Sum Rules and the Szegő Condition for Orthogonal Polynomials on the Real Line

We study the Case sum rules, especially C ₀ , for general Jacobi matrices. We establish situations where the sum rule is valid. Applications include an extension of Shohats theorem to cases with an infinite point spectrum and a proof that if lim n(a _n –1)= and lim nb _n = exist and 2<||, then the Szeg condition fails. Supported in part by NSF grant DMS-9707661.     

A definitive synthesis of D-myo-inositol 1,4,5,6-tetrakisphosphate and its enantiomer D-myo-inositol 3,4,5,6-tetrakisphosphate from a novel butane-2,3-diacetal-protected inositol

New and rapid syntheses of the enantiomeric intracellular signalling molecules d-myo-inositol 1,4,5,6-tetrakisphosphate (1 a) and D-myo-inositol 3,4,5,6-tetrakisphosphate (1 b) are described. The synthetic strategy employs the novel butane-2,3-diacetal-protected (BDA-protected) myo-inositol (+/-)-3 ab, directly accessible from myo-inositol on a large scale, and an optical resolution with diastereoisomeric (R)-(-)-acetylmandelate esters. The X-ray crystal structure of (+/-)-4, an unusual side product of acid-catalysed reaction of myo-inositol with butanedione is also presented, and the absolute configurations of 1 a and 1 b are definitively assigned by conversion of key precursors into (+)-bornesitol and L-iditol hexaacetate, respectively. Biological activity of synthetic 1 b was confirmed in comparison with the natural polyphosphate.     

A mechanistic dichotomy in ruthenium-catalyzed propargyl alcohol reactivity: a novel hydrative diyne cyclization

The cycloisomerization of diyne-ols catalyzed by [CpRu(CH3CN)3]PF6 to 2-vinyl-1-acylcycloalkenes proceeds via a ruthenacyclopentadiene involving initial ionization of the tertiary or secondary alcohol, followed by readdition. In the case of primary alcohols, a competing pathway wherein water first adds would appear to occur. The feasibility of this proposed minor pathway was tested in the reaction of diynes in the presence of water. Quite excitingly, cyclization comcommittant with addition of water to form 1-acylcycloalkenes occurs. This proves to be general process to form five- and six-membered rings. Interestingly, hydrative cyclization of Z-5-decen-2,8-diyne to 1-acetyl-2-ethyl-cyclohexa-1,4-diene occurs without isomerization of the double bonds. Furthermore, the epoxide of the same substrate cyclizes without opening of the strained epoxide. Unsymmetrically substituted diynes cyclize with remarkable chemoselectivity wherein water attacks the less hindered alkynes. beta-branching of any kind gives only a single product. Remarkably, even competing methyl versus ethyl still effects a 2.5:1 selectivity in favoring water addition to the methyl-bearing alkyne. Alcohols can replace water and provide enol ethers. Strong mechanistic evidence suggests two reaction manifolds indeed operate, depending upon the presence of propargyl alcohols and the degree of substitution on the hydroxyl-bearing carbon.     

Nanometer Calibration Particles: What is Available and What is Needed?

Information on available polystyrene calibration spheres is presented regarding the particle diameter, uncertainty in the size, and the width of the size distribution for particles in a size range between 20 and 100nm. The use of differential mobility analysis for measuring the single primary calibration standard in this size range, 100nm NIST Standard Reference Material®1963, is described along with the key factors in the uncertainty assessment. The issues of differences between international standards and traceability to the NIST Standard are presented. The lack of suitable polystyrene spheres in the 20–40nm size range will be discussed in terms of measurement uncertainty and width of the size distributions. Results on characterizing a new class of molecular particles known as dendrimers will be described and the possibilities of using these as size calibration standards for the size range from 3 to 15nm will be discussed.     

Use of cyclodextrin‐modified gold nanoparticles for enantioseparations of drugs and amino acids based on pseudostationary phase‐capillary electrochromatography

The application of chemical‐modified gold nanoparticles (GNPs) as chiral selector for the enantioseparation based on pseudostationary phase‐CEC (PSP‐CEC) is presented. GNPs modified by thiolated β‐CD were characterized by NMR and FT‐IR. The nanoparticle size was determined to be of 9.5 nm (+2.5 nm) by Transmission Electron Microscopy (TEM) and UV spectra. Four pairs of dinitrophenyl‐labeled amino acid enantiomers (DL‐Val, Leu, Glu and Asp) and three pairs of drug enantiomers (RS‐chlorpheniramine, zopiclone and carvedilol) were analyzed by using modified GNPs as the chiral selector in PSP‐CEC. Good theoretical plate number (up to 2.4×10⁵ per meter) and separation resolution (up to 4.7) were obtained even with low concentration of modified GNPs (0.8–1.4 mg/mL). The corresponding concentration of β‐CD in the buffer was only 0.30?0.53 mM, which was much lower than the optimum concentration of 15 mM if pure β‐CD was used as chiral selector. Our results showed that thiolated β‐CD modified GNPs have more sufficient interaction with the analytes, resulting in significant enhancement of enantioseparation. The study shed light on potential usage of chemical modified GNPs as chiral selector for enantioseparation based on PSP‐CEC.     

Thermal degradation and flame retardance in copolymers of methyl methacrylate with diethyl(methacryloyloxymethyl)phosphonate

Methyl methacrylate (MMA) has been free radically copolymerized, both in bulk and in solution, with diethyl(methacryloyloxymethyl)phosphonate (DEMMP), to give polymers which are significantly flame retarded when compared with PMMA, as indicated by the results of limiting oxygen index (LOI) measurements, UL 94 tests, and the results of cone calorimetric experiments. The physical and mechanical properties of the copolymers are similar to those of PMMA, except that the bulk copolymers are slightly crosslinked, and are better than those of PMMA flame retarded to a similar extent by some phosphate and phosphonate additives. Examination of the some of the gaseous products of pyrolysis and combustion, and of chars produced on burning, show that flame retardation occurs in the copolymers by both a condensed-phase and a vapour-phase mechanism. The condensed-phase mechanism is shown to involve generation of phosphorus acid species followed by reaction of these with MMA units giving rise to methacrylic acid units. The methacrylic acid units subsequently form anhydride links, which probably impede depolymerization of the remaining MMA sequences, resulting in evolution of less MMA (the major fuel when MMA-based polymers burn). By undergoing decarboxylation, leading to interchain cyclisation and, eventually, to aromaticisation, the anhydride units are probably also the principal precursors to char.     

Bivariate distributions characterized by one family of conditionals and conditional percentile or mode functions

It is well known that full knowledge of all conditional distributions will typically serve to completely characterize a bivariate distribution. Partial knowledge will often suffice. For example, knowledge of the conditional distribution of X given Y and the conditional mean of Y given X is often adequate to determine the joint distribution of X and Y. In this paper, we investigate the extent to which a conditional percentile function or a conditional mode function (of Y given X), together with knowledge of the conditional distribution of X given Y will determine the joint distribution. Finally, using this methodology a new characterization of the classical bivariate normal distribution is given.