On lattice point weak b-visibility

  • Allie Aird
  • Alberto Alonso
  • Samuel I Cooper
  • Patrick M Crossley
  • Pamela E Harris
  • George Kuliner
  • Robert J Marino
  • Eric S Piato
  • Barbara J Schweitzer SUNY Geneseo
  • Reuben Kaufman

Abstract




For a fixed b ∈ Z+ , a point (r, s) ∈ Z × Z is b-visible from the origin if there exists a power function f (x) = axb with a ∈ Q such that f(0)=0 and f(r)=s, and no other point in the integer lattice belongs to the graph of f. In this article, we extend the definition of b-visibility given by Goins, Harris, Kubik, and Mbirika to the study of weak visibility. For a fixed b ∈ Z+ , we say that a point Q = (h, k) in the array ∆m,n = {1, 2, . . . , m} × {1, 2, . . . , n} is weakly b-visible from a point P = (r, s) ∈ Z+ × Z+ such that P ∈ ∆m,n if no other point in ∆m,n lies on the curve f(x)= ((s−k)/(r−h)b) (x − h)b + k between Q and P . In this paper we give necessary and sufficient conditions for determining if a point in ∆m,n is weakly b-visible by an external point. We also show that for any point P = (r, s) with r > m and s > n, there exists a b ≥ 1 such that every point in ∆m,n is weakly b-visible from P . Our last result considers a fixed b > 1 and specifies the coordinates of a point P that weakly b-views every point in ∆m,n, and as a corollary we provide a way to determine the coordinates of the closest point to the array satisfying such a condition. We conclude by providing a few directions for future research.




Published
2020-06-29
How to Cite
AIRD, Allie et al. On lattice point weak b-visibility. Minnesota Journal of Undergraduate Mathematics, [S.l.], v. 5, n. 1, june 2020. ISSN 2378-5810. Available at: <https://mjum.math.umn.edu/index.php/mjum/article/view/90>. Date accessed: 20 oct. 2020.