Hausdorff measures and packing measures of limit sets of CIFSs of generalized complex continued fractions

ABSTRACT

We consider a certain family of CIFSs of the generalized complex continued fractions with a complex parameter space. We show that for each CIFS of the family, the Hausdorff measure of the limit set of the CIFS with respect to the Hausdorff dimension is zero and the packing measure of the limit set of the CIFS with respect to the Hausdorff dimension is positive (main result). This is a new phenomenon of infinite CIFSs which cannot hold in finite CIFSs. We prove the main result by showing some estimates for the unique conformal measure of each CIFS of the family and by using some geometric observations.     

Charge transfer during H/H− collisions with Cu(100) and Cu(111) surfaces

We study the H and H⁻ survival probabilities during collisions with Cu(100) and Cu(111) surfaces, at energies ranging from 0.5 to 5 keV and exit angles ranging from 20° to 90°. Calculations are performed with the Wave‐Packet Propagation method adapted to ion‐surface interactions. The projectile survival probability depends on the perpendicular velocity and the copper face being investigated. Projectile's interaction time with the surface and the distance of closest approach are important factors that influence the survival. The H⁻ survival on Cu(100) is much smaller than on Cu(111) but only at low velocities, while becoming higher or comparable to Cu(111) for higher velocities. For very fast collisions, the copper surface behaves like a jellium, and the electron involved in charge transfer does not “feel” the particularities of the surface band structure anymore. While the H survival on Cu(100) seems to not depend on energy and exit angle, the H survival on Cu(111) is both energy and angle dependent, and it is smaller. The study of partial density of states indicates that strong atom‐surface interactions at short distances and the role played by surface states are important factors in determining the neutral fractions obtained after scattering.     

Comparison of chiral NMR solvating agents for the enantiodifferentiation of amines

Chiral, enantiomerically pure organic-soluble acids are often used as NMR chiral solvating agents (CSAs) for analyzing the enantiopurity of amines. However, the reports that describe CSAs for amines provide limited comparisons to other previously reported reagents. As such, it is difficult to know which among the many CSAs to pick when studying a new amine. We report a comparison of thirteen commercially available CSAs for the analysis of primary, secondary and tertiary amines in chloroform-d. (R)-α-methoxyphenylacetic acid, (R)-(-)-O-acetylmandelic acid, (R)-2-methoxy-2-(1-naphthyl)propionic acid and (R)-(-)-1,1′-binaphthyl-2,2′-diylhydrogenphosphate are identified as the best four to use as a starting point for the analysis of a new amine.     

Contiguous relations, continued fractions and orthogonality

We examine a special linear combination of balanced very-well-poised basic hypergeometric series that is known to satisfy a transformation. We call this and show that it satisfies certain three-term contiguous relations. From two of these contiguous relations for we obtain fifty-six pairwise linearly independent solutions to a three-term recurrence that generalizes the recurrence for Askey-Wilson polynomials. The associated continued fraction is evaluated using Pincherle's theorem. From this continued fraction we are able to derive a discrete system of biorthogonal rational functions. This ties together Wilson's results for rational biorthogonality, Watson's -analogue of Ramanujan's Entry 40 continued fraction, and a conjecture of Askey concerning the latter. Some new -series identities are also obtained. One is an important three-term transformation for 's which generalizes all the known two- and three-term transformations. Others are new and unexpected quadratic identities for these very-well-poised 's.

     

Rapid prediction and optimization of concentration conditions for preparative fractions by solid-phase extraction

SPE is an effective tool for concentrating preparative fractions isolated from a complex sample. To guarantee high efficiency and recovery of concentration, the concentration conditions could be optimized by predicting the breakthrough volume (V(B)). In this study, a method of predicting V(B )of unknown compounds in preparative fractions at any isocratic mobile phase composition with the analytical retention parameters a and c is described. The a and c values and the relationship between half peak width (W(1/2)) and retention time of a model analyte were measured using the analytical elution mode on an SPE column, and the V(B )and retention volume (V(R)) predicted with the a and c values were validated with breakthrough experiments. However, it is impossible to measure the a and c values of multiple compounds in a complex system directly on an SPE column with a low number of theoretical plates. The correlation of the a and c values between the SPE and analytical columns was developed so that the analytical data could be transferred to the SPE column. With the calculated a and c values, we could optimize the concentration conditions on the basis of the predicted V(B )and the volume of the preparative fraction.     

Determination of volatile and non-volatile nickel and vanadium compounds in crude oil using electrothermal atomic absorption spectrometry after oil fractionation into saturates, aromatics, resins and asphaltenes

In recent work, it has been shown that electrothermal atomic absorption spectrometry (ET AAS) can be used to differentiate between volatile and non-volatile nickel and vanadium compounds in crude oil. In the present work, the distribution of these two groups of compounds over different fractions of crude oil was investigated. For this purpose two crude oil samples were separated in two steps: firstly, the asphaltenes were precipitated with n-heptane, and secondly, the maltenes were loaded on a silica column and eluted with solvents of increasing polarity. The four fractions of maltenes eluted from silica column were: F1, saturated and light aromatics; F2, polyaromatics; F3, resins; and F4, polar compounds. Fractions F1 and F2 were further investigated using gas chromatography, and all fractions were characterized by CHN analysis, confirming the increase of aromatics in the fractions 2, 3, 4 and asphaltenes. For the determination of Ni and V by ET AAS, oil-in-water emulsions were prepared. The speciation analysis was carried out measuring without chemical modifier (stable compounds) and with 20 μg palladium (total Ni and V) and the volatile fraction was calculated by difference. The limits of detection were 0.02 μg g⁻¹ and 0.06 μg g⁻¹, for Ni and V, respectively, based on an emulsion of 2 g of oil in 10 mL. The volatile species of Ni and V were associated with fractions F3 and F4, while only thermally stable Ni and V was precipitated in part together with the asphaltenes.     

Direct asymmetric aldol reactions catalyzed by nanocrystalline copper(II) oxide

The direct asymmetric aldol reactions of aromatic and heteroaromatic aldehydes with acetone to afford chiral β-hydroxy carbonyl compounds in good yields and good to moderate enantioselectivities are realized using nanocrystalline copper(II) oxide in the presence of (1S,2S)-(−)-1,2-diphenylethylenediamine at −30 °C. The catalyst can be reused for four cycles with consistent activity and enantioselectivity.     

On the convergence of continued fractions at Runckel’s points

We consider the limit periodic continued fractions of Stieltjes type

Density functional theory calculations of novel Rh(I) diphosphinite catalysts

Novel Rh(I) diphosphinite catalysts [Rh((R,R)-3,4-(bis(O-diphenylphosphino)-1,2,5,6-tetra-O-methyl-chiro-inositol)]+ ([Rh-CANDYPHOS]+) and [Rh((R,R)-3,4-(bis(O-diphenylphosphino)-1,2,5,6-tetra-O-ethyl-chiro-inositol)]+ ([Rh-EtCP]+) have been prepared utilizing naturally-occurring resources. Potential energy surfaces for the catalyzed asymmetric hydrogenation of the prochiral enamides methyl-(Z)-α-acetamido cinnamate, methyl-(Z)-α-acetamido cinnamic acid and dimethyl itaconate have been surveyed using density function theory (DFT) methods. Key transition states were identified from previous [Rh((R,R)-DUPHOS)]+ studies for the two diastereoisomeric manifolds 1 and 2. Transition state energies were found starting from models based on (1) the X-ray structure of the active complex (CANDYPHOS)(η⁴-(Z,Z)-cyclo-octa-1,5-diene)-rhodium(I) tetrafluoroborate CHCl₃ solvate [3] and (2) models in which the complex (without substrate) started with C2 molecular symmetry. The difficulties encountered in calculations of the transition state energies of large cations are outlined and limitations noted. Transition state enthalpy values are compared with the observed experimental free energy differences results and previous studies 1 and 2. The predictive aspects of the calculations appear to be limited with the starting models playing an important part in the absolute value of the final energies.