Dylan Helliwell
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View article: Cassini sets in taxicab geometry
Cassini sets in taxicab geometry Open
Given two points $p$ and $q$ in the plane and a nonnegative number $r$, the Cassini oval is the set of points $x$ that satisfy $d(x, p) d(x, q) = r^2$. In this paper, we study this set using the taxicab metric. We find that these sets have…
View article: A Poincaré ball model for taxicab hyperbolic geometry
A Poincaré ball model for taxicab hyperbolic geometry Open
Taxicab space is a modification of Euclidean space that uses an alternative notion of distance. Similarly, the Poincaré ball is a model of hyperbolic geometry that consists of a subset of Euclidean space with an alternative notion of dista…
View article: Further study on domains and quasihyperbolic distances
Further study on domains and quasihyperbolic distances Open
We establish constructive geometric tools for determining when a domain is $L^{s}$ -averaging and obtain upper and lower bounds for the $L^{s}$ -integrals of the quasihyperbolic distance. We also construct examples that are helpful…
View article: Further study on domains and quasihyperbolic distances
Further study on domains and quasihyperbolic distances Open
We establish constructive geometric tools for determining when a domain is L s - averaging and obtain upper and lower bounds for the L s - integrals of the quasihyperbolic distance. We also construct examples which are helpful to understan…
View article: Further Study on Domains and Quasihyperbolic Distances
Further Study on Domains and Quasihyperbolic Distances Open
We establish constructive geometric tools for determining when a domain is $L^s$-averaging and obtain upper and lower bounds for the $L^s$-integrals of the quasihyperbolic distance. We also construct examples which are helpful to understan…
View article: A new perspective on taxicab conic sections
A new perspective on taxicab conic sections Open
We explore taxicab conic sections from the perspective of slicing taxicab cones by planes, as opposed to the more well-studied approach from the perspective of distance formulations. After establishing a significant amount of structural fr…
View article: A new perspective on taxicab conic sections
A new perspective on taxicab conic sections Open
We explore taxicab conic sections from the perspective of slicing taxicab cones by planes, as opposed to the more well-studied approach from the perspective of distance formulations. After establishing a significant amount of structural f…
View article: Bach Flow on Homogeneous Products
Bach Flow on Homogeneous Products Open
Qualitative behavior of Bach flow is established on compact four-dimensional\nlocally homogeneous product manifolds. This is achieved by lifting to the\nhomogeneous universal cover and, in most cases, capitalizing on the resultant\ngroup s…