Self-Similar Structures and Fractal Transforms in Approximation Theory

Abstract

An overview is given of the methods for treating complicated problems without small parameters, when the standard perturbation theory based on the existence of small parameters becomes useless. Such complicated problems are typical of quantum physics, many-body physics, physics of complex systems, and various aspects of applied physics and applied mathematics. A general approach for dealing with such problems has been developed, called Self-Similar Approximation Theory. A concise survey of the main ideas of this approach is presented, with the emphasis on the basic notion of group self-similarity. The techniques are illustrated by examples from quantum field theory.

32 pages, Latex

References in corpus (1)

• Self-Similar Extrapolation of Asymptotic Series and Forecasting for Time Series

Cited by in corpus (30)

• Cold Bosons in Optical Lattices
• Principal problems in Bose-Einstein condensation of dilute gases
• Physics of risk and uncertainty in quantum decision making
• Representative statistical ensembles for Bose systems with broken gauge symmetry
• Summation of Power Series by Self-Similar Factor Approximants
• Method of self-similar factor approximants
• Applicability of the Linear delta Expansion for the lambda phi^4 Field Theory at Finite Temperature in the Symmetric and Broken Phases
• Theory of cold atoms: Bose-Einstein statistics
• Bose-Einstein condensation temperature of weakly interacting atoms
• Optimal trap shape for a Bose gas with attractive interactions
• Self-Similar Law of Energy Release before Materials Fracture
• Calculation of critical exponents by self-similar factor approximants
• Highly Accurate Critical Exponents from Self-Similar Variational Perturbation Theory
• Self-similar factor approximants for evolution equations and boundary-value problems
• Continued functions and perturbation series: Simple tools for convergence of diverging series in $O(n)$-symmetric ${\varphi }^4$ field theory at weak coupling limit
• Reliability of the optimized perturbation theory in the 0-dimensional $O(N)$ scalar field model
• Self-similar continued root approximants
• Interplay Between Approximation Theory and Renormalization Group
• Optimization of self-similar factor approximants
• Self-similar extrapolation in quantum field theory
• Self-similarly corrected Pade approximants for nonlinear equations
• Critical Temperature in Weakly Interacting Multicomponent Field Theory
• From Coherent Modes to Turbulence and Granulation of Trapped Gases
• Extrapolation from hypergeometric functions, continued functions and Borel-Leroy transformation; Resummation of perturbative renormalization functions from field theories
• Operator Method for Nonperturbative Calculation of the Thermodynamic Values in Quantum Statistics. Diatomic Molecular Gas
• Self-similar extrapolation of nonlinear problems from small-variable to large-variable limit
• Continued functions and Borel-Leroy transformation: Resummation of six-loop ε-expansions from different universality classes
• Describing phase transitions in field theory by self-similar approximants
• Extrapolation of perturbation-theory expansions by self-similar approximants
• From Asymptotic Series to Self-Similar Approximants