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We have performed a numerical investigation of the influence of disorder on the dynamical non-equilibrium evolution of a 3D site-diluted Ising model from a low-temperature initial state with magnetization m
0 = 1. It is shown that two-time dependences of the autocorrelation and integrated response functions for systems with spin concentrations p = 1.0, 0.95, 0.8, 0.6 and 0.5 demonstrate ageing properties with anomalous slowing-down relaxation and violation of the fluctuation-dissipation ratio. It was revealed that during non-equilibrium critical dynamics in the long-time regime the autocorrelation functions for diluted systems are extremely slow due to the pinning of domain walls on impurity sites. We have found that the fluctuation-dissipation ratio for diluted systems with spin concentration p < 1 while the pure system is characterized by . The autocorrelation function power-law delay becomes the same as for the time dependence of the magnetization in the critical point and is characterized by exponent . Also, for diluted systems we reveal memory effects for critical evolution in the ageing regime with realization of cyclic temperature change and quenching at .

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A Monte Carlo simulation of the non-equilibrium behavior of multilayer magnetic structures Co/Cu(100)/Co and Pt/Co/Cu(100)/Co/Pt characterizing different types of magnetic anisotropy is realized. Simulation of transport properties gives possibility to reveal a nontrivial aging effects in the magnetoresistance of these structures and influence of initial states on two-time dependence of magnetoresistance.

Monte Carlo study of non-equilibrium critical behavior of three-dimensional Heisenberg model in isotropic case and with anisotropy of easy axis type is carried out. Relaxational Glauber-like dynamics of these models with evolution from high-temperature initial state is investigated. Realization of aging is demonstrated for two-time dependence of the autocorrelation function and dynamical susceptibility. Asymptotic fluctuation-dissipation ratios are determined for isotropic X ∞ = 0.383(6) and anisotropic X ∞ = 0.392(7) Heisenberg models. Significant influence of easy-axis anisotropy on non-equilibrium critical behavior of the 3D Heisenberg model leading to characteristics typical for the 3D Ising model is revealed.

A Monte Carlo simulation of the non-equilibrium behavior of multilayer magnetic nanostructure Co/Cu(100)/Co consisting of alternating magnetic and nonmagnetic nanolayers is carried out. Analysis of calculated two-time autocorrelation function for structure relaxing from both high-temperature and low-temperature initial states reveals aging characterized by a slowing down of correlation characteristics with increase of the waiting time. It is shown that, in contrast to bulk magnetic systems, the aging effects in nanostructure arise not only at the ferromagnetic ordering temperature T c but also within a wide temperature range at T ≤ T c . For evolution from high-temperature initial state, the study of dependence of aging characteristics on thickness N of cobalt films reveals a weakening of the aging with increasing N at the critical temperatures T c (N) and an opposite tendency at temperatures T < T c (N) with strengthening of aging with increasing N for considered N ≤ 9 ML. This phenomenon is connected with increasing correlation and relaxation times in nanostructures when temperature is decreased. For case of the low-temperature initial state, it is shown that correlation times are two-three orders of magnitude smaller than those in the evolution from a high-temperature initial state at the same t w values. In this case, time behavior of the autocorrelation function doesn't depend considerably on temperature for T s ≤ T c and thickness N of cobalt films. Simulation of transport properties in Co/Cu(100)/Co structure permitted to calculate temperature dependence of its equilibrium magnetoresistance values. For the first time, it was revealed influence of non-equilibrium behavior on the magnetoresistance with demonstration of nontrivial aging effects. It has been shown that the magnetoresistance reaches plateau in asymptotic long-time regime with values , which depend on type of initial state, thickness of cobalt films, and temperature.

The Monte Carlo simulation of the critical behavior of multilayer structures based on anisotropic Heisenberg model is performed. The influence of the uniaxial anisotropy on the critical behavior of the thin Heisenberg-like film is described. The investigation of non-equilibrium critical behavior of multilayer structure which correspond to the nanoscale superlattice Co/Cu demonstrates that the aging effects can be observed in a wider temperature range than for bulk magnetic systems.

A Monte Carlo numerical simulation of the specific features of nonequilibrium critical behavior is carried out for the two-dimensional structurally disordered XY model during its evolution from a low-temperature initial state. On the basis of the analysis of the two-time dependence of autocorrelation functions and dynamic susceptibility for systems with spin concentrations of p = 1.0, 0.9, and 0.6, aging phenomena characterized by a slowing down of the relaxation system with increasing waiting time and the violation of the fluctuation–dissipation theorem (FDT) are revealed. The values of the universal limiting fluctuation–dissipation ratio (FDR) are obtained for the systems considered. As a result of the analysis of the two-time scaling dependence for spin–spin and connected spin autocorrelation functions, it is found that structural defects lead to subaging phenomena in the behavior of the spin–spin autocorrelation function and superaging phenomena in the behavior of the connected spin autocorrelation function.

The Monte Carlo simulation of the critical behavior of low-dimensional magnets and multilayer structures based on anisotropic Heisenberg mode are presented. The aging and clustering effects are revealed in the critical relaxation of 2d XY-model from non-equilibrium initial states. The investigation of non-equilibrium critical behavior of multilayer structure which correspond to the nanoscale superlattice Co/Cr demonstrates that the aging can be observed in a wider temperature range than for bulk magnetic systems.

1
Laboratoire de Physique des Matériaux,* Université Henri Poincaré Nancy I, B.P. 239, F-54506 Vandœuvre lès Nancy Cedex, France*2
Isaac Newton Institute of Mathematical Sciences, 20 Clarkson Road, Cambridge CB3 0EH, UK
3
Dipartamento di Fisica/INFN—Sezione di Firenze, Università di Firenze, I-50019 Sesto Fiorentino, Italy
4
INFM-SMC-CNR and Dipartamento di Fisica, Università di Roma 'La Sapienza', Piazzale A. Moro 2, I-00185 Roma, Italy
5
Institut für Theoretische Physik I, Universität Erlangen-Nürnberg, Staudtstraße 7B3, D-91058 Erlangen, Germany
6
Department of Physics, Virginia Polytechnic Institute and State University, Blacksburg, VA 24061-0435, USA

The glass transition, extensively studied in dense fluids, polymers, or
colloids, corresponds to a dramatic evolution of equilibrium transport
coefficients upon a modest change of control parameter, like temperature or
pressure. A similar phenomenology is found in many systems evolving far from
equilibrium, such as driven granular media, active and living matter. While
many theories compete to describe the glass transition at thermal equilibrium,
very little is understood far from equilibrium. Here, we solve the dynamics of
a specific, yet representative, class of glass models in the presence of
nonthermal driving forces and energy dissipation, and show that a dynamic
arrest can take place in these nonequilibrium conditions. While the location of
the transition depends on the specifics of the driving mechanisms, important
features of the glassy dynamics are insensitive to details, suggesting that an
`effective' thermal dynamics generically emerges at long time scales in
nonequilibrium systems close to dynamic arrest.

Magnetization relaxation is investigated in a structurally ordered magnetic Co/Cr superlattice. Tailored nanoscale periodicity creates mesoscopic spatial magnetic correlations with slow relaxation dynamics when quenching the system into a nonequilibrium state. Magnetization transients are measured after exposing the heterostructure to a magnetic set field for various waiting times. Scaling analysis reveals an asymptotic power-law behavior in accordance with a full aging scenario. The temperature dependence of the relaxation exponent shows pronounced anomalies at the equilibrium phase transitions of the antiferromagnetic superstructure and the ferromagnetic to paramagnetic transition of the Co layers. The latter leaves only weak fingerprints in the equilibrium magnetic behavior but gives rise to a prominent change in nonequilibrium properties. Our findings suggest scaling analysis of nonequilibrium data as a probe for weak equilibrium phase transitions.

Monte Carlo simulations of the short-time dynamic behavior are reported for
three-dimensional Heisenberg model with long-range correlated disorder at
criticality, in the case corresponding to linear defects. The static and
dynamic critical exponents are determined for systems starting from an ordered
initial state. The obtained values of the exponents are in a good agreement
with results of the field-theoretic description of the critical behavior of
this model in the two-loop approximation.

We investigate the nonequilibrium behaviour of the
d-dimensional Ising model with purely dissipative dynamics during its critical relaxation from a
magnetized initial configuration. The universal scaling forms of the two-time response and
correlation functions of the magnetization are derived within the field-theoretical
approach and the associated scaling functions are computed up to first order in the
-expansion
( = 4 − d). Ageing behaviour is clearly displayed and the associated universal fluctuation–dissipation
ratio tends to for long times. These results are confirmed by Monte Carlo simulations of the
two-dimensional Ising model with Glauber dynamics, from which we find . The crossover to the case of relaxation from a disordered state is discussed and the
crossover function for the fluctuation–dissipation ratio is computed within the Gaussian
approximation.

Recently the series for two renormalization group functions (corresponding to the anomalous dimensions of the fields and ) of the three-dimensional field theory have been extended to next order (seven loops) by Murray and Nickel. We examine the influence of these additional terms on the estimates of critical exponents of the N-vector model, using some new ideas in the context of the Borel summation techniques. The estimates have slightly changed, but remain within the errors of the previous evaluation. Exponents such as (related to the field anomalous dimension), which were poorly determined in the previous evaluation of Le Guillou-Zinn-Justin, have seen their apparent errors significantly decrease. More importantly, perhaps, summation errors are better determined.
The change in exponents affects the recently determined ratios of amplitudes and we report the corresponding new values.
Finally, because an error has been discovered in the last order of the published expansions (order ), we have also re-analysed the determination of exponents from the -expansion.
The conclusion is that the general agreement between -expansion and three-dimensional series has improved with respect to Le Guillou-Zinn-Justin.

This review presents the effective temperature notion as defined from the
deviations from the equilibrium fluctuation-dissipation theorem in out of
equilibrium systems with slow dynamics. The thermodynamic meaning of this
quantity is discussed in detail. Analytic, numeric and experimental
measurements are surveyed. Open issues are mentioned.

We present the experimental observation of the fluctuation-dissipation
theorem (FDT) violation in an assembly of interacting magnetic nanoparticles in
the low temperature superspin glass phase. The magnetic noise is measured with
a two-dimension electron gas Hall probe and compared to the out of phase ac
susceptibility of the same ferrofluid. For "intermediate" aging times of the
order of 1 h, the ratio of the effective temperature $T_{\rm eff}$ to the bath
temperature T grows from 1 to 6.5 when T is lowered from $T_g$ to 0.3 $T_g$,
regardless of the noise frequency. These values are comparable to those
measured in an atomic spin glass as well as those calculated for a Heisenberg
spin glass.

Monte Carlo simulations of the short-time dynamic behavior are reported for three-dimensional weakly site-diluted Ising model with spin concentrations p=0.95 and 0.8 at criticality. In contrast to studies of the critical behavior of the pure systems by the short-time dynamics method, our investigations of site-diluted Ising model have revealed three stages of the dynamic evolution characterizing a crossover phenomenon from the critical behavior typical for the pure systems to behavior determined by the influence of disorder. The static and dynamic critical exponents are determined with the use of the corrections to scaling for systems starting separately from ordered and disordered initial states. The obtained values of the exponents demonstrate a universal behavior of weakly site-diluted Ising model in the critical region. The values of the exponents are compared to results of numerical simulations which have been obtained in various works and, also, with results of the renormalization-group description of this model.

We consider how the Pad'e-Borel, Pad'e-Borel-Leroy, and conformal mapping summation methods for asymptotic series can be used to calculate the dynamical critical exponent for homogeneous and disordered Ising-like systems.

We use simple models (the Ising model in one and two dimensions,
and the spherical model in arbitrary dimension) to put to the
test some recent ideas on the slow dynamics of nonequilibrium
systems. In this review the focus is on the temporal evolution
of two-time quantities and on the violation of the
fluctuation-dissipation theorem, with special emphasis given to
nonequilibrium critical dynamics.

We discuss universal and non-universal critical exponents of a three dimensional Ising system in the presence of weak quenched disorder. Both experimental, computational, and theoretical results are reviewed. Special attention is paid to the results obtained by the field theoretical renormalization group approach. Different renormalization schemes are considered putting emphasis on analysis of divergent series obtained.

The critical behavior of the disordered ferromagnetic Ising model is studied numerically by the Monte Carlo method in a wide range of variation of concentration of nonmagnetic impurity atoms. The temperature dependences of correlation length and magnetic susceptibility are determined for samples with various spin concentrations and various linear sizes. The finite-size scaling technique is used for obtaining scaling functions for these quantities, which exhibit a universal behavior in the critical region; the critical temperatures and static critical exponents are also determined using scaling corrections. On the basis of variation of the scaling functions and values of critical exponents upon a change in the concentration, the conclusion is drawn concerning the existence of two universal classes of the critical behavior of the diluted Ising model with different characteristics for weakly and strongly disordered systems.

The computer simulations of the critical dynamics of the structurally disordered three-dimensional Ising model are performed by the damage spreading method. For the systems with spin densities p = 0.6 and 0.8, we calculate the critical temperature and dynamic critical exponent z characterizing the behavior of relaxation properties near the critical point. The analysis of the results demonstrates the nonuniversality of the critical dynamics in the disordered Ising model. To interpret such dynamics, we introduce the concept of two universal types of critical behavior corresponding to weak and strong disorder, respectively.

It is considered the non-equilibrium critical evolution of three-dimensional pure and site-diluted spin systems described by Ising model from high-temperature initial state with displaying some features, such as ageing and violation of the fluctuation–dissipation theorem. The Monte Carlo simulation results obtained for systems with the spin concentration varying in a wide range are given.

The specific features of a nonequilibrium critical behavior in the three-dimensional structurally disordered Ising model have been studied numerically by the Monte Carlo method. An analysis of the two-time dependence of the autocorrelation function and the dynamic susceptibility for systems with spin concentrations p = 0.8 and 0.6 has revealed the aging effects, which are characterized by a slowing down of the relaxation of the system with an increase in the waiting time, and the violation of the fluctuation-dissipation theorem. The values of the universal limit of fluctuation-dissipation ratio for the considered systems have been obtained using the Monte Carlo method. It has been shown that the presence of structural defects in the system leads to an enhancement of the aging effects and to an increase of the values of the limit of fluctuation-dissipation ratio.

The values of a new universal parameter characterizing a nonequilibrium critical behavior, namely, the fluctuation-dissipation ratio specifying a fundamental relation between the dynamic response function and the correlation function, are calculated for the disordered three-dimensional Ising model. The analysis of the two-time dependence for autocorrelation functions and the ac susceptibility for the systems with spin densities p = 1.0, 0.8, and 0.6 shows the aging effects characterized by the anomalous slowing of relaxation in the system with the growth of the waiting time and the violation of the fluctuation-dissipation theorem. To improve the accuracy of the ac susceptibility calculations, the “thermal bath” technique has been used without introducing the applied magnetic field in the simulation. It has been shown that the structural defects lead to the pronounced enhancement of the aging effects.

A field-theory description of the static and dynamic critical behavior of systems with quenched defects obeying power law correlations ∼|x|-a for large separations x is given. Directly, for three-dimensional systems and for different values of the correlation parameter, 2<~a<~3, a renormalization analysis of the scaling functions in the two-loop approximation is carried out, and the fixed points corresponding to the stability of various types of critical behavior are identified. The obtained results essentially differ from results evaluated by a double ɛ,δ expansion. The static and dynamic critical exponents in the two-loop approximation are calculated with the use of the Padé-Borel summation technique.

As a model for a phase transition in an inhomogeneous system, we consider a system where the local transition temperature varies in space, with a correlation function obeying a power law ∼x-a for large separations x. We extend the Harris criterion for this case, finding that for a<d (where d is the spatial dimension) the disorder is irrelevant if aν-2>0, while if a>d we recover the usual Harris criterion: The disorder is irrelevant if dν-2=-α>0. An m-vector system of this type is studied with the use of a renormalization-group expansion in ε=4-d and δ=4-a. We find a new long-range-disorder fixed point in addition to the short-range-disorder and pure fixed points found previously. The crossover between fixed points is found to follow the extended Harris criterion. The new fixed point has complex eigenvalues, leading to oscillating corrections to scaling, and has a correlation-length exponent ν=2/a. We argue that this new scaling relation is exact and applies more generally than just to the specific model. We show that the extended Harris criterion also applies to percolation with long-range-correlated site or bond-occupation probabilities, so that the scaling law should be obeyed by such systems. Results for the percolation properties of the triangular Ising model are in agreement with these predictions.

A cumulant expansion is used to calculate the transition temperature of Ising models with random-bond defects. For a concentration, x, of missing interactions in the simple-square Ising model the author finds -Tc-1 dTc/dx mod x=0=1.329 compared with the mean-field value of one. If the interactions are independent random variable with a width delta J/J identical to epsilon , the result is -Tc-1 dTc/d epsilon 2 mod epsilon =0=0.312 compared with the mean-field results of zero. An approximation yields the specific heat in the critical regime as C approximately C0/(1+x gamma 2C0), where gamma is a constant and C0 is the unperturbed specific heat at a renormalized temperature. Thus, the specific heat divergence is broadened over a temperature interval Delta T, with Delta T/Tc approximately x(1 alpha )/, where alpha is the critical exponent for the specific heat, and a maximum value of order x-1 is attained. Heuristic arguments show that this smoothing effect occurs if alpha >0.

Static and dynamic critical behaviour is studied for spin systems with quenched impurities that are correlated along an epsilon d-dimensional 'line' and randomly distributed in d- epsilon d dimensions. These impurities make the system anisotropic and modify its critical properties. Critical exponents are calculated at the second order of an expansion in epsilon and epsilon d (with epsilon =4-d). Although renormalisation-group functions depend on the ratio epsilon d/( epsilon + epsilon d), critical exponents are found not to involve this ratio. Pade-like approximants are used to give numerical estimates for the exponents.

We review the progress made in dynamic bulk critical behaviour in equilibrium in the last 25 years since the review of Halperin and Hohenberg. We unify the presentation of the theoretical background by restricting ourselves to the field-theoretic renormalization group method. The main results obtained in the different universality classes are presented. This contains the critical dynamics near the gas–liquid transition in pure fluids (model H), the plait point and consolute point in mixtures (model H'), the superfluid transition in 4He (model F) and 4He–3He mixtures (model F'), the Curie point (model J) and Neel point (model G) in Heisenberg magnets and the superconducting transition. In comparison with experimental results, it became clear that in most cases one has to consider apart from the universal asymptotic critical behaviour also the non-universal effective behaviour. Either because it turned out to be inevitable due to a small dynamical transient exponent inhibiting the system to reach the asymptotics (e.g., at the superfluid transition) or because one is interested in the region further away from the phase transition like in pure fluids and mixtures at their gas–liquid or demixing transition. The calculation of the critical dynamics is adequate in most cases only in two-loop order. We review these results and present the solution to unreasonable features found for some models. Thus, we consider model C where relaxational and diffusive dynamics are coupled and the scaling properties and the limit to a purely relaxational model (model A) have not been understood. In general for models where the order parameter couples to other conserved densities time scale ratios between the kinetic coefficients of the order parameter and the conserved densities play an important role. Their fixed-point values and the approach to the fixed point are changed considerably in two-loop order compared to their values in one-loop order. These considerations are relevant for the explanation of the dynamical critical shape functions of systems such as superfluid helium (model F) and the isotropic antiferromagnet (model G). As far as possible, the comparison of results obtained by the renormalization group theory with numerical simulations has been made.

These lectures give an introduction to Monte Carlo simulations of classical statistical physics systems and their statistical
analysis. After briefly recalling a few elementary properties of phase transitions, the concept of importance sampling Monte
Carlo methods is discussed and illustrated by a few standard local update algorithms (Metropolis, heat-bath, Glauber). Then
emphasis is placed on thorough analyses of the generated data paying special attention to the choice of estimators, autocorrelation
times and statistical error analysis. This leads to the phenomenon of critical slowing down at continuous phase transitions.
For illustration purposes, only the two-dimensional Ising model will be needed. To overcome the slowing-down problem, non-local
cluster algorithms have been developed which will be discussed next. Then the general tool of reweighting techniques will
be explained. This paves the way to introduce simulated and parallel tempering methods which are very useful for simulations
of complex, possibly disordered systems. Finally, also the important alternative approach using multicanonical ensembles is
briefly outlined.

We study the critical relaxation properties of Model A (purely dissipative relaxation) starting from a macroscopically prepared initial state characterised by non-equilibrium values for order parameter and correlations. Using a renormalisation group approach we observe that even (macroscopically)early stages of the relaxation process display universal behaviour governed by a new, independent “initial slip” exponent. For large times, the system crosses over to the well-known long-time relaxation behaviour.
The new exponent is calculated toO(ε2) in ε=4−d, whered is the spatial dimension of the system. The initial slip scaling form of general correlation and response functions as well as the order parameter is derived, exploiting a short-time operator expansion. The leading scaling behaviour is determined by initial states with sharp values of the order parameter. Non-vanishing correlations generate corrections to scaling.

The evolution of the magnetic properties of Fe/Cr superlattices with a decrease in the nominal thickness of the iron layers
down to atomic dimensions at which these layers are not continuous has been analyzed. Investigations have been carried out
with multilayer samples with Fe-layer thicknesses in a range of 2–6 Å and Cr-layer thicknesses of 10 and 20 Å. It has been
found that the system with various Fe-layer thicknesses and at various temperatures exhibits various magnetic phases—superparamagnetic,
magnetically ordered, and nonergodic—characterized by the dependence of the magnetization of the sample on its magnetic prehistory.
It has been shown that the observed nonergodic phase has the properties of a spin glass. A qualitative phase diagram of the
magnetic states of the system has been obtained.

An introductory review of the central ideas in the modern theory of
dynamic critical phenomena is followed by a more detailed account
of recent developments in the field. The concepts of the conventional
theory, mode-coupling, scaling, universality, and the renormalization
group are introduced and are illustrated in the context of a simple
exampleâthe phase separation of a symmetric binary fluid. The renormalization
group is then developed in some detail, and applied to a variety
of systems. The main dynamic universality classes are identified
and characterized. It is found that the mode-coupling and renormalization
group theories successfully explain available experimental data at
the critical point of pure fluids, and binary mixtures, and at many
magnetic phase transitions, but that a number of discrepancies exist
with data at the superfluid transition of 4He.

Using recently developed histogram techniques and an ultrafast multispin coding simulation algorithm, we have investigated the critical behavior of the d=3 simple-cubic Ising model. We have studied lattice sizes ranging from L=8 to 96 using between 3×106 and 12×106 Monte Carlo steps (complete lattice updates). By accurately measuring the finite-size behavior of several different thermodynamic quantities, we are able to determine the critical properties with a precision comparable to that obtained with Monte Carlo renormalization-group and sophisticated series-expansion techniques. The best estimate of the inverse critical temperature from our analysis is Kc=0.221 659 5±0.000 002 6. The advantages of the histogram technique are discussed, as are the potential problems that can arise at this level of resolution.

The theory of phase ordering dynamics -- the growth of order through domain coarsening when a system is quenched from the homogeneous phase into a broken-symmetry phase -- is reviewed, with the emphasis on recent developments. Interest will focus on the scaling regime that develops at long times after the quench. How can one determine the growth laws that describe the time-dependence of characteristic length scales, and what can be said about the form of the associated scaling functions? Particular attention will be paid to systems described by more complicated order parameters than the simple scalars usually considered, e.g. vector and tensor fields. The latter are needed, for example, to describe phase ordering in nematic liquid crystals, on which there have been a number of recent experiments. The study of topological defects (domain walls, vortices, strings, monopoles) provides a unifying framework for discussing coarsening in these different systems.

Description of nonequilibrium critical behaviour in disordered systems by the theoretical methods

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Ageing properties of critical systems

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Slow relaxation and nonequilibrium dynamics in condensed matter, Les Houches, Ecole d'Ete de Physique Theorique

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Violation of the fluctuation-dissipation theorem in glassy systems: basic notions and the numerical evidence

- A Crisanti
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A. Crisanti and F. Ritort, Violation of the fluctuation-dissipation theorem in glassy systems: basic
notions and the numerical evidence, 2003 J. Phys. A: Math. Gen. 36 R181.

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V. V. Prudnikov, P. V. Prudnikov, A. N. Vakilov, A. S. Krinitsyn, and M. V. Rychkov, in
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by R. R. Nazirov (KDU publ., Moscow, 2009), Vol 1, p. 240-263 (in Russian).