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.     

Dimerisation of Dipiperidinoacetylene: Convenient Access to Tetraamino-1,3-Cyclobutadiene and Tetraamino-1,2-Cyclobutadiene Metal Complexes

The reaction of 1,2-dipiperidinoacetylene ( 1 ) with 0.5 equivalents of SnCl₂ or GeCl₂⋅dioxane afforded the 1,2,3,4-tetrapiperidino-1,3-cyclobutadiene tin and germanium dichloride complexes 2 a and 2 b , respectively. A competing redox reaction was observed with excess amounts of SnCl₂, which produced a tetrapiperidinocyclobutadiene dication with two trichlorostannate(II) counterions. Heating neat 1 to 110 °C for 16 h cleanly produced the dimer 1,3,4,4-tetrapiperidino-3-buten-1-yne ( 3 ); its reaction with stoichiometric amounts of SnCl₂ or GeCl₂⋅dioxane furnished the 1,3,4,4-tetrapiperidino-1,2-cyclobutadiene tin and germanium dichloride complexes 4 a and 4 b , respectively. Transition-metal complexes containing this novel four-membered cyclic bent allene (CBA) ligand were prepared by reaction of 3 with [(tht)AuCl], [RhCl(CO)₂]₂, and [(Me₃N)W(CO)₅] to form [(CBA)AuCl] ( 5 ), [(CBA)RhCl(CO)₂] ( 6 ), and [(CBA)W(CO)₅] ( 7 ). The molecular structures of all compounds 2 – 7 were determined by X-ray diffraction analyses, and density functional theory (DFT) calculations were carried out to rationalise the formation of 3 and 4 a .     

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.