We solve the equilibrium meanfield equation of state of Ising ferromagnet （obtained from Bragg-Williams theory） by Newton-Raphson method. The number of iterations required to get a convergent solution （within a spe...We solve the equilibrium meanfield equation of state of Ising ferromagnet （obtained from Bragg-Williams theory） by Newton-Raphson method. The number of iterations required to get a convergent solution （within a specified accuracy） of equilibrium magnetisation, at any particular temperature, is observed to diverge in a power law fashion as the temperature approaches the critical value. This is identified as the critical slowing down. The exponent is also estimated. This value of the exponent is compared with that obtained from analytic solution. Besides this, the numerical results are also compared with some experimental results exhibiting satisfactory degree of agreement. It is observed from this study that the information of the invariance of time scale at the critical point is present in the meanfield equilibrium equation of state of Ising ferromagnet.展开更多
Scoring in a basketball game is a highly dynamic, non-linear process. NBA teams try to be more and more competitive each season. For instance, they incorporate into their rosters the best players in the world. This an...Scoring in a basketball game is a highly dynamic, non-linear process. NBA teams try to be more and more competitive each season. For instance, they incorporate into their rosters the best players in the world. This and other mechanisms concur to make the scoring process in NBA games exciting and rarely predictable. This paper is to study the behavior of timing and scoring in basketball gaines. The authors analyze all the games in five NBA regular seasons （2005-06, 2006-07, 2007-08, 2008-09, 2009-10）, for a total of 6150 games. Scoring does not behave uniformly; therefore, the authors also analyze the distributions of the differences in points in the basketball games. To further analyze the behavior of the tail of the distribution, the authors also carry out a semilog-plot and a log-log plot to verify whether this trend approaches a Poisson distribution or a PL. This paper reveals different areas of behavior related to the score, with specific instances of time that could be considered tipping points of the game. The presence of these critical points suggests that there are phase transitions where the dynamic scoring of the games varies significantly.展开更多
文摘We solve the equilibrium meanfield equation of state of Ising ferromagnet （obtained from Bragg-Williams theory） by Newton-Raphson method. The number of iterations required to get a convergent solution （within a specified accuracy） of equilibrium magnetisation, at any particular temperature, is observed to diverge in a power law fashion as the temperature approaches the critical value. This is identified as the critical slowing down. The exponent is also estimated. This value of the exponent is compared with that obtained from analytic solution. Besides this, the numerical results are also compared with some experimental results exhibiting satisfactory degree of agreement. It is observed from this study that the information of the invariance of time scale at the critical point is present in the meanfield equilibrium equation of state of Ising ferromagnet.
文摘Scoring in a basketball game is a highly dynamic, non-linear process. NBA teams try to be more and more competitive each season. For instance, they incorporate into their rosters the best players in the world. This and other mechanisms concur to make the scoring process in NBA games exciting and rarely predictable. This paper is to study the behavior of timing and scoring in basketball gaines. The authors analyze all the games in five NBA regular seasons （2005-06, 2006-07, 2007-08, 2008-09, 2009-10）, for a total of 6150 games. Scoring does not behave uniformly; therefore, the authors also analyze the distributions of the differences in points in the basketball games. To further analyze the behavior of the tail of the distribution, the authors also carry out a semilog-plot and a log-log plot to verify whether this trend approaches a Poisson distribution or a PL. This paper reveals different areas of behavior related to the score, with specific instances of time that could be considered tipping points of the game. The presence of these critical points suggests that there are phase transitions where the dynamic scoring of the games varies significantly.
