57 research outputs found
Dynamics of tournaments: the soccer case
A random walk-like model is considered to discuss statistical aspects of
tournaments. The model is applied to soccer leagues with emphasis on the
scores. This competitive system was computationally simulated and the results
are compared with empirical data from the English, the German and the Spanish
leagues and showed a good agreement with them. The present approach enabled us
to characterize a diffusion where the scores are not normally distributed,
having a short and asymmetric tail extending towards more positive values. We
argue that this non-Gaussian behavior is related with the difference between
the teams and with the asymmetry of the scores system. In addition, we compared
two tournament systems: the all-play-all and the elimination tournaments.Comment: To appear in EPJ
Time-Fractional KdV Equation: Formulation and Solution using Variational Methods
In this work, the semi-inverse method has been used to derive the Lagrangian
of the Korteweg-de Vries (KdV) equation. Then, the time operator of the
Lagrangian of the KdV equation has been transformed into fractional domain in
terms of the left-Riemann-Liouville fractional differential operator. The
variational of the functional of this Lagrangian leads neatly to Euler-Lagrange
equation. Via Agrawal's method, one can easily derive the time-fractional KdV
equation from this Euler-Lagrange equation. Remarkably, the time-fractional
term in the resulting KdV equation is obtained in Riesz fractional derivative
in a direct manner. As a second step, the derived time-fractional KdV equation
is solved using He's variational-iteration method. The calculations are carried
out using initial condition depends on the nonlinear and dispersion
coefficients of the KdV equation. We remark that more pronounced effects and
deeper insight into the formation and properties of the resulting solitary wave
by additionally considering the fractional order derivative beside the
nonlinearity and dispersion terms.Comment: The paper has been rewritten, 12 pages, 3 figure
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