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.     

Synthesis and liquid crystalline behavior of bulky poly(methacrylamide)s

N‐(Bis(4‐(2‐ethylhexyloxy)phenyl)(phenyl)‐methyl)methacrylamide was synthesized and polymerized via reversible addition‐fragmentation chain‐transfer (RAFT) polymerization. The chain‐transfer agent (4‐cyano‐4‐(phenylcarbonothioylthio) pentanoic acid (CPADB)), combined with a chiral additive, and a radical initiator yielded polymers with dispersities between 1.2 and 1.4. At low concentrations, the polymers are soluble in hexanes and chloroform while at higher concentrations they swell in these solvents. Characterization of the polymers by wide‐angle X‐ray scattering (WAXS) revealed an interplanar distance of 19.0 Å. The WAXS data combined with polarized optical microscopy support a lamellar crystallization and lyotropic liquid crystalline behavior in hexanes and chloroform. © 2015 Wiley Periodicals, Inc. J. Polym. Sci., Part A: Polym. Chem. 2015 , 53, 2563–2568     

Synthesis of tertiary amine‐based pH‐responsive polymers by RAFT Polymerization

Well‐defined tertiary amine‐based pH‐responsive homopolymers and block copolymers were synthesized via reversible addition‐fragmentation chain transfer (RAFT) polymerization using 4‐cyanopentanoic acid dithiobenzoate (CPAD) as the RAFT agent for homopolymers and a poly(ethylene glycol) (PEG) macro‐RAFT agent for the block copolymers. ¹H NMR and gel permeation chromatography results confirmed the successful synthesis of these homopolymers and block copolymers. Kinetics studies indicated that the formation of both the homopolymers and the block copolymers were well defined. The pKa titration experiments suggested that the homopolymers and the related block copolymers have a similar pKa. The dynamic light scattering investigation showed that all of the block copolymers underwent a sharp transition from unimers to micelles around their pKa and the hydrodynamic diameter (D_h) was not only dependent on the molecular weight but also on the composition of the block copolymers. The polymer solution of PEG‐b‐PPPDEMA formed the largest micelle compare to the PEG‐b‐PDPAEMA and PEG‐b‐PDBAEMA with a similar molecular weight. © 2015 Wiley Periodicals, Inc. J. Polym. Sci., Part A: Polym. Chem. 2015 , 53, 1010–1022     

Synthesis and characterization of double hydrophilic block copolymers containing semi-rigid and flexible segments

3-Methyl-(E)-stilbene (3MSti) and 4-(diethylamino)-(E)-stilbene (DEASti) monomers are synthesized and polymerized separately with maleic anhydride (MAn) in a strictly alternating fashion using reversible addition-fragmentation chain transfer (RAFT) polymerization techniques. The optimal RAFT chain transfer agents (CTAs) for each copolymerization affect the reaction kinetics and CTA compatibilities. Psuedo-first order polymerization kinetics are demonstrated for the synthesis of poly((3-methyl-(E)-stilbene)-alt-maleic anhydride) (3MSti-alt-MAn) with a thiocarbonylthio CTA (methyl 2-(dodecylthiocarbonothioylthio)−2-methylpropionate, TTCMe). In contrast, a dithioester CTA (cumyl dithiobenzoate, CDB) controls the synthesis of poly((4-(diethylamino)-(E)-stilbene)-alt-maleic anhydride) (DEASti-alt-MAn) with pseudo-first order polymerization kinetics. DEASti-alt-MAn is chain extended with 4-acryloylmorpholine (ACMO) to synthesize diblock copolymers and subsequently converted to a double hydrophilic polyampholyte block copolymers (poly((4-(diethylamino)-(E)-stilbene)-alt-maleic acid))-b-acryloylmorpholine) (DEASti-alt-MA)-b-ACMO) via acid hydrolysis. The isoelectric point and dissociation behavior of these maleic acid-containing copolymers are determined using ζ-potential and acid–base titrations, respectively. © 2014 Wiley Periodicals, Inc. J. Polym. Sci., Part A: Polym. Chem. 2015 , 53, 219–227     

Synthesis of polymer with defined fluorescent end groups via reversible addition fragmentation transfer polymerization for characterizing the conformations of polymer chains in solutions

A new type of chain transfer agent used in reversible addition fragmentation chain transfer (RAFT) polymerization named 9‐anthracenylmethyl (4‐cyano‐4‐(N‐carbazylcarbodithioate) pentanoate) (ACCP) was synthesized with a total yield over 75% by the incorporation of both fluorescent donor and acceptor chromophores. Polymerization of heterotelechelic α,ω end‐labeled dye‐functionalized polystyrene (PS), poly(methyl methacrylate) (PMMA), and poly(n‐butyl methacrylate) (PBMA) with adjustable molecular weights and narrow polydispersity could be conducted by a one‐pot procedure through RAFT polymerization with this bischromophore chain transfer agent. The polymerizations demonstrated “living” controlled characteristics. By taking advantage of the characteristic fluorescence resonance energy transfer (FRET) response between the polymer chain terminals, the variation of chain dimensions in solution from the dilute region to the semidilute region can be monitored by changes in the ratio of the fluorescence intensities of the carbazolyl group to the anthryl group, which lends itself to potential applications in characterizing chain dimensions in solutions for thermodynamic or dynamic studies. © 2016 Wiley Periodicals, Inc. J. Polym. Sci., Part A: Polym. Chem. 2016 , 54, 2413–2420