scholarly journals Padé and Post-Padé Approximations for Critical Phenomena

Symmetry ◽  
2020 ◽  
Vol 12 (10) ◽  
pp. 1600 ◽  
Author(s):  
Simon Gluzman
Keyword(s):  
Critical Phenomena ◽  
Critical Points ◽  
Critical Index ◽  
Extrapolation Methods ◽  
Padé Approximations ◽  
Pade Approximations ◽  
Self Similar ◽  
Direct Extrapolation

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.

scholarly journals Critical Indices and Self-Similar Power Transform

Axioms ◽  
2021 ◽  
Vol 10 (3) ◽  
pp. 162
Author(s):  
Simon Gluzman
Keyword(s):  
Critical Phenomena ◽  
Random Media ◽  
Critical Index ◽  
Power Transformation ◽  
Scaling Exponent ◽  
Critical Indices ◽  
Correction To Scaling ◽  
Self Similar ◽  
Additional Constraints ◽  
Sensitivity Derivative

“Odd” factor approximants of the special form suggested by Gluzman and Yukalov (J. Math. Chem. 2006, 39, 47) are amenable to optimization by power transformation and can be successfully applied to critical phenomena. The approach is based on the idea that the critical index by itself should be optimized through the parameters of power transform to be calculated from the minimal sensitivity (derivative) optimization condition. The critical index is a product of the algebraic self-similar renormalization which contributes to the expressions the set of control parameters typical to the algebraic self-similar renormalization, and of the power transform which corrects them even further. The parameter of power transformation is, in a nutshell, the multiplier connecting the critical exponent and the correction-to-scaling exponent. We mostly study the minimal model of critical phenomena based on expansions with only two coefficients and critical points. The optimization appears to bring quite accurate, uniquely defined results given by simple formulas. Many important cases of critical phenomena are covered by the simple formula. For the longer series, the optimization condition possesses multiple solutions, and additional constraints should be applied. In particular, we constrain the sought solution by requiring it to be the best in prediction of the coefficients not employed in its construction. In principle, the error/measure of such prediction can be optimized by itself, with respect to the parameter of power transform. Methods of calculation based on optimized power-transformed factors are applied and results presented for critical indices of several key models of conductivity and viscosity of random media, swelling of polymers, permeability in two-dimensional channels. Several quantum mechanical problems are discussed as well.

scholarly journals Algebraic self-similar renormalization in the theory of critical phenomena

1997 ◽  
Vol 55 (4) ◽  
pp. 3983-3999 ◽  
Author(s):  
S. Gluzman ◽  
V. I. Yukalov
Keyword(s):  
Critical Phenomena ◽  
Self Similar

Padé approximations to the logarithm II: Identities, recurrences, and symbolic computation

2006 ◽  
Vol 11 (2) ◽  
pp. 139-158 ◽  
Author(s):  
Kathy Driver ◽  
Helmut Prodinger ◽  
Carsten Schneider ◽  
J. A. C. Weideman
Keyword(s):  
Symbolic Computation ◽  
Padé Approximations ◽  
Pade Approximations

Dynamic Stability of Hingeless Rotor Blade in Hover Using Padé Approximations

AIAA Journal ◽  
2016 ◽  
Vol 54 (5) ◽  
pp. 1769-1777 ◽  
Author(s):  
Ahmed A. Hussein ◽  
Muhammad R. Hajj ◽  
Samir M. Elkholy ◽  
Gamal M. ELbayoumi
Keyword(s):  
Dynamic Stability ◽  
Rotor Blade ◽  
Padé Approximations ◽  
Pade Approximations
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