Asymptotic upper bounds for the coefficients in the Chebyshev series expansion for a general order integral of a function

The usual way to determine the asymptotic behavior of the Chebyshev coefficients for a function is to apply the method of steepest descent to the integral representation of the coefficients. However, the procedure is usually laborious. We prove an asymptotic upper bound on the Chebyshev coefficients for the integral of a function. The tightness of this upper bound is then analyzed for the case , the first integral of a function. It is shown that for geometrically converging Chebyshev series the theorem gives the tightest upper bound possible as . For functions that are singular at the endpoints of the Chebyshev interval, , the theorem is weakened. Two examples are given. In the first example, we apply the method of steepest descent to directly determine (laboriously!) the asymptotic Chebyshev coefficients for a function whose asymptotics have not been given previously in the literature: a Gaussian with a maximum at an endpoint of the expansion interval. We then easily obtain the asymptotic behavior of its first integral, the error function, through the application of the theorem. The second example shows the theorem is weakened for functions that are regular except at . We conjecture that it is only for this class of functions that the theorem gives a poor upper bound.

     

The green synthesis of fine particles of gold using an aqueous extract of <Emphasis Type="Italic">Monotheca buxifolia</Emphasis> (Flac.)

This study deals with the synthesis and physicochemical investigation of gold nanoparticles using an aqueous extract of Monotheca buxifolia (Flac.). On the treatment of aqueous solution of tetrachloroauric acid with the plant extract, gold nanoparticles were rapidly fabricated. The synthesized particles were characterized by UV–Vis spectrophotometry (UV), Fourier transform infrared spectroscopy (FTIR), energy dispersive X-ray (EDX) and Scanning electron microscopy (SEM). The formation of AuNPs was confirmed by noting the change in color through visual observations as well as via UV–Vis spectroscopy. UV?Vis spectrum of the aqueous medium containing gold nanoparticles showed an absorption peak at around 540 nm. FTIR was used to identify the chemical composition of gold nanoparticles and Au-capped plant extract. The presence of elemental gold was also confirmed through EDX analysis. SEM analysis of the gold nanoparticles showed that they have a uniform spherical shape with an average size in the range of 70–78 nm. This green system showed to be better capping and stabilizing agent for the fine particles. Further, the antioxidant activity of Monotheca buxifolia (Flac.) extract and Au-capped with the plant extract was also evaluated using FeCl₃/K₃[Fe(CN)]₆ in vitro assay.     

Spectra of Tukey types of ultrafilters on Boolean algebras

Extending recent investigations on the structure of Tukey types of ultrafilters on \({\mathcal{P}(\omega)}\) to Boolean algebras in general, we classify the spectra of Tukey types of ultrafilters for several classes of Boolean algebras, including interval algebras, tree algebras, and pseudo-tree algebras.     

The C-terminal ester of membrane anchored peptide ion channels affects anion transport

Five heptapeptide derivatives, [CH3(CH2)17]2NCOCH2OCH2CO-Gly-Gly-Gly-Pro-Gly-Gly-Gly-OR, in which R = ethyl, 2-propyl, heptyl, benzyl, and cyclohexylmethyl, were found to transport chloride anion through a phospholipid bilayer to varying extents dependent on the identity of R. It was concluded that the R group is a membrane anchor for the synthetic chloride channels.     

Copolymerization of ethylene with polar monomers: chain propagation and side reactions. A DFT theoretical study using zwitterionic Ni(II) and Pd(II) catalysts

Calculations utilizing anionic substituted derivates of the cationic N(wedge)N--Ni(II) and Pd(II) diimine Brookhart complex have been carried out on the barriers of ethylene and acrylonitrile insertion into a M- methyl, propyl and CH(CN)Et bond for M = Ni, Pd. The possibility of side reactions such as chelate formation with the polar functionality and oligomerization of the active species after acrylonitrile insertion are explored. The diimine ring system N--N = -NR' 'CR(1)CR(2)NR' ' with R' ' = 2,6-C(6)H(3)(i-Pr)(2) and R(1),R(2) = Me was functionalized by adding one or two anionic groups (BF(3)(-), etc.) in place of i-Pr on the aryl rings or by replacing one Me diimine backbone group (R(1)) with BH(3)(-). The objective of this investigation is computationally to design catalysts for ethylene/acrylonitrile copolymerization that have activities that are comparable to that of the cationic Ni(II) diimine or at least the Pd(II) diimine Brookhart system for ethylene homopolymerization. Complexes that might meet this objective are discussed.     

Synthesis and biological evaluation of fluorescently labeled epothilone analogs for tubulin binding studies

The two fluorescently labeled epothilones 14 and 15 have been synthesized using a modification of Nicolaou's macrolactonization and Stille coupling strategy. The cytotoxicities of the compounds were 6.1 and 2.7 μg/mL, respectively, against the A2870 ovarian cancer cell line, and 0.5 and 1.0 μg/mL, respectively, against the PC-3 prostate cancer cell line. The critical concentration of tubulin was 0.5 and 1.0 μM in the presence of 14 and 15, respectively, compared with 0.3 μM for paclitaxel. The fluorescent properties of the two molecules in solution and bound to microtubules are described.     

Green Synthesis of Silver Nanoparticles Using an Aqueous Extract of <Emphasis Type="Italic">Monotheca buxifolia</Emphasis> (Flac.) Dcne

This study deals with the synthesis and physicochemical investigation of silver nanoparticles using an aqueous extract of Monotheca buxifolia (Flac.). On the treatment of aqueous solution of silver nitrate with the plant extract, silver nanoparticles were rapidly fabricated. The synthesized particles were characterized by using UV–visible spectrophotometry (UV), Fourier transform infrared spectroscopy (FTIR), Energy dispersive X-ray (EDX) and Scanning electron microscopy (SEM). The formation of AgNPs was confirmed by noting the change in colour through visual observations as well as via UV–Vis spectroscopy. UV–Vis spectrum of the aqueous medium containing silver nanoparticles showed an absorption peak at around 440 nm. FTIR was used to identify the chemical composition of silver nanoparticles and Ag-capped plant extract. The presence of elemental silver was also confirmed through EDX analysis. The SEM analysis of the silver nanoparticles showed that they have a uniform spherical shape with an average size in the range of 40–78 nm. This green system showed better capping and stabilizing agent for the fine particles. Further, in vitro the antioxidant activity of Monotheca buxifolia (Flac.) and Ag-capped with the plant was also evaluated using FeCl₃/K₃Fe (CN)₆ essay.     

Size-dependent dissociation pH of thiolate ligands from cadmium chalcogenide nanocrystals

A method, pseudo steady-state titration, is introduced for determining the precipitation pH of nanocrystals coated by electron-donating ligands. CdSe nanocrystals coated with hydrophilic deprotonated thiol (thiolate) ligands were studied systematically. For comparison, CdTe and CdS nanocrystals coated with the same types of ligands were also examined. The results show that the precipitation of the nanocrystals is caused by the dissociation of the nanocrystal-ligand coordinating bonds from the nanocrystal surface. The ligands are removed from the surface due to protonation in a relatively low pH range, between 2 and 7 depending on the size, approximately within the quantum confinement size regime, and chemical composition (band gap) of the nanocrystals. In contrast, the redispersion of the nanocrystals was found to be solely determined by the deprotonation of the ligands. The size-dependent dissociation pH of the ligands was tentatively used as a means for determining the size-dependent free energy associated with the formation of a nanocrystal-ligand coordinating bond.     

D-nuclear quadrupole coupling in the rotational spectrum of 15N2…DF and an interpretation of the hyperfine coupling constants χ aaD and DaaHF within a series of complexes B…HF

Hyperfine structure in the J = 1 ← 0 and J = 2 ← 1 transitions of ¹⁵N₂…DF, arising from D-nuclear quadrupole coupling and D, ¹⁹F nuclear-spin—nuclear-spin coupling, has been observed and measured by using pulsed-nozzle Fourier-transform microwave spectroscopy. The following rotational, centrifugal distortion and hyperfine coupling constants were determined: β_o = 3091.9004 MHz, D_J = 14.75 kHz, χ_aa^D = 278.6(38) kHz and D_aa^HF = ?38.5(38) kHz. An interpretation is presented for the variation of the coupling, constants χ_aa^D and D_aa^HF along the series B…DF and B…HF, where B = Ar, Kr, Xe, ¹⁵N₂, CO, PH₃, H₂S, HC¹⁵N and H₂O.     

Cluster-containing carbon-rich molecules: Reactions of ruthenium cluster carbonyls with {Au(PR3)}2(μ-CCCC) (R = Ph, tol)

Complexes containing C₄ ligands attached to one or two AuRu₃ clusters by conventional σ, 2π interactions have been obtained from reactions between (R₃P)AuC≡CC≡CAu(PR₃) (R = Ph, tol) or Au(C≡CC≡CH){P(tol)₃} and either Ru₃(CO)₁₂, Ru₃(CO)₁₀(NCMe)₂ or Ru₃(μ-dppm)(CO)₁₀. The X-ray determined structures of {(R₃P)AuRu₃(CO)₉}₂(μ₃,η²:μ₃,η²-C₂C₂) [R = Ph (1) (three solvates), tol (2)], AuRu₃{μ₃,η²-C₂C≡CAu(PPh₃)}(CO)₉(PPh₃) (3) and {(Ph₃P)AuRu₃(μ-dppm)(CO)₇} (μ₃,η²:μ₃,η²-C₂C₂){Ru₃(μ-H)(μ-dppm)(CO)₇} (4) are reported.