Relevant bibliographies by topics / Approximations and expansions – Approximations and expansions – Padé; approximation

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  1. Journal articles
  2. Dissertations / Theses
  3. Books
  4. Book chapters
  5. Conference papers

Academic literature on the topic 'Approximations and expansions – Approximations and expansions – Padé; approximation'

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Published: 4 June 2021
Last updated: 5 February 2022

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Journal articles on the topic "Approximations and expansions – Approximations and expansions – Padé; approximation"

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Avram, Florin, Andras Horváth, Serge Provost, and Ulyses Solon. "On the Padé and Laguerre–Tricomi–Weeks Moments Based Approximations of the Scale Function W and of the Optimal Dividends Barrier for Spectrally Negative Lévy Risk Processes." Risks 7, no. 4 (December 11, 2019): 121. http://dx.doi.org/10.3390/risks7040121.

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This paper considers the Brownian perturbed Cramér–Lundberg risk model with a dividends barrier. We study various types of Padé approximations and Laguerre expansions to compute or approximate the scale function that is necessary to optimize the dividends barrier. We experiment also with a heavy-tailed claim distribution for which we apply the so-called “shifted” Padé approximation.
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Capozziello, S., R. D’Agostino, and O. Luongo. "High-redshift cosmography: auxiliary variables versus Padé polynomials." Monthly Notices of the Royal Astronomical Society 494, no. 2 (April 7, 2020): 2576–90. http://dx.doi.org/10.1093/mnras/staa871.

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ABSTRACT Cosmography becomes non-predictive when cosmic data span beyond the redshift limit z ≃ 1. This leads to a strong convergence issue that jeopardizes its viability. In this work, we critically compare the two main solutions of the convergence problem, i.e. the y-parametrizations of the redshift and the alternatives to Taylor expansions based on Padé series. In particular, among several possibilities, we consider two widely adopted parametrizations, namely y1 = 1−a and $y_2=\arctan (a^{-1}-1)$, being a the scale factor of the Universe. We find that the y2-parametrization performs relatively better than the y1-parametrization over the whole redshift domain. Even though y2 overcomes the issues of y1, we get that the most viable approximations of the luminosity distance dL(z) are given in terms of Padé approximations. In order to check this result by means of cosmic data, we analyse the Padé approximations up to the fifth order, and compare these series with the corresponding y-variables of the same orders. We investigate two distinct domains involving Monte Carlo analysis on the Pantheon Superovae Ia data, H(z) and shift parameter measurements. We conclude that the (2,1) Padé approximation is statistically the optimal approach to explain low- and high-redshift data, together with the fifth-order y2-parametrization. At high redshifts, the (3,2) Padé approximation cannot be fully excluded, while the (2,2) Padé one is essentially ruled out.
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Alkhalifah, Tariq. "Prestack wavefield approximations." GEOPHYSICS 78, no. 5 (September 1, 2013): T141—T149. http://dx.doi.org/10.1190/geo2012-0532.1.

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The double-square-root (DSR) relation offers a platform to perform prestack imaging using an extended single wavefield that honors the geometrical configuration between sources, receivers, and the image point, or in other words, prestack wavefields. Extrapolating such wavefields, nevertheless, suffers from limitations. Chief among them is the singularity associated with horizontally propagating waves. I have devised highly accurate approximations free of such singularities which are highly accurate. Specifically, I use Padé expansions with denominators given by a power series that is an order lower than that of the numerator, and thus, introduce a free variable to balance the series order and normalize the singularity. For the higher-order Padé approximation, the errors are negligible. Additional simplifications, like recasting the DSR formula as a function of scattering angle, allow for a singularity free form that is useful for constant-angle-gather imaging. A dynamic form of this DSR formula can be supported by kinematic evaluations of the scattering angle to provide efficient prestack wavefield construction. Applying a similar approximation to the dip angle yields an efficient 1D wave equation with the scattering and dip angles extracted from, for example, DSR ray tracing. Application to the complex Marmousi data set demonstrates that these approximations, although they may provide less than optimal results, allow for efficient and flexible implementations.
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Gluzman, Simon. "Padé and Post-Padé Approximations for Critical Phenomena." Symmetry 12, no. 10 (September 25, 2020): 1600. http://dx.doi.org/10.3390/sym12101600.

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We discuss and apply various direct extrapolation methods for calculation of the critical points and indices from the perturbative expansions my means of Padé-techniques and their various post-Padé extensions by means of root and factor approximants. Factor approximants are applied to finding critical points. Roots are employed within the context of finding critical index. Additive self-similar approximants are discussed and DLog additive recursive approximants are introduced as their generalization. They are applied to the problem of interpolation. Several examples of interpolation are considered.
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ANDRIANOV, IGOR VASIL'EVICH, VICTOR ISAAKOVICH OLEVS'KYY, and JAN AWREJCEWICZ. "ANALYTICAL PERTURBATION METHOD FOR CALCULATION OF SHELLS BASED ON 2D PADÉ APPROXIMANTS." International Journal of Structural Stability and Dynamics 13, no. 07 (August 23, 2013): 1340003. http://dx.doi.org/10.1142/s0219455413400038.

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Calculations of nonlinear displacements and vibrations of inhomogeneous loaded shells with developable principal surface by means of different analytical methods are represented. It is shown that solutions to these methods are the expansions of exact solution in the Taylor series for an independent variable, and in the particular case — for the powers of a natural parameter. A method that provides a polynomial asymptotic approximation of the exact solution of the general form and its meromorphic continuation based on 1D and 2D Padé approximations is proposed. Calculations of nonlinear deformation and stability of elastic flexible circular cylindrical shell under uniform external pressures and of free oscillations of simply supported stringer shell demonstrate the efficiency and accuracy of the proposed method.
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Kaneda, Yukio, Koji Gotoh, and Takashi Ishihara. "Taylor Expansions and Padé Approximations of Lagrangian and Eulerian Two-Time Velocity Correlations in Turbulence." Journal of the Physical Society of Japan 67, no. 4 (April 15, 1998): 1075–78. http://dx.doi.org/10.1143/jpsj.67.1075.

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Ebeling, Werner, Heidi Reinholz, and Gerd Röpke. "Hydrogen, helium and lithium plasmas at high pressures." European Physical Journal Special Topics 229, no. 22-23 (December 2020): 3403–31. http://dx.doi.org/10.1140/epjst/e2020-000066-6.

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AbstractThe equations of state (EoS) and other thermodynamic properties of plasmas of the light elements H, He, and Li, are calculated using inverted fugacity expansions. Fugacity expansions are known as an alternative to density expansions but show often an inferior convergence. If, however, the inversion can be solved, the fugacity representations may be very efficient. In particular, the contributions of deeply bound states are included in the fugacity expansion in a very effective way. The mathematical problems on nonlinearity connected with the inversion of fugacities to densities are reduced to solvable algebraic problems. The inversion of fugacities to densities is solved separately for two density regions: (i) In the low density, non-degenerate region we consider ring contributions describing screening effects and ladder contributions describing bound state formation. (ii) In the high density, degenerate region the electrons are described by the Fermi–Dirac distribution. Hartree–Fock contributions and Pauli blocking have to be taken into account. The ions are considered as classical, strongly correlated subsystem eventually forming a Wigner lattice. We solve the inversion problem for each of the regions. Near the crossing point, the separate solutions are connected to each other, either by smooth concatenation at the crossing point or by Padé approximations.
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MADSEN, P. A., H. B. BINGHAM, and HUA LIU. "A new Boussinesq method for fully nonlinear waves from shallow to deep water." Journal of Fluid Mechanics 462 (July 10, 2002): 1–30. http://dx.doi.org/10.1017/s0022112002008467.

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A new method valid for highly dispersive and highly nonlinear water waves is presented. It combines a time-stepping of the exact surface boundary conditions with an approximate series expansion solution to the Laplace equation in the interior domain. The starting point is an exact solution to the Laplace equation given in terms of infinite series expansions from an arbitrary z-level. We replace the infinite series operators by finite series (Boussinesq-type) approximations involving up to fifth-derivative operators. The finite series are manipulated to incorporate Padé approximants providing the highest possible accuracy for a given number of terms. As a result, linear and nonlinear wave characteristics become very accurate up to wavenumbers as high as kh = 40, while the vertical variation of the velocity field becomes applicable for kh up to 12. These results represent a major improvement over existing Boussinesq-type formulations in the literature. A numerical model is developed in a single horizontal dimension and it is used to study phenomena such as solitary waves and their impact on vertical walls, modulational instability in deep water involving recurrence or frequency downshift, and shoaling of regular waves up to breaking in shallow water.
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Dunsby, Peter K. S., and Orlando Luongo. "On the theory and applications of modern cosmography." International Journal of Geometric Methods in Modern Physics 13, no. 03 (March 2016): 1630002. http://dx.doi.org/10.1142/s0219887816300026.

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Cosmography represents an important branch of cosmology which aims to describe the universe without the need of postulating a priori any particular cosmological model. All quantities of interest are expanded as a Taylor series around here and now, providing in principle, a way of directly matching with cosmological data. In this way, cosmography can be regarded a model-independent technique, able to fix cosmic bounds, although several issues limit its use in various model reconstructions. The main purpose of this review is to focus on the key features of cosmography, emphasizing both the strategy for obtaining the observable cosmographic series and pointing out any drawbacks which might plague the standard cosmographic treatment. In doing so, we relate cosmography to the most relevant cosmological quantities and to several dark energy models. We also investigate whether cosmography is able to provide information about the form of the cosmological expansion history, discussing how to reproduce the dark fluid from the cosmographic sound speed. Following this, we discuss limits on cosmographic priors and focus on how to experimentally treat cosmographic expansions. Finally, we present some of the latest developments of the cosmographic method, reviewing the use of rational approximations, based on cosmographic Padé polynomials. Future prospects leading to more accurate cosmographic results, able to better reproduce the expansion history of the universe, are also discussed in detail.
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Geer, James F., and Carl M. Andersen. "Hybrid Pade´-Galerkin Technique for Differential Equations." Applied Mechanics Reviews 46, no. 11S (November 1, 1993): S255—S265. http://dx.doi.org/10.1115/1.3122644.

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A three-step hybrid analysis technique, which successively uses the regular perturbation expansion method, the Pade´ expansion method, and then a Galerkin approximation, is presented and applied to some model boundary value problems. In the first step of the method, the regular perturbation method is used to construct an approximation to the solution in the form of a finite power series in a small parameter ε associated with the problem. In the second step of the method, the series approximation obtained in step one is used to construct a Pade´ approximation in the form of a rational function in the parameter ε. In the third step, the various powers of ε which appear in the Pade´ approximation are replaced by new (unknown) parameters {δj}. These new parameters are determined by requiring that the residual formed by substituting the new approximation into the governing differential equation is orthogonal to each of the perturbation coordinate functions used in step one. The technique is applied to model problems involving ordinary or partial differential equations. In general, the technique appears to provide good approximations to the solution even when the perturbation and Pade´ approximations fail to do so. The method is discussed and topics for future investigations are indicated.
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Dissertations / Theses on the topic "Approximations and expansions – Approximations and expansions – Padé; approximation"

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Pachon, Ricardo. "Algorithms for polynomial and rational approximation." Thesis, University of Oxford, 2010. http://ora.ox.ac.uk/objects/uuid:f268a835-46ef-45ea-8610-77bf654b9442.

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Robust algorithms for the approximation of functions are studied and developed in this thesis. Novel results and algorithms on piecewise polynomial interpolation, rational interpolation and best polynomial and rational approximations are presented. Algorithms for the extension of Chebfun, a software system for the numerical computation with functions, are described. These algorithms allow the construction and manipulation of piecewise smooth functions numerically with machine precision. Breakpoints delimiting subintervals are introduced explicitly, implicitly or automatically, the latter method combining recursive subdivision and edge detection techniques. For interpolation by rational functions with free poles, a novel method is presented. When the interpolation nodes are roots of unity or Chebyshev points the algorithm is particularly simple and relies on discrete Fourier transform matrices, which results in a fast implementation using the Fast Fourier Transform. The method is generalised for arbitrary grids, which requires the construction of polynomials orthogonal on the set of interpolation nodes. The new algorithm has connections with other methods, particularly the work of Jacobi and Kronecker, Berrut and Mittelmann, and Egecioglu and Koc. Computed rational interpolants are compared with the behaviour expected from the theory of convergence of these approximants, and the difficulties due to truncated arithmetic are explained. The appearance of common factors in the numerator and denominator due to finite precision arithmetic is characterised by the behaviour of the singular values of the linear system associated with the rational interpolation problem. Finally, new Remez algorithms for the computation of best polynomial and rational approximations are presented. These algorithms rely on interpolation, for the computation of trial functions, and on Chebfun, for the location of trial references. For polynomials, the algorithm is particularly robust and efficient, and we report experiments with degrees in the thousands. For rational functions, we clarify the numerical issues that affect its application.
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Hennessy, Matthew Gregory. "Mathematical problems relating to the fabrication of organic photovoltaic devices." Thesis, University of Oxford, 2014. http://ora.ox.ac.uk/objects/uuid:7753abec-bb6e-4d8a-aa5b-b527c5beb49b.

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The photoactive component of a polymeric organic solar cell can be produced by drying a mixture consisting of a volatile solvent and non-volatile polymers. As the solvent evaporates, the polymers demix and self-assemble into microscale structures, the morphology of which plays a pivotal role in determining the efficiency of the resulting device. Thus, a detailed understanding of the physical mechanisms that drive and influence structure formation in evaporating solvent-polymer mixtures is of high scientific and industrial value. This thesis explores several problems that aim to produce novel insights into the dynamics of evaporating solvent-polymer mixtures. First, the role of compositional Marangoni instabilities in slowly evaporating binary mixtures is studied using the framework of linear stability theory. The analysis is non-trivial because evaporative mass loss naturally leads to a time-dependent base state. In the limit of slow evaporation compared to diffusion, a separation of time scales emerges in the linear stability problem, allowing asymptotic methods to be applied. In particular, an asymptotic solution to linear stability problems that have slowly evolving base states is derived. Using this solution, regions of parameter space where an oscillatory instability occurs are identified and used to formulate appropriate conditions for observing this phenomenon in future experiments. The second topic of this thesis is the use of multiphase fluid models to study the dynamics of evaporating solvent-polymer mixtures. A two-phase model is used to assess the role of compositional buoyancy and to examine the formation of a polymer-rich skin at the free surface. Then, a three-phase model is used to conduct a preliminary investigation of the link between evaporation and phase separation. Finally, this thesis explores the dynamics of a binary mixture that is confined between two horizontal walls using a diffusive phase-field model and its sharp-interface and thin-film approximations. We first determine the conditions under which a homogeneous mixture undergoes phase separation to form a metastable bilayer. We then present a novel mechanism for generating a repeating lateral sequence of alternating A-rich and B-rich domains from this bilayer.
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Gyurko, Lajos Gergely. "Numerical methods for approximating solutions to rough differential equations." Thesis, University of Oxford, 2008. http://ora.ox.ac.uk/objects/uuid:d977be17-76c6-46d6-8691-6d3b7bd51f7a.

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The main motivation behind writing this thesis was to construct numerical methods to approximate solutions to differential equations driven by rough paths, where the solution is considered in the rough path-sense. Rough paths of inhomogeneous degree of smoothness as driving noise are considered. We also aimed to find applications of these numerical methods to stochastic differential equations. After sketching the core ideas of the Rough Paths Theory in Chapter 1, the versions of the core theorems corresponding to the inhomogeneous degree of smoothness case are stated and proved in Chapter 2 along with some auxiliary claims on the continuity of the solution in a certain sense, including an RDE-version of Gronwall's lemma. In Chapter 3, numerical schemes for approximating solutions to differential equations driven by rough paths of inhomogeneous degree of smoothness are constructed. We start with setting up some principles of approximations. Then a general class of local approximations is introduced. This class is used to construct global approximations by pasting together the local ones. A general sufficient condition on the local approximations implying global convergence is given and proved. The next step is to construct particular local approximations in finite dimensions based on solutions to ordinary differential equations derived locally and satisfying the sufficient condition for global convergence. These local approximations require strong conditions on the one-form defining the rough differential equation. Finally, we show that when the local ODE-based schemes are applied in combination with rough polynomial approximations, the conditions on the one-form can be weakened. In Chapter 4, the results of Gyurko & Lyons (2010) on path-wise approximation of solutions to stochastic differential equations are recalled and extended to the truncated signature level of the solution. Furthermore, some practical considerations related to the implementation of high order schemes are described. The effectiveness of the derived schemes is demonstrated on numerical examples. In Chapter 5, the background theory of the Kusuoka-Lyons-Victoir (KLV) family of weak approximations is recalled and linked to the results of Chapter 4. We highlight how the different versions of the KLV family are related. Finally, a numerical evaluation of the autonomous ODE-based versions of the family is carried out, focusing on SDEs in dimensions up to 4, using cubature formulas of different degrees and several high order numerical ODE solvers. We demonstrate the effectiveness and the occasional non-effectiveness of the numerical approximations in cases when the KLV family is used in its original version and also when used in combination with partial sampling methods (Monte-Carlo, TBBA) and Romberg extrapolation.
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Books on the topic "Approximations and expansions – Approximations and expansions – Padé; approximation"

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1968-, Arvesú Jorge, and Lopez Lagomasino Guillermo 1948-, eds. Recent advances in orthogonal polynomials, special functions, and their applications: 11th International Symposium on Orthogonal Polynomials, Special Functions, and Their Applications, August 29-September 2, 2011, Universidad Carlos III de Madrid, Leganes, Spain. Providence, R.I: American Mathematical Society, 2012.

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Jensen, J. L. Saddlepoint approximations, Edgeworth expansions and normal approximations: From independence to dependence. Aarhus C, Denmark: Dept. of Theoretical Statistics, Institute of Mathematics, University of Aarhus, 1993.

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Wong, R. Asymptotic approximations of integrals. Boston: Academic Press, 1989.

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Wong, R. Asymptotic approximations of integrals. Philadelphia: Society for Industrial and Applied Mathematics, 2001.

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service), SpringerLink (Online, ed. Algebraic Approximation: A Guide to Past and Current Solutions. Basel: Springer Basel AG, 2012.

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Pardalos, P. M. Nonlinear Analysis: Stability, Approximation, and Inequalities. New York, NY: Springer New York, 2012.

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1939-, Schumaker Larry L., and SpringerLink (Online service), eds. Approximation Theory XIII: San Antonio 2010. New York, NY: Springer Science+Business Media, LLC, 2012.

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Dai, Feng. Approximation Theory and Harmonic Analysis on Spheres and Balls. New York, NY: Springer New York, 2013.

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1962-, Petras Knut, ed. Quadrature theory: The theory of numerical integration on a compact interval. Providence, R.I: American Mathematical Society, 2011.

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Ervedoza, Sylvain. Numerical Approximation of Exact Controls for Waves. New York, NY: Springer New York, 2013.

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Book chapters on the topic "Approximations and expansions – Approximations and expansions – Padé; approximation"

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Gonchar, A. A., E. A. Rakhmanov, and S. P. Suetin. "On the Rate of Convergence of Padé Approximants of Orthogonal Expansions." In Progress in Approximation Theory, 169–90. New York, NY: Springer New York, 1992. http://dx.doi.org/10.1007/978-1-4612-2966-7_7.

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Andrianov, Igor, and Anatoly Shatrov. "Padé Approximation to Solve the Problems of Aerodynamics and Heat Transfer in the Boundary Layer." In Mathematical Theorems - Boundary Value Problems and Approximations. IntechOpen, 2020. http://dx.doi.org/10.5772/intechopen.93084.

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In this chapter, we describe the applications of asymptotic methods to the problems of mathematical physics and mechanics, primarily, to the solution of nonlinear singular perturbed problems. We also discuss the applications of Padé approximations for the transformation of asymptotic expansions to rational or quasi-fractional functions. The applications of the method of matching of internal and external asymptotics in the problem of boundary layer of viscous gas by means of Padé approximation are considered.
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"INNER-OUTER EXPANSIONS." In Applications of Padé Approximation Theory in Fluid Dynamics, 47–53. WORLD SCIENTIFIC, 1994. http://dx.doi.org/10.1142/9789814354370_0004.

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Carlip, Steven. "The weak field approximation." In General Relativity, 66–71. Oxford University Press, 2019. http://dx.doi.org/10.1093/oso/9780198822158.003.0008.

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The Einstein field equations are a complicated set of coupled partial differential equations, which are usually too complicated to find exact solutions. This chapter introduces a simple approximation for weak fields. It discusses the lowest order solution, which gives back Newtonian gravity, and the next order, which includes “gravitomagnetic” or “frame-dragging” effects. The chapter briefly discusses higher order approximations, expansions around a curved background, and the evidence that gravitational energy itself gravitates. It concludes with a brief description of an alternative derivation of the Einstein field equations, starting from flat spacetime and “bootstrapping” the gravitational self-interaction.
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Conference papers on the topic "Approximations and expansions – Approximations and expansions – Padé; approximation"

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Mansour, Kamyar. "Symbolic Calculation for Free Convection in a Circular Cavity With Periodic Heat Generation." In ASME 2009 Heat Transfer Summer Conference collocated with the InterPACK09 and 3rd Energy Sustainability Conferences. ASMEDC, 2009. http://dx.doi.org/10.1115/ht2009-88382.

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We consider the two-dimensional problem of steady natural convection in a circular cavity with periodic heat generation filled with viscous fluid subject to cosine temperature variation on the boundary. The solution is expanded for low Rayleigh number and extended to 16 terms by computer. Analysis of these expansions allows the exact computation for arbitrarily accuracy up to 50000 figures. Although the range of the radius of convergence is small but pade approximation leads our result to be good even for higher value of the similarity parameter.
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Mansour, Kamyar. "Natural Convection With Porous Medium in a Narrow Horizontal Cylindrical Annulus With Linear Heat Generation." In ASME 2011 9th International Conference on Nanochannels, Microchannels, and Minichannels. ASMEDC, 2011. http://dx.doi.org/10.1115/icnmm2011-58125.

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We consider the two-dimensional problem of steady natural convection in a narrow (micro) horizontal cylindrical annulus filled with porous medium due with linear volumetric heat generation. The solution is expanded in powers of a single combined similarity parameter, which is the product of the gap ratio to the power of two, and Rayleigh number. The series is extended by means of symbolic calculation up to 28 terms. Analysis of these expansions allows the exact computation for arbitrarily accuracy up to 50000 figures. Although the range of the radius of convergence is small, but pade approximation leads our results to be good even for much higher value of the similarity parameter.
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Mansour, Kamyar. "Symbolic Calculation of Laminar Convection in Uniformly Heated Horizontal Pipe at High Prandtl Number." In ASME 2010 3rd Joint US-European Fluids Engineering Summer Meeting collocated with 8th International Conference on Nanochannels, Microchannels, and Minichannels. ASMEDC, 2010. http://dx.doi.org/10.1115/fedsm-icnmm2010-31062.

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We consider fully developed steady laminar flow through a uniformly heated horizontal pipe is simplified by assuming infinite Prandtl number. The solution is expanded in powers of a single combined similarity parameter which is the product of the Prandtl, Rayleigh, and Reynolds numbers and the series extended by means of symbolic calculation up to 16 terms. Analysis of these expansions allows the exact computation for arbitrarily accuracy up to 50000 figures. Although the range of exactness is almost the same order of the radius of convergence but Pade approximation lead our result to be good even for much higher value of the similarity parameter.
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Mansour, Kamyar. "Symbolic Calculation for Free Convection in a Circular Cavity With Porous Material." In ASME 2009 Heat Transfer Summer Conference collocated with the InterPACK09 and 3rd Energy Sustainability Conferences. ASMEDC, 2009. http://dx.doi.org/10.1115/ht2009-88502.

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We consider the two-dimensional problem of steady natural convection in a circular cavity filled with porous material due to a cosine temperature variation on the boundary. We use Darcy’s law for this cavity filled with porous material. The solution is governed by dimensionless parameter Darcy-Rayleigh number. The solution is expanded for low Darcy-Rayleigh number and extended to 18 terms by computer. Analysis of these expansions allows the exact computation for arbitrarily accuracy up to 50000 figures. Although the range of the radius of convergence is small but Pade approximation leads our result to be good even for higher value of the similarity parameter.
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Mansour, Kamyar. "Natural Convection in a Micro Size Horizontal Cylindrical Annulus With Constant Volumetric Heat Flux." In ASME 2008 6th International Conference on Nanochannels, Microchannels, and Minichannels. ASMEDC, 2008. http://dx.doi.org/10.1115/icnmm2008-62249.

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We consider the two-dimensional problem of steady natural convection in a narrow (Micro size) horizontal cylindrical annulus filled with viscous fluid and constant volumetric heat flux. The solution is expanded in powers of a single combined similarity parameter, which is the product of the gap ratio to the power of four, and Grashof number and the series extended by means of symbolic calculation up to 14 terms. Analysis of these expansions allows the exact computation for arbitrarily accuracy up to 50000 figures. Although the range of the radius of convergence is almost zero but Pade approximation lead our result to be good even for much higher value of the similarity parameter.
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Mansour, Kamyar. "Natural Convection in a Micro Size Horizontal Cylindrical Annulus With Periodic Volumetric Heat Flux." In ASME 2009 7th International Conference on Nanochannels, Microchannels, and Minichannels. ASMEDC, 2009. http://dx.doi.org/10.1115/icnmm2009-82162.

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We consider the two-dimensional problem of steady natural convection in a narrow (Micro size) Horizontal Cylindrical annulus filled with viscous fluid and periodic volumetric heat flux. The solution is expanded in powers of a single combined similarity parameter, which is the product of the Gap ratio to the power of four, and Rayleigh number and the series extended by means of symbolic calculation up to 16 terms. Analysis of these expansions allows the exact computation for arbitrarily accuracy up to 50000 figures. Although the range of the radius of convergence is almost zero but Pade approximation lead our result to be good even for much higher value of the similarity parameter.
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Mansour, Kamyar. "Natural Convection With Porous Medium in a Narrow Horizontal Cylindrical Annulus With Constant Volumetric Heat Flux." In ASME 2008 First International Conference on Micro/Nanoscale Heat Transfer. ASMEDC, 2008. http://dx.doi.org/10.1115/mnht2008-52273.

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We consider the two-dimensional problem of steady natural convection in a narrow Horizontal Cylindrical annulus filled with porous medium due to a constant temperature variation on the outer and adiabatic conditions at the inner boundaries with constant volumetric heat flux The solution is expanded in powers of a single combined similarity parameter, which is the product of the Gap ratio to the power of two, and Rayleigh number. The series is extended by means of symbolic calculation up to 28 terms. Analysis of these expansions allows the exact computation for arbitrarily accuracy up to 50000 figures. Although the range of the radius of convergence is small, but Pade approximation leads our results to be good even for much higher value of the similarity parameter.
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Mansour, Kamyar. "Natural Convection With Porous Medium in a Narrow Horizontal Cylindrical Annulus With Periodic Heat Generation." In ASME 2009 Second International Conference on Micro/Nanoscale Heat and Mass Transfer. ASMEDC, 2009. http://dx.doi.org/10.1115/mnhmt2009-18031.

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We consider the two-dimensional problem of steady natural convection in a narrow horizontal cylindrical annulus filled with porous medium due to a constant temperature variation on the outer and at the inner boundaries with periodic volumetric heat flux. The solution is expanded in powers of a single combined similarity parameter, which is the product of the Gap ratio to the power of two, and Rayleigh number. The series is extended by means of symbolic calculation up to 28 terms. Analysis of these expansions allows the exact computation for arbitrarily accuracy up to 50000 figures. Although the range of the radius of convergence is small, but Pade approximation leads our results to be good even for much higher value of the similarity parameter.
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Mansour, Kamyar. "The Adiabatic Conditions on Inner Boundary for Natural Convection With Porous Medium in a Narrow Horizontal Cylindrical Annulus." In ASME 2007 InterPACK Conference collocated with the ASME/JSME 2007 Thermal Engineering Heat Transfer Summer Conference. ASMEDC, 2007. http://dx.doi.org/10.1115/ipack2007-33086.

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Abstract:
We consider the two-dimensional problem of steady natural convection in a narrow Horizontal Cylindrical annulus filled with porous medium due to a cosine temperature variation on the outer and adiabatic conditions at the inner boundaries The solution is expanded in powers of a single combined similarity parameter, which is the product of the Gap ratio to the power of two and Rayleigh number and the series extended by means of symbolic calculation up to 28 terms. Analysis of these expansions allows the exact computation for arbitrarily accuracy up to 50000 figures. Although the range of the radius of convergence is small but Pade approximation leads our result to be good even for much higher value of the similarity parameter.
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10

Mansour, Kamyar. "Natural Convection With Porous Medium in a Narrow Horizontal Cylindrical Annulus With Constant Heat Generation." In ASME 2010 8th International Conference on Nanochannels, Microchannels, and Minichannels collocated with 3rd Joint US-European Fluids Engineering Summer Meeting. ASMEDC, 2010. http://dx.doi.org/10.1115/fedsm-icnmm2010-30114.

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Abstract:
We consider the two-dimensional problem of steady natural convection in a narrow horizontal cylindrical annulus filled with porous medium due to a constant temperature variation on the outer and adiabatic conditions at the inner boundaries with constant volumetric heat flux. The solution is expanded in powers of a single combined similarity parameter, which is the product of the gap ratio to the power of two, and Rayleigh number. The series is extended by means of symbolic calculation up to 28 terms. Analysis of these expansions allows the exact computation for arbitrarily accuracy up to 50000 figures. Although the range of the radius of convergence is small, but Pade approximation leads our results to be good even for much higher value of the similarity parameter.
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