TY - JOUR

T1 - On Pinned Billiard Balls and Foldings

AU - Athreya, Jayadev S.

AU - Burdzy, Krzysztof

AU - Duarte, Mauricio

N1 - Publisher Copyright:
© 2023 Department of Mathematics, Indiana University. All rights reserved.

PY - 2023

Y1 - 2023

N2 - We consider systems of “pinned balls,” that is, balls that have fixed positions and pseudo-velocities. Pseudo-velocities change according to the same rules as those for velocities of totally elastic collisions between moving balls. The times of possible pseudo-collisions for different pairs of pinned balls are chosen in an exogenous way. We give an explicit upper bound for the maximum number of pseudo-collisions for a system of n pinned balls in a d-dimensional space, in terms of n, d, and the locations of ball centers. We also study foldings, that is, mappings that formalize the idea of folding a piece of paper along a crease. We prove a theorem on foldings that implies the number of pseudo-collisions of a finite number of pinned balls must be finite.

AB - We consider systems of “pinned balls,” that is, balls that have fixed positions and pseudo-velocities. Pseudo-velocities change according to the same rules as those for velocities of totally elastic collisions between moving balls. The times of possible pseudo-collisions for different pairs of pinned balls are chosen in an exogenous way. We give an explicit upper bound for the maximum number of pseudo-collisions for a system of n pinned balls in a d-dimensional space, in terms of n, d, and the locations of ball centers. We also study foldings, that is, mappings that formalize the idea of folding a piece of paper along a crease. We prove a theorem on foldings that implies the number of pseudo-collisions of a finite number of pinned balls must be finite.

UR - http://www.scopus.com/inward/record.url?scp=85167893576&partnerID=8YFLogxK

U2 - 10.1512/IUMJ.2023.72.9350

DO - 10.1512/IUMJ.2023.72.9350

M3 - Article

AN - SCOPUS:85167893576

SN - 0022-2518

VL - 72

SP - 897

EP - 925

JO - Indiana University Mathematics Journal

JF - Indiana University Mathematics Journal

IS - 3

ER -