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Globe-hopping [pre-print]
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Chistikov, Dmitry and Paterson, Mike (2018) Globe-hopping [pre-print]. Working Paper. Coventry, UK: University of Warwick. (Unpublished)
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Abstract
We consider the grasshopper problem (Goulko and Kent, 2017) on the sphere. We find a lawn (subset L of the spherical surface) with the following properties:
- its area is half of the area of the sphere,
- it is antipodal, i.e., every point of the sphere is in L if and only if the opposite point is not in L, and
- the probability that a grasshopper’s jump stays in L is strictly greater than that for the lawn S consisting of the southern hemisphere.
Let the sphere have radius 1 and the jump size be θ. All distances here are spherical (determined by the length of great circle arcs). Assuming θ ∈ (0, π/2), our results hold if θ/π is irrational or if θ/π = p/q where p, q ∈ N, the fraction is irreducible, and p is even. If p is odd, our results may or may not hold depending on the specific values of p and q; in particular, jumps of size π/q, q ∈ N, present an open problem.
Item Type: | Working or Discussion Paper (Working Paper) | ||||
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Subjects: | Q Science > QA Mathematics | ||||
Divisions: | Faculty of Science, Engineering and Medicine > Science > Computer Science | ||||
Library of Congress Subject Headings (LCSH): | Sphere, Circle | ||||
Publisher: | University of Warwick | ||||
Place of Publication: | Coventry, UK | ||||
Official Date: | 7 December 2018 | ||||
Dates: |
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Institution: | University of Warwick | ||||
Status: | Not Peer Reviewed | ||||
Publication Status: | Unpublished | ||||
Access rights to Published version: | Restricted or Subscription Access | ||||
Date of first compliant deposit: | 12 December 2018 | ||||
Date of first compliant Open Access: | 12 December 2018 | ||||
Version or Related Resource: | https://wrap.warwick.ac.uk/136860 | ||||
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