Related papers: Dan Reznik's identities and more

The most general solution to the Einstein equations in $4=3+1$ dimensions in the asymptotical limit close to the cosmological singularity under the BKL (Belinski-Khalatnikov-Lifshitz) hypothesis, for which space gradients are neglected and…

Establishing global well-posedness and convergence toward equilibrium of the Boltzmann equation with specular reflection boundary condition has been one of the central questions in the subject of kinetic theory. Despite recent significant…

We apply the technique of K\'aroly Bezdek and Daniel Bezdek to study billiard trajectories in convex bodies, when the length is measured with a (possibly asymmetric) norm. We prove a lower bound for the length of the shortest closed…

We focus on the outer length billiard dynamics, acting on the exterior of a strictly-convex planar domain. We first show that ellipses are totally integrable. We then provide an explicit representation of first order terms for the formal…

Billiards tables - a minimal model for particles moving in a confined region - are known to present classical (and quantum) different features according to their shape, ranging from strongly chaotic to integrable dynamics. Here we consider…

Consider a periodic tiling of a plane by equal triangles obtained from the equilateral tiling by a linear transformation. We study a following tiling billiard: a ball follows straight segments and bounces of the boundaries of the tiles into…

This work presents a framework for billiards in convex domains on two dimensional Riemannian manifolds. These domains are contained in connected, simply connected open subsets which are totally normal. In this context, some basic properties…

In an ordinary billiard system trajectories of a Hamiltonian system are elastically reflected after a collision with a hypersurface (scatterer). If the scatterer is a submanifold of codimension more than one, we say that the billiard is…

We present some foundational results about the outer length billiard system, including its generating function and the invariant area form. We describe the limiting behavior of the orbits far away from the billiard table: the orbits of the…

In this text we study billiards on ovals and investigate some consequences of a rotational symmetry of the boundary on the dynamics. As it simplifies some calculations, the symmetry helps to obtain the results. We focus on periodic orbits…

The structure of the semiclassical trace formula can be used to construct a quasi-classical evolution operator whose spectrum has a one-to-one correspondence with the semiclassical quantum spectrum. We illustrate this for marginally…

Euclidean billiard partitions were recently introduced by Andrews, Dragovic and Radnovic in their study of periodic trajectories of ellipsoidal billiards in the Euclidean space. They are integer partitions into distinct parts such that (E1)…

The classical inner and outer billiards can be formulated in variational terms, with length and area as the respective generating functions. The other two combinations, ``inner with area'' and ``outer with length,'' are more recently…

We use both Abel's lemma on summation by parts and Zeilberger's algorithm to find recurrence relations for definite summations. The role of Abel's lemma can be extended to the case of linear difference operators with polynomial…

In our paper we study periodic geodesic motion on multidimensional ellipsoids with elastic impacts along confocal quadrics. We show that the method of isoperiodic deformation is applicable.

We study approximations of billiard systems by lattice graphs. It is demonstrated that under natural assumptions the graph wavefunctions approximate solutions of the Schroedinger equation with energy rescaled by the billiard dimension. As…

A correspondence between the orbits of a system of 2 complex, homogeneous, polynomial ordinary differential equations with real coefficients and those of a polygonal billiard is displayed. This correspondence is general, in the sense that…

In the present paper we introduce a new generating function for outer billiards in the plane. Using this generating function, we prove the following rigidity result: if the vicinity of the smooth convex plane curve $\gamma$ of positive…

Consider billiard dynamics in a strictly convex domain, and consider a trajectory that begins with the velocity vector making a small positive angle with the boundary. Lazutkin proved that in two dimensions, it is impossible for this angle…

We study outer length billiards; our main results are as follows. We prove 3- and 4-periodic versions of the Ivrii conjecture. We show that, for every period $n\ge 3$, there exists a functional space of billiard tables that possess…