Summation of power series by self-similar factor approximants

The problem of extrapolating the series in powers of small variables to the region of large variables is addressed. Such a problem is typical of quantum theory and statistical physics. A method of extrapolation is developed based on self-similar factor and root approximants, suggested earlier by the authors. It is shown that these approximants and their combinations can effectively extrapolate power series to the region of large variables, even up to infinity. Several examples from quantum and statistical mechanics are analyzed, illustrating the approach.

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Self-similar extrapolation of nonlinear problems from small-variable to large-variable limit

International Journal of Modern Physics B ◽

10.1142/s0217979220502082 ◽

2020 ◽

Vol 34 (21) ◽

pp. 2050208

Author(s):

V. I. Yukalov ◽

E. P. Yukalova

Keyword(s):

Asymptotic Series ◽

Nonlinear Problems ◽

Numerical Convergence ◽

Physical Problems ◽

Variable Limit ◽

Small Parameters ◽

Self Similar ◽

Similar Approximation ◽

Similar Factor ◽

Class Of Functions

Complicated physical problems are usually solved by resorting to perturbation theory leading to solutions in the form of asymptotic series in powers of small parameters. However, finite, and even large values of the parameters, are often of main physical interest. A method is described for predicting the large-variable behavior of solutions to nonlinear problems from the knowledge of only their small-variable expansions. The method is based on self-similar approximation theory resulting in self-similar factor approximants. The latter can well approximate a large class of functions, rational, irrational, and transcendental. The method is illustrated by several examples from statistical and condensed matter physics, where the self-similar predictions can be compared with the available large-variable behavior. It is shown that the method allows for finding the behavior of solutions at large variables when knowing just a few terms of small-variable expansions. Numerical convergence of approximants is demonstrated.

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Rapid Multimodal Image Registration Based on the Local Edge Histogram

Mathematical Problems in Engineering ◽

10.1155/2021/5598177 ◽

2021 ◽

Vol 2021 ◽

pp. 1-9

Author(s):

Dong Zhao

Keyword(s):

Image Registration ◽

Edge Information ◽

Multimodal Image Registration ◽

Image Blurring ◽

Self Similar ◽

Similar Factor ◽

Local Edge ◽

Multimodality Registration ◽

Edge Histogram ◽

Multimodal Images

Due to significant differences in imaging mechanisms between multimodal images, registration methods have difficulty in achieving the ideal effect in terms of time consumption and matching precision. Therefore, this paper puts forward a rapid and robust method for multimodal image registration by exploiting local edge information. The method is based on the framework of SURF and can simultaneously achieve real time and accuracy. Due to the unpredictability of multimodal images’ textures, the local edge descriptor is built based on the edge histogram of neighborhood around keypoints. Moreover, in order to increase the robustness of the whole algorithm and maintain the SURF’s fast characteristic, saliency assessment of keypoints and the concept of self-similar factor are presented and introduced. Experimental results show that the proposed method achieves higher precision and consumes less time than other multimodality registration methods. In addition, the robustness and stability of the method are also demonstrated in the presence of image blurring, rotation, noise, and luminance variations.

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Fractals in the Large

Canadian Journal of Mathematics ◽

10.4153/cjm-1998-036-5 ◽

1998 ◽

Vol 50 (3) ◽

pp. 638-657 ◽

Cited By ~ 28

Author(s):

Robert S. Strichartz

Keyword(s):

Invariant Measure ◽

Power Series ◽

Metric Space ◽

Iterated Function System ◽

Invariant Set ◽

Invariant Sets ◽

Iterated Function ◽

Positive Weights ◽

Self Similar ◽

Expansive Maps

AbstractA reverse iterated function system (r.i.f.s.) is defined to be a set of expansive maps ﹛T1,…, Tm﹜ on a discrete metric space M. An invariant set F is defined to be a set satisfying , and an invariant measure μ is defined to be a solution of for positive weights pj. The structure and basic properties of such invariant sets and measures is described, and some examples are given. A blowup ℱ of a self-similar set F in ℝn is defined to be the union of an increasing sequence of sets, each similar to F. We give a general construction of blowups, and show that under certain hypotheses a blowup is the sum set of F with an invariant set for a r.i.f.s. Some examples of blowups of familiar fractals are described. If μ is an invariant measure on ℤ+ for a linear r.i.f.s., we describe the behavior of its analytic transform, the power series on the unit disc.

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RECONSTRUCTING GENERALIZED EXPONENTIAL LAWS BY SELF-SIMILAR EXPONENTIAL APPROXIMANTS

International Journal of Modern Physics C ◽

10.1142/s012918310300470x ◽

2003 ◽

Vol 14 (04) ◽

pp. 509-527 ◽

Cited By ~ 4

Author(s):

S. GLUZMAN ◽

D. SORNETTE ◽

V. I. YUKALOV

Keyword(s):

Power Series ◽

Padé Approximants ◽

The Self ◽

Pade Approximants ◽

Self Similar

We apply the technique of self-similar exponential approximants based on successive truncations of simple continued exponentials to reconstruct functional laws of the quasi-exponential class from the knowledge of only a few terms of their power series. Comparison with the standard Padé approximants shows that, in general, the self-similar exponential approximants provide significantly better reconstructions.

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Self-similar factor approximants

Physical Review E ◽

10.1103/physreve.67.026109 ◽

2003 ◽

Vol 67 (2) ◽

Cited By ~ 43

Author(s):

S. Gluzman ◽

V. I. Yukalov ◽

D. Sornette

Keyword(s):

Self Similar ◽

Similar Factor

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Application of nonstationary self-similar variables for solving the three-dimensional problem of the decay of a special discontinuity

Вычислительные технологии ◽

10.25743/ict.2021.26.5.005 ◽

2021 ◽

pp. 52-64

Author(s):

Сергей Петрович Баутин ◽

Сергей Львович Дерябин

Keyword(s):

Cauchy Problem ◽

Power Series ◽

Initial Conditions ◽

Three Dimensional ◽

Physical Space ◽

Initial Moment ◽

Algebraic Equations ◽

Gas Dynamics Equations ◽

Self Similar ◽

Isentropic Flows

Построение в физическом пространстве решения задачи о распаде специального разрыва, т.е. трехмерных изэнтропических течений политропного газа, возникающих после мгновенного разрушения в начальный момент времени непроницаемой стенки, отделяющей неоднородный движущийся газ от вакуума. В задаче учитывается действие силы тяжести и силы Кориолиса. В систему уравнений газовой динамики введена автомодельная особенность в переменную, которая выводит с поверхности раздела. Для полученной системы поставлена задача Коши с данными на звуковой характеристике. Решение задачи строилось в виде степенных рядов. Часть коэффициентов рядов определялась при решении алгебраических уравнений, а часть из решений - обыкновенных дифференциальных уравнений. Методом мажорант доказана сходимость построенных рядов. Построенное решение позволяет задавать начальные условия для разностной схемы при численном моделировании решений данной характеристической задачи Коши The aim of this study is to construct a solution to the problem of the decay of a special discontinuity in physical space. The problem reduces to finding of three-dimensional isentropic flows of a polytropic gas that occur after the instantaneous destruction of an impermeable wall separating an inhomogeneous moving gas from a vacuum at the initial moment of time. The problem takes into account the forces of gravity and Coriolis. Research methods. In the system of gas dynamics equations, a self-similar feature is introduced in a variable that outputs from the initial interface. For the resulting system, the Cauchy problem is formulated using conditions on the sound characteristic. The solution to this problem is constructed in the form of power series. The coefficients of the series are partly determined by solving algebraic equations, another part can be found as solutions of ordinary differential equations. The convergence of the constructed series is proved by the Majorant method The results obtained in the work. In the form of a convergent power series, solutions to the problem of the decay of a special discontinuity in physical space are constructed. Conclusions. The solution constructed in physical space allows setting the initial conditions for the numerical simulation of this characteristic Cauchy problem using a difference scheme.

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