The Spectrum of the Billiard Laplacian of a Family of Random Billiards

Random billiards are billiard dynamical systems for which the reflection law giving the post-collision direction of a billiard particle as a function of the pre-collision direction is specified by a Markov (scattering) operator P. Billiards with microstructure are random billiards whose Markov operator is derived from a “microscopic surface structure” on the boundary of the billiard table. The microstructure in turn is defined in terms of what we call a billiard cellQ, the shape of which completely determines the operator P. This operator, defined on an appropriate Hilbert space, is bounded self-adjoint and, for the examples considered here, a Hilbert-Schmidt operator. A central problem in the statistical theory of such random billiards is to relate the geometric characteristics of Q and the spectrum of P. We show, for a particular family of billiard cell shapes parametrized by a scale invariant curvature K (Fig. 2), that the billiard Laplacian P−I is closely related to the ordinary spherical Laplacian, and indicate, by partly analytical and partly numerical means, how this provides asymptotic information about the spectrum of P for small values of K. It is shown, in particular, that the second moment of scattering about the incidence angle closely approximates the spectral gap of P.     

Existence of mild solutions for fractional integrodifferential equations of Sobolev type with nonlocal conditions

This paper is concerned with the fractional integrodifferential equations of Sobolev type with nonlocal condition in a separable Banach space. With the help of the theory of propagation family as well as the theory of the measures of noncompactness and the condensing maps, we obtain the existence result of mild solutions for above equations. Two examples are given to illustrate the existence theorem.     

Robustness of Mann's algorithm for nonexpansive mappings

Mann's algorithm is proved robust in the sense that appropriately small perturbations do not alter the convergence of the algorithm. We prove this for nonexpansive mappings in a Banach space setting, which extends the initial work of Combettes [P.L. Combettes, On the numerical robustness of the parallel projection method in signal synthesis, IEEE Signal Process. Lett. 8 (2001) 45-47] where projections are considered in the Hilbert space framework.     

Extremal Solutions of a Class of Nonlinear Integro-Differential Equations in Banach spaces

The monotone iterative technique is applied to a class of nonlinear first order integro-differential equations in Banach spaces. First a linear system with a ``small' nonlinear perturbation is solved using Banach's Contraction Principle and the technique of Green's function. Then based upon a comparison result, the existence of minimal and maximal solutions is proved.

     

Current in Periodic Lorentz Gases with Twists

We study electrical current in two-dimensional periodic Lorentz gas in the presence of a twist force on the scatterers. In this deterministic system, billiard orbits are still geodesics between collisions, but do not reflect elastically when reaching the boundary. When the horizon is finite, i.e. the free flights between collisions are bounded, the resulting current J is proportional to the strength of the twist force measured by ε. We also prove the existence of a unique SRB measure, for which the Pesin entropy formula and Young’s expression for the fractal dimension are valid.     

Fixed points of semigroups of Lipschitzian mappings defined on nonconvex domains

Certain fixed point theorems are established for nonlinear semigroups of Lipschitzian mappings defined on nonconvex domains in Hilbert and Banach spaces. Some known results are thus generalized.     

中国城市航空网络的实证研究与分析

以城市为节点，城市间直航线路为边，实证地研究了中国城市航空网络的拓扑性质.研究表明，中国城市航空网络是一个小世界网络，具有短的平均路径长度和大的簇系数，且其度分布服从双段幂律分布.它的度度相关性质与世界航空网络和北美航空网络都不相同.当度较小时，世界航空网络和北美航空网络都是正相关的，但中国城市航空网络未表现出度度相关性；而对于度较大的节点，世界航空网络中其邻点平均度几乎是一个常值，但中国城市航空网络却呈现出负相关性.以往的实证研究暗示，节点具有明确几何位置的网络，如计算机互联网、电力网络等，不表现层次性.但是中国城市航空网络展现出明显的层次性，表明地理因素对其结构演化的影响并不强烈.进一步地，以城市间直航计划每周提供的座位数为边权，研究了网络的含权性质，发现该网络节点度权之间是幂律相关的，相关指数为1.37.