Series Expansion Of Functions Research Articles

Algebraic independence results for the sixteen families of q-series

The sixteen families of q-series containing the Ramanujan functions were listed by I.J. Zucker (SIAM J. Math. Anal. 10:192–206, 1979), which are generated from the Fourier series expansions of the Jacobian elliptic functions or some of their squares. This paper discusses algebraic independence properties for these q-series. We determine all the sets of q-series such that, at each algebraic point, the values of the q-series in the set are algebraically independent over ℚ. We also present several algebraic relations over ℚ for two or three of these q-series.

Generalizations of Chromatic Derivatives and Series Expansions

Chromatic series expansions of bandlimited functions provide an alternative representation to the Whittaker-Shannon-Kotel'nikov sampling series. Chromatic series share similar properties with Taylor series insofar as the coefficients of the expansions, which are called chromatic derivatives, are based on the ordinary derivatives of the function. Chromatic derivatives are linear combinations of ordinary derivatives in which the coefficients of the combinations are related to a system of orthonormal polynomials. But unlike Taylor series, chromatic series have more useful applications in signal processing because they represent bandlimited functions more efficiently than Taylor series. The goal of this article is to generalize the notion of chromatic derivatives and then extend chromatic series expansions to a larger class of signals than the class of bandlimited signals. In particular, we extend chromatic series to signals given by integral transforms other than the Fourier transform, such as the Laplace and Hankel transforms.

Fourier-Bessel Expansions with Arbitrary Radial Boundaries

Series expansion of single variable functions is represented in Fourier-Bessel form with unknown coefficients. The proposed series expansions are derived for arbitrary radial boundaries in problems of circular domain. Zeros of the generated transcendental equation and the relationship of orthogonality are employed to find the unknown coefficients. Several numerical and graphical examples are explained and discussed.

Laplace series expansion of the internal potential of a homogeneous circular torus

The internal potential of a homogeneous circular torus first is represented by a series expansion in spherical functions (Laplace series). Exact analytical formulas for the coefficients of this series are derived and it is shown that they can be expressed through the standard Gauss hypergeometric function depending only on the geometric parameter of the torus. Convergence of the series is proved and the radius of convergence is determined. The relation of the radius with the torus geometrical parameter is found. A special spherical shell, where the problem of expansion of the torus potential should be solved in additional investigations, is detected.

On numerical calculation of probabilities according to Dirichlet distribution

The main difficulty in numerical solution of probabilistic constrained stochastic programming problems is the calculation of the probability values according to the underlying multivariate probability distribution. In addition, when we are using a nonlinear programming algorithm for the solution of the problem, the calculation of the first and second order partial derivatives may also be necessary. In this paper we give solutions to the above problems in the case of Dirichlet distribution. For the calculation of the cumulative distribution function values, the Lauricella function series expansions will be applied up to 7 dimensions. For higher dimensions we propose the hypermultitree bound calculations and a variance reduction simulation procedure based on these bounds. There will be given formulae for the calculation of the first and second order partial derivatives, too. The common property of these formulae is that they involve only lower dimensional cumulative distribution function calculations. Numerical test results will also be presented.

On convexity of MQAM's and MPAM's bit error probability functions

AbstractFor 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.

Holomorphic transforms with application to affine processes

In a rather general setting of Itô–Lévy processes we study a class of transforms (Fourier for example) of the state variable of a process which are holomorphic in some disc around time zero in the complex plane. We show that such transforms are related to a system of analytic vectors for the generator of the process, and we state conditions which allow for holomorphic extension of these transforms into a strip which contains the positive real axis. Based on these extensions we develop a functional series expansion of these transforms in terms of the constituents of the generator. As application, we show that for multi-dimensional affine Itô–Lévy processes with state dependent jump part the Fourier transform is holomorphic in a time strip under some stationarity conditions, and give log-affine series representations for the transform.

Holomorphic Transforms with Application to Affine Processes

In a rather general setting of Ito-Levy processes we study a class of transforms (Fourier for example) of the state variable of a process which are holomorphic in some disc around time zero in the complex plane. We show that such transforms are related to a system of analytic vectors for the generator of the process, and we state conditions which allow for holomorphic extension of these transforms into a strip which contains the positive real axis. Based on these extensions we develop a functional series expansion of these transforms in terms of the constituents of the generator. As application, we show that for multidimensional affine Ito-Levy processes with state dependent jump part the Fourier transform is holomorphic in a time strip under some stationarity conditions, and give log-affine series representations for the transform.

Convergence domain of series expansions of multivariate algebraic functions

Convergence domain of series expansions of multivariate algebraic functions

Laplace series expansion of the potential of a homogeneous circular torus

The external potential of a homogeneous circular torus is represented by a series expansion in spherical functions (Laplace series). Exact analytical formulas are derived for the coefficients of this series, which can be expressed in terms of Legendre polynomials depending only on the geometrical parameter of the torus. The convergence of the series is proved and the radius of convergence is determined. The resultant expressions are verified numerically.

Solution to Boundary-value Problems in Fabrication of High-precision Reflector Panels

The fabrication of high-precision panels for the compact antenna test range (CATR) with a sandwich construction of two aluminum skin-plates and one aluminum middle plate, which are bonded to two aluminum honeycomb core-layers poses a lot of tricky problems. Of them, the force analysis of individual skin-layers and the springback calculation of sandwich are of utmost importance. Under reasonable assumptions, by using Fourier expansion of stress function and power series expansion of deflection function, two boundary-value problems of skin-layer and sandwich, i.e. force analysis and springback calculation, are solved. An analytical solution of stress function in series form and an approximate solution of deflection function are obtained. Compared with finite element method (FEM), the generalized double Fourier series method used in individual skin force analysis is superior in accuracy. Compared with experimental results, the weighted residual method used in springback calculation of sandwich-structured panel is acceptable from view of precision. Based on calculation results, processing parameters to panel fabrication are determined more efficient and accurate.

A study of transient flows of Newtonian fluids through micro-annuals with a slip boundary

In this paper, we study the pressure gradient driven transient flow of an incompressible Newtonian liquid in micro-annuals under a Navier slip boundary condition. By using the Fourier series expansion in time and Bessel functions in space, an exact solution is derived and is shown to include some existing known results as special cases. By analysing the exact solution, it is found that the influences of boundary slip on the flow behaviour are qualitatively different for different types of pressure fields driving the flow. For pressure fields with a constant pressure gradient, the flow rate increases with the increase in the slip parameter l almost linearly when l is large; while, for pressure fields with a wave form pressure gradient within a certain frequency range, as the slip parameter l increases, the amplitude of the flow rate increases first and then approaches a constant value when l becomes sufficiently large. It is also found that to achieve a given flow rate, one could have different designs and the graphs for the design are presented and discussed in this paper.

A complex variable solution for stress and displacement field around a lined circular tunnel at great depth

AbstractBy applying the series expansion technique in the complex variable method established by Muskhelishvili, the plane elasticity problem for the stress and displacement field around a lined circular tunnel in conjunction with the consideration of misfit and interaction between the liner and the surrounding geomaterial is dealt with. The tunnel is assumed to be driven in a homogeneous and isotropic geomaterial. The coefficients in the Laurent series expansion of the stress functions are determined. The complex potentials in the liner and the surrounding geomaterial are explicitly derived, respectively. As an example, the case of a lined circular tunnel located in an isotropic initial stress field but subjected to uniform internal pressure is numerically considered. Numerical results indicate that the installation of tunnel liner can reduce the influences of the tunnel excavation on the in situ displacement and stress fields. However, the relative thickness and rigidity of the liner should be in an appropriate range. In addition, the effect of the tunnel excavation upon the displacement field is more significant than that upon the stress field. As far as the stress field in the surrounding geomaterial is concerned, when the ratio between the cover depth of tunnel and the tunnel radius is larger than 5, the results for the stress field in the paper are applicable. When the ratio between the tunnel depth and the tunnel radius is larger than 20, the results are applicable for the displacement field. Copyright © 2008 John Wiley & Sons, Ltd.

Space-Time-Frequency Characterization of 3D Nonisotropic MIMO Multicarrier Propagation Channels Employing Directional Antennas

Channel models for outdoor wireless systems usually assume two-dimensional (2D) random scattering media. In the practical outdoor wireless channels, the impact of the wave propagation in the third-dimension is definitely important; especially when the communication system efficiently exploits potentials of multiple antennas. In this paper, we propose a new model for multiple-input multiple-output (MIMO) multicarrier propagation channels in a three-dimensional (3D) environment. Specifically, the proposed model describes the cross-correlation function (CCF) between two subchannels of an outdoor MIMO channel employing directional antennas and in the presence of nonisotropic wave propagation in 3D space. The derived CCF consists of some correlation terms. Each correlation term is in the form of a linear series expansion of averaged Bessel functions of the first kind with different orders. In practice, each correlation term has a limited number of Bessel components. Our numerical evaluations show the impact of different parameters of the propagation environment as well as the employed antennas on the resulting CCF. Using the proposed CCF, we also establish simple formulas to approximate the coherence time, the coherence bandwidth and the spatial coherence of such channels. The numerical curve fitting results fit to the empirical results reported in the channel modeling literature.

Solutions of integral and integro-differential equation systems by using differential transform method

In this study, differential transform method (DTM) is applied to both integro-differential and integral equation systems. The method is further expanded with a formulation to treat Fredholm integrals. If the system considered has a solution in terms of the series expansion of known functions, this powerful method catches the exact solution. So as to show this capability and robustness, some systems of integral and integro-differential equations are solved as numerical examples.

Special power series expansions of monogenic functions around surfaces of codimension two

AbstractIn this paper we will focus on power series expansions around special surfaces of codimension two, in particular around spheres and products of spheres. This will include a version of the Cauchy–Kowalewski extension theorem around these surfaces. This higher codimension result was only obtained so far in the flat case (see Partial Differential Equations with Complex Analysis. Pitman Research Notes Mathematical Series, vol. 262. Longman Science and Technology: Harlow, 1992; 61–92; Mathematics and its Applications, vol. 53. Kluwer Academic Publishers Group: Dordrecht, 1992). Copyright © 2008 John Wiley & Sons, Ltd.

An alternative method for calculation of Rayleigh and Love wave phase velocities by using three-component records on a single circular array without a central station

SUMMARY A method for the computation of phase velocities of surface waves from microtremor waveforms is shown. The technique starts from simultaneous three-component records obtained in a circular array without a central station. Then, Fourier spectra of vertical, radial and tangential components of motion are calculated for each station and considered as complex-valuated functions of the azimuthal coordinate. A couple of intermediate real physical quantities, B and C, can be defined from the 0- and ±1-order coefficients of the Fourier series expansion of such functions. Finally, phase velocities of Rayleigh and Love waves can be retrieved from B and C by solving respective one-unknown equations. The basic assumption is the possibility of expanding the wavefield as a sum of plane surface waves with Rayleigh and Love wavenumbers being univocal functions of the circular frequency. The method is tested in synthetic ambient noise wavefields confirming its reliability and robustness for passive seismic surveying.

Analytical Solution for the Stationary Response of Alluvial Valley Containing Multiple Circular‐Arc Layers to Incident Plane P Waves

AbstractBy the method of Fourier‐Bessel series expansion of wave functions, this paper presents an analytical solution to the stationary dynamic response of an alluvial valley containing arbitrary number of circular‐arc layers to the incidence of plane P waves. Taking the three‐layer valleys as examples and using the spatial distributions of stationary ground motion amplitudes, the influences of the layering of deposit media in the valley on the ground motion are studied in a broad frequency band.

New method to obtain small parameter power series expansions of Mathieu radial and angular functions

First published in Mathematics of Computation in Vol 78 Issue 265, published by the American Mathematical Society.

Fast protein tertiary structure retrieval based on global surface shape similarity

Characterization and identification of similar tertiary structure of proteins provides rich information for investigating function and evolution. The importance of structure similarity searches is increasing as structure databases continue to expand, partly due to the structural genomics projects. A crucial drawback of conventional protein structure comparison methods, which compare structures by their main-chain orientation or the spatial arrangement of secondary structure, is that a database search is too slow to be done in real-time. Here we introduce a global surface shape representation by three-dimensional (3D) Zernike descriptors, which represent a protein structure compactly as a series expansion of 3D functions. With this simplified representation, the search speed against a few thousand structures takes less than a minute. To investigate the agreement between surface representation defined by 3D Zernike descriptor and conventional main-chain based representation, a benchmark was performed against a protein classification generated by the combinatorial extension algorithm. Despite the different representation, 3D Zernike descriptor retrieved proteins of the same conformation defined by combinatorial extension in 89.6% of the cases within the top five closest structures. The real-time protein structure search by 3D Zernike descriptor will open up new possibility of large-scale global and local protein surface shape comparison.