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Bessel Function (bessel + function)
Selected AbstractsComparative study of the least squares approximation of the modified Bessel functionINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN BIOMEDICAL ENGINEERING, Issue 8 2008Jianguo XinArticle first published online: 14 DEC 200 Abstract The least squares problem of the modified Bessel function of the second kind has been considered in this study with the Fourier series, Tchebycheff and Legendre approximation. Numerical evidence shows that the Gibbs phenomenon exists in the approximation with the truncated Fourier series, thus, giving poor convergence results compared with the other polynomial bases. For the latter two cases, the Legendre series perform better than Tchebycheff series in terms of the ,2 norm of the relative errors for each order of the polynomial approximation, and the ratio of the ,2 norm of the relative errors from the corresponding approximation seems to be a constant value of 1.3. Copyright © 2006 John Wiley & Sons, Ltd. [source] Orientational analysis of planar fibre systems observed as a Poisson shot-noise processJOURNAL OF MICROSCOPY, Issue 1 2007SALME KÄRKKÄINEN Summary We consider two-dimensional fibrous materials observed as a digital greyscale image. The problem addressed is to estimate the orientation distribution of unobservable thin fibres from a greyscale image modelled by a planar Poisson shot-noise process. The classical stereological approach is not straightforward, because the point intensities of thin fibres along sampling lines may not be observable. For such cases, Kärkkäinen et al. (2001) suggested the use of scaled variograms determined from grey values along sampling lines in several directions. Their method is based on the assumption that the proportion between the scaled variograms and point intensities in all directions of sampling lines is constant. This assumption is proved to be valid asymptotically for Boolean models and dead leaves models, under some regularity conditions. In this work, we derive the scaled variogram and its approximations for a planar Poisson shot-noise process using the modified Bessel function. In the case of reasonable high resolution of the observed image, the scaled variogram has an approximate functional relation to the point intensity, and in the case of high resolution the relation is proportional. As the obtained relations are approximative, they are tested on simulations. The existing orientation analysis method based on the proportional relation is further experimented on images with different resolutions. The new result, the asymptotic proportionality between the scaled variograms and the point intensities for a Poisson shot-noise process, completes the earlier results for the Boolean models and for the dead leaves models. [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] A novel analytical solution for constant-head test in a patchy aquiferINTERNATIONAL JOURNAL FOR NUMERICAL AND ANALYTICAL METHODS IN GEOMECHANICS, Issue 12 2006Shaw-Yang Yang Abstract A mathematical model describing the hydraulic head distribution for a constant-head test performed in a well situated at the centre of a patchy aquifer is presented. The analytical solution for the mathematical model is derived by the Laplace transforms and the Bromwich integral method. The solution for the hydraulic head has been shown to satisfy the governing equations, related boundary conditions, and continuity requirements for the hydraulic head and flow rate at the interface of the patch and outer regions. An efficient numerical approach is proposed to evaluate the solution, which has an integral covering an integration range from zero to infinity and an integrand consisting the product and square of the Bessel functions. This solution can be used to produce the curves of dimensionless hydraulic head against dimensionless time for investigating the effect of the contrast of formation properties on the dimensionless hydraulic head distribution. Define the ratio of outer-region transmissivity to patch-region transmissivity as ,. The dimensionless hydraulic head for ,=0.1 case is about 2.72 times to that for ,=10 case at dimensionless large time (e.g. ,,106) when the dimensionless distance (,) equals 10. The results indicate that the hydraulic head distribution highly depends on the hydraulic properties of two-zone formations. Copyright © 2006 John Wiley & Sons, Ltd. [source] An integral transform generating elementary functions in Clifford analysisMATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 6 2006G. Laville Abstract In this paper we introduce a real integral transform which links trigonometric and Bessel functions. This allows us to construct a monogenic pseudo-exponential in Clifford analysis. There is a deep difference between odd and even dimensions. Copyright © 2006 John Wiley & Sons, Ltd. [source] Sound propagation in a channel with a discontinuity in the form of a semicircleMATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 15 2003Volkmar Weise Abstract For the description of wave propagation in ducts with variable cross-sections it is necessary to determine reflection and transmission at the points of discontinuity in the cross-sectional contour. Analytic solutions are well known for rectangular and circular duct cross-sections with rotational symmetric arrangements. The aim of this paper is the description of sound propagation in cylindrical ducts with semicircular discontinuities and thus non-rotationally symmetric discontinuities of the cross-sectional contour. In particular, excitation with higher azimuthal and radial modes also will be included. This solution is achieved by using the orthogonality properties of the trigonometric and Bessel functions. Copyright © 2003 John Wiley & Sons, Ltd. [source] Comparison of the ENEAR peculiar velocities with the PSCz gravity fieldMONTHLY NOTICES OF THE ROYAL ASTRONOMICAL SOCIETY, Issue 3 2001Adi Nusser We present a comparison between the peculiar velocity field measured from the ENEAR all-sky Dn,, catalogue and that derived from the galaxy distribution of the IRAS Point Source Catalog Redshift Survey (PSCz). The analysis is based on a modal expansion of these data in redshift space by means of spherical harmonics and Bessel functions. The effective smoothing scale of the expansion is almost linear with redshift reaching 1500 km s,1 at 3000 km s,1. The general flow patterns in the filtered ENEAR and PSCz velocity fields agree well within 6000 km s,1, assuming a linear biasing relation between the mass and the PSCz galaxies. The comparison allows us to determine the parameter where , is the cosmological density parameter and b is the linear biasing factor. A likelihood analysis of the ENEAR and PSCz modes yields in good agreement with values obtained from Tully,Fisher surveys. [source] | ||||||||||