Third-order Coefficients Research Articles

Chromium-doped forsterite: dispersion measurement with white-light interferometry.

Using a Michelson white-light interferometer, we measure the group-delay dispersion and third-order dispersion coefficients, d2(phi)/d(omega)2 and d3(phi)/d(omega)3, of chromium-doped forsterite (Cr:Mg2SiO4) over wavelengths of 1050-1600 nm for light polarized along both the c and b crystal axes. In this interval, the second-order dispersion for the c axis ranges from 35 fs2/mm to -14 fs2/mm, and the third-order dispersion ranges from 36 fs3/mm to 142 fs3/mm. For the b axis the second-order dispersion ranges from 35 fs2/mm to -15 fs2/mm and the third-order from 73 fs3/mm to 185 fs3/mm. Our data are relevant for the development of optimized dispersion compensation tools for Cr:Mg2SiO4 femtosecond lasers. These measurements help to clarify previously published results and show some significant discrepancies that existed, especially in the third-order dispersion. Our results should furthermore be useful to build up an analytic expression for the index of refraction of chromium forsterite.

A laser-induced fluorescence study of the O( [formula omitted]) + CF 3I and IO + NO reactions : Evidence for an association channel for O + CF 3I at low temperatures

The reaction O( 3 P)+ CF 3 I → k 1 products was monitored by the detection of the product IO using laser-induced fluorescence at 445 nm, with generation of O( 3 P ) via 248 nm photolysis of O 3 or 193 nm photolysis of N 2O. The temporal evolution of IO as a function of [CF 3I] was used to determine the rate coefficient, k 1, in the temperature range 223–523 K at a variety of pressures. At 223 K, a pressure dependence of k 1 was observed, indicative of an association channel, and for N 2 bath gas the third-order rate coefficient was estimated as k assoc=(5.5±2)×10 −31 cm 6 molecule −1 s −1. For T>223 K, k 1 was found to be independent of pressure and was described by the Arrhenius expression: k 1(T)=(9.9±1.0)×10 −12 exp − (1930±800) RT cm 3 molecule −1 s −1 The rate coefficient for the reaction IO+ NO → k 2 products was measured at T=294 K to be k 2=(1.82±0.1)×10 −11 cm 3 molecule −1 s −1, and independent of pressure.

Fuzzy-Hybrid Neural Network Based ECG Beat Recognition Using Three Different Types of Feature Sets

This paper represents the application on the fuzzy-hybrid neural network for electrocardiographic (ECG) beat recognition and classification. We proposed the approach to ECG beat recognition that is based on QRS complex classification. Instead of the original QRS waveform, we have used three different types of QRS features sets. The main objective of this work was to develop a technique which was less sensitive to the morphological variation of the QRS waveform. Linear predictive coefficients (LPC), third-order cumulant-based auto regressive coefficient (AR), and the variance of the wavelet transform detail coefficients of the isolated QRS complexes are used as the features. It will be shown that the wavelet transform based approach is better than the other techniques. The main properties of the proposed method are simplicity, moderate recognition rate, and fast computation time.

Novel Ion-Lens Configurations: A Further Step to 2-nm Ion-Beam Lithography

Composite axially symmetric immersion ion lenses are considered that consist of an electrostatic and a magnetic lens. For the first time, their performance is evaluated over the entire range of operating conditions: from the case of a zero magnetic field to the case of a zero ion energy on the target. Operating conditions are characterized in terms of τ = Wt/W0, where W0 is the energy of an ion at the boundary of the region in which the trajectories are parallel to the axis and Wt is that on the target. For the first time, simple analytical approximations are derived for Cc/r, Cs/r, f/r, and NI, where Cc is the chromatic-aberration coefficient, Cs is the third-order spherical-aberration coefficient, f is the focal distance, NI is the magnetomotive force of the coil, and r is the outer radius of the coil. The behavior of the four quantities is explored as a function of τ. The following conclusions are drawn: (i) The aberrations are maximum for a zero magnetic field. (ii) The aberration coefficients decrease monotonically with increasing NIand decreasing τ, the lens changing from an accelerating to a decelerating one. (iii) If τ → , then Cs/r – τ1/4, Cc/r – τ1/6, f/r– τ1/3, and NI – τ–1/2. (iv) The lenses are suitable for resistless heavy-ion projection lithography and can provide 20 × 1011 pixels of area 2 × 2 nm2 for an exposed area of 3 × 3 mm2. (v) Used in heavy-ion microprobe systems, the lenses could enable resistless lithography over much larger areas than existing equipment.

Modification of the formulas for third-order aberration coefficients.

New parameters for calculation of third-order aberration coefficients (Seidel aberration coefficients) are introduced. The formulas for Seidel aberration coefficients are linear in these new variables. With these new variables it is possible to calculate the shape and the refractive index of the glass of the individual lenses of the optical system, which was not possible before.

Two-dimensional rhombohedral grid coordination polymers [M(bbbt)(2)(NCS)(2)](n)(M = Co, Mn, or Cd; bbbt = 1,1'-(1,4-butanediyl) bis-1H-benzotriazole): synthesis, crystal structures, and third-order nonlinear optical properties.

In this paper, treatment of 1,1'-(1,4-butanediyl) bis-1H-benzotriazole (bbbt) and KSCN with Co(II), Mn(II), or Cd(II) afforded three two-dimensional rhombohedral grid coordination polymers [M(bbbt)(2)(NCS)(2)](n)(M = Co, 1; Mn, 2; Cd, 3). The two-dimensional rhombohedral grids are parallel to the crystallographic ac plane. The rhombohedral grid consists of 44-membered rings of M(4)(bbbt)(4), and gives the dimensions of 12.913 x 10.764 A for polymer 1, 13.106 x 10.797 A for polymer 2, and 13.256 x 10.870 A for polymer 3. The three polymers' third-order nonlinear optical (NLO) properties were determined by Z-scan technique in DMF solution. The results show that all three polymers show very large NLO absorption and strong NLO refraction properties. The third-order NLO absorptive coefficients alpha(2) are 5.4 x 10(-9) m W(-1) for polymer 1, 5.2 x 10(-9) m W(-1) for polymer 2, and 5.0 x 10(-9) m W(-1) for polymer 3. The alpha(2) values are larger than those of all the reported cluster compounds. The NLO refractive index values n(2) of the three polymers are 5.73 x 10(-19), 3.55 x 10(-19), and 3.07 x 10(-19) m(2) W(-1), respectively. Their hyperpolarizability gamma values are calculated to be 2.40 x 10(-30) esu for polymer 1, 1.52 x 10(-30) esu for polymer 2, and 1.50 x 10(-30) esu for polymer 3. The gamma values are comparable to those of clusters and better than those of organometallic compounds, semiconductors, and fullerene.

Perturbative expansions from Monte Carlo simulations at weak coupling: Wilson loops and the static-quark self-energy

Perturbative coefficients for Wilson loops and the static-quark self-energy are extracted from Monte Carlo simulations at weak coupling. The lattice volumes and couplings are chosen to ensure that the lattice momenta are all perturbative. Twisted boundary conditions are used to eliminate the effects of lattice zero modes and to suppress nonperturbative finite-volume effects due to Z(3) phases. Simulations of the Wilson gluon action are done with both periodic and twisted boundary conditions, and over a wide range of lattice volumes (from $3^4$ to $16^4$) and couplings (from $\beta \approx 9$ to $\beta \approx 60$). A high precision comparison is made between the simulation data and results from finite-volume lattice perturbation theory. The Monte Carlo results are shown to be in excellent agreement with perturbation theory through second order. New results for third-order coefficients for a number of Wilson loops and the static-quark self-energy are reported.

Higher-Order Eigensensitivity Analysis of a Defective Matrix

Under the assumption that all of the eigenvalues of the eigenproblem for the first-order perturbation coefficients of the eigenvalues are simple and nonzero, a direct method is given for calculating the first- to third-order perturbation coefficients of the eigenvalues and the first- to second-order perturbation coefficients of the eigenvectors of a defective matrix. The method is extended to the case where one of the first-order perturbation coefficients of the eigenvalues associated with the lowest-order Jordan blocks is zero. Numerical examples show the validity of the method.

Third-order aberration coefficients of anamorphic GRIN optical systems: a one-dimensional refractive index distribution

Third-order geometrical aberration coefficients of anamorphic optical systems with a one-dimensional refractive index distribution are obtained in an explicit form.

High-order combined geometric-chromatic aberrations of magnetic electron lenses using differential algebra

In this paper, high-order geometric and chromatic aberrations of magnetic electron lenses are dealt with together by using the differential algebraic method. High-order geometric-chromatic aberration analytic expressions are obtained and arbitrary order combined aberrations can be calculated. As an example, a Glaser's bell-shaped magnetic electron lens has been studied and the second- and third-order combined geometric-chromatic aberration coefficients have been calculated. The aberration figures of the combined second-order aberrations are also presented. This developed method can be of great utility in high-order combined aberration analysis and correction for high-resolution charged particle optical systems.

Dispersion management for solitons in a Korteweg-de Vries system.

The existence of "dispersion-managed solitons," i.e., stable pulsating solitary-wave solutions to the nonlinear Schrodinger equation with periodically modulated and sign-variable dispersion is now well known in nonlinear optics. Our purpose here is to investigate whether similar structures exist for other well-known nonlinear wave models. Hence, here we consider as a basic model the variable-coefficient Korteweg-de Vries equation; this has the form of a Korteweg-de Vries equation with a periodically varying third-order dispersion coefficient, that can take both positive and negative values. More generally, this model may be extended to include fifth-order dispersion. Such models may describe, for instance, periodically modulated waveguides for long gravity-capillary waves. We develop an analytical approximation for solitary waves in the weakly nonlinear case, from which it is possible to obtain a reduction to a relatively simple integral equation, which is readily solved numerically. Then, we describe some systematic direct simulations of the full equation, which use the soliton shape produced by the integral equation as an initial condition. These simulations reveal regions of stable and unstable pulsating solitary waves in the corresponding parametric space. Finally, we consider the effects of fifth-order dispersion. (c) 2002 American Institute of Physics.

The Morphology, Environment, and Interstellar Medium of Early‐Type Galaxies as a Means for Studying the Evolution of Galaxies Outside of Clusters

The main goal of this work is to study the properties of isolated early-type galaxies with the hope of learning about their formation and evolution. With this goal in mind, B and I optical and near-infrared images of four different samples Ks of early-type galaxies in contrasting galactic density environments have been secured with the Steward Observatory 90 and 61 inch telescopes. The four early-type galaxy samples consist of a sample extracted from the Catalog of Isolated Galaxies (V. E. Karachentseva 1973, Commun. Spec. Astrophys. Obs., 8, 3), a sample of isolated galaxies previously studied by G. Fasano & C. Bonoli (1989, AA most of the galaxies in this sample have been studied by T. Wiklind, F. Combes, & C. Henkel 1995, AA T. Wiklind & C. Henkel 1995, AA and W. K. Huchtmeier, L. J. Sage, & C. Henkel 1995, A&A, 300, 675). This data set is used to examine whether differences in the galactic environment are related to differences in key properties of early-type galaxies. Multicolor surface photometry analysis is performed on the four samples, and the results are used in several contexts. For most of the galaxies in the Karachentseva sample, surface photometry is reported for the first time. Likewise, data for Ks most of the objects are also reported for the first time. The light profile of the galaxies is fitted with Sersic profile functions (J. L. Sersic 1968, Atlas de 1/n I(r) p I exp { b [(r/r ) 1]} e n e Galaxias Australes [Cordoba: Observatorio Astronomico]). We have found that the correlation between the logarithm of the Sersic structure parameter n and the Sersic effective radius re previously discussed by other investigations (e.g., N. Caon, M. Cappacioli, & M. D’Onofrio 1993, MNRAS, 265, 1013 and A. Graham, T. R. Lauer, M. Colles, & M. Postman 1996, ApJ, 465, 534) appears to be valid over a fairly wide range of galaxy sizes. Furthermore, we have found that this correlation does not appear to be strongly dependent on other properties, such as environment or ISM content. Since the de Vaucouleurs profile function is a special case (when ) of the Sersic profile n p 4 function, some of the limitations of the widely accepted de Vaucouleurs function to describe the light profile of elliptical galaxies are discussed. The B I color index, ellipticity, position angle, and thirdand fourth-order Fourier coefficients are derived from the surface photometry. It is confirmed that the third-order coefficients signal the presence of dust and that the coefficient of cos 4v gives structural information on the galaxies. Furthermore, the subtraction of model elliptical galaxies has revealed peculiarities in the morphology, such as disks, spiral arms, dust lanes, multiple nuclei, merger signatures, etc. As an example of model subtraction, we present in Figure 1 the I-band image of the galaxy KIG 83 (UGC 1503) and in Figure 2 the ratio of the original image and a model image. Whereas spiral features can barely be seen in Figure 1, such features clearly stand out in Figure 2. The results of the surface photometry as well as the morphological analysis are used to compare the properties among the different samples. The largest and most relevant sample in this investigation consists of 39 isolated galaxies, and it contains ∼25% of the galaxies classified as elliptical galaxies in the Karachentseva catalog. According to our morphological analysis, only approximately one-half of the galaxies classified as elliptical galaxies in that catalog appear to be correctly classified, a result that may imply a reduction in the percentage of elliptical galaxies in the Karachentseva catalog to ∼6% of the total population of isolated galaxies. This result suggests that the overall percentage of elliptical galaxies that exist in isolated regions of space is smaller by a factor of ∼2 than currently accepted values. Most of the misclassified galaxies turned out to have spiral arms, but a significant number of perturbed galaxies or merger candidates were also found among the galaxies in the Karachentseva catalog. It is argued in the thesis that the fraction of merger candidates among isolated elliptical galaxies can be used to empirically constrain models for the evolution of compact groups into isolated galaxies (J. Saucedo & J. Bieging 2001, Ap&SS, in press). Aside from the optical and infrared studies, the results of a search for the CO(1–0) molecular line for six of the isolated early-type galaxies in the Karachentseva catalog taken at the

COMPUTATION OF CONVECTION-DIFFUSION PROBLEMS DIFFERENCED WITH HIGH-ORDER UPWIND SCHEMES

In this work, we investigate preconditioning techniques for the conjugate gradient method used in convection-diffusion problems differenced by a third-order upwind-biased scheme. To compute the preconditioning matrix, we use the incomplete lower-upper (ILU) decomposition method for matrices computed by the first-order and the third-order differencing schemes. Low-level fill-in is introduced to improve the convergence. For convection-dominant problems, we also investigate the effectiveness of the domain decomposition technique for the computation of the preconditioning matrix. In the computation of the transport of a passive scalar in the driven-cavity flow, we find that the ILU decomposition of the third-order coefficient matrix gives best performance if the fill-in level is high enough. However, in convection-dominant problems, for which a third-order scheme is most appropriate, the common ILU decomposition of the penta-diagonal coefficient matrix of a first-order scheme can achieve similar performance in combination with domain decomposition with a small overlap.

Application of the matrizant method to calculate the third-order aberration coefficients for a sector magnetic field including fringing-field effects

The matrizant method is used to study the nonlinear dynamics of charged particles in magnetic sector analyzers. The calculations of matrizants (transfer matrices) allow for both fringing-field effects due to the stray field and higher harmonics of the sector magnetic field (up to the third order). For the rectangular distribution of the field components along the optical axis, analytical expressions for the aberration coefficients (including dispersion ones) are derived up to the third order. In the simulation of real fields with a nonzero stray-field width, the smooth distribution of the field components is employed. The aberration coefficients for this distribution were calculated by means of a conservative numerical method.

Third-order transport properties of ions in electrostatic fields

Third-order transport properties are calculated through a three-temperature kinetic theory for ions moving in gases under the action of a homogeneous electrostatic field. The analytic procedure is found to be in good agreement with independently obtained numerical results, as well as with calculations from molecular dynamics simulation of the ionic motion. Finally, a relation which enables the calculation of all components of third-order transport coefficients from characteristic integrals of velocity correlation functions is suggested and tested successfully.

Polynomial families and Schrödinger equation: one example for nonhypergeometric type of correspondence

We discuss the explicit construction of the Schrodinger equations admitting representation through some family of general nonorthogonal polynomials. The specific choice of the third-order polynomial coefficient functions, that lead to quasi-solvable families of Schrodinger potentials, is considered in detail.

Relationship between fifth-order aberration coefficients in rotationallysymmetric electron optical system

Relationships between aberration coefficients up to the fifth orderhave been derived using symplectic conditions for a rotationally symmetricelectron optical system. These relationships show that the third- andfifth-order coefficients correlate and some of the fifth-order aberration coefficientscan be calculated by the third-order coefficients. Using these relationships, wecross-checked a computer code for the fifth-order aberration analysis with adifferential algebra method.

Pressure Dependence of Effective Pair Potentials in AgBr Determined by Extended X-Ray Absorption Fine Structure

The pressure dependence of extended X-ray absorption fine structure (EXAFS) Debye-Waller factors in AgBr has been investigated using the cumulant expansion method. The Br K-edge EXAFS spectra were measured in the transmission mode under high pressure (≤9.1 GPa) at room temperature using a cubic anvil type apparatus (MAX90) and synchrotron radiation from the Photon Factory, Tsukuba. The effective pair potentials, V(u)=au2/2+bu3/3!, were evaluated and the potential coefficient a at 2.1, 4.2 and 6.1 GPa are 1.59(4), 1.75(4) and 1.91(4) eV/Å-2, respectively. The energies of the third-order anharmonic potential coefficient b maintain nearly constant values with pressure though the third-order cumulant σ3 decreases with increasing in pressure.

Eigensensitivity Analysis of a Defective Matrix

The formulas for calculating the first- to third-order perturbation coefficients of the eigenvalues and the first-order perturbation coefficients of eigenvectors of a defective matrix are derived by use of the direct perturbation method for the case where the eigenvalue problem for the first-order perturbation coefficients of the eigenvalues of the defective matrix has repeated eigenvalues. Associated with νth-order Jordan blocks of the matrix, the perturbation equations are derived by expanding the perturbed eigenvalues and eigenvectors in power series of η= ∈ 1/ν , where ∈ is the small parameter of the problem, substituting the series into the perturbed eigenvalue problem and comparing the powers of η. The kth-order perturbation equations for k = 1,., ν - 1 can be solved formally. The solutions contain some undetermined quantities. By use of the solvability conditions of the kth-order equations for k = v, ν + 1,..., the undetermined quantities in the solutions can be calculated in turns. When we run all orders of the Jordan blocks, the problem is solved. The numerical example shows the validity of the method.

First-order Markov chain approach to wind speed modelling

Hourly wind speed data in northwestern region of Turkey are simulated by using transition matrix approach of the first-order Markov chain method. For this purpose, the wind speed time series is divided into various states depending on the arithmetic average and the standard deviation. Once the control and validation of the model is confirmed, it is then used for generating synthetic series of various lengths of any desired duration. Wind speed measurement parameters are used to generate synthetic series with the preservation of the statistical parameters and the first-order autocorrelation coefficient. It is observed that for short periods, the parametric results obtained from the synthetic time series are close to the measured values. It is concluded in this paper that first-order Markov chain, despite its simplicity, accounts for more than 90% of the statistical parameters in the synthetic wind speed time series at most of the 10 stations considered. However, the discrepancy of the results is due to the use of first-order Markov chain as a first approximation. Since, hourly wind speed data are used in addition to the first-order autocorrelation coefficient second and even third-order autocorrelation coefficients are also significant. Therefore, in the future hourly wind speed modelling at least second- or preferably third-order, Markov chain must be tried.