Related papers: Cutting sequence and Sturmian sequence in billiard

Suppose that we are given two independent sets $I_b$ and $I_r$ of a graph such that $|I_b|=|I_r|$, and imagine that a token is placed on each vertex in $I_b$. Then, the sliding token problem is to determine whether there exists a sequence…

This paper studies sequential quantum games under the assumption that the moves of the players are drawn from groups and not just plain sets. The extra group structure makes possible to easily derive some very general results characterizing…

We consider classical dynamical properties of a particle in a constant gravitational force and making specular reflections with circular, elliptic or oval boundaries. The model and collision map are described and a detailed study of the…

In this paper, we are interested in the speed of convergence of the stochastic billiard evolving in a convex set K. This process can be described as follows: a particle moves at unit speed inside the set K until it hits the boundary, and is…

A system of two masses connected with a weightless rod (called dumbbell in this paper) interacting with a flat boundary is considered. The sharp bound on the number of collisions with the boundary is found using billiard techniques. In…

In "Playing Pool with $\pi$", Galperin invented an extraordinary method to learn the digits of $\pi$ by counting the collisions of billiard balls. Here I demonstrate an exact isomorphism between Galperin's bouncing billiards and Grover's…

We study the collision dynamics of a spinning cue ball approaching a static object ball with equal mass on a plane, common in billiards. While typical collisions in billiards are nearly perfectly elastic, with a restitution coefficient…

This paper proposes a new mechanism for pruning a search game-tree in computer chess. The algorithm stores and then reuses chains or sequences of moves, built up from previous searches. These move sequences have a built-in forward-pruning…

We provide a method to straighten each billiard trajectory in a flat disk. As an application, we show that for each point on the disk and for almost all directions, the closure of the corresponding billiard trajectory contains a vertex. We…

We study a class of planar billiards having the remarkable property that their phase space consists up to a set of zero measure of two invariant sets formed by orbits moving in opposite directions. The tables of these billiards are tubular…

We study diffractive effects in two dimensional polygonal billiards. We derive an analytical trace formula accounting for the role of non-classical diffractive orbits in the quantum spectrum. As an illustration the method is applied to a…

We consider a non-minimal billiard trajectory inside the cube. We study the language of the associated orbit when the map is coded with three letters associated to three non-parallel faces of the cube.

We describe an algorithm for the sequential sampling of entries in multiway contingency tables with given constraints. The algorithm can be used for computations in exact conditional inference. To justify the algorithm, a theory relates…

We introduce CUT, the class of 2-player partition games. These are NIM type games, played on a finite number of heaps of beans. The rules are given by a set of positive integers, which specifies the number of allowed splits a player can…

Let $\ell \geq 2$ be an integer. For each $\eps >0$ remove from $\R^2$ the union of discs of radius $\eps$ centered at the integer lattice points $(m,n$, with $m\nequiv n\mod{\ell}$. Consider a point-like particle moving linearly at unit…

We give topological lower bounds on the number of periodic and closed trajectories in strictly convex smooth billiards. We use variational reduction admitting a finite group of symmetries and apply topological approach based on equivariant…

Dynamical properties are studied for escaping particles, injected through a hole in an oval billiard. The dynamics is considered for both static and periodically moving boundaries. For the static boundary, two different decays for the…

The distribution of distances in the star graph $ST_n$, ($1<n\in\Z$), is established, and subsequently a threaded binary tree is obtained that realizes an orientation of $ST_n$ whose levels are given by the distances to the identity…

The orienteering problem is a well-studied and fundamental problem in transportation science. In the problem, we are given a graph with prizes on the nodes and lengths on the edges, together with a budget on the overall tour length. The…

In this paper we investigate the existence of closed billiard trajectories in not necessarily smooth convex bodies. In particular, we show that if a body $K\subset \mathbb{R}^d$ has the property that the tangent cone of every non-smooth…