The coagulation–fragmentation equation describes the concentration fi(t) of particles of size i ∈ N/{0} at time t ⩾ 0 in a spatially homogeneous infinite system of particles subjected to coalescence and break–up. We show that when the rate of fragmentation is sufficiently stronger than that of coalescence, (fi(t))i ∈ N/{0} tends to a unique equilibrium as t tends to infinity. Although we suppose that the initial datum is sufficiently small, we do not assume a detailed balance (or reversibility) condition. The rate of convergence we obtain is, furthermore, exponential.
{"title":"Exponential trend to equilibrium for discrete coagulation equations with strong fragmentation and without a balance condition","authors":"N. Fournier, S. Mischler","doi":"10.1098/rspa.2004.1294","DOIUrl":"https://doi.org/10.1098/rspa.2004.1294","url":null,"abstract":"The coagulation–fragmentation equation describes the concentration fi(t) of particles of size i ∈ N/{0} at time t ⩾ 0 in a spatially homogeneous infinite system of particles subjected to coalescence and break–up. We show that when the rate of fragmentation is sufficiently stronger than that of coalescence, (fi(t))i ∈ N/{0} tends to a unique equilibrium as t tends to infinity. Although we suppose that the initial datum is sufficiently small, we do not assume a detailed balance (or reversibility) condition. The rate of convergence we obtain is, furthermore, exponential.","PeriodicalId":20722,"journal":{"name":"Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2004-09-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79650152","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
A random fuzzy variable is a function from a possibility space to the set of random variables. In this paper, some mathematical properties of a random fuzzy variable are studied. First, some properties of chance measure of random fuzzy events are given. Then, several continuity theorems for random fuzzy sequences are investigated. Furthermore, the concept of chance distribution for a random fuzzy variable is presented, and some properties of chance distribution are obtained.
{"title":"Continuity theorems and chance distribution of random fuzzy variables","authors":"Yuanguo Zhu, Baoding Liu","doi":"10.1098/rspa.2004.1280","DOIUrl":"https://doi.org/10.1098/rspa.2004.1280","url":null,"abstract":"A random fuzzy variable is a function from a possibility space to the set of random variables. In this paper, some mathematical properties of a random fuzzy variable are studied. First, some properties of chance measure of random fuzzy events are given. Then, several continuity theorems for random fuzzy sequences are investigated. Furthermore, the concept of chance distribution for a random fuzzy variable is presented, and some properties of chance distribution are obtained.","PeriodicalId":20722,"journal":{"name":"Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2004-09-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72989323","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
For several decades the propagation characteristics of acoustic pulses (attenuation and sound speed) have been inverted in attempts to measure the size distributions of gas bubbles in liquids. While this has biomedical and industrial applications, most notably it has been attempted in the ocean for defence and environmental purposes, where the bubbles are predominantly generated by breaking waves. Such inversions have required assumptions, and the state–of–the–art technique still assumes that the bubbles undergo linear, steady–state monochromatic pulsations in the free field, without interacting. The measurements always violate, to a greater or lesser extent, these assumptions. The errors incurred by the use of such assumptions have been difficult to quantify, but are expected to be most severe underneath breakers in the surf zone, where the void fraction is greatest. Very few measurements have been made in this important region of the ocean. This paper provides a method by which attenuation can be predicted through clouds of bubbles which need not be homogeneous, nor restricted to linear steady–state monochromatic pulsations. To allow inversion of measured surf zone attenuations to estimate bubble populations with current computational facilities, this model is simplified such that the bubble cloud is assumed to be homogeneous and the bubbles oscillating in steady state (although still nonlinearly). The uses of the new methods for assessing the errors introduced in using state–of–the–art inversions are discussed, as are their implications for oceanographic and industrial nonlinear bubble counters, for biomedical contrast agents, and for sonar target detection in the surf zone.
{"title":"Propagation through nonlinear time-dependent bubble clouds and the estimation of bubble populations from measured acoustic characteristics","authors":"T. Leighton, S. D. Meers, P. White","doi":"10.1098/rspa.2004.1298","DOIUrl":"https://doi.org/10.1098/rspa.2004.1298","url":null,"abstract":"For several decades the propagation characteristics of acoustic pulses (attenuation and sound speed) have been inverted in attempts to measure the size distributions of gas bubbles in liquids. While this has biomedical and industrial applications, most notably it has been attempted in the ocean for defence and environmental purposes, where the bubbles are predominantly generated by breaking waves. Such inversions have required assumptions, and the state–of–the–art technique still assumes that the bubbles undergo linear, steady–state monochromatic pulsations in the free field, without interacting. The measurements always violate, to a greater or lesser extent, these assumptions. The errors incurred by the use of such assumptions have been difficult to quantify, but are expected to be most severe underneath breakers in the surf zone, where the void fraction is greatest. Very few measurements have been made in this important region of the ocean. This paper provides a method by which attenuation can be predicted through clouds of bubbles which need not be homogeneous, nor restricted to linear steady–state monochromatic pulsations. To allow inversion of measured surf zone attenuations to estimate bubble populations with current computational facilities, this model is simplified such that the bubble cloud is assumed to be homogeneous and the bubbles oscillating in steady state (although still nonlinearly). The uses of the new methods for assessing the errors introduced in using state–of–the–art inversions are discussed, as are their implications for oceanographic and industrial nonlinear bubble counters, for biomedical contrast agents, and for sonar target detection in the surf zone.","PeriodicalId":20722,"journal":{"name":"Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2004-09-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80399969","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
An asymptotic formula is found that links the behaviour of the Jacobi polynomial Pnα,β)(z) in the interval of orthogonality [–1,1] with that outside the interval. The two infinite series involved in this formula are shown to be exponentially improved asymptotic expansions. The method used in this paper can also be adopted in other cases of orthogonal polynomials such as Hermite and Laguerre.
{"title":"Uniform asymptotic expansion of the Jacobi polynomials in a complex domain","authors":"R. Wong, Yuqiu Zhao","doi":"10.1098/rspa.2004.1296","DOIUrl":"https://doi.org/10.1098/rspa.2004.1296","url":null,"abstract":"An asymptotic formula is found that links the behaviour of the Jacobi polynomial Pnα,β)(z) in the interval of orthogonality [–1,1] with that outside the interval. The two infinite series involved in this formula are shown to be exponentially improved asymptotic expansions. The method used in this paper can also be adopted in other cases of orthogonal polynomials such as Hermite and Laguerre.","PeriodicalId":20722,"journal":{"name":"Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2004-09-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81818628","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
For a nonlinear composite, a bound on its effective energy density does not induce a corresponding bound on its constitutive relation, because differentiating a bound on a function does not automatically bound its derivative. In this work, a method introduced by Milton and Serkov for bounding directly the constitutive relation is refined by employing a linear comparison material, in a way similar to that employed by the present authors to obtain bounds of ‘Hashin–Shtrikma’ type for the effective energy of a nonlinear composite. The original Milton–Serkov approach produces bounds with a close relationship to the classical energy bounds of Voigt and Reuss type. The bounds produced in the present implementation are closely related to bounds of Hashin–Shtrikman type for the composite. Indeed, in the case of a linear composite, the present method delivers the Hashin–Shtrikman bounds in their generalized form, valid for any two–point statistics. It is demonstrated for nonlinear examples that the approximate constitutive relation that is obtained by differentiating the energy bound can be on the boundary of the bounding set for the exact constitutive relation, but a simple counterexample is presented to show that this is not always the case.
{"title":"Bounds for the effective constitutive relation of a nonlinear composite","authors":"D. Talbot, John R. Willis","doi":"10.1098/rspa.2004.1309","DOIUrl":"https://doi.org/10.1098/rspa.2004.1309","url":null,"abstract":"For a nonlinear composite, a bound on its effective energy density does not induce a corresponding bound on its constitutive relation, because differentiating a bound on a function does not automatically bound its derivative. In this work, a method introduced by Milton and Serkov for bounding directly the constitutive relation is refined by employing a linear comparison material, in a way similar to that employed by the present authors to obtain bounds of ‘Hashin–Shtrikma’ type for the effective energy of a nonlinear composite. The original Milton–Serkov approach produces bounds with a close relationship to the classical energy bounds of Voigt and Reuss type. The bounds produced in the present implementation are closely related to bounds of Hashin–Shtrikman type for the composite. Indeed, in the case of a linear composite, the present method delivers the Hashin–Shtrikman bounds in their generalized form, valid for any two–point statistics. It is demonstrated for nonlinear examples that the approximate constitutive relation that is obtained by differentiating the energy bound can be on the boundary of the bounding set for the exact constitutive relation, but a simple counterexample is presented to show that this is not always the case.","PeriodicalId":20722,"journal":{"name":"Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2004-09-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76552702","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper, a one–dimensional stability analysis of weakly curved, quasi–steady detonation waves is performed using a numerical shooting method, for an idealized detonation with a single irreversible reaction. Neutral stability boundaries are determined and shown in an activation temperature–curvature diagram, and the dependence of the complex growth rates on curvature is investigated for several cases. It is shown that increasing curvature destabilizes detonation waves, and hence curved detonations can be unstable even when the planar front is stable. Even a small increase in curvature can significantly destabilize the wave. It is also shown that curved detonations are always unstable sufficiently near the critical curvature above which there are no underlying quasi–steady solutions.
{"title":"One-dimensional linear stability of curved detonations","authors":"S. Watt, G. Sharpe","doi":"10.1098/rspa.2004.1290","DOIUrl":"https://doi.org/10.1098/rspa.2004.1290","url":null,"abstract":"In this paper, a one–dimensional stability analysis of weakly curved, quasi–steady detonation waves is performed using a numerical shooting method, for an idealized detonation with a single irreversible reaction. Neutral stability boundaries are determined and shown in an activation temperature–curvature diagram, and the dependence of the complex growth rates on curvature is investigated for several cases. It is shown that increasing curvature destabilizes detonation waves, and hence curved detonations can be unstable even when the planar front is stable. Even a small increase in curvature can significantly destabilize the wave. It is also shown that curved detonations are always unstable sufficiently near the critical curvature above which there are no underlying quasi–steady solutions.","PeriodicalId":20722,"journal":{"name":"Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2004-09-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79099698","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We study the effects of oversized solute atoms on the diffusion of clusters of self–interstitial atoms produced in metals by high–energy irradiation. We use kinetic Monte Carlo (KMC) simulations in model body–centred cubic iron, and include elastic interactions between the defects. We show that elastic repulsion between solute atoms and the clusters can confine the latter to one–dimensional segments. The easy direction of motion of each cluster is assumed to rotate infrequently, allowing it to escape to a new confined segment. The consequences of the confinement for the effective diffusivity of the cluster and its rate of reaction with other small static spherical sinks are explored both by KMC simulations and by an analytic theory. It is shown that the predictions of the theory agree very well with the computer simulations. We suggest some of the possible consequences of these findings for the design of alloys that are more resistant to the effects of high–energy radiation damage.
{"title":"Confinement of interstitial cluster diffusion by oversized solute atoms","authors":"T. Hudson, S. Dudarev, A. Sutton","doi":"10.1098/rspa.2004.1289","DOIUrl":"https://doi.org/10.1098/rspa.2004.1289","url":null,"abstract":"We study the effects of oversized solute atoms on the diffusion of clusters of self–interstitial atoms produced in metals by high–energy irradiation. We use kinetic Monte Carlo (KMC) simulations in model body–centred cubic iron, and include elastic interactions between the defects. We show that elastic repulsion between solute atoms and the clusters can confine the latter to one–dimensional segments. The easy direction of motion of each cluster is assumed to rotate infrequently, allowing it to escape to a new confined segment. The consequences of the confinement for the effective diffusivity of the cluster and its rate of reaction with other small static spherical sinks are explored both by KMC simulations and by an analytic theory. It is shown that the predictions of the theory agree very well with the computer simulations. We suggest some of the possible consequences of these findings for the design of alloys that are more resistant to the effects of high–energy radiation damage.","PeriodicalId":20722,"journal":{"name":"Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2004-09-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79664631","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The problem of heat and mass transfer within a porous catalytic pellet in which an irreversible first–order exothermic reaction occurs is a much–studied problem in chemical–reactor engineering. The system is described by two coupled reaction–diffusion equations for the temperature and the degree of reactant conversion. The Galerkin method is used to obtain a semi–analytical model for the pellet problem with both one– and two–dimensional slab geometries. This involves approximating the spatial structure of the temperature and reactant–conversion profiles in the pellet using trial functions. The semi–analytical model is obtained by averaging the governing partial differential equations. As the Arrhenius law cannot be integrated explicitly, the semi–analytical model is given by a system of integrodifferential equations. The semi–analytical model allows both steady–state temperature and conversion profiles and steady–state diagrams to be obtained as the solution to sets of transcendental equations (the integrals are evaluated using quadrature rules). Both the static and dynamic multiplicity of the semi–analytical model is investigated using singularity theory and a local stability analysis. An example of a stable limit cycle is also considered in detail. Comparison with numerical solutions of the governing reaction–diffusion equations and with other results in the literature shows that the semi–analytical solutions are extremely accurate.
{"title":"Semi-analytical solutions for one- and two-dimensional pellet problems","authors":"T. Marchant, M. I. Nelson","doi":"10.1098/rspa.2004.1286","DOIUrl":"https://doi.org/10.1098/rspa.2004.1286","url":null,"abstract":"The problem of heat and mass transfer within a porous catalytic pellet in which an irreversible first–order exothermic reaction occurs is a much–studied problem in chemical–reactor engineering. The system is described by two coupled reaction–diffusion equations for the temperature and the degree of reactant conversion. The Galerkin method is used to obtain a semi–analytical model for the pellet problem with both one– and two–dimensional slab geometries. This involves approximating the spatial structure of the temperature and reactant–conversion profiles in the pellet using trial functions. The semi–analytical model is obtained by averaging the governing partial differential equations. As the Arrhenius law cannot be integrated explicitly, the semi–analytical model is given by a system of integrodifferential equations. The semi–analytical model allows both steady–state temperature and conversion profiles and steady–state diagrams to be obtained as the solution to sets of transcendental equations (the integrals are evaluated using quadrature rules). Both the static and dynamic multiplicity of the semi–analytical model is investigated using singularity theory and a local stability analysis. An example of a stable limit cycle is also considered in detail. Comparison with numerical solutions of the governing reaction–diffusion equations and with other results in the literature shows that the semi–analytical solutions are extremely accurate.","PeriodicalId":20722,"journal":{"name":"Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2004-08-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85613218","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We determine the effects of a constant contact angle on (i) the natural slosh frequencies in rectangular and cylindrical containers, and (ii) the nonlinear sloshing in a two–dimensional rectangular container. In analysing the former we use perturbation and other methods to get some explicit results which are then checked and extended using a new modal method somewhat similar to that of Perko. The modal method is then used to calculate the nonlinear slosh response in (ii). It is seen that the response depends significantly on the contact angle.
{"title":"The effects of the contact angle on sloshing in containers","authors":"R. Kidambi, P. Shankar","doi":"10.1098/rspa.2004.1281","DOIUrl":"https://doi.org/10.1098/rspa.2004.1281","url":null,"abstract":"We determine the effects of a constant contact angle on (i) the natural slosh frequencies in rectangular and cylindrical containers, and (ii) the nonlinear sloshing in a two–dimensional rectangular container. In analysing the former we use perturbation and other methods to get some explicit results which are then checked and extended using a new modal method somewhat similar to that of Perko. The modal method is then used to calculate the nonlinear slosh response in (ii). It is seen that the response depends significantly on the contact angle.","PeriodicalId":20722,"journal":{"name":"Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2004-08-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72724242","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The possibility of using long–range ultrasonic guided–wave non–destructive testing for the inspection of metallic structures coated with attenuative layers is extremely attractive. However, the material damping may cause severe attenuation of the acoustic signal, resulting in reduction of the test range. This paper addresses the dispersion of shear horizontal waves propagating in metallic plates coated with viscoelastic layers. It is shown that the bilayer modes can be viewed as an interaction between the free elastic plate modes and the modes of the viscoelastic layer if it were clamped on its lower surface. For low–attenuation materials the interaction is strong and the trajectories of the bilayer dispersion curves jump between the two families as the frequency increases. However, as the material damping increases the interaction becomes weak and most of the energy is confined in either the metallic plate or in the attenuative layer, and modes no longer jump. The guided–wave attenuation is related to the strain energy in the viscoelastic layer per unit in–plane power flow by means of an energy–balance argument. The guided–wave attenuation of the modes whose energy travels primarily in the elastic plate exhibits periodic peaks in the frequency domain, which occur at roughly equally spaced critical frequencies. These frequencies are close to the through–thickness resonance frequencies of the clamped–free viscoelastic layer if it is considered to be elastic. Minima of the guided–wave attenuation occur around the Love transition frequencies.
{"title":"On the nature of shear horizontal wave propagation in elastic plates coated with viscoelastic materials","authors":"F. Simonetti, P. Cawley","doi":"10.1098/rspa.2004.1284","DOIUrl":"https://doi.org/10.1098/rspa.2004.1284","url":null,"abstract":"The possibility of using long–range ultrasonic guided–wave non–destructive testing for the inspection of metallic structures coated with attenuative layers is extremely attractive. However, the material damping may cause severe attenuation of the acoustic signal, resulting in reduction of the test range. This paper addresses the dispersion of shear horizontal waves propagating in metallic plates coated with viscoelastic layers. It is shown that the bilayer modes can be viewed as an interaction between the free elastic plate modes and the modes of the viscoelastic layer if it were clamped on its lower surface. For low–attenuation materials the interaction is strong and the trajectories of the bilayer dispersion curves jump between the two families as the frequency increases. However, as the material damping increases the interaction becomes weak and most of the energy is confined in either the metallic plate or in the attenuative layer, and modes no longer jump. The guided–wave attenuation is related to the strain energy in the viscoelastic layer per unit in–plane power flow by means of an energy–balance argument. The guided–wave attenuation of the modes whose energy travels primarily in the elastic plate exhibits periodic peaks in the frequency domain, which occur at roughly equally spaced critical frequencies. These frequencies are close to the through–thickness resonance frequencies of the clamped–free viscoelastic layer if it is considered to be elastic. Minima of the guided–wave attenuation occur around the Love transition frequencies.","PeriodicalId":20722,"journal":{"name":"Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2004-08-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83941559","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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