| 1. |
- Bajo, Esme, et al.
(author)
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Weighted Ehrhart theory: Extending Stanley's nonnegativity theorem
- 2024
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In: Advances in Mathematics. - : Elsevier BV. - 0001-8708 .- 1090-2082. ; 444
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Journal article (peer-reviewed)abstract
- We generalize R. P. Stanley's celebrated theorem that the h⁎-polynomial of the Ehrhart series of a rational polytope has nonnegative coefficients and is monotone under containment of polytopes. We show that these results continue to hold for weighted Ehrhart series where lattice points are counted with polynomial weights, as long as the weights are homogeneous polynomials decomposable as sums of products of linear forms that are nonnegative on the polytope. We also show nonnegativity of the h⁎-polynomial as a real-valued function for a larger family of weights. We explore the case when the weight function is the square of a single (arbitrary) linear form. We show stronger results for two-dimensional convex lattice polygons and give concrete examples showing tightness of the hypotheses. As an application, we construct a counterexample to a conjecture by Berg, Jochemko, and Silverstein on Ehrhart tensor polynomials.
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| 2. |
- Versteegh, Marijn A. M., et al.
(author)
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Giant Rydberg excitons in Cu2O probed by photoluminescence excitation spectroscopy
- 2021
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In: PHYSICAL REVIEW B. - : American Physical Society (APS). - 2469-9950 .- 2469-9969. ; 104:24
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Journal article (peer-reviewed)abstract
- Rydberg excitons are, with their ultrastrong mutual interactions, giant optical nonlinearities, and very high sensitivity to external fields, promising for applications in quantum sensing and nonlinear optics at the singlephoton level. To design quantum applications it is necessary to know how Rydberg excitons and other excited states relax to lower-lying exciton states. Here, we present photoluminescence excitation spectroscopy as a method to probe transition probabilities from various excitonic states in cuprous oxide. We show giant Rydberg excitons at T = 38 mK with principal quantum numbers up to n = 30, corresponding to a calculated diameter of 3 mu m.
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