The dual of a constacyclic code,self dual,reversible constacyclic codes and constacyclic codes with complementary duals over finite local Frobenius non-chain rings with nilpotency index 3

Over finite local Frobenius non-chain rings with nilpotency index 3 and when the length of the codes is relatively prime to the characteristic of the residue field of the ring, the structure of the dual of $\gamma$-constacyclic codes is established and the algebraic characterization of self-dual $\gamma$-constacyclic codes, reversible $\gamma$-constacyclic codes and $\gamma$-constacyclic codes with complementary dual are given. Generators for the dual code are obtained from those of the original constacyclic code.     

Synthesis of Seven-Membered Benzolactones by Nickel-Catalyzed C−H Coupling of Benzamides with Oxetanes

A NiCl₂(PEt₃)₂-catalyzed regioselective C−H coupling of 8-aminoquinoline-derived benzamides with oxetanes has been developed. The reaction proceeds with concomitant removal of the 8-aminoquinoline auxiliary to directly form the corresponding seven-membered benzolactones, which frequently occur in natural products and bioactive molecules. Additionally, no stereochemical erosion is observed during the course of the reaction, and the use of enantioenriched and substituted oxetane thus provides a new avenue to the optically active benzolactone.     

On the variety of triangles for a hyper-Kähler fourfold constructed by Debarre and Voisin

We study the similarities between the Fano varieties of lines on a cubic fourfold, a hyper-Kähler fourfold studied by Beauville and Donagi, and the hyper-Kähler fourfold constructed by Debarre and Voisin in [3]. We exhibit an analog of the notion of “triangle” for these varieties and prove that the 6-dimensional variety of “triangles” is a Lagrangian subvariety in the cube of the constructed hyper-Kähler fourfold.     

Modular class primes in the Sundaram sieve

The purpose of this paper is to consider analogues of the twin-prime conjecture in various classes within modular rings.     

Commuting traces of multilinear maps on invertible elements

Let R be a unital simple ring. Under some technical restrictions, we characterize m-linear mappings G:R^m→R satisfying [G(u,…,u),u]?=?0 for all unit u∈R.