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We are left with the zeroth and pth coefficients, which both equal 1, yielding the desired equation. Thus in characteristic p the freshman's dream is a valid identity. This result demonstrates that exponentiation by p produces an endomorphism , known as the Frobenius endomorphism of the ring.
A tropical counterpart of the Abel–Jacobi map can be applied to a crystal design. [21] The weights in a weighted finite-state transducer are often required to be a tropical semiring. Tropical geometry can show self-organized criticality. [22] Tropical geometry has also found applications in several topics within theoretical high energy physics.
Chapter four studies tropical connections to the Grassmannian, neighbor joining in the space of metric trees, and matroids. chapter five considers tropical analogues of some of the important concepts in linear algebra, and chapter six connects tropical varieties to toric varieties and polyhedral geometry. [1] [2] [3]
The concept of Rustic Modern interior design received increased attention after 2000, especially in the State of California. [3] This incarnation of the design method includes design elements alternatively known as "kitsch" that represent eras of design in the 20th century such as chalkboards or twig racks alongside modern conveniences like reduced energy lighting and other state-of-the-art ...
"Beach chic" was the title of an article in 2006 by the Times fashion editor Lisa Armstrong about shopping for accessories to accompany a bikini. [1] These included a "cover-up" (e.g. a kaftan), flat sandals, a hat, a fake tan and - with the comforting footnote, "No, you will not look like a WAG [wife or girlfriend of a footballer]" - denture cleaner to whiten finger-nails.
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Mathematics in art: Albrecht Dürer's copper plate engraving Melencolia I, 1514. Mathematical references include a compass for geometry, a magic square and a truncated rhombohedron, while measurement is indicated by the scales and hourglass. [1] Wireframe drawing [2] of a vase as a solid of revolution [2] by Paolo Uccello. 15th century
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