Prime divisors of sparse integers

We obtain a lower bound on the number of prime divisors of integers whose g-ary expansion contains a fixed number of nonzero digits. This revised version was published online in August 2006 with corrections to the Cover Date.     

关于体上分块矩阵的群逆

本文利用分块矩阵方法．研究了体上两个矩阵乘积的群逆的存在性及表示形式，给出了体上两个矩阵乘积群逆存在的充分必要条件和表示形式．并且在一定条件下．给出了体上分块矩阵的群逆存在性及表示形式．     

A NUMERICALLY STABLE BLOCK MODIFIED GRAM-SCHMIDT ALGORITHM FOR SOLVING STIFF WEIGHTED LEAST SQUARES PROBLEMS

Recently, Wei in proved that perturbed stiff weighted pseudoinverses and stiff weighted least squares problems are stable, if and only if the original and perturbed coefficient matrices A and A^- satisfy several row rank preservation conditions. According to these conditions, in this paper we show that in general, ordinary modified Gram-Schmidt with column pivoting is not numerically stable for solving the stiff weighted least squares problem. We then propose a row block modified Gram-Schmidt algorithm with column pivoting, and show that with appropriately chosen tolerance, this algorithm can correctly determine the numerical ranks of these row partitioned sub-matrices, and the computed QR factor R^- contains small roundoff error which is row stable. Several numerical experiments are also provided to compare the results of the ordinary Modified Gram-Schmidt algorithm with column pivoting and the row block Modified Gram-Schmidt algorithm with column pivoting.     

Control of the block copolymer morphology in templated epoxy thermosets

It has been found that by the addition of low concentrations of an amphiphilic block copolymer to an epoxy resin, novel disordered morphologies can be formed and preserved through curing. This article will focus on characterizing the influence of the block copolymer and casting solvent on the templated morphology achieved in the thermoset sample. The ultimate goal of this work is to determine the parameters that would control the microphase morphology produced. Epoxy resins blended with a series of amphiphilic block copolymers based on hydrogenated polyisoprene (polyethylene-alt-propylene or PEP) and polyethylene oxide (PEO), specifically, were investigated. In this article, the cure-induced order–order phase transition from the spherical to wormlike micelle morphology will also be discussed. It is proposed that the formation of the wormlike micelle structure from the spherical micelle structure is similar to the phase transition behavior that occurs in dilute block copolymer solutions as a function of the influence of the solvent on micelle morphology. © 2007 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 45: 3338–3348, 2007     

Synthesis,dynamic light scattering,and luminescence properties of copolymers containing iridium(III) bisterpyridine in the side chain

Supramolecular block‐random copolymers containing [Ir(terpy)₂]³⁺ in the side chain were synthesized via postfunctionalization of a P(S‐b‐AC_terpy) block copolymer. Absorbance and emission spectra compared to a model compound show that the polymer backbone has a minor effect on the polymer absorbance but produces a larger shift for the phosphorescence signals to higher wavelength. Dynamic light scattering of the metal complex containing copolymer studied in various solvents showed monomodal aggregation with decreasing aggregate size as the solvent dielectric constant increased. The copolymer precursor P(S‐b‐AC_terpy) shows multimodal aggregation in different solvents with the major population consisting of single chains. This difference in behavior between the two polymers is attributed to the electrolytic nature of the complex and the amphiphilicity induced by the charged metal complex. Supramolecular copolymers like these will continue to have interesting self‐organizational properties and may find applications in multicomponent systems for photoinduced charge separation processes. © 2006 Wiley Periodicals, Inc. J Polym Sci Part A: Polym Chem 45: 1109–1121, 2007     

Effect of core-shell micelle formation on the redox properties of phenothiazine-labeled poly(ethyl glycidy ether)-block-poly(ethylene oxide)

Redox properties of phenothiazine-labeled poly(ethyl glycidy ether)-block-poly(ethylene oxide) (PT-EGE_n-b-EO_m) are reversibly changed by core-shell micelle formation. In the temperature range higher than the critical micellization temperature (cmt), the anodic potential of PT group positively shifts and concomitantly its anodic current decrease, or levels off compared to those of the reference polymer PT-EO_m without the thermo-responsive EGE_n segment. The former alteration is caused by incorporation of hydrophobic PT groups into a core of the micelle and the latter by the decrease in the diffusion coefficient of PT groups due to formation of the core-shell micelles. The cmt value and the temperature-dependent alteration in the redox properties strongly depend on the polymer structure, especially the length of thermo-responsive EGE_n segment. The electrochemically determined hydrodynamic radii of the polymer aggregates seem to be overestimated, compared to the values reported for the aggregates of other thermo-responsive polymers with similar molecular weights, implying the presence of electrochemically inactive PT groups in the copolymers having longer thermo-responsive segments.     

Glass transitions and mixed phases in block SBS

Differential scanning calorimetry (DSC) does not allow for easy determination of the glass‐transition temperature (T_g) of the polystyrene (PS) block in styrene–butadiene–styrene (SBS) block copolymers. Modulated DSC (MDSC), which deconvolutes the standard DSC signal into reversing and nonreversing signals, was used to determine the (T_g) of both the polybutadiene (PB) and PS blocks in SBS. The T_g of the PB block was sharp, at ?92 °C, but that for the PS blocks was extremely broad, from ?60 to 125 °C with a maximum at 68 °C because of blending with PB. PS blocks were found only to exist in a mixed PS–PB phase. This concurred with the results from dynamic mechanical analysis. Annealing did not allow for a segregation of the PS blocks into a pure phase, but allowed for the segregation of the mixed phase into two mixed phases, one that was PB‐rich and the other that was PS‐rich. It is concluded that three phases coexist in SBS: PB, PB‐rich, and PS‐rich phases. © 2004 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 43: 276–279, 2005     

Diblock copolymers based on allyl methacrylate: Synthesis,characterization, and chemical modification

Different diblock copolymers constituted by one segment of a monomer supporting a reactive functional group, like allyl methacrylate (AMA), were synthesized by atom transfer radical polymerization (ATRP). Bromo‐terminated polymers, like polystyrene (PS), poly(methyl methacrylate) (PMMA), and poly(butyl acrylate) (PBA) were employed as macroinitiators to form the other blocks. Copolymerizations were carried out using copper chloride with N,N,N′,N″,N″‐pentamethyldiethylenetriamine (PMDETA) as the catalyst system in benzonitrile solution at 70 °C. At the early stage, the ATRP copolymerizations yielded well‐defined linear block copolymers. However, with the polymerization progress a change in the macromolecular architecture takes place due to the secondary reactions caused by the allylic groups, passing to a branched and/or star‐shaped structure until finally yielding gel at monomer conversion around 40% or higher. The block copolymers were characterized by means of size exclusion chromatography (SEC), ¹H NMR spectroscopy, and differential scanning calorimetry (DSC). In addition, one of these copolymers, specifically P(BA‐b‐AMA), was satisfactorily modified through osmylation reaction to obtain the subsequent amphiphilic diblock copolymer of P(BA‐b‐DHPMA), where DHPMA is 2,3‐dihydroxypropyl methacrylate; demonstrating the feasibility of side‐chain modification of the functional obtained copolymers. © 2007 Wiley Periodicals, Inc. J Polym Sci Part A: Polym Chem 45: 3538–3549, 2007     

A Tin(IV) Porphyrin with Two Axial Organometallic NCN‐Pincer Platinum Units

A tin(IV) porphyrin was combined with two axial NCN‐pincer platinum(II) fragments by utilizing the oxophilicity of the apical positions on the tin atom and the acidic nature of the NCN‐pincer platinum derived benzoic acid. The solid‐state structure determined by X‐ray crystallography revealed some close contacts between the pincer complexes and the meso‐p‐tolyl subsitutents of the porphyrin. It was shown by ¹H NMR spectroscopy that these close contacts were not present in solution and that this compound can potentially act as a novel building block for supramolecular architectures.     

两亲性嵌段聚合物的同步辐射小角x射线散射研究

应用同步辐射小角x射线散射(SAXS)方法研究了不同聚合条件下苯乙烯对乙烯基苯甲酸两亲性嵌段聚合物的聚集行为,结果发现该聚合物在选择性溶液中自组装形成胶束.胶束的形态和结构取决于嵌段聚合物的组成、浓度以及溶剂的性质等因素 关键词： 小角x射线散射 两亲性嵌段聚合物 分形维数 粒径