arXiv subjects

S. Gluzman

Publications and source records attributed to S. Gluzman.

At least 37 records · Page 2Linked to original sources

arXiv2003

Summation of Power Series by Self-Similar Factor Approximants

A novel method of summation for power series is developed. The method is based on the self-similar approximation theory. The trick employed is in transforming, first, a series expansion into a product expansion and in applying the self-similar renormalization to the latter rather to the former. This results in self-similar factor approximants extrapolating the sought functions from the region of asymptotically small variables to their whole domains. The method of constructing crossover formulas, interpolating between small and large values of variables is also analysed. The techniques are illustrated on different series which are typical of problems in statistical mechanics, condensed-matter physics, and, generally, in many-body theory.

cond-mat.stat-mech↗

arXiv2002

Self-similar Approximants of the Permeability in Heterogeneous Porous Media from Moment Equation Expansions

We use a mathematical technique, the self-similar functional renormalization, to construct formulas for the average conductivity that apply for large heterogeneity, based on perturbative expansions in powers of a small parameter, usually the log-variance $σ_Y^2$ of the local conductivity. Using perturbation expansions up to third order and fourth order in $σ_Y^2$ obtained from the moment equation approach, we construct the general functional dependence of the transport variables in the regime where $σ_Y^2$ is of order 1 and larger than 1. Comparison with available numerical simulations give encouraging results and show that the proposed method provides significant improvements over available expansions.

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arXiv2002

Self-Similar Factor Approximants

The problem of reconstructing functions from their asymptotic expansions in powers of a small variable is addressed by deriving a novel type of approximants. The derivation is based on the self-similar approximation theory, which presents the passage from one approximant to another as the motion realized by a dynamical system with the property of group self-similarity. The derived approximants, because of their form, are named the self-similar factor approximants. These complement the obtained earlier self-similar exponential approximants and self-similar root approximants. The specific feature of the self-similar factor approximants is that their control functions, providing convergence of the computational algorithm, are completely defined from the accuracy-through-order conditions. These approximants contain the Pade approximants as a particular case, and in some limit they can be reduced to the self-similar exponential approximants previously introduced by two of us. It is proved that the self-similar factor approximants are able to reproduce exactly a wide class of functions which include a variety of transcendental functions. For other functions, not pertaining to this exactly reproducible class, the factor approximants provide very accurate approximations, whose accuracy surpasses significantly that of the most accurate Pade approximants. This is illustrated by a number of examples showing the generality and accuracy of the factor approximants even when conventional techniques meet serious difficulties.

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arXiv2002

Reconstructing Generalized Exponential Laws by Self-Similar Exponential Approximants

We apply the technique of self-similar exponential approximants based on successive truncations of 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.

cond-mat↗

arXiv2001

Log-periodic route to fractal functions

Log-periodic oscillations have been found to decorate the usual power law behavior found to describe the approach to a critical point, when the continuous scale-invariance symmetry is partially broken into a discrete-scale invariance (DSI) symmetry. We classify the `Weierstrass-type'' solutions of the renormalization group equation F(x)= g(x)+(1/m)F(g x) into two classes characterized by the amplitudes A(n) of the power law series expansion. These two classes are separated by a novel ``critical'' point. Growth processes (DLA), rupture, earthquake and financial crashes seem to be characterized by oscillatory or bounded regular microscopic functions g(x) that lead to a slow power law decay of A(n), giving strong log-periodic amplitudes. In contrast, the regular function g(x) of statistical physics models with ``ferromagnetic''-type interactions at equibrium involves unbound logarithms of polynomials of the control variable that lead to a fast exponential decay of A(n) giving weak log-periodic amplitudes and smoothed observables. These two classes of behavior can be traced back to the existence or abscence of ``antiferromagnetic'' or ``dipolar''-type interactions which, when present, make the Green functions non-monotonous oscillatory and favor spatial modulated patterns.

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arXiv2000

Functional Renormalization Prediction of Rupture

We develop theoretical formulas for the prediction of the rupture of systems which are known to exhibit a critical behavior, based solely on the knowledge of the early time evolution of an observable, such as the acoustic emission rate as a function of time or of stress. From the parameterization of such early time evolution in terms of a low-order polynomial, we use the functional renormalization approach introduced by Yukalov and Gluzman to transform this polynomial into a function which is asymptotically a power law. The value of the critical time tc,conditioned on the assumption that tc exists, is thus determined from the knowledge of the coefficients of the polynomials. We test with success this prediction scheme with respect to the order of the polynomials and as a function of noise.

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arXiv2000

Self-Consistent Theory of Rupture by Progressive Diffuse Damage

We analyze a self-consistent theory of crack growth controlled by a cumulative damage variable d(t) dependent on stress history. As a function of the damage exponent $m$, which controls the rate of damage dd/dt \propto sigma^m as a function of local stress $σ$, we find two regimes. For 0 < m < 2, the model predicts a finite-time singularity. This retrieves previous results by Zobnin for m=1 and by Bradley and Wu for 0 < m < 2. To improve on this self-consistent theory which neglects the dependence of stress on damage, we apply the functional renormalization method of Yukalov and Gluzman and find that divergences are replaced by singularities with exponents in agreement with those found in acoustic emission experiments. For m =2 and m > 2, the rupture dynamics is not defined without the introduction of a regularizing scheme. We investigate three regularization schemes involving respectively a saturation of damage, a minimum distance of approach to the crack tip and a fixed stress maximum. In the first and third schemes, the finite-time singularity is replaced by a crack dynamics defined for all times but which is controlled by either the existence of a microscopic scale at which the stress is regularized or by the maximum sustainable stress. In the second scheme, a finite-time singularity is again found. In the first two schemes within this regime m > 2, the theory has no continuous limit.

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arXiv2000

Self-Similar Crossover in Statistical Physics

An analytical method is advanced for constructing interpolation formulae for complicated problems of statistical mechanics, in which just a few terms of asymptotic expansions are available. The method is based on the self-similar approximation theory, being its variant where control functions are defined from asymptotic crossover conditions. Several examples from statistical physics demonstrate that the suggested method results in rather simple and surprisingly accurate formulae.

cond-mat↗

arXiv1999

Fundamental Framework for Technical Analysis

Starting from the characterization of the past time evolution of market prices in terms of two fundamental indicators, price velocity and price acceleration, we construct a general classification of the possible patterns characterizing the deviation or defects from the random walk market state and its time-translational invariant properties. The classification relies on two dimensionless parameters, the Froude number characterizing the relative strength of the acceleration with respect to the velocity and the time horizon forecast dimensionalized to the training period. Trend-following and contrarian patterns are found to coexist and depend on the dimensionless time horizon. The classification is based on the symmetry requirements of invariance with respect to change of price units and of functional scale-invariance in the space of scenarii. This ``renormalized scenario'' approach is fundamentally probabilistic in nature and exemplifies the view that multiple competing scenarii have to be taken into account for the same past history. Empirical tests are performed on on about nine to thirty years of daily returns of twelve data sets comprising some major indices (Dow Jones, SP500, Nasdaq, DAX, FTSE, Nikkei), some major bonds (JGB, TYX) and some major currencies against the US dollar (GBP, CHF, DEM, JPY). Our ``renormalized scenario'' exhibits statistically significant predictive power in essentially all market phases. In constrast, a trend following strategy and trend + acceleration following strategy perform well only on different and specific market phases. The value of the ``renormalized scenario'' approach lies in the fact that it always finds the best of the two, based on a calculation of the stability of their predicted market trajectories.

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arXiv1999

Weighted Fixed Points in Self-Similar Analysis of Time Series

The self-similar analysis of time series is generalized by introducing the notion of scenario probabilities. This makes it possible to give a complete statistical description for the forecast spectrum by defining the average forecast as a weighted fixed point and by calculating the corresponding a priori standard deviation and variance coefficient. Several examples of stock-market time series illustrate the method.

cond-mat↗

arXiv1998

Booms and Crashes in Self-Similar Markets

Sharp changes in time series representing market dynamics are studied by means of the self--similar analysis suggested earlier by the authors. These sharp changes are market booms and crashes. Such crises phenomena in markets are analogous to critical phenomena in physics. A simple classification of the market crisis phenomena is given.

cond-mat.stat-mech↗

arXiv1998

Unified Approach to Crossover Phenomena

A general analytical method is developed for describing crossover phenomena of arbitrary nature. The method is based on the algebraic self-similar renormalization of asymptotic series, with control functions defined by crossover conditions. The method can be employed for such difficult problems for which only a few terms of asymptotic expansions are available, and no other techniques are applicable. As an illustration, analytical solutions for several important physical problems are presented.

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arXiv1998

Self-Similar Exponential Approximants

An approach is suggested defining effective sums of divergent series in the form of self-similar exponential approximants. The procedure of constructing these approximants from divergent series with arbitrary noninteger powers is developed. The basis of this construction is the self-similar approximation theory. Control functions governing the convergence of exponentially renormalized series are defined from stability and fixed-point conditions and from additional asymptotic conditions when the latter are available. The stability of the calculational procedure is checked by analyzing cascade multipliers. A number of physical examples for different statistical systems illustrate the generality and high accuracy of the approach.

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arXiv1998

Self-Similar Interpolation in Quantum Mechanics

An approach is developed for constructing simple analytical formulae accurately approximating solutions to eigenvalue problems of quantum mechanics. This approach is based on self-similar approximation theory. In order to derive interpolation formulae valid in the whole range of parameters of considered physical quantities, the self-similar renormalization procedure is complimented here by boundary conditions which define control functions guaranteeing correct asymptotic behaviour in the vicinity of boundary points. To emphasize the generality of the approach, it is illustrated by different problems that are typical for quantum mechanics, such as anharmonic oscillators, double-well potentials, and quasiresonance models with quasistationary states. In addition, the nonlinear Schrödinger equation is considered, for which both eigenvalues and wave functions are constructed.

cond-mat↗

arXiv1998

Resummation Methods for Analyzing Time Series

An approach is suggested for analyzing time series by means of resummation techniques of theoretical physics. A particular form of such an analysis, based on the algebraic self-similar renormalization, is developed and illustrated by several examples from the stock market time series.

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arXiv1998

Renormalization Group Analysis of October Market Crashes

The self-similar analysis of time series, suggested earlier by the authors, is applied to the description of market crises. The main attention is payed to the October 1929, 1987 and 1997 stock market crises, which can be successfully treated by the suggested approach. The analogy between market crashes and critical phenomena is emphasized.

cond-mat.stat-mech↗

arXiv1998

Fixed Points in Self-Similar Analysis of Time Series

Two possible definitions of fixed points in the self-similar analysis of time series are considered. One definition is based on the minimal-difference condition and another, on a simple averaging. From studying stock market time series, one may conclude that these two definitions are practically equivalent. A forecast is made for the stock market indices for the end of March 1998.

cond-mat.stat-mech↗

arXiv1997

Critical Indices as Limits of Control Functions

A variant of self-similar approximation theory is suggested, permitting an easy and accurate summation of divergent series consisting of only a few terms. The method is based on a power-law algebraic transformation, whose powers play the role of control functions governing the fastest convergence of the renormalized series. A striking relation between the theory of critical phenomena and optimal control theory is discovered: The critical indices are found to be directly related to limits of control functions at critical points. The method is applied to calculating the critical indices for several difficult problems. The results are in very good agreement with accurate numerical data.

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