On lattice point weak b-visibility

Submitted article with Allie Aird, Alberto Alonso, Samuel I. Cooper, Patrick M. Crossley, George Kuliner, Robert J. Marino, Eric S. Piato, and Barbara J. Schweitzer.

Abstract

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 ∈ N, 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)∈ ∆ if no other point in ∆ lies on the curve f(x) = [(s−k)/(r−h)^b](x-h)^b +k between Q and P. Using this definition, 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≥1such 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 coordinate

s of the closest point to the array satisfying such a condition. We end the article by providing a few directions for future research.

SUNY Geneseo Math Research Weekend January 26-27, 2018

I was invited to be the research leader for the 2018 SUNY Geneseo Math Research Weekend. During this weekend of events students and I collaborated on a new research project titled "On lattice point weak b-visibility." The manuscript of results has now been submitted for publication!