Half Inverse Riordan Arrays and Their Related Vertical Recurrence Relation

We discuss two different procedures to study the half Riordan arrays and their inverses. One of the procedures shows that every Riordan array is the half Riordan array of a unique Riordan array. It is well known that every Riordan array has its half Riordan array. Therefore, this paper answers the converse question: Is every Riordan array the half Riordan array of some Riordan arrays? In addition, this paper shows that the vertical recurrence relation of the column entries of the half Riordan array is equivalent to the horizontal recurrence relation of the original Riordan array''s row entries.     

A symbolic operator approach to several summation formulas for power series II

Here expounded is a kind of symbolic operator method that can be used to construct many transformation formulas and summation formulas for various types of power series including some old ones and more new ones.     

On the Applications of the Linear Recurrence Relationships to Pseudoprimes

We present here some results on the applications of linear recursive sequences of order $2$ to the Fermat pseudoprimes, Fibonacci pseudoprimes, and Dickson pseudoprimes.     

Recognition of essential purines by the U1A protein

Background  

The RNA recognition motif (RRM) is one of the largest families of RNA binding domains. The RRM is modulated so that individual proteins containing RRMs can specifically recognize RNA targets with diverse sequences and structures. Understanding the principles governing this specificity will be important for the rational modification and design of RRM-RNA complexes.     

The Sheffer group and the Riordan group

We define the Sheffer group of all Sheffer-type polynomials and prove the isomorphism between the Sheffer group and the Riordan group. An equivalence of the Riordan array pair and generalized Stirling number pair is also presented. Finally, we discuss a higher dimensional extension of Riordan array pairs.     

Structure and dynamics of a dense dipolar system in an electric field and their relevance to electrorheological fluids

Results of computer simulations of a dense system of dipolar spheres in an electric field are summarized. Dissipative and Hamiltonian dynamics algorithms have been used to find energy minima of the system for varying particle densities. The structures obtained by these simulations are in reasonable agreement with experimentally observed structures in electrorheological (ER) fluids. Qualitative agreement is also obtained with the limited available experimental observations on the dynamics of ER fluids.     

Bismuth compounds in organic synthesis. Bismuth nitrate catalyzed chemoselective synthesis of acylals from aromatic aldehydes

Aromatic aldehydes are smoothly converted into the corresponding acylals in good yields in the presence of 3-10 mol% Bi(NO₃)₃·5H₂O. Ketones are not affected under the reaction conditions. The relatively non-toxic nature of the catalyst, its ease of handling, easy availability and low cost make this procedure especially attractive for large-scale synthesis.     

Iron(III) p-toluenesulfonate catalyzed synthesis of homoallyl ethers from acetals and aldehydes

Iron(III) p-toluenesulfonate, Fe(OTs)₃·6H₂O, is an inexpensive, versatile and commercially available catalyst for the allylation of acetals using allyltrimethylsilane to yield homoallyl ethers in moderate to good yields. The one-pot conversion of aldehydes to homoallyl ethers using alkoxysilanes has also been accomplished using Fe(OTs)₃·6H₂O as a catalyst. The use of mild reaction conditions and a relatively non-corrosive catalyst make this method an attractive option for the synthesis of a range of homoallyl ethers.