Extensions of Padé approximants

Baker, George A.; Graves‐Morris, P. R.

doi:10.1017/cbo9780511530074.009

Cited by 601 publications

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“…M = N → ∞. In some cases, this allows the application of known mathematical theorems that ensure convergence [15,16,20]. This has been used for the study of certain Green functions [17,18,19].…”

Section: Higher Order Pas In the Lsmmentioning

confidence: 99%

Some remarks on the Padé unitarization of low-energy amplitudes

2008

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We present a critical analysis of Padé-based methods for the unitarization of low energy amplitudes. We show that the use of certain Padé Approximants to describe the resonance region may lead to inaccurate determinations. In particular, we find that in the Linear Sigma Model the unitarization of the low energy amplitude through the inverse amplitude method produces essentially incorrect results for the mass and width of the sigma. Alternative sequences of Padés are studied and we find that the diagonal sequences (i.e., [N/N ]) have much better convergence properties.

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Section: Higher Order Pas In the Lsmmentioning

confidence: 99%

Some remarks on the Padé unitarization of low-energy amplitudes

2008

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“…Still, it is possible to reconstruct the full energy dependence approximately from the known expansions at higher orders. Using a seminumerical method based on Padé approximations [35,36,37,38,39], the correlators of all four currents were reconstructed at O(α 2 s ) [5,7,40]. Moreover, it was demonstrated in [41] that in spite of the rather low amount of infor-mation available at O(α 3 s ) it is still viable to reconstruct the vector correlator and predict expansion coefficients with decent accuracy.…”

Section: Introductionmentioning

confidence: 99%

Reconstruction of heavy quark current correlators at

et al. 2009

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We construct approximate formulas for the O(α 3 s ) QCD contributions to vector, axial-vector, scalar and pseudo-scalar quark current correlators, which are valid for arbitrary values of momenta and masses. The derivation is based on conformal mapping and the Padé approximation procedure and incorporates known expansions in the low energy, threshold and high energy regions. We use our results to estimate additional terms in these expansions.

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“…The integral in (38) is known as a Stieltjes integral [72]. This formula is typically derived and used for the case of complex constants: σ 1 , σ 2 , and σ * .…”

Section: E Analytical Continuation Methodsmentioning

confidence: 99%

Measures of microstructure to improve estimates and bounds on elastic constants and transport coefficients in heterogeneous media

Berryman

2006

Mechanics of Materials

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The most commonly discussed measures of microstructure in composite materials are the spatial correlation functions, which in a porous medium measure either the grain-to-grain correlations, or the pore-to-pore correlations in space. Improved bounds based on this information such as the Beran-Molyneux bounds for bulk modulus and the Beran bounds for conducticity are well-known.It is first shown here how to make direct use of this information to provide estimates that always lie between these upper and lower bounds for any microstructure whenever the microgeometry parameters are known. Then comparisons are made between these estimates, the bounds, and two new types of estimates. One new estimate for elastic constants makes use of the PeselnickMeister bounds (based on Hashin-Shtrikman methods) for random polycrystals of laminates to generate self-consistent values that always lie between the bounds. A second new type of estimate for conductivity assumes that measurements of formation factors (of which there are at least two distinct types in porous media, associated respectively with pores and grains) are available, and computes new bounds based on this information. The paper compares and contrasts these various methods in order to clarify just what microstructural information and how precisely that information needs to be known in order to be useful for estimating material constants in random and heterogeneous media. * berryman1@llnl.gov

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Extensions of Padé approximants

Cited by 601 publications

References 0 publications

Some remarks on the Padé unitarization of low-energy amplitudes

Some remarks on the Padé unitarization of low-energy amplitudes

Reconstruction of heavy quark current correlators at

Measures of microstructure to improve estimates and bounds on elastic constants and transport coefficients in heterogeneous media

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