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``Information and contact geometric description of expectation variables exactly derived from master equations'',
Physica Scripta, XX YY ( arXiv : 1805.10592 )
Published: Sep/2019
abstract : In this paper a class of dynamical systems describing expectation variables exactly derived from continuous-time master equations is introduced and studied from the viewpoint of differential geometry, where such master equations consist of a set of appropriately chosen Markov kernels. To geometrize such dynamical systems for expectation variables, information geometry is used for expressing equilibrium states, and contact geometry is used for nonequilibrium states. Here time-developments of the expectation variables are identified with contact Hamiltonian vector fields on a contact manifold. Also, it is shown that the convergence rate of this dynamical system is exponential. Duality emphasized in information geometry is also addressed throughout.
``Expectation variables on a para-contact metric manifold exactly derived from master equations'',
Geometric Science of Information pp 239-247, GSI 19 ( arXiv:1905.05939 )
Published: Aug/2019
abstract : Based on information and para-contact metric geometries, in this paper a class of dynamical systems is formulated for describing time-development of expectation variables. Here such systems for expectation variables are exactly derived from continuous-time master equations describing nonequilibrium processes.
``Aspects of Quantum Energy and Stress in Inhomogeneous Unbounded Dielectric Continua'',
Reviews in Mathematical Physics,31,1950002 [50pages], (2018) (arXiv:1402.6582)
Published: Oct/2018
abstract : This article addresses a number of issues associated with the problem of calculating contributions from the electromagnetic quantum induced energy and stress in a stationary material with an inhomogeneous polarizability. After briefly reviewing the conventional approaches developed by Lifshitz el al and more recent attempts by others, we emphasize the need to accommodate the effects due to the classical constitutive properties of the material in any experimental attempt to detect such contributions. Attention is then concentrated on a particular system composed of an ENZ-type (epsilon-near-zero) meta-material, chosen to have an anisotropic and inhomogeneous permittivity confined in an infinitely long perfectly conducting open waveguide. This permits us to deduce from the source-free Maxwell's equations a complete set of harmonic electromagnetic evanescent eigen-modes and eigen-frequencies. Since these solutions prohibit the existence of asymptotic scattering states in the guide an alternative regularization scheme, based on the Euler-Maclaurin formula, enables us to prescribe precise criteria for the extraction of finite quantum expectation values from regularized mode sums together with error bounds on these values. This scheme is used to derive analytic results for regularized energy densities in the guide. The criteria are exploited to construct a numerical scheme that is bench-marked by comparing its output with the analytic results derived from the special properties of the inhomogeneous ENZ medium.
``Hessian-information geometric formulation of Hamiltonian systems and generalized Toda’s dual transform'',
J Phys A: Math.Theor. 51 324001 [23pages], (2018) ( arXiv:1801.04759 )
Published : Jul. /2018 ( online )
abstract : In this paper a class of classical Hamiltonian systems is geometrically formulated. This class is such that a Hamiltonian can be written as the sum of a kinetic energy function and a potential energy function. In addition, these energy functions are assumed strictly convex. For this class of Hamiltonian systems Hessian and information geometric formulation is given. With this formulation, a generalized Toda’s dual transform is proposed, where his original transform was used in deriving his integrable lattice system. Then a relation between the generalized Toda’s dual transform and the Legendre transform of a class of potential energy functions is shown. As an extension of this formulation, dissipation-less electric circuit models are also discussed in the geometric viewpoint above.
``Maps on statistical manifolds exactly reduced from the Perron-Frobenius equations for solvable chaotic maps'',
J. Math. Phys. 59, 032701 [13 pages], (2018) ( arXiv:1707.03607 )
Published : Mar. /2018 ( online )
abstract : Maps on a parameter space for expressing distribution functions are exactly derived from the Perron- Frobenius equations for a generalized Boole transform family. Here the generalized Boole transform family is a one-parameter family of maps where it is defined on a subset of the real line and its probability distribution function is the Cauchy distribution with some parameters. With this reduction some relations between the statistical picture and the orbital one are shown. From the viewpoint of information geometry, the parameter space can be identified with a statistical manifold, and then it is shown that the derived maps can be characterized. Also, with an induced symplectic structure from a statistical structure, symplectic and information geometric aspects of the derived maps are discussed.
``Contact geometric description of distributed-parameter port-Hamiltonian systems with respect to Stokes-Dirac structures and its information geometry'',
( arXiv:1702.06369 )
Published : XX /201X ( online )
abstract : This paper studies distributed-parameter systems on Riemannian manifolds with respect to Stokes-Dirac structures in a language of contact geometry with fiber bundles. For the class where energy functionals are quadratic, it is shown that distributed-parameter port-Hamiltonian systems with respect to Stokes-Dirac structures on one, two, and three dimensional Riemannian manifolds are written in terms of contact Hamiltonian vector fields on bundles. Their fiber spaces are contact manifolds and base spaces are Riemannian manifolds. In addition, for a class of distributed-parameter port-Hamiltonian systems, information geometry induced from contact manifolds and convex energy functionals is introduced and briefly discussed.
``Maxwell's equations in media as a contact Hamiltonian vector field and its information geometry -- An approach with a bundle whose fiber is a contact manifold --'',
( arXiv:1702.05746 )
Published : XX /201X ( online )
abstract : It is shown that Maxwell's equations in media without source can be written as a contact Hamiltonian vector field restricted to a Legendre submanifold, where this submanifold is in a fiber space of a bundle and is generated by either electromagnetic energy functional or co-energy functional. Then, it turns out that Legendre duality for this system gives the induction oriented formulation of Maxwell's equations and field intensity oriented one. Also, information geometry of the Maxwell fields is introduced and discussed.
``Contact geometric descriptions of vector fields on dually flat spaces and their applications in electric circuit models and nonequilibrium statistical mechanics'',
J. Math. Phys. 57, 102702[40pages], (2016). ( arXiv:1512.00950 )
Published : Oct /2016 ( online )
abstract : Contact geometry has been applied to various mathematical sciences, and it has been proposed that a contact manifold and a strictly convex function induce a dually flat space that is used in information geometry. Here, such a dually flat space is related to a Legendre submanifold in a contact manifold. In this paper contact geometric descriptions of vector fields on dually flat spaces are proposed on the basis of the theory of contact Hamiltonian vector fields. Based on these descriptions, two ways of lifting vector fields on Legendre submanifolds to contact manifolds are given. For some classes of these lifted vector fields, invariant measures in contact manifolds and stability analysis around Legendre submanifolds are explicitly given. Throughout this paper, Legendre duality is explicitly stated. In addition, to show how to apply these general methodologies to applied mathematical disciplines, electric circuit models and some examples taken from nonequilibrium statistical mechanics are analyzed.
``The dynamics of compact laser pulses'',
J. Phys.A: Math. Theor. 49, 265203 (11pages), (2016) ( arXiv:1501.07753 )
Published : May/2016 ( online )
abstract : We discuss the use of a class of exact finite energy solutions to the vacuum source-free Maxwell equations as models for multi- and single cycle laser pulses in classical interaction with relativistic charged point particles. These compact solutions are classified in terms of their chiral content and their influence on particular charge configurations in space. The results of such classical interactions motivate a phenomenological quantum description of a propagating laser pulse in a medium in terms of an effective quantum Hamiltonian.
Shin-itiro GOTO, Robin W. TUCKER and Timothy J. WALTON,
``Classical dynamics of free electromagnetic laser pulses''
Nuclear Inst. and Methods in Physics Research, B ( arXiv:1508.05191 )
Published: February/2015
abstract : We discuss a class of exact finite energy solutions to the vacuum source-free Maxwell field equations as models for multi- and single cycle laser pulses in classical interaction with relativistic charged test particles. These solutions are classified in terms of their chiral content based on their influence on particular charge configurations in space. Such solutions offer a computationally efficient parameterization of compact laser pulses used in laser-matter simulations and provide a potential means for experimentally bounding the fundamental length scale in the generalized electrodynamics of Bopp, Landé and Podolsky.
``Legendre submanifolds in contact manifolds as attractors and geometric nonequilibrium thermodynamics''
J. Math. Phys. 56, 073301 (30pages) (2015), (arXiv:1412.5780)
Published: Jul/2015
abstract : It has been proposed that equilibrium thermodynamics is described on Legendre submanifolds in contact geometry. It is shown in this paper that Legendre submanifolds embedded in a contact manifold can be expressed as attractors in phase space for a certain class of contact Hamiltonian vector fields. By giving a physical interpretation that points outside the Legendre submanifold can represent nonequilibrium states of thermodynamic variables, in addition to that points of a given Legendre submanifold can represent equilibrium states of the variables, this class of contact Hamiltonian vector fields is physically interpreted as a class of relaxation processes, in which thermodynamic variables achieve an equilibrium state from a nonequilibrium state through a time evolution, a typical nonequilibrium phenomenon. Geometric properties of such vector fields on contact manifolds are characterized after introducing a metric tensor field on a contact manifold. It is also shown that a contact manifold and a strictly convex function induce a lower dimensional dually flat space used in information geometry where a geometrization of equilibrium statistical mechanics is constructed. Legendre duality on contact manifolds is explicitly stated throughout. cited by :
``On the computation of Casimir stresses in open media and Lifshitz theory''
J. Phys. A: Math. Theor. 46, 405301 (32pp) (2013) (arXiv:1308.2884)
Corrigendum: (2013 J. Phys. A: Math. Theor. 46 405301)
abstract : A classification of the electromagnetic modes on open and closed spatial domains containing media with piecewise homogeneous permittivities is used to facilitate the derivation of quantum induced Casimir stresses in dielectrics. By directly exploiting the complex analytic properties of solutions of the macroscopic Maxwell equations for open systems it is shown how regular expressions for such stresses can be expressed in terms of double integrals involving either real or pure imaginary frequencies associated with harmonic modes in conformity with the Lifshitz theory for separated planar dielectric half-spaces. The derivation is self-contained without recourse to the Krein formula for a density of states or mode regularization and offers a more direct approach to other open systems.
``Numerical Regularization of Electromagnetic Quantum Fluctuations in Inhomogeneous Dielectric Media''
Phys. Rev. A85, 034103 (2012) (arXiv:1201.1160)
abstract : Electromagnetic Casimir stresses are of relevance to many technologies based on mesoscopic devices such as microelectromechanical systems embedded in dielectric media, Casimir induced friction in nanomachinery, microfluidics, and molecular electronics. Computation of such stresses based on cavity QED generally requires numerical analysis based on a regularization process. The scheme described below has the potential for wide applicability to systems involving realistic inhomogeneous media. From a knowledge of the spectrum of the stationary modes of the electromagnetic field the scheme is illustrated by estimating numerically the Casimir stress on opposite faces of a pair of perfectly conducting planes separated by a vacuum and the change in this result when the region between the plates is filled with an incompressible inhomogeneous nondispersive dielectric. cited by :
Shin-itiro GOTO, Robin W. TUCKER and Timothy J. WALTON,
``Quantum Electromagnetic Fluctuations in Inhomogeneous Dielectric Media''
Proc. SPIE 8072, 80720O (2011) (arXiv:1107.1521)
abstract : A new mathematical and computational technique for calculating quantum vacuum expectation values of energy and momentum densities associated with electromagnetic fields in bounded domains containing inhomogeneous media is discussed. This technique is illustrated by calculating the mode contributions to the difference in the vacuum force expectation between opposite ends of an inhomogeneous dielectric non-dispersive medium confined to a perfectly conducting rigid box.
``The Electrodynamics of Inhomogeneous Rotating Media and the Abraham and Minkowski Tensors I : General Theory'',
Proc. Roy. Soc. A, Vol. 467, No. 2125, pp59--78,(2011). (arXiv:1003.1637)
See also Data Supplement
abstract : This is paper I of a series of two papers, offering a self-contained analysis of the role of electromagnetic stress-energy-momentum tensors in the classical description of continuous polarizable perfectly insulating media. While acknowledging the primary role played by the total stress-energy-momentum tensor on spacetime we argue that it is meaningful and useful in the context of covariant constitutive theory to assign preferred status to particular parts of this total tensor, when defined with respect to a particular splitting. The relevance of tensors, associated with the electromagnetic fields that appear in Maxwell's equations for polarizable media, to the forces and torques that they induce has been a matter of some debate since Minkowski, Einstein and Laub, and Abraham considered these issues over a century ago. The notion of a force density that arises from the divergence of these tensors is strictly defined relative to some inertial property of the medium. Consistency with the laws of Newtonian continuum mechanics demands that the total force density on any element of a medium be proportional to the local linear acceleration field of that element in an inertial frame and must also arise as part of the divergence of the total stress-energy-momentum tensor. The fact that, unlike the tensor proposed by Minkowski, the divergence of the Abraham tensor depends explicitly on the local acceleration field of the medium as well as the electromagnetic field sets it apart from many other terms in the total stress-energy-momentum tensor for a medium. In this paper, we explore how electromagnetic forces or torques on moving media can be defined covariantly in terms of a particular 3-form on those spacetimes that exhibit particular Killing symmetries. It is shown how the drive-forms associated with translational Killing vector fields lead to explicit expressions for the electromagnetic force densities in stationary media subject to the Minkowski constitutive relations and these are compared with other models involving polarizable media in electromagnetic fields that have been considered in the recent literature.
``The Electrodynamics of Inhomogeneous Rotating Media and the Abraham and Minkowski Tensors II : Applications'',
Proc. Roy. Soc. A Vol. 467, No. 2125, pp79--98,(2011)., (arXiv:1003.1642)
See also Data Supplement
abstract : Applications of the covariant theory of drive-forms are considered for a class of perfectly insulating media. The distinction between the notions of `classical photons' in homogeneous bounded and unbounded stationary media and in stationary unbounded magneto-electric media is pointed out in the context of the Abraham, Minkowski and symmetrized Minkowski electromagnetic stress-energy-momentum tensors. Such notions have led to intense debate about the role of these (and other) tensors in describing electromagnetic interactions in moving media. In order to address some of these issues for material subject to the Minkowski constitutive relations, the propagation of harmonic waves through homogeneous and inhomogeneous, isotropic plane-faced slabs at rest is first considered. To motivate the subsequent analysis on accelerating media, two classes of electromagnetic modes that solve Maxwell's equations for uniformly rotating homogeneous polarizable media are enumerated. Finally it is shown that, under the influence of an incident monochromatic, circularly polarized, plane electromagnetic wave, the Abraham and symmetrized Minkowski tensors induce different time-averaged torques on a uniformly rotating materially inhomogeneous dielectric cylinder. We suggest that this observation may offer new avenues to explore experimentally the covariant electrodynamics of more general accelerating media.
``Amplitude equations for a linear wave equation in a weakly curved pipe''
J. Phys. A:Math. Theor. 42, 445205 (2009) (14pp) (arXiv:0910.0549)
abstract : We study boundary effects in a linear wave equation with Dirichlet-type conditions in a weakly curved pipe. The coordinates in our pipe are prescribed by a given small curvature with finite range, with the pipe's cross section being circular. Based on the straight pipe case, a perturbative analysis by which the boundary value conditions are exactly satisfied is employed. As such an analysis, we decompose the wave equation into a set of ordinary differential equations perturbatively. We show the conditions when secular terms due to the curved boundary appear in the naive peturbative analysis. In eliminating such a secularity with a singular perturbation method, we derive amplitude equations and show that the eigenfrequencies in time are shifted due to the curved boundary.
cited by :
``Synchronization of chaotic semiconductor lasers by optical injection with random phase modulation'',
Opt. Quant. Electron 41, Number 3, pp137--149 (2009).
abstract : It is shown that identical synchronization of two chaotic semiconductor lasers can be achieved by injection of a common optical signal with randomly varying phase. An optical signal with randomly modulated phase is injected into two semiconductor lasers which have chaotic oscillations due to optical feedback. Strong correlation between complex intensity oscillations of the two lasers is observed even though the intensity of the common injection signal is constant. Characteristic properties of this type of synchronization are shown, in particular, the dependence of the synchronization threshold on the injection strength and the rate of phase modulation, and the dependence of the intensity correlation on the difference in phase of optical feedback.
cited by :
``Electromagnetic Fields Produced by Moving Sources in a Curved Beam Pipe''
Journal of Mathematical Physics 50, 063510 (2009), 45 pages
abstract : A new geometrical perturbation scheme is developed in order to calculate the electromagnetic fields produced by charged sources in prescribed motion moving in a non-straight perfectly conducting beam pipe. The pipe is regarded as a perturbed infinitely long hollow right-circular cylinder. The perturbation maintains the pipe's circular cross-section while deforming its axis into a planar space-curve with, in general, non-constant curvature. Various charged source models are considered including a charged bunch and an off-axis point particle. In the ultra-relativistic limit this permits a calculation of the longitudinal wake potential in terms of powers of the product of the pipe radius and the arbitrarily varying curvature of the axial space-curve. Analytic expressions to leading order are presented for beam pipes with piecewise defined constant curvature modelling pipes with straight segments linked by circular arcs of finite length. The language of differential forms is used throughout and to illustrate the power of this formalism a pedagogical introduction is developed by deriving the theory ab-initio from Maxwell's equations expressed intrinsically as a differential system on (Minkowski) spacetime.
cited by :
``Conditional Lyapunov Exponent Depending on Spectrum of Input Noise in Common-Noise-Induced Synchronization''
IEICE A Trans. on Fundamentals., 91-A No.9 pp2535-2539 (2008)
abstract : We study the synchronization of dynamical systems induced by common additional colored noise. In particular, we consider the special case that the external input noise is generated by a linear second-order differential equation forced by Gaussian white noise. So the frequency spectrum of this noise is not constant. In the case that noise-free dynamics is chaotic, we find examples where the synchronization is enhanced when the peak of the input noise is close to the peak of noise-free dynamics in frequency space. In the case that noise-free dynamics is non-chaotic, we do not observe this phenomenon.
``Numerical analysis on chaos synchronization in semiconductor lasers subject to a common drive signal''
(IEEJ Transactions on Electronics, Information and Systems, Vol. 128 No.5 pp768--774, 2008.)
abstract : We numerically observe chaos synchronization of two semiconductor lasers commonly driven by a chaotic semiconductor laser subject to optical feedback. We observe strongly correlated chaos synchronization between the two response lasers even when the correlation between the drive and response laser is low. We show that the cross correlation between the response is larger than that between drive and response over a wide parameter region.
``Renormalization Reductions for Systems with Delay''
Prog. Theor. Phys. 118. No.2 (2007), pp211--227.
abstract : The renormalization method, which is a type of perturbation method, is extended to a tool to study weakly nonlinear time-delay systems. For systems with order-one delay, we show that the renormalization method leads to reduced systems without delay. For systems with order-one delay and long delay, we propose an extended renormalization method which leads to reduced systems with delay. In some examples, the validity of our perturbative results is confirmed analytically and numerically. We also compare our reduced equations with reduced equations obtained with another perturbation method.
cited by :
``From an Unstable Periodic Orbit to the Lyapunov Exponent and a Macroscopic Variable --Periodic orbit dependencies--''
Prog. Theor. Phys. 118. No.1 (2007), pp25--33.
abstract : We study the problem of determining which periodic orbits in phase space can predict the largest Lyapunov exponent and the expectation values of macroscopic variables in a Hamiltonian system with many degrees of freedom. We also attempt to elucidate the manner in which these orbits yield such predictions. The model which we use in this paper is a discrete nonlinear Schr""odinger equation. Using a method based on the modulational estimate of a periodic orbit, we predict the largest Lyapunov exponent and the expectation value of a macroscopic variable. We show that (i) the predicted largest Lyapunov exponent generally depends on the periodic orbit which we employ, and (ii) the predicted expectation value of the macroscopic variable does not depend on the periodic orbit, at least in a high energy regime. In addition, the physical meanings of these dependencies are considered.
cited by :
``Common-chaotic-signal induced synchronization in semiconductor lasers''
Optics Express, vol.15. No. 7, (2007),pp3974--3980.
abstract : We experimentally and numerically observe synchronization of two semiconductor lasers commonly driven by a chotic semiconductor laser subject to optical feedback. Under condition that the relaxation oscillation frequency is matched between the two response lasers, but mismatched between the drive and the two response lasers, we show that it is possible to observe strongly correlated synchronization between the two response lasers even when the correlation between the drive and response lasers is low. We also show that the cross correlation between the two responses is larger than that between drive and responses over a wide parameter region.
cited by :
``Analytical Expression for Low-Dimensional Resonance Islands in a 4-dimensional Symplectic Map''
Prog. Theor. Phys. vol. 115. No. 2. (2006), pp251--258.
abstract : We study 2- and 4-dimensional nearly integrable symplectic maps using a singular perturbation method. Resonance island structures in the 2- and 4-dimensional maps are obtained. The validity of these perturbative results are confirmed numerically.
cited by :
``Non-universal finite size effects with universal infinite-size free energy''
Physica A 354 (2005) pp312-322.
abstract : We study finite size effects in a family of systems in which a parameter controls interaction-range. In the long-range regime where the infinite-size free energy is universal and indicates a second-order phase transition, we show that the finite size effects are not universal but depend on the interaction-range. The finite size effects are observed through discrepancies between time-averages of macroscopic variables in Hamiltonian dynamics and canonical averages of ones with infinite degrees of freedom. For the subcritical regime, it is numerically shown that convergences towards the canonical averages become slower as the interaction-range becomes shorter. For the supercritical regime, the relation to a pair of the discrepancies is theoretically predicted and numerically confirmed.
See also Prog. Theor. Phys. Supp. Vol. 162, (2006),pp. 97--103., a proceeding of a conference.
cited by :
``Renormalization Analysis of Resonance Structure in a 2-D Symplectic Map'',
Prog. Theor. Phys. vol. 111. No. 4. (2004), pp463--474.
abstract : A symplecticity-preserving RG analysis is carried out to study the resonance structure near an elliptic fixed point of a prototype symplectic map in two dimensions. Through analysis of fixed points of a reduced RG map, the topology of the resonance structure, such as a chain of resonant islands, can be determined analytically. The application of this analysis to the H�non map is also presented.
cited by :
``Liouville operator approach to symplecticity-preserving renormalization group method'',
Physica D 194, No. 4.(2004), pp175--186.
abstract : We present a method to construct symplecticity-preserving renormalization group maps by using the Liouville operator. The resultant RG maps accurately reproduce the long-time behavior of the original symplectic maps even when a resonant island chain appears.
cited by :
``Random Wandering around Homoclinic-Like Manifolds in a Symplectic Map Chain'',
Prog. Theor. Phys. vol. 107. No. 4. (2002), pp. 637--654.
abstract : We present a method to construct a symplecticity preserving renormalization group map of a chain of weakly nonlinear symplectic maps and obtain a general reduced symplectic map describing its long-time behavior. It is found that the modulational instability in the reduced map triggers random wandering of orbits around some homoclinic-like manifolds. This behavior is understood as Bernoulli shifts.
cited by :
``Regularized Renormalization Group Reduction of Symplectic Maps'',
J. Phys. Soc. Jpn. vol. 70, No. 1. (2001), pp. 49--54.
abstract : By means of the perturbative renormalization group method, we study a long-time behaviour of some symplectic discrete maps near elliptic and hyperbolic fixed points. It is shown that a naive renormalization group (RG) map breaks the symplectic symmetry and fails to describe a long-time behaviour. In order to preserve the symplectic symmetry, we present a regularization procedure, which gives a regularized symplectic RG map describing an approximate long-time behaviour successfully.
cited by :
``Asymptotic Expansions of Unstable and Stable Manifolds in Time-Discrete Systems'',
Prog. Theor. Phys. vol. 105, No. 1. (2001), pp. 99--107.
abstract : By means of an updated renormalization method, we construct asymptotic expansions for unstable manifolds of hyperbolic fixed points in the double-well map and the dissipative H�non map, both of which exhibit strong homoclinic chaos. In terms of the asymptotic expansion, a simple formulation is presented to give the first homoclinic point in the double-well map. Even a truncated expansion of the unstable manifold is shown to reproduce the well-known many-leaved (fractal) structure of the strange attractor in the H�non map.
cited by :
``Dynamics near Resonance Junctions in Hamiltonian Systems'',
Prog. Theor. Phys. vol. 102, No.5. (1999), pp. 953--963.
abstract : An approximate Poincar� map near equally strong multiple resonances is reduced by means of the method of averaging. Near the resonance junction of three degrees of freedom, we find that some homoclinic orbits (“whiskers”) in single resonance lines survive and form nearly periodic orbits, each of which looks like a pair of homoclinic orbits.
cited by :
``Lie-Group Approach to Perturbative Renormalization Group Method'',
Prog. Theor. Phys. vol.102, No.3. (1999),pp. 471--497.
abstract : A new Lie-group approach to the perturbative renormalization group (RG) method is developed to obtain asymptotic solutions of both autonomous and non-autonomous ordinary differential equations. Reduction of some partial differential equations to typical RG equations is also achieved with this approach, and a simple recipe for providing RG equations is presented.
cited by :