Continued Roots, Power Transform and Critical Properties

Simon Gluzman

doi:10.3390/sym13081525

Abstract

We consider the problem of calculation of the critical amplitudes at infinity by means of the self-similar continued root approximants. Region of applicability of the continued root approximants is extended from the determinate (convergent) problem with well-defined conditions studied before by Gluzman and Yukalov (Phys. Lett. A 377 2012, 124), to the indeterminate (divergent) problem my means of power transformation. Most challenging indeterminate for the continued roots problems of calculating critical amplitudes, can be successfully attacked by performing proper power transformation to be found from the optimization imposed on the parameters of power transform. The self-similar continued roots were derived by systematically applying the algebraic self-similar renormalization to each and every level of interactions with their strength increasing, while the algebraic renormalization follows from the fundamental symmetry principle of functional self-similarity, realized constructively in the space of approximations. Our approach to the solution of the indeterminate problem is to replace it with the determinate problem, but with some unknown control parameter b in place of the known critical index β. From optimization conditions b is found in the way making the problem determinate and convergent. The index β is hidden under the carpet and replaced by b. The idea is applied to various, mostly quantum-mechanical problems. In particular, the method allows us to solve the problem of Bose-Einstein condensation temperature with good accuracy.

Highlights

Power transformation [1] could be combined with continued root approximants [2]
The problem of finding the critical amplitudes is equivalent to calculating the constant value of the functions at infinity, another old problem existing in applied mathematics [3]
After standard calculations with 3 + 1 optimization scheme we find rather good estimate β 3+1 (0.271135) = 0.67592, bringing the error of 1.39%, Following the scheme of indeterminate problem and the “inverse” scheme we obtain with the power-transformed self-similar continued roots β 1 (1) = 0.632456, β 2 (1) = 0.581431, β 3 (1) = 0.589432, β 4 (1) = 0.607582, β 5 (1) = 0.625906, with the best estimate giving an error of 6%

Summary

Introduction

Power transformation [1] could be combined with continued root approximants [2].Through its application we intend to extend the applicability of the continued roots to a broader class of problems of finding the critical amplitudes and indices.Generally speaking, the problem of finding the critical amplitudes is equivalent to calculating the constant value of the functions at infinity, another old problem existing in applied mathematics [3]. Power transformation [1] could be combined with continued root approximants [2]. Through its application we intend to extend the applicability of the continued roots to a broader class of problems of finding the critical amplitudes and indices. The problem of finding the critical amplitudes is equivalent to calculating the constant value of the functions at infinity, another old problem existing in applied mathematics [3]. The self-similar continued roots were derived in [2] by systematically applying the algebraic self-similar renormalization to each and every level of interactions with their strength increasing, while the algebraic renormalization follows from the functional selfsimilarity principle implied in the space of approximations, as shown in [5]. Power transformation can be viewed as an algebraic transformation applied to the continued root approximant as a whole

Objectives

Findings

Discussion

Conclusion