The statistics of continued fractions for polynomials over a finite field

Given a finite field of order and polynomials of degrees respectively, there is the continued fraction representation . Let denote the number of such pairs for which and for . We give both an exact recurrence relation, and an asymptotic analysis, for . The polynomial associated with the recurrence relation turns out to be of P-V type. We also study the distribution of . Averaged over all and as above, this presents no difficulties. The average value of is , and there is full information about the distribution. When is fixed and only is allowed to vary, we show that this is still the average. Moreover, few pairs give a value of that differs from this average by more than

     

The photoactivity of porous Tio2 anodized at a high voltage

The photoactivity of TiO₂ produced by high-voltage anodization as described by Marchenoir et al. was compared to single-crystal TiO₂. The anodic oxide film was found to be highly porous. The photocurrent and quantum yield of the anodic sample is markedly larger on the long-wavelength side of the action spectrum as compared to sinle-crystal rutile. We ascribe this behavior either to the formation of anatase “pockets” in the film or to the incomplete oxidation of Ti during the anodization process.     

The determination of thiols with diphenylpicrylhydrazyl as a spectrophotometric reagent

Diphenylpicrylhydrazyl (DPPH), a stable, intensely purple free radical, is used as a reagent in the quantitative determination of various aromatic and aliphatic thiols by indirect spectrophotometric analysis. Plots of degree of reaction vs. time show that thiophenol and its derivatives react more quickly than aliphatic thiols with DPPH. Calibration plots are linear over the concentration range 0.05-3.00 x 10(-5)M thiol. The average relative error is in the range 1-2% and the absolute standard deviations range up to 0.50 x 10(-6)M.     

Deprotonated diindolylmethanes as dianionic analogues of scorpionate bis(pyrazolyl)borate ligands: synthesis and structural characterization of representative titanocene and zirconocene complexes

Deprotonation of di(3-methylindol-2-yl)phenylmethane (L₂H₂) or with two equivalents of ⁿBuLi, followed by reactions with Cp₂TiCl₂ or Cp₂ZrCl₂ yielded complexes . Compounds 1-4 were characterized by NMR spectroscopy, and compounds 1, 3, and 4 were further analyzed by X-ray crystallography and elemental analysis. The molecular structures of 1, 3, and 4 illustrate that chelating di(3-methylindol-2-yl)methanes have a structural relationship to coordinated bis(pyrazolyl)borates.     

Recent comparability of oceanographic nutrients data: results of a 2003 intercomparison exercise using reference materials.

An intercomparison exercise was conducted using the recently developed Reference Material for Nutrients in Seawater (RMNS). Discrepancies of reported values among laboratories were greater than the homogeneity of RMNS samples and the reported analytical precision of nutrients. The variability of in-house standards of the participating laboratories might be the most likely source of interlaboratory discrepancies. Therefore, the use of common reference materials, i.e. certified RM, is essential to establish and improve the comparability of nutrient data of the world's oceans.     

Effects of solvent on the relative stability of mono and di-aluminium aryloxide complexes of bipyridines: anomalous behavior of [(tBu)2Al(OPh)]2(mu-4,4-bipy)

The temperature dependence of the solution equilibrium constants for [((t)Bu)(2)Al(OPh)]2(mu-4,4'-bipy)(1a), [((t)Bu)2Al(OPh)](2)(mu-bipetha)(2a, bipetha = 1,2-bis(4-pyridyl)ethane), and [((t)Bu)(2)Al(OPh)]2(mu-bipethe)(3a, bipethe =trans-1,2-bis(4-pyridyl)ethylene) in C6D6 and CDCl(3) allow for the determination of DeltaH and DeltaS for the dissociation of one Al(tBu)2OPh moiety from the bridging ligand, i.e., 2[(tBu)2AL(OPh)]2(mu-L)<==>(K1)2AL(OPh)(tBu)2(L)+[(tBu)2Al(mu-OPh)]2. For compounds and the DeltaH values in C6D6[99(2) kJ mol(-1)(2a) and 109(5) kJ mol(-1)(3a)] and CDCl3[115(5) kJ mol(-1)(2a) and 139(7) kJ mol(-1)(3a)] were found to be inversely proportional with the dielectric constant of the solvent. In contrast, the DeltaH value for 1a in CDCl3 is surprisingly small [14.9(7) kJ mol(-1)] and does not fit with the trends adopted by the bipetha and bipethe derivatives or the value obtained in C6D6[110(2) kJ mol(-1)]. Unlike the other compounds and the C6D6 solutions, the CDCl3 solution of 1a allows for the observation of a second equilibrium 2Al(OPh)(tBu)2(L)<==>(K2)[(tBu)2Al(mu-OPh)]2+2L, for which the DeltaH has been determined [4.5(3) kJ mol(-1)]. This result suggests that in CDCl3 bonding of the second Al(tBu)2OPh moiety to Al(OPh)(tBu)2(4,4'-bipy)(1b) is stabilized by the presence of the first aluminium, which is counter to ab initio calculations that predicts the aluminium in Al(OPh)((t)Bu)2(L) should destabilize the Al-N interaction with a second Al(tBu)2OPh group. The BDE for dissociation of both Al(tBu)2OPh moieties from 1a-3a, and the energy of formation of hydrogen bond interactions with CHCl3, has been calculated by ab initio methods, and no unusual effects are inherent in 1a.     

Synthesis, relaxometric and photophysical properties of a new pH-responsive MRI contrast agent: the effect of other ligating groups on dissociation of a p-nitrophenolic pendant arm

Two gadolinium(III) chelates, GdNP-DO3A (1-methlyene-(p-NitroPhenol)-1,4,7,10-tetraazacycloDOdecane-4,7,10-triAcetate) and GdNP-DO3AM (1-methlyene(p-NitroPhenol)-1,4,7,10-tetraazacycloDOdecane-4,7,10-triacetAMide), containing a single nitrophenolic pendant arm plus either three acetate or three amide pendant arms were synthesized and characterized. The properties of the gadolinium, terbium, and dysprosium complexes of these ligands were examined as a function of pH. The extent and mechanism of the changes in water relaxivity with pH of each gadolinium complex was found to differ substantially for the two complexes. The water relaxivity of Gd(NP-DO3A) increases from 4.1 mM(-1) s(-1) at pH 9 to 7.0 mM(-1) s(-1) at pH 5 as a result of acid-catalyzed dissociation of the nitrophenol from the lanthanide. The nitrophenol group in Gd(NP-DO3AM) does not dissociate from the metal center even at pH 5; therefore, the very modest increase in relaxivity in this complex must be ascribed to an increase in prototropic exchange rate of the bound water and/or phenolic protons.     

Microwave-assisted synthesis of N,N'-diaryl cyanoguanidines

[reaction: see text] A mild, efficient, and high-yielding method for the synthesis of N,N'-diaryl cyanoguanidines from their corresponding thioureas under microwave-assisted conditions is described. A series of cyanoguanidines were synthesized containing both electron-donating and electron-withdrawing substituents. The reactions were facilitated by the use of polar solvents along with moderate temperatures.     

Iron‐Catalyzed Stereoselective Cross‐Coupling Reactions of Stereodefined Enol Carbamates with Grignard Reagents

A practical and highly stereoselective iron‐catalyzed cross‐coupling reaction of stereodefined enol carbamates and Grignard reagents to yield tri‐ and tetrasubstituted acrylates is reported. A facile method for the stereoselective generation of these enol carbamates has also been developed.