Counting Sums of Three Squares

Let (x) be the error term associated with the asymptotic formulafor the counting function on the set of numbers that are sumsof three squares. We give a formula for (x) and use it to provethat (x) has average order and normal order 3 1og x/(16 1og2).     

Pyridyl Pyrrolide Boron Complexes: The Facile Generation of Thermally Activated Delayed Fluorescence and Preparation of Organic Light‐Emitting Diodes

The electron positive boron atom usually does not contribute to the frontier orbitals for several lower‐lying electronic transitions, and thus is ideal to serve as a hub for the spiro linker of light‐emitting molecules, such that the electron donor (HOMO) and acceptor (LUMO) moieties can be spatially separated with orthogonal orientation. On this basis, we prepared a series of novel boron complexes bearing electron deficient pyridyl pyrrolide and electron donating phenylcarbazolyl fragments or triphenylamine. The new boron complexes show strong solvent‐polarity dependent charge‐transfer emission accompanied by a small, non‐negligible normal emission. The slim orbital overlap between HOMO and LUMO and hence the lack of electron correlation lead to a significant reduction of the energy gap between the lowest lying singlet and triplet excited states (ΔE_T‐S) and thereby the generation of thermally activated delay fluorescence (TADF).     

Are the radical centers in peptide radical cations mobile? The generation, tautomerism, and dissociation of isomeric alpha-carbon-centered triglycine radical cations in the gas phase

The mobility of the radical center in three isomeric triglycine radical cations[G(*)GG](+), [GG(*)G](+), and [GGG(*)](+) has been investigated theoretically via density functional theory (DFT) and experimentally via tandem mass spectrometry. These radical cations were generated by collision-induced dissociations (CIDs) of Cu(II)-containing ternary complexes that contain the tripeptides YGG, GYG, and GGY, respectively (G and Y are the glycine and tyrosine residues, respectively). Dissociative electron transfer within the complexes led to observation of [Y(*)GG](+), [GY(*)G](+), and [GGY(*)](+); CID resulted in cleavage of the tyrosine side chain as p-quinomethide, yielding [G(*)GG](+), [GG(*)G](+), and [GGG(*)](+), respectively. Interconversions between these isomeric triglycine radical cations have relatively high barriers (> or = 44.7 kcal/mol), in support of the thesis that isomerically pure [G(*)GG](+), [GG(*)G](+), and [GGG(*)](+) can be experimentally produced. This is to be contrasted with barriers < 17 kcal/mol that were encountered in the tautomerism of protonated triglycine [Rodriquez C. F. et al. J. Am. Chem. Soc. 2001, 123, 3006-3012]. The CID spectra of [G(*)GG](+), [GG(*)G](+), and [GGG(*)](+) were substantially different, providing experimental proof that initially these ions have distinct structures. DFT calculations showed that direct dissociations are competitive with interconversions followed by dissociation.     

Metal‐Sensitive Hydrothermal Synthesis and Structural Characterization of a Mixed‐Metal and Porous Coordination Polymer Containing 2,3‐Pyridinedicarboxylate

The hydrothermal reaction of 2,3‐pyridinedicarboxylic acid (2,3‐H₂pda) with a mixture of Cd(NO₃)₂ and Ni(NO₃)₂ afforded a coordination polymer, [CdNi(2,3‐pda)₂(H₂O)₃]_∞ ( 1 ); in contrast, that with a mixture of Cd(NO₃)₂ and Zn(NO₃)₂ surprisingly produced a discrete molecule, trans‐[Cd(3‐pa)₂(H₂O)₄] ( 2 ) (3‐pa^? = 3‐pyridinecarboxylate). Since a direct reaction between a single metal salt, Cd(NO₃)₂ or Zn(NO₃)₂, and 3‐pyridinecarboxylic acid (3‐Hpa) under similar hydrothermal conditions yielded different coordination polymers containing 3‐pa^?, it appears that the apparently thermal decarboxylation from ligated 2,3‐pda^2? to 3‐pa^? occurs after complexation of both metal cations, Cd(II) and Zn(II). A new coordination mode, formed for 2,3‐pda^2? in structure 1 , appears to help formation of microporous channels by piling up the observed 2D hydrogen‐bonded heteropolynuclear layers. Each channel apparently consists of two interpenetrating 6³ Cd(II) and Ni(II) nets.     

Direct Electrochemistry of Glucose Oxidase at Carbon Nanotube‐gold Colloid Modified Electrode with Poly(diallyldimethylammonium chloride) Coating

Glucose oxidase showed direct electrochemical transfer at glassy carbon electrodes immobilized with carbon nanotube‐gold colloid (CNT‐Au) composites with poly(diallydimethylammonium chloride) (PDDA) coatings. The modified electrode (GC/CNT/Au/PDDA‐GOD) was employed for the amperometric determination of glucose. Under optimal conditions, the biosensor displayed linear response to glucose from 0.5 to 5 mM with a sensitivity of 2.50 mA M^?1 at an applied potential of ?0.3 V (vs. Ag|AgCl reference).     

Imaging the reaction dynamics of OH + CD4. 3. Isotope effects

Isotope effects in the reaction of hydroxyl radical with methane are investigated in a crossed-beam experiment. By exploiting different combinations of OH/OD + CH4/CD4/CHD3, a total of eight isotopically variant reactions are examined to decipher the dynamical consequence of both primary and secondary isotope effects. The most prominent observation is from isotopic substitution of the transferred atom (primary isotope effect), which yields more stretch-excited water product for the D atom case than H atom. The opposite is found, however, for excitations in bending and combination modes. The secondary isotope effects, from either substituted hydroxyl radical or the three H atoms that are not being abstracted, are relatively minor and manifest themselves in spreading the vibrational distribution of the water coproducts.     

Effect of complex formation on the half wave potentials of redox active ligands

A general equation is derived for the shift in half wave potential, ΔE , for ligand reduction on complexation with a metal. The formation of several complexes with differing stoichiometry and protonation states is considered explicitly. The dependence of ΔE on metal ion concentration, holding the ligand concentration constant, can be used to determine stability constants or their ratios. The meaning of ΔE depends on the stoichiometric changes following ligand oxidation state change, in a manner analogous to the case where the metal is the redox site.     

Amphiphilic diblock copolymers based on poly(2‐ethyl‐2‐oxazoline) and poly(4‐substituted‐ε‐caprolactone): Synthesis,characterization, and cellular uptake

Amphiphilic diblock copolymers with various block compositions were synthesized on poly(2‐ethyl‐2‐oxazoline) (PEtOz) as a hydrophilic block and poly(4‐methyl‐ε‐caprolactone) (PMCL) or poly(4‐phenyl‐ε‐caprolactone) (PBCL) as a hydrophobic block. These PEtOz‐b‐PMCL and PEtOz‐b‐PBCL copolymers consisting of soft domains of amorphous PEtOz and PM(B)CL had no melting endothermal peaks but displayed T_g. The lower critical solution temperature (LCST) values for the PEtOz‐b‐PMCL, and the PEtOz‐b‐PBCL aqueous solution were observed to shift to lower temperature than PEtOz homopolymers. Their aqueous solutions were characterized using fluorescence techniques and dynamic light scattering (DLS). The block copolymers formed micelles with critical micelle concentrations (CMCs) in the range 0.6–11.1 mg L^?1 in an aqueous phase. As the length of the hydrophobic PMCL or PBCL blocks elongated, lower CMC values were generated. The mean diameters of the micelles were between 127 and 318 nm, with PDI in the range of 0.06–0.21, suggesting nearly monodisperse size distributions. The drug entrapment efficiency and drug‐loading content of micelles depend on block polymer compositions. In vitro cell viability assay showed that PEtOz‐b‐PMCL has low cytotoxicity. Doxorubicin hydrochloride (DOX)‐loaded micelles facilitated human cervical cancer (HeLa) cell uptake of DOX; uptake was completed within 2 h, and DOX was able to reach intracellular compartments and enter the nuclei by endocytosis. © 2013 Wiley Periodicals, Inc. J. Polym. Sci., Part A: Polym. Chem. 2013, 51, 2769–2781     

Formation of microporous poly(hydroxybutyric acid) membranes for culture of osteoblast and fibroblast

Microporous membranes of a biodegradable polymer, poly(hydroxybutyric acid) (PHB), were prepared by a phase‐inversion process and their cell compatibility was evaluated in vitro. A ternary system, ethanol/chloroform/PHB, was employed to prepare the membranes, wherein ethanol and chloroform were served as the nonsolvent and solvent for PHB, respectively. In the phase‐inversion process, the polymer dissolution temperature was varied from 80 to 120°C to yield membranes with specific morphologies, such as globular particles, porous channels, etc. Moreover, cell viability was examined on the formed membranes. Two cell lines, osteoblast hFOB1.19 and fibroblast L929, were cultured in vitro. It was found that these two types of cells exhibited different responses on different membranes: the hFOB1.19 cells showed significant increase in cell proliferation with increase in surface roughness, whereas the L929 cells demonstrated an opposite trend, preferring to attach and grow on a flat surface. PHB membranes with different morphologies exhibit different cell compatibilities, which may be useful means for the architectural design of materials for tissue engineering. Copyright © 2009 John Wiley & Sons, Ltd.