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.     

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.     

Octadecanuclear Gear Wheels by Self‐Assembly of Self‐Assembled “Double Saddle”‐Type Coordination Entities: Molecular “Rangoli”

A series of self‐assembled “double saddle”‐type trinuclear complexes of [Pd₃L′₃ L ₂] formulation have been synthesized by complexation of a series of cis‐protected palladium(II) components with a slightly divergent “E‐shaped” non‐chelating tridentate ligand, 1,1′‐(pyridine‐3,5‐diyl)bis(3‐(pyridin‐3‐yl)urea ( L ). The cis‐protecting agents L′ employed in the study are ethylenediamine (en), tetramethylethylenediamine (tmeda), 2,2′‐bipyridine (bpy), and 1,10‐phenanthroline (phen), for 1 , 2 , 3 , and 4 , respectively. The crystal structures of [Pd₃(tmeda)₃( L )₂](NO₃)₆ ( 2 ), [Pd₃(bpy)₃( L )₂](NO₃)₆ ( 3 ), and [Pd₃(phen)₃( L )₂](NO₃)₆ ( 4 ) unequivocally support the new architecture. Two of the “double saddle”‐type complexes ( 3 and 4 ) are suitably crafted with π surfaces at the strategically located cis‐protecting sites to facilitate intermolecular π–π interactions in the solid state. As a consequence, six units of the 3 (or 4 ) are assembled, by means of six‐pairs of π–π stacking interactions, in a circular geometry to form an octadecanuclear molecular ring of [(Pd₃L′₃ L ₂)₆] composition. The overall arrangement of the rings in the crystal packing is equated with the traditional Indian art form rangoli.     

Shimalactone Biosynthesis Involves Spontaneous Double Bicyclo-Ring Formation with 8π-6π Electrocyclization

Shimalactones A and B are neuritogenic polyketides possessing characteristic oxabicyclo[2.2.1]heptane and bicyclo[4.2.0]octadiene ring systems that are produced by the marine fungus Emericella variecolor GF10. We identified a candidate biosynthetic gene cluster and conducted heterologous expression analysis. Expression of ShmA polyketide synthase in Aspergillus oryzae resulted in the production of preshimalactone. Aspergillus oryzae and Saccharomyces cerevisiae transformants expressing ShmA and ShmB produced shimalactones A and B, thus suggesting that the double bicyclo-ring formation reactions proceed non-enzymatically from preshimalactone epoxide. DFT calculations strongly support the idea that oxabicyclo-ring formation and 8π-6π electrocyclization proceed spontaneously after opening of the preshimalactone epoxide ring through protonation. We confirmed the formation of preshimalactone epoxide in vitro, followed by its non-enzymatic conversion to shimalactones in the dark.     

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.     

Left dihedral codes over finite chain rings

Let R be a finite commutative chain ring, $D_{2n}$ be the dihedral group of size 2n and $R[D_{2n}]$ be the dihedral group ring. In this paper, we completely characterize left ideals of $R[D_{2n}]$ (called left $D_{2n}$-codes) when $\gcd (char(R),n)=1$. In this way, we explore the structure of some skew-cyclic codes of length 2 over R and also over $R\times S$, where S is an isomorphic copy of R. As a particular result, we give the structure of cyclic codes of length 2 over R. In the case where $R={\mathbb{F}}_{p^m}$ is a Galois field, we give a classification for left $D_{2N}$-codes over ${\mathbb{F}}_{p^m}$, for any positive integer N. In both cases we determine dual codes and identify self-dual ones.     

Polybenzimidazole dendrimer containing triazine rings-based high-temperature proton exchange membranes with high performances over a wide humidity range

A high-temperature proton exchange membrane with high proton conductivity over a wide humidity range still remains a challenge. PBI dendrimer containing triazine rings (TPBI) was synthesized to approach this aim considering its high content of hygroscopic terminal groups and of larger free volume. A novel proton conductor previously synthesized (zirconium 3-sulfopropyl phosphonate, ZrSP) was doped due to its good proton conductivity over a wide humidity range. TPBI was post-crosslinked with a tetrafunctional epoxy resin (N,N,N′,N′-tetraglycidyl-4,4′-diaminodiphenylmethane, TGDDM) to enhance the mechanical stability at low cross-linking degrees, which allowed high doping levels of ZrSP, and thus, high conductivity. The prepared membranes (TPBI-TGDDM/ZrSP) showed good thermal stability, high proton conductivity over wide humidity range, and good dimensional stability. At suitable degrees of branching, TPBI-TGDDM/ZrSP exhibited superior mechanical property, oxidative stability, methanol barrier property, and membrane selectivity than its linear analog (mPBI-TGDDM/ZrSP). As ZrSP instead of PA was applied as the proton conductor, TPBI-TGDDM/ZrSP showed good durability of proton conductivity, especially in comparison with TPBI-TGDDM/PA, which highly retarded decline in conductivity caused by PA leaking. The proton conductivity at 180 °C of TPBI(20)-TGDDM(10)/ZrSP(50) achieved 142, 84.2 and 23.6 mS cm^?1 at 100%, 50%, and 0 RH, respectively.