On Pairs of Diagonal Quintic Forms

We demonstrate that a pair of additive quintic equations in at least 34 variables has a nontrivial integral solution, subject only to an 11-adic solubility hypothesis. This is achieved by an application of the Hardy–Littlewood method, for which we require a sharp estimate for a 33.998th moment of quintic exponential sums. We are able to employ p-adic iteration in a form that allows the estimation of such a mean value over a complete unit square, thereby providing an approach that is technically simpler than those of previous workers and flexible enough to be applied to related problems.     

Facile syntheses of surface-functionalized micelles and shell cross-linked nanoparticles

Block copolymer micelles and shell cross-linked nanoparticles (SCKs) presenting Click-reactive functional groups on their surfaces were prepared using two separate synthetic strategies, each employing functionalized initiators for the controlled radical polymerization of acrylate and styrenic monomers to afford amphiphilic block copolymers bearing an alkynyl or azido group at the α-terminus. The first route for the synthesis of the azide-functionalized nanostructures was achieved via sequential nitroxide-mediated radical polymerization (NMP) of tert-butyl acrylate and styrene, originating from a benzylic chloride-functionalized initiator, followed by deprotection of the acrylic acids, supramolecular assembly of the block copolymer in water and conversion of the benzylic chloride to a benzylic azide. In contrast, the second strategy utilized an alkynyl-functionalized reversible addition fragmentation transfer (RAFT) agent directly for the RAFT-based sequential polymerization of tetrahydropyran acrylate and styrene, followed by selective cleavage of the tetrahydropyran esters to give the α-alkynyl-functionalized block copolymers. These Click-functionalized polymers, with the functionality located at the hydrophilic polymer termini, were then self-assembled using a mixed-micelle methodology to afford surface-functionalized “Clickable” micelles in aqueous solutions. The optimum degree of incorporation of the Click-functionalized polymers was investigated and determined to be ca. 25%, which allowed for the synthesis of well-defined surface-functionalized nanoparticles after cross-linking selectively throughout the shell layer using established amidation chemistry. Functionalization of the chain ends was shown to be an efficient process under standard Click conditions and the resulting functional groups revealed a more “solution-like” environment when compared to the functional group randomly inserted into the hydrophilic shell layer. © 2006 Wiley Periodicals, Inc. J Polym Sci Part A: Polym Chem 44: 5203–5217, 2006     

Crystallization of Arthrobacter sp. strain 1C N-(1-D-carboxyethyl)-L-norvaline dehydrogenase and its complex with NAD+

The novel NAD+-linked opine dehydrogenase from a soil isolate Arthrobacter sp. strain 1C belongs to an enzyme superfamily whose members exhibit quite diverse substrate specificites. Crystals of this opine dehydrogenase, obtained in the presence or absence of co-factor and substrates, have been shown to diffract to beyond 1.8 ? resolution. X-ray precession photographs have established that the crystals belong to space group P21212, with cell parameters a = 104.9, b = 80.0, c = 45.5 ? and a single subunit in the asymmetric unit. The elucidation of the three-dimensional structure of this enzyme will provide a structural framework for this novel class of dehydrogenases to enable a comparison to be made with other enzyme families and also as the basis for mutagenesis experiments directed towards the production of natural and synthetic opine-type compounds containing two chiral centres.     

Disk morphology and disk-to-cylinder tunability of poly(acrylic acid)-b-poly(methyl acrylate)-b-polystyrene triblock copolymer solution-state assemblies

Disk and cylindrical micellar assemblies were formed through self-organization of poly(acrylic acid)-b-poly(methyl acrylate)-b-polystyrene (PAA-b-PMA-b-PS) amphiphilic triblock copolymers with organic diamines as counterions in water/ tetrahydrofuran (THF) solvent mixtures. The system was investigated by means of transmission electron microscopy and cryogenic transmission electron microscopy. It was found that the assembled-state morphologies could be modified by alteration of the type and concentration of cationic diamine counterion undergoing interaction with the negatively charged, polyelectrolyte PAA corona block, the relative amount of water in the water/THF mixture, and the hydrophobic block chain length. Multivalency of the organic amine counterion was critical for disk formation. It was further demonstrated that a single block copolymer underwent disc-to-cylindrical micellar transitions reversibly with variation in the relative water/THF ratio. The ability to form disks beginning from either THF-rich or water-rich solutions indicated that the disk morphology was thermodynamically stable and that THF was important in keeping the micellar structure from becoming kinetically frozen. The nanoassemblies were produced having low size dispersities and were stable for at least one month. Intermediate structures between disks and cylinders were also observed, indicating two distinct kinetic pathways between the two micelle structures.     

Investigation of segmental reorganization within amphiphilic block polymer nanoparticles derived from shell crosslinked micelle templates: Shell crosslinked knedel-like inversion

The construction of nanoscopic materials by synthetic methodologies that iterate covalent and supramolecular interactions has been developed over the past three decades as a powerful method to afford complex functional materials. Indeed, the present study was nearly lost in the archives of dissertation research completed in 2001, which revealed nanoscale conformational dynamics in the segmental reorganization, and partial inversion, of topologically shell crosslinked knedel-like (SCK) nanoparticles. © 2019 Wiley Periodicals, Inc. J. Polym. Sci. 2020 , 58, 204–214     

Near-Optimal Mean Value Estimates for Multidimensional Weyl Sums

We obtain sharp estimates for multidimensional generalisations of Vinogradov’s mean value theorem for arbitrary translation-dilation invariant systems, achieving constraints on the number of variables approaching those conjectured to be the best possible. Several applications of our bounds are discussed.     

A polarity‐activation strategy for the high incorporation of 1‐alkenes into functional copolymers via RAFT copolymerization

A new synthetic methodology for the preparation of copolymers having high incorporation of 1‐alkene together with multifunctionalities has been developed by polarity‐activated reversible addition‐fragmentation chain transfer (RAFT) copolymerization. This approach provides well‐defined alternating poly(1‐decene‐alt‐maleic anhydride), expanding the monomer types for living copolymerizations. Although neither 1‐decene (DE) nor maleic anhydride (MAn) has significant reactivity in RAFT homopolymerization, their copolymers have been synthesized by RAFT copolymerizations. The controlled characteristics of DE‐MAn copolymerizations were verified by increased copolymer molecular weights during the copolymerization process. Ternary copolymers of DE and MAn, with high conversion of DE, could be obtained by using additive amounts (5 mol %) of vinyl acetate or styrene (ST), demonstrating further enhanced monomer reactivities and complex chain structures. When ST was selected as the third monomer, copolymers with block structures were obtained, because of fast consumption of ST in the copolymerization. Moreover, a wide variety of well‐defined multifunctional copolymers were prepared by RAFT copolymerizations of various functional 1‐alkenes with MAn. For each copolymerization, gel permeation chromatography analysis showed that the resulting copolymer had well‐controlled M_n values and fairly low polydispersities (PDI = 1.3–1.4), and ¹H and ¹³C NMR spectroscopies indicated strong alternating tendency during copolymerization with high incorporation of 1‐alkene units, up to 50 mol %. © 2008 Wiley Periodicals, Inc. J Polym Sci Part A: Polym Chem 46: 3488–3498, 2008