How Many Timesteps for a Cycle? Analysis of the Wisdom-Holman Algorithm

The Wisdom-Holman algorithm is an effective method for numerically solving nearly integrable systems. It takes into account the exact solution of the integrable part. If the nearly integrable system is the solar system, for example, the Wisdom-Holman algorithm uses the solution consisting of Keplerian orbits obtained when the interplanetary interactions are ignored. The effectiveness of the algorithm lies in its ability to take long timesteps. We use the Duffing oscillator and Kepler's problem with forcing to deduce how long those timesteps can be. For nearly Keplerian orbits, the timesteps must be at least six per orbital period even when the orbital eccentricity is zero. High eccentricity of the Keplerian orbits constrains the algorithm and forces it to take shorter timesteps. The analysis is applied to the solar system and other problems.     

Fast head-related transfer function measurement via reciprocity

An efficient method for head-related transfer function (HRTF) measurement is presented. By applying the acoustical principle of reciprocity, one can swap the speaker and the microphone positions in the traditional (direct) HRTF measurement setup, that is, insert a microspeaker into the subject's ear and position several microphones around the subject, enabling simultaneous HRTF acquisition at all microphone positions. The setup used for reciprocal HRTF measurement is described, and the obtained HRTFs are compared with the analytical solution for a sound-hard sphere and with KEMAR manikin HRTF obtained by the direct method. The reciprocally measured sphere HRTF agrees well with the analytical solution. The reciprocally measured and the directly measured KEMAR HRTFs are not exactly identical but agree well in spectrum shape and feature positions. To evaluate if the observed differences are significant, an auditory localization model based on work by J. C. Middlebrooks [J. Acoust. Soc. Am. 92, 2607-2624 (1992)] was used to predict where a virtual sound source synthesized with the reciprocally measured HRTF would be localized if the directly measured HRTF were used for the localization. It was found that the predicted localization direction generally lies close to the measurement direction, indicating that the HRTFs obtained via the two methods are in good agreement.     

β-Cyclodextrin Mediated Synthesis of Syndiotactic-rich Polystyrene

Stereoregular polystyrene (PS) was prepared using various molarequivalents of -cyclodextrin (-CD) by polymerisingstyrene in the presence of -CD. Stereoregular (isotactic,atactic and syndiotactic) distributions of the preparedpolystyrene polymers were determined from terminal modelBernoullian statistics using ¹³C NMR data. Inclusioncomplexation of styrene by -cyclodextrin was detected byUV-Visible spectroscopy, which gave a binding constant value of31606 ± 3350 M^-1 for the 1 : 1 complex. With anincrease in the styrene : -CD ratios, the proportionof syndiotactic polymers increased. The glass transitiontemperature (T_g) of the polymer also increased along withmelting temperature (T_m) at higher styrene : -CDratios. In addition, the molecular weight of the polymers prepareddecreased with increase in the -CD concentration.     

A structural study of the naringin-β-cyclodextrin complex

The structure of the inclusion complex formed between naringin (naringenin-7-O--neohesperidoside) and-cyclodextrin (BCD) was studied in detail by UV and NMR spectroscopic techniques and potentiometry. A binding constant value of 1016±150M^–1 was arrived at from UV studies. Potentiometric studies showed that pK values of 4-OH and 5-OH were affected by and-cyclodextrins. One-dimensional difference NOE and spin-lattice relaxation time (T ₁) measurements indicated that the aglycone protion was affected more than the neohesperidoside portion. TheT ₁ values analysed for local motions indicated that _c values of complexed naringin was higher than that of free naringin. The internal rotation calculated for different groups showed _i values for the phenolic and dihydrobenzopyran portion decrease by a factor of 2. Also a value of 0.12–0.17 observed for the aglycone portion indicated that the coupling between guest and host is weak. All the studies have shown that the disposition in which the phenol group at 2 is inside the BCD cavity with 4-keto and 5-OH hydrogen bonded to the secondary hydroxyl groups at the rim of the wider end of the BCD cavity is the most probable one.     

Inclusion of Ring A of Cholesterol Inside the β-Cyclodextrin Cavity: Evidence from Oxidation Reactions and Structural Studies

The disposition of cholesterol inside the -cyclodextrin cavity(-CD) was deduced from oxidation of cholesterol secondary alcoholgroups by Ca(OCl)₂ and H₂O₂ in thepyridine–acetic acid system. The amount of cholest-4-ene-3-one formedwas found to be proportional to the concentration of -cyclodextrin,resulting in 56.1% of ketone. The oxidation rate was enhanced by-cyclodextrin and its methyl, polymer and 1 : 1copper(II)–-cyclodextrin derivatives. Detailed investigationsinvolving UV-visible, ¹³C- and ¹H-NMR(T₁, 1D NOE and ROESY) spectroscopic studies were carried out.A binding constant value of 15,385 ± 1500 M^-2 wasobtained for the 2 : 1heptakis-2,6-di-O-methyl--cyclodextrin(DM-CD) : cholesterolcomplex in chloroform from UV studies. Proton and solid state¹³C-CP MAS spectra of the -CD–cholesterol mixtureshowed large magnitude shifts for the protons from the wider end of the-CD cavity as well as those of ring A and ring B of cholesterol. Both1D NOE and ROESY measurements indicated the proximity between ring A andring B protons of cholesterol and the wider end protons of -CD andDM-CD. Besides, analysis of _c,_i and tau;_m from T₁measurements showed not only a lowering of rotational motions but a value of 0.016–0.048 for some of the cholesterol protons, typical of aweak complex. Based on these studies, a probable structure for the 2 : 1complex involving two molecules of -CD/DM-CD was proposed withportions of ring A and ring B being present inside the wider end of the-CD/DM-CD cavity and ring D and the side chain attached atposition 17, projecting into the wider end of the secondCD/DM-CD molecule.