Chebyshev expansions and rational approximations

Let A(z) = A_m(z) + a_mz^mB(z,m) where A_m(z) is a polynomial in z of degree m-1. Suppose A(z) and B(z,m) are approximated by main diagonal Padé approximations of order n and r respectively. Suppose that the number of operations needed to evaluate both sides of the above equations by means of the Padé approximations and polynomial noted are the same. Thus 4n = 3m + 4r. We address ourselves to the question of which procedure is more efficient? That is, which procedure produces the smallest error? A variant of this problem is the situation where A(z) and B(z,m) are approximated by their representations in infinite series of Chebyshev polynomials of the first kind truncated after n and r terms respectively. Here n = m + r.Let F(z) have two different series type representations in overlapping or completely disjoint regions of the complex z-plane. Suppose that for each representation there is a sequence of rational approximations of the same type, say of the Padé class, which converge for |arg z| < π except possibly for some finite set of points. Assume that the number of machine operations required to make evaluations using the noted approximations are the same. Again, we ask which procedure is best? Other variants are studied.General answers to the above questions are not known. Instead, we illustrate the ideas for a number of the rather common special functions.     

A note on evaluation of coefficients in the polynomials of Padé approximants by solving systems of linear equations

This is a sequel to my previous paper concerning the determination of the coefficients in the polynomials which define Padé fractions, where the coefficients are found by solving systems of linear equations. The present note uses the same models and computer as before, but the computations are far more extensive so as to reveal more pointedly the effects of round off error in the coefficients as the order of the system increases. Only the main diagonal Padé entries are studied numerically. The numerics are achieved using two routines in LINPACK one of which evaluates a condition number for the matrix. This is advantageous if one suspects ill conditioning. In our previous paper, it was shown that though the relative errors in the numerator and denominator polynomials increase as the order of the system increases, the effect on the Padé approximant is virtually nil at least for the size of the variable x and the order n considered. Of course, ultimately so as to render the without bound, we should expect so much contamination due to round off so as to render the results meaningless. All of this is illustrated with numerics. Heuristic procedures based on the numerical data developed are presented to warn of deterioration due to round off.     

The determination of normal paraffins in petroleum products

A normal paraffin, outside the carbon number range of the sample, is quantitatively added to the sample as an internal standard, and 10–100 μl of the mixture are injected into a small absorber unit which contains 400 mg of activated molecular sieve. The temperature of the absorber unit is raised from ambient to 300° at about 15°/min while nitrogen is passed. This elutes the non-linear hydrocarbons but leaves the normal paraffins absorbed. The sieve is removed from the absorber unit and the paraffins are released by destroying the sieve structure with hydrofluoric acid. A pellet of potassium hydroxide is then added to neutralise the excess acid and the released normal paraffins are extracted in 0.3 ml of isooctane. This isooctane solution is examined on a programmed-temperature chromatograph; the chromatogram obtained consists of a solvent peak followed by well-resolved peaks representing the normal paraffins. These are easily measured and the concentrations of the normal paraffins in the original sample are calculated against the internal standard.     

An EBIC equation for solar cells

When an electron beam of a scanning electron microscope (SEM) impinges on an N-P junction, the generation of electron-hole pairs by impact ionization causes a characteristic short circuit current I_SC to flow. The I_SC, i.e. EBIC (electron beam induced current) depends strongly on the configuration used to investigate the cell's response. In this paper we consider the case where the plane of the junction is perpendicular to the surface. An EBIC equation amenable to numerical computations is derived as a function of cell thickness, source depth, surface recombination velocity, diffusion length, and distance of the junction to the beam-cell interaction point for a cell with an ohmic contact at its back surface. It is shown that the EBIC equation presented here is more general and easier to use than those previously reported. The effects of source depth, ohmic contact, and diffusion length on the normalized EBIC characteristic are discussed.     

A structurally guided method for the decomposition of expression in music performance

A method for separating, profiling, and quantifying the contributions of different structural components to expressive musical performance is described. The method is demonstrated through its application to a set of expert piano performances of a short piece from the classical period. The results show that the output of the method aids in the understanding of how the different structural components in a piece of music combine in the generation of an expressive performance. A second demonstration applies the method to performances at different tempi to illustrate its effectiveness in pinpointing the structural features responsible for small but statistically significant differences between performances. The method is compared with other approaches to the analysis and modeling of musical performance, and a number of potential applications are identified.     

Gas phase infrared multiple-photon dissociation spectra of methanol, ethanol and propanol proton-bound dimers, protonated propanol and the propanol/water proton-bound dimer

The infrared multiphoton dissociation (IRMPD) spectra of three homogenous proton-bound dimers are presented and the major features are assigned based on comparisons with the neutral alcohol and with density functional theory calculations. As well, the IRMPD spectra of protonated propanol and the propanol/water proton-bound dimer (or singly hydrated protonated propanol) are presented and analysed. Two primary IRMPD photoproducts were observed for each of the alcohol proton bound dimers and were found to vary with the frequency of the radiation impinging upon the ions. For example, when the proton-bound dimer absorbs weakly a larger amount of S(N)2 product, protonated ether and water, are observed. When the proton-bound dimer absorbs more strongly, an increase in the simple dissociation product, protonated alcohol and neutral alcohol, is observed. With the aid of RRKM calculations this frequency dependence of the branching ratio is explained by assuming that photon absorption is faster than dissociation for these species and that only a few photons extra are necessary to make the higher-energy dissociation channel (simple cleavage) competitive with the lower energy (S(N)2) reaction channel.     

Morphology of polythiophene and polyphenyl films produced via surface polymerization by ion-assisted deposition

Conducting polymer films are grown by either mass-selected or non-mass-selected, hyperthermal thiophene ions coincident on a surface with a thermal beam of organic monomers of either alpha-terthiophene (3T) or p-terphenyl (3P) neutrals. Previous experiments verified polymerization of both 3T and 3P by 200 eV C(4)H(4)S(+) during surface polymerization by ion-assisted deposition (SPIAD). A wide variety of structures are observed by scanning electron microscopy to form in the SPIAD polythiophene and polyphenyl films. These structures include microscale islands, lamellar structures, fractal-like growth patterns, and nanoscale crystallites. Some of the deposited films diffract X-rays while others show electron micrographs of crystallites. The variation of these patterns with deposition conditions clearly indicate that ion-induced polymerization mediates film morphology through control of ion energy and ion/neutral ratio. Furthermore, these ion-assisted events mediate important thermal processes such as sublimation.     

Lost heat capacity and entropy in the helical magnet MnSi

The heat capacity of MnSi at B = 0 and B = 4 T was measured in the temperature range 2.5-100 K. To analyze the data, calculations of the phonon spectrum and phonon density of states in MnSi were performed. The calculated phonon frequencies were confirmed by means of inelastic neutron scattering. The analysis of the data suggests the existence of negative contributions to the heat capacity and entropy of MnSi at T > T(c) that may imply a specific ordering in the spin subsystem in the paramagnetic phase of MnSi.