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.     

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.     

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.     

Phosphoryl‐Rich Flame‐Retardant Ions (FRIONs): Towards Safer Lithium‐Ion Batteries

The functionalized catecholate, tetraethyl (2,3‐dihydroxy‐1,4‐phenylene)bis(phosphonate) (H₂‐DPC), has been used to prepare a series of lithium salts Li[B(DPC)(oxalato)], Li[B(DPC)₂], Li[B(DPC)F₂], and Li[P(DPC)₃]. The phosphoryl‐rich character of these anions was designed to impart flame‐retardant properties for their use as potential flame‐retardant ions (FRIONs), additives, or replacements for other lithium salts for safer lithium‐ion batteries. The new materials were fully characterized, and the single‐crystal structures of Li[B(DPC)(oxalato)] and Li[P(DPC)₃] have been determined. Thermogravimetric analysis of the four lithium salts show that they are thermally stable up to around 200 °C. Pyrolysis combustion flow calorimetry reveals that these salts produce high char yields upon combustion.     

From Ion‐Like Ethylzinc Aluminates to [EtZn(arene)2]+[Al(ORF)4]− Salts

Attempts to prepare previously unknown simple and very Lewis acidic [RZn]⁺[Al(OR^F)₄]^? salts from ZnR₂, AlR₃, and HO?R^F delivered the ion‐like RZn(Al(OR^F)₄) (R=Me, Et; R^F=C(CF₃)₃) with a coordinated counterion, but never the ionic compound. Increasing the steric bulk in RZn⁺ to R=CH₂CMe₃, CH₂SiMe₃, or Cp*, thus attempting to induce ionization, failed and led only to reaction mixtures including anion decomposition. However, ionization of the ion‐like EtZn(Al(OR^F)₄) compound with arenes yielded the [EtZn(arene)₂]⁺[Al(OR^F)₄]^? salts with arene=toluene, mesitylene, or o‐difluorobenzene (o‐DFB)/toluene. In contrast to the ion‐like EtZn(η³‐C₆H₆)(CHB₁₁Cl₁₁), which co‐crystallizes with one benzene molecule, the less coordinating nature of the [Al(OR^F)₄]^? anion allowed the ionization and preparation of the purely organometallic [EtZn(arene)₂]⁺ cation. These stable materials have further applications as, for example, initiators of isobutene polymerization. DFT calculations to compare the Lewis acidities of the zinc cations to those of a large number of organometallic cations were performed on the basis of fluoride ion affinity. The complexation energetics of EtZn⁺ with arenes and THF was assessed and related to the experiments.     

A comparative study of truly incompressible and weakly compressible SPH methods for free surface incompressible flows

In this paper, the performance of the incompressible SPH (ISPH) method and an improved weakly compressible SPH (IWCSPH) method for free surface incompressible flows are compared and analyzed. In both methods, the Navier–Stokes equations are solved, and no artificial viscosity is used. The ISPH algorithm in this paper is based on the classical SPH projection method with common treatments on solid boundaries and free surfaces. The IWCSPH model includes some advanced corrective algorithms in density approximation and solid boundary treatment (SBT). In density approximation, the moving least squares (MLS) approach is applied to re‐initialize density every several steps to obtain smoother and more stable pressure fields. An improved coupled dynamic SBT algorithm is implemented to obtain stable pressure values near solid wall areas and, thus, to minimize possible numerical oscillations brought in by the solid boundaries. Three representative numerical examples, including a benchmark test for hydrostatic pressure, a dam breaking problem and a liquid sloshing problem, are comparatively analyzed with ISPH and IWCSPH. It is demonstrated that the present IWCSPH is more attractive than ISPH in modeling free surface incompressible flows as it is more accurate and more stable with comparable or even less computational efforts. Copyright © 2013 John Wiley & Sons, Ltd.     

Complications in the Castagnoli-Cushman reaction: An unusual course of reaction between cyclic anhydrides and sterically hindered indolenines

Attempted reaction of indolenines (which represent rather sterically hindered cyclic imines) with a series of dicarboxylic acid anhydrides yielded no expected product, the Castagnoli-Cushman lactam. Instead, products presumably formed via N-acyliminium species trapping by a carboxylate anion. Among them, hydrolytically labile 2:2 adducts of an indolenine and a cyclic anhydride, containing a 16-membered cyclic core, are particularly intriguing. This result contradicts the recently reported successful Castagnoli-Cushman reaction of indolenines with homophthalic anhydride suggesting a mechanistic switch in the course of the reaction.     

Generalized matching theorems for closed coverings of convex sets

In this paper we use fixed point and coincidence theorems due to Park [8] to give matching theorems concerning closed coverings of nonempty convex sets in a real topological vector space. Our new results extend previously given ones due to Ky Fan [2], [3], Shih [10], Shih and Tan [11], and Park [7].