Double bubble theorem (original) (raw)
Na teoria matemática das superfícies mínimas, a conjectura da bolha dupla afirma que a forma que envolve e separa dois volumes dados e tem a mínima possível é uma bolha dupla padrão - duas superfícies esféricas encontrando-se em ângulos de 2/3π em um círculo comum. Isso é agora um teorema, como uma demonstração que foi publicada em 2002.
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| dbo:abstract | In the mathematical theory of minimal surfaces, the double bubble theorem states that the shape that encloses and separates two given volumes and has the minimum possible surface area is a standard double bubble: three spherical surfaces meeting at angles of 120° on a common circle. The double bubble theorem was formulated and thought to be true in the 19th century, and became a "serious focus of research" by 1989, but was not proven until 2002. The proof combines multiple ingredients. Compactness of rectifiable currents (a generalized definition of surfaces) shows that a solution exists. A symmetry argument proves that the solution must be a surface of revolution, and it can be further restricted to having a bounded number of smooth pieces. Jean Taylor's proof of Plateau's laws describes how these pieces must be shaped and connected to each other, and a final case analysis shows that, among surfaces of revolution connected in this way, only the standard double bubble has locally-minimal area. The double bubble theorem extends the isoperimetric inequality, according to which the minimum-perimeter enclosure of any area is a circle, and the minimum-surface-area enclosure of any single volume is a sphere. Analogous results on the optimal enclosure of two volumes generalize to weighted forms of surface energy, to Gaussian measure of surfaces, and to Euclidean spaces of any dimension. (en) Na teoria matemática das superfícies mínimas, a conjectura da bolha dupla afirma que a forma que envolve e separa dois volumes dados e tem a mínima possível é uma bolha dupla padrão - duas superfícies esféricas encontrando-se em ângulos de 2/3π em um círculo comum. Isso é agora um teorema, como uma demonstração que foi publicada em 2002. (pt) Теорема о двойном пузыре гласит, что стандартный двойной пузырь (то есть три сферические шапки, сходящиеся под углом 120° на общей граничной окружности) имеет минимальную площадь среди всех плёнок, охватывающих и разделяющих два данных объема. Доказательство сочетает в себе несколько ингредиентов.Компактность спрямляемых потоков (обобщенных поверхностей) показывает, что решение существует.Симметрия используется для доказательсва, что решение должно быть поверхностью вращения, и имеете ограниченное число гладких кусков.Далее доказывается, что среди поверхностей вращения только стандартный двойной пузырь имеет локально минимальную площадь. Теорема о двойном пузыре обобщает изопериметрическое неравенство, согласно которому оболочка с минимальным периметром любой области представляет собой круг, а оболочка с минимальной площадью поверхности любого отдельного объема представляет собой сферу. (ru) |
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| dbp:title | Double Bubble (en) |
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| rdfs:comment | Na teoria matemática das superfícies mínimas, a conjectura da bolha dupla afirma que a forma que envolve e separa dois volumes dados e tem a mínima possível é uma bolha dupla padrão - duas superfícies esféricas encontrando-se em ângulos de 2/3π em um círculo comum. Isso é agora um teorema, como uma demonstração que foi publicada em 2002. (pt) In the mathematical theory of minimal surfaces, the double bubble theorem states that the shape that encloses and separates two given volumes and has the minimum possible surface area is a standard double bubble: three spherical surfaces meeting at angles of 120° on a common circle. The double bubble theorem was formulated and thought to be true in the 19th century, and became a "serious focus of research" by 1989, but was not proven until 2002. (en) Теорема о двойном пузыре гласит, что стандартный двойной пузырь (то есть три сферические шапки, сходящиеся под углом 120° на общей граничной окружности) имеет минимальную площадь среди всех плёнок, охватывающих и разделяющих два данных объема. Теорема о двойном пузыре обобщает изопериметрическое неравенство, согласно которому оболочка с минимальным периметром любой области представляет собой круг, а оболочка с минимальной площадью поверхности любого отдельного объема представляет собой сферу. (ru) |
| rdfs:label | Double bubble theorem (en) Conjectura da bolha dupla (pt) Теорема о двойном пузыре (ru) |
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