Home About us Contact
|
Home About us Contact | ||
|
|
||
Series Expansion (series + expansion)
Kinds of Series Expansion Selected AbstractsPhysical foundations, models, and methods of diffusion magnetic resonance imaging of the brain: A reviewCONCEPTS IN MAGNETIC RESONANCE, Issue 5 2007Ludovico Minati Abstract The foundations and characteristics of models and methods used in diffusion magnetic resonance imaging, with particular reference to in vivo brain imaging, are reviewed. The first section introduces Fick's laws, propagators, and the relationship between tissue microstructure and the statistical properties of diffusion of water molecules. The second section introduces the diffusion-weighted signal in terms of diffusion of magnetization (Bloch,Torrey equation) and of spin-bearing particles (cumulant expansion). The third section is dedicated to the rank-2 tensor model, the bb -matrix, and the derivation of indexes of anisotropy and shape. The fourth section introduces diffusion in multiple compartments: Gaussian mixture models, relationship between fiber layout, displacement probability and diffusivity, and effect of the b -value. The fifth section is devoted to higher-order generalizations of the tensor model: singular value decompositions (SVD), representation of angular diffusivity patterns and derivation of generalized anisotropy (GA) and scaled entropy (SE), and modeling of non-Gaussian diffusion by means of series expansion of Fick's laws. The sixth section covers spherical harmonic decomposition (SHD) and determination of fiber orientation by means of spherical deconvolution. The seventh section presents the Fourier relationship between signal and displacement probability (Q -space imaging, QSI, or diffusion-spectrum imaging, DSI), and reconstruction of orientation-distribution functions (ODF) by means of the Funk,Radon transform (Q -ball imaging, QBI). © 2007 Wiley Periodicals, Inc. Concepts Magn Reson Part A 30A: 278,307, 2007. [source] Three-dimensional thermoelastic stresses in off-axis oriented single crystals with hexagonal symmetryCRYSTAL RESEARCH AND TECHNOLOGY, Issue 3 2007K. Böttcher Abstract A three-dimensional (3D) thermoelastic stress analysis is carried out on a single crystal with axisymmetric geometry but with a hexagonal crystallographic symmetry. The crystallographic orientation is off-axis with respect to the cylindrical coordinate system. By applying a Fourier series expansion with respect to the rotational angle , of the cylindrical coordinates, the 3D boundary value problem is reduced to a sequence of 2D ones on the meridian plane, which are solved by the finite-element method. In our example, the off-axis orientation is towards a direction of high symmetry, and therefore only four of the six stress tensor components are non-zero. In the end, the stress tensor is projected onto the slip system of the crystal. (© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) [source] Optimal Portfolio Allocation under Higher MomentsEUROPEAN FINANCIAL MANAGEMENT, Issue 1 2006Eric Jondeau C22; C51; G12 Abstract We evaluate how departure from normality may affect the allocation of assets. A Taylor series expansion of the expected utility allows to focus on certain moments and to compute the optimal portfolio allocation numerically. A decisive advantage of this approach is that it remains operational even for a large number of assets. While the mean-variance criterion provides a good approximation of the expected utility maximisation under moderate non-normality, it may be ineffective under large departure from normality. In such cases, the three-moment or four-moment optimisation strategies may provide a good approximation of the expected utility. [source] Body-wave traveltime and amplitude shifts from asymptotic travelling wave couplingGEOPHYSICAL JOURNAL INTERNATIONAL, Issue 2 2006F. Pollitz SUMMARY We explore the sensitivity of finite-frequency body-wave traveltimes and amplitudes to perturbations in 3-D seismic velocity structure relative to a spherically symmetric model. Using the approach of coupled travelling wave theory, we consider the effect of a structural perturbation on an isolated portion of the seismogram. By convolving the spectrum of the differential seismogram with the spectrum of a narrow window taper, and using a Taylor's series expansion for wavenumber as a function of frequency on a mode dispersion branch, we derive semi-analytic expressions for the sensitivity kernels. Far-field effects of wave interactions with the free surface or internal discontinuities are implicitly included, as are wave conversions upon scattering. The kernels may be computed rapidly for the purpose of structural inversions. We give examples of traveltime sensitivity kernels for regional wave propagation at 1 Hz. For the direct SV wave in a simple crustal velocity model, they are generally complicated because of interfering waves generated by interactions with the free surface and the Mohorovi,i, discontinuity. A large part of the interference effects may be eliminated by restricting the travelling wave basis set to those waves within a certain range of horizontal phase velocity. [source] Traveltime approximation for a reflected wave in a homogeneous anisotropic elastic layerGEOPHYSICAL JOURNAL INTERNATIONAL, Issue 1 2002M. Zillmer Summary An approximation to the traveltime field is calculated for an elastic wave that propagates in a homogeneous anisotropic layer and is reflected at a plane boundary. The traveltime is approximated by a Taylor series expansion with the third derivative of the traveltime being taken into account. The coefficients of the series refer to the seismic ray, which is locally the fastest ray. Simple formulae are obtained for orthorhombic media in the crystal coordinate system, which relate the traveltimes of the reflected waves to the elastic constants of the medium. A numerical example is presented for wave propagation in orthorhombic olivine, which is a constituent of the Earth's mantle. A second example is given by an isotropic host rock with a set of parallel cracks, which is an important model for wave propagation in the Earth's crust. The elastic parameters can be determined by measuring the reflection times as a function of source,receiver offset. The approximate traveltime,distance curves are compared with traveltimes obtained from seismic ray tracing. [source] MODFLOW 2000 Head Uncertainty, a First-Order Second Moment MethodGROUND WATER, Issue 3 2003Harry S. Glasgow A computationally efficient method to estimate the variance and covariance in piezometric head results computed through MODFLOW 2000 using a first-order second moment (FOSM) approach is presented. This methodology employs a first-order Taylor series expansion to combine model sensitivity with uncertainty in geologic data. MOD-FLOW 2000 is used to calculate both the ground water head and the sensitivity of head to changes in input data. From a limited number of samples, geologic data are extrapolated and their associated uncertainties are computed through a conditional probability calculation. Combining the spatially related sensitivity and input uncertainty produces the variance-covariance matrix, the diagonal of which is used to yield the standard deviation in MODFLOW 2000 head. The variance in piezometric head can be used for calibrating the model, estimating confidence intervals, directing exploration, and evaluating the reliability of a design. A case study illustrates the approach, where aquifer transmis-sivity is the spatially related uncertain geologic input data. The FOSM methodology is shown to be applicable for calculating output uncertainty for (1) spatially related input and output data, and (2) multiple input parameters (trans-missivity and recharge). [source] Solution of clamped rectangular plate problemsINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN BIOMEDICAL ENGINEERING, Issue 10 2004Robert L. Taylor Abstract In this brief note, we present an efficient scheme for determining very accurate solutions to the clamped rectangular plate problem. The method is based upon the classical double cosine series expansion and an exploitation of the Sherman,Morrison,Woodbury formula. If the cosine expansion involves M terms and N terms in the two plate axes directions, then the classical method for this problem involves solving a system of (MN) × (MN) equations. Our proposal reduces the problem down to a system of well-conditioned N × N equations (or M × M when M < N). Numerical solutions for rectangular plates with various side ratios are presented and compared to the solution generated via Hencky's method. Corrections to classical results and additional digits for use in finite-element convergence studies are given. As an application example, these are used to show the rate of convergence for thin plate finite-element solutions computed using the Bogner,Fox,Schmit element. Copyright © 2004 John Wiley & Sons, Ltd. [source] Stability and accuracy analysis of a discrete model reference adaptive controller without and with time delayINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 9 2010Oreste S. Bursi Abstract Adaptive control techniques can be applied to dynamical systems whose parameters are unknown. We propose a technique based on control and numerical analysis approaches to the study of the stability and accuracy of adaptive control algorithms affected by time delay. In particular, we consider the adaptive minimal control synthesis (MCS) algorithm applied to linear time-invariant plants, due to which, the whole controlled system generated from state and control equations discretized by the zero-order-hold (ZOH) sampling is nonlinear. Hence, we propose two linearization procedures for it: the first is via what we term as physical insight and the second is via Taylor series expansion. The physical insight scheme results in useful methods for a priori selection of the controller parameters and of the discrete-time step. As there is an inherent sampling delay in the process, a fixed one-step delay in the discrete-time MCS controller is introduced. This results in a reduction of both the absolute stability regions and the controller performance. Owing to the shortcomings of ZOH sampling in coping with high-frequency disturbances, a linearly implicit L-stable integrator is also used within a two degree-of-freedom controlled system. The effectiveness of the methodology is confirmed both by simulations and by experimental tests. Copyright © 2009 John Wiley & Sons, Ltd. [source] A second-order homogenization procedure for multi-scale analysis based on micropolar kinematicsINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 12 2007Ragnar Larsson Abstract The paper presents a higher order homogenization scheme based on non-linear micropolar kinematics representing the macroscopic variation within a representative volume element (RVE) of the material. On the microstructural level the micro,macro kinematical coupling is introduced as a second-order Taylor series expansion of the macro displacement field, and the microstructural displacement variation is gathered in a fluctuation term. This approach relates strongly to second gradient continuum formulations, presented by, e.g. Kouznetsova et al. (Int. J. Numer. Meth. Engng 2002; 54:1235,1260), thus establishing a link between second gradient and micropolar theories. The major difference of the present approach as compared to second gradient formulations is that an additional constraint is placed on the higher order deformation gradient in terms of the micropolar stretch. The driving vehicle for the derivation of the homogenized macroscopic stress measures is the Hill,Mandel condition, postulating the equivalence of microscopic and macroscopic (homogenized) virtual work. Thereby, the resulting homogenization procedure yields not only a stress tensor, conjugated to the micropolar stretch tensor, but also the couple stress tensor, conjugated to the micropolar curvature tensor. The paper is concluded by a couple of numerical examples demonstrating the size effects imposed by the homogenization of stresses based on the micropolar kinematics. Copyright © 2006 John Wiley & Sons, Ltd. [source] Boundary element formulation for 3D transversely isotropic cracked bodiesINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 4 2004M. P. Ariza Abstract The boundary traction integral representation is obtained in elasticity when the classical displacement representation is differentiated and combined according to Hooke's law. The use of both traction and displacement integral representations leads to a mixed (or dual) formulation of the BEM where the discretization effort for crack problems is much smaller than in the classical formulation. A boundary element analysis of three-dimensional fracture mechanics problems of transversely isotropic solids based on the mixed formulation is presented in this paper. The hypersingular and strongly singular kernels appearing in the formulation are regularized by using two terms of the displacement series expansion and one term of the traction expansion, at the collocation point. All the remaining integrals are analytically evaluated or transformed by means of Stokes' theorem into regular or weakly singular integrals, which are numerically computed. The method is general and can be used for elements of any shape including quarter-point crack front elements. No change of co-ordinates is required for the integration. The formulation as presented in this paper is something as clear, general and easy to handle as the classical BE formulation. It is used in combination with three-dimensional quadratic and quarter-point elements to obtain accurate results for several different crack problems. Cracks in boundless and finite transversely isotropic domains are studied. The meshes are simple and include only discretization of the crack and the external boundary. The obtained results are in good agreement with those existing in the literature. Copyright © 2004 John Wiley & Sons, Ltd. [source] A low-order, hexahedral finite element for modelling shells,INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 7 2004Samuel W. Key Abstract A thin, eight-node, tri-linear displacement, hexahedral finite element is the starting point for the derivation of a constant membrane stress resultant, constant bending stress resultant shell finite element. The derivation begins by introducing a Taylor series expansion for the stress distribution in the isoparametric co-ordinates of the element. The effect of the Taylor series expansion for the stress distribution is to explicitly identify those strain modes of the element that are conjugate to the mean or average stress and the linear variation in stress. The constant membrane stress resultants are identified with the mean stress components, and the constant bending stress resultants are identified with the linear variation in stress through the thickness along with in-plane linear variations of selected components of the transverse shear stress. Further, a plane-stress constitutive assumption is introduced, and an explicit treatment of the finite element's thickness is introduced. A number of elastic simulations show the useful results that can be obtained (tip-loaded twisted beam, point-loaded hemisphere, point-loaded sphere, tip-loaded Raasch hook, and a beam bent into a ring). All of the gradient/divergence operators are evaluated in closed form providing unequivocal evaluations of membrane and bending strain rates along with the appropriate divergence calculations involving the membrane stress and bending stress resultants. The fact that a hexahedral shell finite element has two distinct surfaces aids sliding interface algorithms when a shell folds back on itself when subjected to large deformations. Published in 2004 by John Wiley & Sons, Ltd. [source] A two-step Taylor-characteristic-based Galerkin method for incompressible flows and its application to flow over triangular cylinder with different incidence anglesINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 11 2010Yan Bao Abstract An alternative characteristic-based scheme, the two-step Taylor-characteristic-based Galerkin method is developed based on the introduction of multi-step temporal Taylor series expansion up to second order along the characteristic of the momentum equation. Contrary to the classical characteristic-based split (CBS) method, the current characteristic-based method does not require splitting the momentum equation, and segregate the calculation of the pressure from that of the velocity by using the momentum,pressure Poisson equation method. Some benchmark problems are used to examine the effectiveness of the proposed algorithm and to compare with the original CBS method, and the results show that the proposed method has preferable accuracy with less numerical dissipation. We further applied the method to the numerical simulation of flow around equilateral triangular cylinder with different incidence angles in free stream. In this numerical investigation, the flow simulations are carried out in the low Reynolds number range. Instantaneous streamlines around the cylinder are used as a means to visualize the wake region behind, and they clearly show the flow pattern around the cylinder in time. The influence of incidence angle on flow characteristic parameters such as Strouhal number, Drag and Lift coefficients are discussed quantitatively. Copyright © 2009 John Wiley & Sons, Ltd. [source] Development of a class of multiple time-stepping schemes for convection,diffusion equations in two dimensionsINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 12 2006R. K. Lin Abstract In this paper we present a class of semi-discretization finite difference schemes for solving the transient convection,diffusion equation in two dimensions. The distinct feature of these scheme developments is to transform the unsteady convection,diffusion (CD) equation to the inhomogeneous steady convection,diffusion-reaction (CDR) equation after using different time-stepping schemes for the time derivative term. For the sake of saving memory, the alternating direction implicit scheme of Peaceman and Rachford is employed so that all calculations can be carried out within the one-dimensional framework. For the sake of increasing accuracy, the exact solution for the one-dimensional CDR equation is employed in the development of each scheme. Therefore, the numerical error is attributed primarily to the temporal approximation for the one-dimensional problem. Development of the proposed time-stepping schemes is rooted in the Taylor series expansion. All higher-order time derivatives are replaced with spatial derivatives through use of the model differential equation under investigation. Spatial derivatives with orders higher than two are not taken into account for retaining the linear production term in the convection,diffusion-reaction differential system. The proposed schemes with second, third and fourth temporal accuracy orders have been theoretically explored by conducting Fourier and dispersion analyses and numerically validated by solving three test problems with analytic solutions. Copyright © 2006 John Wiley & Sons, Ltd. [source] A novel finite point method for flow simulationINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 12 2002M. Cheng Abstract A novel finite point method is developed to simulate flow problems. The mashes in the traditional numerical methods are supplanted by the distribution of points in the calculation domain. A local interpolation based on the properties of Taylor series expansion is used to construct an approximation for unknown functions and their derivatives. An upwind-dominated scheme is proposed to efficiently handle the non-linear convection. Comparison with the finite difference solutions for the two-dimensional driven cavity flow and the experimental results for flow around a cylinder shows that the present method is capable of satisfactorily predicting the flow separation characteristic. The present algorithm is simple and flexible for complex geometric boundary. The influence of the point distribution on computation time and accuracy of results is included. Copyright © 2002 John Wiley & Sons, Ltd. [source] The design of improved smoothing operators for finite volume flow solvers on unstructured meshesINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 8 2001Benjamin de Foy Abstract Spatial operators used in unstructured finite volume flow solvers are analysed for accuracy using Taylor series expansion and Fourier analysis. While approaching second-order accuracy on very regular grids, operators in common use are shown to have errors resulting in accuracy of only first-, zeroth- or even negative-order on three-dimensional tetrahedral meshes. A technique using least-squares optimization is developed to design improved operators on arbitrary meshes. This is applied to the fourth-order edge sum smoothing operator. The improved numerical dissipation leads to a much more accurate prediction of the Strouhal number for two-dimensional flow around a cylinder and a reduction of a factor of three in the loss coefficient for inviscid flow over a three-dimensional hump. Copyright © 2001 John Wiley & Sons, Ltd. [source] A novel approach to enable decorrelating multiuser detection without matrix inversion operationsINTERNATIONAL JOURNAL OF COMMUNICATION SYSTEMS, Issue 9 2004Hsiao-Hwa Chen Abstract This paper proposes a non-matrix inversion based algorithm to implement decorrelating detection (DD), namely quasi-decorrelating detector (QDD), which uses truncated matrix series expansion to overcome the problems associated with the matrix inversion in DD, such as noise enhancement, computational complexity and matrix singularity, etc. Two alternative QDD implementation schemes are presented in this paper; one is to use multi-stage feedforward filters and the other is to use an nth order single matrix filter (neither of which involves matrix inversion). In addition to significantly reduced computational complexity if compared with DD, the QDD algorithm offers a unique flexibility to trade among MAI suppression, near-far resistance and noise enhancement depending on varying system set-ups. The obtained results show that the QDD outperforms DD in either AWGN or multipath channel if a proper number of feed-forward stages can be used. We will also study the impact of correlation statistics of spreading codes on the QDD's performance with the help of a performance-determining factor derived in the paper, which offers a code-selection guideline for the optimal performance of QDD algorithm. Copyright © 2004 John Wiley & Sons, Ltd. [source] Systematic estimation of state noise statistics for extended Kalman filtersAICHE JOURNAL, Issue 2 2000Jaleel Valappil The successful application of model-based control depends on the information about the states of the dynamic system. State-estimation methods, like extended Kalman filters (EKF), are useful for obtaining reliable estimates of the states from a limited number of measurements. They also can handle the model uncertainties and the effect of unmeasured disturbances. The main issue in applying EKF remains that one needs to specify the confidence in the model in terms of process noise covariance matrix. The information about the model uncertainties can effectively and systematically calculate the process noise covariance matrix for an EKF. Two systematic approaches are used for this calculation. The first is based on a Taylor series expansion of the nonlinear equations around the nominal parameter values, while the second accounts for the nonlinear dependence of the system on the fitted parameters by Monte Carlo simulations that can easily be performed on-line. The value of the process noise covariance matrix obtained is not limited to a diagonal form and depends on the current state of the dynamic system. Thus the a-priori information regarding the uncertainty in the model is utilized and the need for extensive tuning of the EKF is eliminated. The application of these techniques to example processes is also discussed. The accuracy of this methodology is compared very favorably with the traditional methods of trial-and-error tuning of EKF. [source] On a generalized Appell system and monogenic power seriesMATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 4 2010S. Bock Abstract Recently Appell systems of monogenic polynomials in ,3 were constructed by several authors. Main purpose of this paper is the description of another Appell system that is complete in the space of square integrable quaternion-valued functions. A new Taylor-type series expansion based on the Appell polynomials is presented, which can be related to the corresponding Fourier series analogously as in the complex one-dimensional case. These results find applications in the description of the hypercomplex derivative, the monogenic primitive of a monogenic function and the characterization of functions from the monogenic Dirichlet space. Copyright © 2009 John Wiley & Sons, Ltd. [source] Multi-dimensional inhomogeneity indicators and the force on uncharged spheres in electric fieldsMATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 7 2009Dirk Langemann Abstract Uncharged droplets on outdoor high-voltage equipment suffer a non-vanishing total force in non-homogeneous electric fields. Here, the model problem of a spherical test body is considered in arbitrary dimensions. A series expansion of inhomogeneity indicators is proven, which approximates the total force in local terms of the undisturbed electric field. The proof uses the ideas of generalized spherical harmonics without referring to the particular choice of the orthonormal system. The fast converging series expansion establishes a relationship between the solutions of two partial differential equations on different domains. Copyright © 2008 John Wiley & Sons, Ltd. [source] Finite-difference approximation for the u(k) -derivative with O(hM,k+1) accuracy: An analytical expressionNUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS, Issue 5 2006Vadim Dubovsky Abstract An approximation of function u(x) as a Taylor series expansion about a point x0 at M points xi, , i = 1,2,,,M is used where xi are arbitrary-spaced. This approximation is a linear system for the derivatives u(k) with an arbitrary accuracy. An analytical expression for the inverse matrix A,1 where A = [Aik] = (xi , x0)k is found. A finite-difference approximation of derivatives u(k) of a given function u(x) at point x0 is derived in terms of the values u(xi). © 2006 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2006 [source] Fast direct solvers for Poisson equation on 2D polar and spherical geometriesNUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS, Issue 1 2002Ming-Chih Lai Abstract A simple and efficient class of FFT-based fast direct solvers for Poisson equation on 2D polar and spherical geometries is presented. These solvers rely on the truncated Fourier series expansion, where the differential equations of the Fourier coefficients are solved by the second- and fourth-order finite difference discretizations. Using a grid by shifting half mesh away from the origin/poles, and incorporating with the symmetry constraint of Fourier coefficients, the coordinate singularities can be easily handled without pole condition. By manipulating the radial mesh width, three different boundary conditions for polar geometry including Dirichlet, Neumann, and Robin conditions can be treated equally well. The new method only needs O(MN log2N) arithmetic operations for M × N grid points. © 2002 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 18: 56,68, 2002 [source] Stochastic Response of a Continuous System with Stochastic Surface Irregularities to an Accelerated LoadPROCEEDINGS IN APPLIED MATHEMATICS & MECHANICS, Issue 1 2003C.A. Schenk The problem of calculating the second moment characteristics of the response of a general class of nonconservative linear distributed parameter systems with stochastically varying surface roughness, excited by a moving concentrated load, is investigated. In particular the case of an accelerated load is discussed. The surface roughness is modeled as a Gaussian stationary second order process. For the stochastic representation of the surface roughness a orthogonal series expansion of the covariance kernel, the so called Karhunen-Loéve expansion, is applied. The resulting initial/boundary value problem is transformed by eigenfunction expansion into the modal state space. Second moment characteristics of the response are determined numerically by direct integration using a Runge-Kutta method. [source] Molecular shapes from small-angle X-ray scattering: extension of the theory to higher scattering anglesACTA CRYSTALLOGRAPHICA SECTION A, Issue 2 2009V. L. Shneerson A low-resolution shape of a molecule in solution may be deduced from measured small-angle X-ray scattering I(q) data by exploiting a Hankel transform relation between the coefficients of a multipole expansion of the scattered amplitude and corresponding coefficients of the electron density. In the past, the radial part of the Hankel transform has been evaluated with the aid of a truncated series expansion of a spherical Bessel function. It is shown that series truncation may be avoided by analytically performing the radial integral over an entire Bessel function. The angular part of the integral involving a spherical harmonic kernel is performed by quadrature. Such a calculation also allows a convenient incorporation of a molecular hydration shell of constant density intermediate between that of the protein and the solvent. Within this framework, we determine the multipole coefficients of the shape function by optimization of the agreement with experimental data by simulated annealing. [source] Semiparametric competing risks analysisTHE ECONOMETRICS JOURNAL, Issue 2 2007José Canals-Cerdá Summary, In this paper we analyse a semi-parametric estimation technique for competing risks models based on series expansion of the joint density of the unobserved heterogeneity components. This technique allows for unrestricted correlation among the risks. The finite sample behavior of the estimation technique is analysed in a Monte Carlo experiment using an empirically relevant data-generating process. The estimator performs well when compared with the Heckman,Singer estimator. [source] Chiral determination: direct interpretation of convergent-beam electron diffraction patterns using the series expansion of Cowley and MoodieACTA CRYSTALLOGRAPHICA SECTION B, Issue 4 2007Andrew W. S. Johnson Given a small number of structure factors of a known chiral structure of unknown hand, it is shown that the hand can be determined from the sign of the contrast difference of two reflections in a suitably oriented convergent-beam electron diffraction (CBED) pattern. A simple formula for this difference, which takes into account all the significant second-order scattering, is derived using the series expansion of Cowley and Moodie for n -beam diffraction. The reason for the success of a three-beam interpretation is investigated. The method is applied to patterns from thin crystals in which a mirror projection symmetry can be found and its validity is demonstrated by agreement with experiment using samples of known hand. The advantages of recording patterns near major zone axes are discussed as well as some other experimental aspects of chiral determination using CBED. [source] On accurate boundary conditions for a shape sensitivity equation methodINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 2 2006R. Duvigneau Abstract This paper studies the application of the continuous sensitivity equation method (CSEM) for the Navier,Stokes equations in the particular case of shape parameters. Boundary conditions for shape parameters involve flow derivatives at the boundary. Thus, accurate flow gradients are critical to the success of the CSEM. A new approach is presented to extract accurate flow derivatives at the boundary. High order Taylor series expansions are used on layered patches in conjunction with a constrained least-squares procedure to evaluate accurate first and second derivatives of the flow variables at the boundary, required for Dirichlet and Neumann sensitivity boundary conditions. The flow and sensitivity fields are solved using an adaptive finite-element method. The proposed methodology is first verified on a problem with a closed form solution obtained by the Method of Manufactured Solutions. The ability of the proposed method to provide accurate sensitivity fields for realistic problems is then demonstrated. The flow and sensitivity fields for a NACA 0012 airfoil are used for fast evaluation of the nearby flow over an airfoil of different thickness (NACA 0015). Copyright © 2005 John Wiley & Sons, Ltd. [source] On convexity of MQAM's and MPAM's bit error probability functionsINTERNATIONAL JOURNAL OF COMMUNICATION SYSTEMS, Issue 11 2009M. Naeem Abstract For MQAM and MPAM with practical values of M and Gray mapping, we provide a rigorous proof that the associated bit error probability (BEP) functions are convex of the signal-to-noise ratio per symbol. The proof employs Taylor series expansions of the BEP functions' second derivatives and term-by-term comparisons between positive and negative terms. Convexity results are useful for optimizing communication systems as in optimizing adaptive transmission policies. Copyright © 2009 John Wiley & Sons, Ltd. [source] Gaussian approximation of exponential type orbitals based on B functionsINTERNATIONAL JOURNAL OF QUANTUM CHEMISTRY, Issue 2 2009Didier Pinchon Abstract This work gives new, highly accurate optimized gaussian series expansions for the B functions used in molecular quantum mechanics. These functions are generally chosen because of their compact Fourier transform, following Shavitt. The inverse Laplace transform in the square root of the variable is used for Gauss quadrature in this work. Two procedures for obtaining accurate gaussian expansions have been compared for the required extended precision arithmetic. The first is based on Gaussian quadratures and the second on direct optimization. Both use the Maple computer algebra system. Numerical results are tabulated and compared with previous work. Special cases are found to agree before pushing the optimization technique further. The optimal gaussian expansions of B functions obtained in this work are available for reference. © 2008 Wiley Periodicals, Inc. Int J Quantum Chem, 2009 [source] Auxiliary functions for molecular integrals with Slater-type orbitals.INTERNATIONAL JOURNAL OF QUANTUM CHEMISTRY, Issue 1 2008Abstract The Gauss transform of Slater-type orbitals is used to express several types of molecular integrals involving these functions in terms of simple auxiliary functions. After reviewing this transform and the way it can be combined with the shift operator technique, a master formula for overlap integrals is derived and used to obtain multipolar moments associated to fragments of two-center distributions and overlaps of derivatives of Slater functions. Moreover, it is proved that integrals involving two-center distributions and irregular harmonics placed at arbitrary points (which determine the electrostatic potential, field and field gradient, as well as higher order derivatives of the potential) can be expressed in terms of auxiliary functions of the same type as those appearing in the overlap. The recurrence relations and series expansions of these functions are thoroughly studied, and algorithms for their calculation are presented. The usefulness and efficiency of this procedure are tested by developing two independent codes: one for the derivatives of the overlap integrals with respect to the centers of the functions, and another for derivatives of the potential (electrostatic field, field gradient, and so forth) at arbitrary points. © 2007 Wiley Periodicals, Inc. Int J Quantum Chem, 2008 [source] Auxiliary functions for molecular integrals with Slater-type orbitals.INTERNATIONAL JOURNAL OF QUANTUM CHEMISTRY, Issue 9 2006Abstract Many types of molecular integrals involving Slater functions can be expressed, with the ,-function method in terms of sets of one-dimensional auxiliary integrals whose integrands contain two-range functions. After reviewing the properties of these functions (including recurrence relations, derivatives, integral representations, and series expansions), we carry out a detailed study of the auxiliary integrals aimed to facilitate both the formal and computational applications of the ,-function method. The usefulness of this study in formal applications is illustrated with an example. The high performance in numerical applications is proved by the development of a very efficient program for the calculation of two-center integrals with Slater functions corresponding to electrostatic potential, electric field, and electric field gradient. © 2006 Wiley Periodicals, Inc. Int J Quantum Chem, 2006 [source] | ||||||||||||||||||||||