Sums of non-integral powers

We give an approximation to the sum of non-integral powers of positive integers which is a natural generalization of Faulhaber's formula for the sum of integral powers of positive integers.     

Soap-film-like minimal surfaces spanning knots

It is proved that any three-times continuously differentiable, nontrivial knot in three-dimensional euclidean space supports a surface that minimizes area among nearby surfaces but that does not touch all of the supporting knot. This provides a mathematical model of a physical phenomenon occurring in soap-films. The notion of a homotopically spanning surface is defined to determine an appropriate class of admissible surfaces, and it is shown that there is a lower bound on the area of admissible surfaces. The existence of an area minimizing admissible surface is then proved by the direct method based on earlier work of E. R. Reiffenberg.     

Bicyclic and acyclic diamides: comparison of their aqueous phase binding constants with Nd(III), Am(III), Pu(IV), Np(V), Pu(VI), and U(VI)

This report describes affinity measurements for two, water-soluble, methyl-alkylated diamides incorporating the malonamide functionality, N,N,N',N' tetramethylmalonamide (TMMA) and a bicyclic diamide (1a), toward actinide metal cations (An) in acidic nitrate solutions. Ligand complexation to actinides possessing oxidation states ranging from +3 to +6 was monitored through optical absorbance spectroscopy, and formation constants were obtained from the refinement of the spectrophotometric titration data sets. Species analysis gives evidence for the formation of 1, 4, 1, and 2 spectrophotometrically observable complexes by TMMA to An(III, IV, V, and VI), respectively, while for 1a, the respective numbers are 3, 4, 2, and 2. Consistent with the preorganization of 1a toward actinide binding, a significant difference is found in the magnitudes of their respective formation constants at each complexation step. It has been found that the binding affinity for TMMA follows the well-established order An(V) < An(III) < An(VI) < An(IV). However, with 1a, Np(V) forms stronger complexes than Am(III). The complexation of 1a with Np(V) and Pu(VI) at an acidity of 1.0 M is followed by reduction to Np(IV) and Pu(IV), whereas TMMA does not perturb the initial oxidation state for these dioxocations. These measurements of diamide binding affinity mark the first time single-component optical absorbance spectra have been reported for a span of actinide-diamide complexes covering all common oxidation states in aqueous solution.     

The measurement of molecular diversity by receptor site interaction simulation

The assembly of large compound libraries for the purpose of screening against various receptor targets to identify chemical leads for drug discovery programs has created a need for methods to measure the molecular diversity of such libraries. The method described here, for which we propose the acronym RESIS (for Receptor Site Interaction Simulation), relates directly to this use. A database is built of three-dimensional representations of the compounds in the library and a set of three-point three-dimensional theoretical receptor sites is generated based on putative hydrophobic and polar interactions. A series of flexible, three-dimensional searches is then performed over the database, using each of the theoretical sites as the basis for one such search. The resulting pattern of hits across the grid of theoretical receptor sites provides a measure of the molecular diversity of the compound library. This can be conveniently displayed as a density map which provides a readily comprehensible visual impression of the library diversity characteristics. A library of 7500 drug compounds derived from the CIPSLINEPC databases was characterized with respect to molecular diversity using the RESIS method. Some specific uses for the information obtained from application of the method are discussed. A comparison was made of the results from the RESIS method with those from a recently published two-dimensional approach for assessing molecular diversity using sets of compounds from the Maybridge database (MAY).     

Comparative analytical quantitation of clenbuterol in biological matrices using GC-MS and EIA

A simple and sensitive procedure utilizing GC-MS for the identification and quantitation of clenbuterol in biofluids and tissues is described. This improved method utilizes trimethylboroxine for the derivatization of clenbuterol, requires only 1 mL/g of biological sample, and most importantly does not require an extra cleaning step for urine specimens prior to extraction. Linear quantitative response curves have been generated for derivatized clenbuterol over a concentration range of 5-200 ng/mL. The extraction efficiency at four representative points of the standard curve exceeded 90% in both specimen types (plasma and urine). Linear regression analyses of the standard curve in both specimen types exhibited correlation coefficients ranging from 0.997 to 1.000. The Limit of detection (LOD) and Limit of quantitation (LOQ) values for plasma specimens were determined to be 0.5 and 1.5 ng/mL respectively. For urine specimens, LOD and LOQ values were 0.2 and 0.7 ng/microL respectively. Percentage recoveries ranged from 91 to 95% for urine and 89 to 101% for plasma. Precision and accuracy (within-run and between-run) studies reflected a high level of reliability and reproducibility of the method. In addition to its reliability, sensitivity and simplicity, this modified procedure is more efficient and cost effective, requiring less time, only 1 mL of sample, and minimal amounts of extraction solvents. The applicability of the method for the detection and quantitation of clenbuterol in biological tissues of rats treated with the drug was demonstrated successfully. For comparative analysis of clenbuterol in plasma and liver samples, both GC-MS and enzyme immunoassay (EIA) methods are found to be suitable. Due to potential antibody-cross reactivity with EIA, the GC-MS method is the method of choice for most samples because of its specificity. However, the EIA method is considered the method of choice for analysis of clenbuterol found in concentrations below the limits of quantitation by GC-MS due to its sensitivity.     

Weakly coordinating anions: crystallographic and NQR studies of halogen-metal bonding in silver, thallium, sodium, and potassium halomethanesulfonates

35Cl, (79,81)Br, and (127)I NQR (nuclear quadrupole resonance) spectroscopy in conjunction with X-ray crystallography is potentially one of the best ways of characterizing secondary bonding of metal cations such as Ag(+) to halogen donor atoms on the surfaces of very weakly coordinating anions. We have determined the X-ray crystal structure of Ag(O(3)SCH(2)Cl) (a = 13.241(3) A; b = 7.544(2) A; c = 4.925(2) A; orthorhombic; space group Pnma; Z = 4) and compared it with the known structure of Ag(O(3)SCH(2)Br) (Charbonnier, F.; Faure, R.; Loiseleur, H. Acta Crystallogr., Sect. B 1978, 34, 3598-3601). The halogen atom in each is apical (three-coordinate), being weakly coordinated to two silver ions. (127)I NQR studies on Ag(O(3)SCH(2)I) show the expected NQR consequences of three-coordination of iodine: substantially reduced NQR frequencies nu(1) and nu(2) and a fairly small NQR asymmetry parameter eta. The reduction of the halogen NQR frequency of the coordinating halogen atom in Ag(O(3)SCH(2)X) becomes more substantial in the series X = Cl < Br < I, indicating that the coordination to Ag(+) strengthens in this series, as expected from hard-soft acid-base principles. The numbers of electrons donated by the organic iodine atom to Ag(+) have been estimated; these indicate that the bonding to the cation is weak but not insignificant. We have not found any evidence for the bonding of these organohalogen atoms to another soft-acid metal ion, thallium. A scheme for recycling of thallium halide wastes is included.     

Use of Sucrose‐based Epoxy ­Formulations and Cellulosic Fibers in ­the Design,Preparation, and Screening ­of New Composite Insulation Materials

Non‐woven composite insulation materials were generated from cotton, kenaf, jute, polyester, polypropylene, sucrose‐based epoxy formulations, and aluminum foil. The needlepunched fiber batts were rendered flame resistant by use of inorganic reagents and urea. To discover suitable epoxy formulations to bind the cellulose fibers to themselves or to dissimilar surfaces and to make flexible composites, a comparison of the performance of the known epoxy allyl sucroses (EAS), epoxy crotyl sucroses (ECS), and diglycidyl ether of bisphenol‐A (DGEBA) was made. The epoxies were cured with commercial diethylenetriamine (DETA), and UNIREZs‐2142 and 2355®, to discover a formulation with the following characteristics: (a) low cure temperature; (b) low Young's moduli and glass transition temperatures of cured thermosets for flexible composites; (c) ample bond strength between the fabric and the bonded surfaces; and (d) non‐cytotoxicity and non‐mutagenicity of the epoxies. Based on results following these criteria, EAS was selected, and the formulation comprising EAS and UNIREZ‐2355® was deemed suitable to bind fiber batts to surfaces of any type and geometry. ASTM guidelines were used to construct a wooden frame cube (heat box) for the simultaneous rapid screening of cellulosic fiber batts and composites. The new materials were compared against R‐19 fiberglass insulation for their ability to resist heat flow (denoted by relative R‐values) and time taken to approach thermal equilibrium. Plain non‐woven cellulosic fiber batts showed relative R‐values of 4.0 °F ft² hr/Btu per inch thickness (0.27 K m²/W per cm), and took about 2 hr to establish equilibrium heat flow. Commercial fiberglass batts showed relative R‐values­of 2.2 per in (0.15 per cm) and took 1 hr to attain equilibrium heat flow. When 6.25 in (15.9 cm) thick batts of fiberglass were needle punched to a thickness of 1 in (2.54 cm), relative R‐values and equilibrium heat flow times were 4.0 per in (0.27 per cm) and 2 hr, respectively. This denoted that the densities and thermal resistances of non‐conducting materials are raised concurrently. Anisotropic heat flow behavior was observed in cellulosic fiber composites with aluminum foil (shiny side out) bonded on one side. It depended upon whether the aluminum foil side or the fibers side faced the heat source. In the latter orientation the aluminum acted as a heat sink, and in the former orientation the foil acted as a poor heat reflector. The poor performance of these insulation composites was related to the fact that aluminum was directly bonded to the fiber batts and was acting as a heat conductor. When cellulose fiber shims (spacers) were placed between the fiber batts and the aluminum foil, the R‐values of the composites were comparable to those of plain batts but the times taken to approach thermal equilibrium increased to >3 hr, denoting that the foil was acting more as a reflector and less as a conductor. Copyright © 2001 John Wiley & Sons, Ltd.     

On the Lang–Trotter conjecture for two elliptic curves

The Ramanujan Journal - We extend the author’s earlier computation and give coefficient formulas for the (quasimodular) Poincaré square series of weight 3&nbsp;/&nbsp;2 and...