A Framework on Moment Model Reduction for Kinetic Equation

By a further investigation on the structure of the coefficient matrix of the globally hyperbolic regularized moment equations for Boltzmann equation in [Z. Cai, Y. Fan and R. Li, Comm. Math. Sci., 11 (2013), pp. 547-571], we propose a uniform framework to carry out model reduction to general kinetic equations, to achieve certain moment system. With this framework, the underlying reason why the globally hyperbolic regularization in [Z. Cai, Y. Fan and R. Li, Comm. Math. Sci., 11 (2013), pp. 547-571] works is revealed. The even fascinating point is, with only routine calculation, existing models are represented and brand new models are discovered. Even if the study is restricted in the scope of the classical Grad's 13-moment system, new model with global hyperbolicity can be deduced.Comment: 22 page

On a Penrose Inequality with Charge

We construct a time-symmetric asymptotically flat initial data set to the Einstein-Maxwell Equations which satisfies the inequality: m - 1/2(R + Q^2/R) < 0, where m is the total mass, R=sqrt(A/4) is the area radius of the outermost horizon and Q is the total charge. This yields a counter-example to a natural extension of the Penrose Inequality to charged black holes.Comment: Minor revision: some typos; author's address updated; bibliographical reference added; journal information: to appear in Comm. Math. Phy

Efficient simulated tempering with approximated weights: Applications to first-order phase transitions

Simulated tempering (ST) has attracted a great deal of attention in the last years, due to its capability to allow systems with complex dynamics to escape from regions separated by large entropic barriers. However its performance is strongly dependent on basic ingredients, such as the choice of the set of temperatures and their associated weights. Since the weight evaluations are not trivial tasks, an alternative approximated approach was proposed by Park and Pande (Phys. Rev. E {\bf 76}, 016703 (2007)) to circumvent this difficulty. Here we present a detailed study about this procedure by comparing its performance with exact (free-energy) weights and other methods, its dependence on the total replica number $R$ and on the temperature set. The ideas above are analyzed in four distinct lattice models presenting strong first-order phase transitions, hence constituting ideal examples in which the performance of algorithm is fundamental. In all cases, our results reveal that approximated weights work properly in the regime of larger $R$'s. On the other hand, for sufficiently small $R$ its performance is reduced and the systems do not cross properly the free-energy barriers. Finally, for estimating reliable temperature sets, we consider a simple protocol proposed at Comp. Phys. Comm. {\bf 128}, 2046 (2014).Comment: Published online in Comp. Phys. Comm. (2015

Ramanujan's "Lost Notebook" and the Virasoro Algebra

By using the theory of vertex operator algebras, we gave a new proof of the famous Ramanujan's modulus 5 modular equation from his "Lost Notebook" (p.139 in \cite{R}). Furthermore, we obtained an infinite list of $q$-identities for all odd moduli; thus, we generalized the result of Ramanujan.Comment: To appear in Comm. Math. Phy