Gregory Galperin
600 Lincoln Avenue, Charleston IL, 61920-3099
Depatment of Mathematics, Eastern Illinois University
Publications:
Albeverio S., Galperin G. A., Nizhnik I. L., Nizhnik L. P.
Generalized billiards inside an infinite strip with periodic laws of reflection along the strip's boundaries
2005, vol. 10, no. 3, pp. 285-306
Abstract
A constructive description of generalized billiards is given, the billiards being inside an infinite strip with a periodic law of reflection off the strip's bottom and top boundaries. Each of the boundaries is equipped with the same periodic lattice, where the number of lattice's nodes between any two successive reflection points may be prescribed arbitrarily. For such billiards, a full description of the structure of the set of billiard trajectories is provided, the existence of spatial chaos is found, and the exact value of the spatial entropy in the class of monotonic billiard trajectories is found.
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Galperin G. A.
Playing pool with $\pi$ (the number $\pi$ from a billiard point of view)
2003, vol. 8, no. 4, pp. 375-394
Abstract
Counting collisions in a simple dynamical system with two billiard balls can be used to estimate $\pi$ to any accuracy.
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Galperin G. A.
Relationship between Euclidean, Lobachevskian (hyperbolic), and billiard metrics and its application to a billiard problem in $R^d$
2003, vol. 8, no. 4, pp. 441-448
Abstract
A relationship between the three metrics — Billiard, Euclidean, and Lobachevskian (Hyperbolic) — is established in the article. This relationship is applied to a billiard problem on generalized diagonals of a Euclidean multidimensional convex polyhedron.
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Chernov N., Galperin G. A.
Search light in billiard tables
2003, vol. 8, no. 2, pp. 225-241
Abstract
We investigate whether a search light, $S$, illuminating a tiny angle ("cone") with vertex $A$ inside a bounded region $Q \in \mathbb{R}^2$ with the mirror boundary $\partial Q$, will eventually illuminate the entire region $Q$. It is assumed that light rays hitting the corners of $Q$ terminate. We prove that: $(\mathbf{I})$ if $Q =$ a circle or an ellipse, then either the entire $Q$ or an annulus between two concentric circles/confocal ellipses (one of which is $\partial Q$) or the region between two confocal hyperbolas will be illuminated; $(\mathbf{II})$ if $Q =$ a square, or $(\mathbf{III})$ if $Q =$ a dispersing (Sinai) or semidespirsing billiards, then the entire region $Q$ is will be illuminated.
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Galperin G. A., Zvonkine D.
Periodic Billiard Trajectories in Right Triangle
2003, vol. 8, no. 1, pp. 29-44
Abstract
This paper is dedicated to periodic billiard trajectories in right triangles and right-angled tetrahedra. We construct a specific type of periodic trajectories and show that the trajectories of this type fill the right triangle entirely. Then we establish the instability of all known types of periodic trajectories in right triangles. Finally, some of these results are generalized to the $n$-dimensional case and are given a mechanical interpretation.
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Galperin G. A.
Convex Polyhedra without Simple Closed Geodesics
2003, vol. 8, no. 1, pp. 45-58
Abstract
It is proved that, in contrast to the smooth convex surfaces homeomorphic to the two-dimensional sphere, the most of convex polyhedra in three dimensional space are free of closed geodesics without self-intersections.
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