Space of Locally Integrable Functions

It is shown that a Korovkin type theorem for a sequence of linear positive operators acting in weighted space L _p,w (loc) does not hold in all this space and is satisfied only on some subspace.