Universally Optimal Periodic Configurations in the Plane
Fuente:
arXiv
Gespeichert in:
| Hauptverfasser: | , |
|---|---|
| Format: | Preprint |
| Veröffentlicht: |
2023
|
| Schlagworte: | |
| Online-Zugang: | |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| _version_ | 1866914092469452800 |
|---|---|
| author | Hardin, Doug Tenpas, Nathaniel |
| author_facet | Hardin, Doug Tenpas, Nathaniel |
| contents | We develop lower bounds for the energy of configurations in $\mathbb{R}^d$ periodic with respect to a lattice. In certain cases, the construction of sharp bounds can be formulated as a finite dimensional, multivariate polynomial interpolation problem. We use this framework to show a scaling of the equitriangular lattice $A_2$ is universally optimal among all configurations of the form $ω_4+ A_2$ where $ω_4$ is a 4-point configuration in $\mathbb{R}^2$. Likewise, we show a scaling and rotation of $A_2$ is universally optimal among all configurations of the form $ω_6+L$ where $ω_6$ is a 6-point configuration in $\mathbb{R}^2$ and $L=\mathbb{Z} \times \sqrt{3} \mathbb{Z}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2307_15822 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Universally Optimal Periodic Configurations in the Plane Hardin, Doug Tenpas, Nathaniel Classical Analysis and ODEs Mathematical Physics 52C35, 74G65 "Primary" 52C05 "Secondary" We develop lower bounds for the energy of configurations in $\mathbb{R}^d$ periodic with respect to a lattice. In certain cases, the construction of sharp bounds can be formulated as a finite dimensional, multivariate polynomial interpolation problem. We use this framework to show a scaling of the equitriangular lattice $A_2$ is universally optimal among all configurations of the form $ω_4+ A_2$ where $ω_4$ is a 4-point configuration in $\mathbb{R}^2$. Likewise, we show a scaling and rotation of $A_2$ is universally optimal among all configurations of the form $ω_6+L$ where $ω_6$ is a 6-point configuration in $\mathbb{R}^2$ and $L=\mathbb{Z} \times \sqrt{3} \mathbb{Z}$. |
| title | Universally Optimal Periodic Configurations in the Plane |
| topic | Classical Analysis and ODEs Mathematical Physics 52C35, 74G65 "Primary" 52C05 "Secondary" |
| url | https://arxiv.org/abs/2307.15822 |