arXiv subjects

Lizhou Chen

Publications and source records attributed to Lizhou Chen.

3 recordsLinked to original sources

arXiv2006

Proof of Murphy-Cohen Conjecture on One-dimensional Hard Ball Systems

We prove the Murphy and Cohen's conjecture that the maximum number of collisions of n + 1 elastic particles moving freely on a line is n(n+1)/2 if no interior particle has mass less than the arithmetic mean of the masses of its immediate neighbors. In fact, we prove the stronger result that, for the same conclusion, the condition no interior particle has mass less than the geometric mean, rather than the arithmetic mean, of the masses of its immediate neighbors suffices.

math.GM↗

arXiv2006

On Estimates of the Number of Collisions for Billiards in Polyhedral Angles

We obtain an upper bound of the number of collisions of any billiard trajectory in a polyhedral angle in terms of the minimal eigenvalue of a positive definite matrix which characterizes the angle. Elements of the matrix are scalar products between the unit normal vectors of faces of the angle.

math.DS↗

arXiv2005

A Remark of the Sanders-Wang's Theorem on Symmetry-integrability

We extend the integrability analysis for scalar evolution equations of type $$u_t=u_m+f(u,u_1,...,u_{m-1})$$ from the case that the right hand side is a $λ$-homogeneous formal power series to the case that it is a nonhomogeneous formal power series. It is proved that the existence of one nontrivial symmetry implies the existence of infinitely many, more precisely, the orders of the infinite integrable hierarchy must be one of the following cases: $\mathbb{Z}_++1$, $2\mathbb{Z}_++1$, $6\mathbb{Z}_+\pm1$, or $6\mathbb{Z}_++1$. Moreover, if the nonlinear part of the equation is a polynomial of order less than $m-1$, we show that any generalized symmetry is also of polynomial type.

math-ph↗