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- In physics, the poppy-seed bagel theorem concerns interacting particles (e.g., electrons) confined to a bounded surface (or body) when the particles repel each other pairwise with a magnitude that is proportional to the inverse distance between them raised to some positive power . In particular, this includes the Coulomb law observed in Electrostatics and Riesz potentials extensively studied in Potential theory. For such particles, a stable equilibrium state, which depends on the parameter , is attained when the associated potential energy of the system is minimal (the so-called generalized Thomson problem). For large numbers of points, these equilibrium configurations provide a discretization of which may or may not be nearly uniform with respect to the surface area (or volume) of . The poppy-seed bagel theorem asserts that for a large class of sets , the uniformity property holds when the parameter is larger than or equal to the dimension of the set . For example, when the points ("poppy seeds") are confined to the 2-dimensional surface of a torus embedded in 3 dimensions (or "surface of a bagel"), one can create a large number of points that are nearly uniformly spread on the surface by imposing a repulsion proportional to the inverse square distance between the points, or any stronger repulsion. From a culinary perspective, to create the nearly perfect poppy-seed bagel where bites of equal size anywhere on the bagel would contain essentially the same number of poppy seeds, impose at least an inverse square distance repelling force on the seeds. (en)
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- 10272 (xsd:nonNegativeInteger)
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- Near minimal -energy 1000-point configurations on a torus (en)
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- Te020m1kcrop.png (en)
- Te100m1kcrop.png (en)
- Te200m1kcrop.png (en)
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- In physics, the poppy-seed bagel theorem concerns interacting particles (e.g., electrons) confined to a bounded surface (or body) when the particles repel each other pairwise with a magnitude that is proportional to the inverse distance between them raised to some positive power . In particular, this includes the Coulomb law observed in Electrostatics and Riesz potentials extensively studied in Potential theory. For such particles, a stable equilibrium state, which depends on the parameter , is attained when the associated potential energy of the system is minimal (the so-called generalized Thomson problem). For large numbers of points, these equilibrium configurations provide a discretization of which may or may not be nearly uniform with respect to the surface area (or volume) of . Th (en)
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- Poppy-seed bagel theorem (en)
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