Maurice Pierre | Searcharxiv

arXiv2022

Identities of the Fractional Fourier Transform and the Versor Transform

We provide an introduction to the Fractional Fourier Transform $\mathcal{F}_{\theta}$ and draw a connection between it and the unit complex number $e^{i\theta}$. Motivated by this, we define an entirely new object associated with any unit quaternion $e^{i\xi_{1}}\cos\eta+e^{i\xi_{2}}j\sin\eta$, which we call the Versor Transform $\mathcal{V}_{(\xi_{1},\eta,\xi_{2})}$. This transform, which has both the Fourier and Laplace Transforms as special cases, encourages an alternate view of the relationship between them. We also derive several identities for both $\mathcal{F}_{\theta}$ and $\mathcal{V}_{(\xi_{1},\eta,\xi_{2})}$.

math.GM↗

arXiv2019

Maximally Dense Disc Packings on the Plane

Suppose one has a collection of disks of various sizes with disjoint interiors, a packing in the plane, and suppose the ratio of the smallest radius divided by the largest radius lies between $1$ and $q$. In his 1964 book Regular Figures (MR0165423), L\'aszl\'o Fejes T\'oth found a series of packings that were his best guess for the maximum density for any $1 > q > 0.2$. Meanwhile Gerd Blind in (MR0275291, MR0377702) proved that for $1 \ge q > 0.72$, the most dense packing possible is $\pi/\sqrt{12}$, which is when all the disks are the same size. In Regular Figures, the upper bound of the ratio $q$ such that the density of his packings is greater than $\pi/\sqrt{12}$ that Fejes T\'oth found was $0.6457072159...$. Here we improve that upper bound to $0.6585340820...$. Our new packings are based on a perturbation of a triangulated packing that has three distinct sizes of disks, found by Fernique, Hashemi, and Sizova (MR4292755), which is something of a surprise.

math.MG↗