Shifted lattices and asymptotically optimal ellipses
| Autor: | Shiya Liu, Richard S. Laugesen |
|---|---|
| Rok vydání: | 2018 |
| Předmět: |
Algebra and Number Theory
Applied Mathematics Open problem 010102 general mathematics Mathematical analysis Regular polygon Lattice (group) 010103 numerical & computational mathematics Ellipse 01 natural sciences Quadrant (plane geometry) Family of curves Geometry and Topology 0101 mathematics Laplace operator Analysis Eigenvalues and eigenvectors Mathematics |
| Zdroj: | The Journal of Analysis. 26:71-102 |
| ISSN: | 2367-2501 0971-3611 |
| Popis: | Translate the positive-integer lattice points in the first quadrant by some amount in the horizontal and vertical directions, and consider a decreasing concave (or convex) curve in the first quadrant. Construct a family of curves by rescaling in the coordinate directions while preserving area, and identify the curve in the family that encloses the greatest number of the shifted lattice points. We find the limiting shape of this maximizing curve as the area is scaled up towards infinity. The limiting shape depends explicitly on the lattice shift, except that when the shift is too negative, the maximizing curve fails to converge and instead degenerates. Our results handle the p-circle $$x^p+y^p=1$$ when $$p>1$$ (concave) and also when $$0 |
| Databáze: | OpenAIRE |
| Externí odkaz: |
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