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symplectic

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EnglishEdit

EtymologyEdit

A calque of complex, coined by Hermann Weyl in his 1939 book The Classical Groups: Their Invariants and Representations. From Ancient Greek συμπλεκτικός (sumplektikós), from συμ (sum) (variant of σύν (sún)), + πλεκτικός (plektikós) (from πλέκω (plékō)); modelled on complex (from Latin complexus (braided together), from com- (together) + plectere (to weave, braid)). In both cases the suffix is from Proto-Indo-European *plek-.

The symplectic group had previously been called the line complex group.

AdjectiveEdit

symplectic (not comparable)

  1. (group theory, of a group) Whose characteristic abelian subgroups are cyclic.
  2. (mathematics, multilinear algebra, of a bilinear form) That is alternating and nondegenerate.
  3. (mathematics, multilinear algebra, of a vector space) That is equipped with an alternating nondegenerate bilinear form.
  4. (mathematics) Of or pertaining to (the geometry of) a differentiable manifold equipped with a closed nondegenerate bilinear form.
    • 1995, V. I. Arnold, Some remarks on symplectic monodromy of Milnor fibrations, Helmut Hofer, Clifford H. Taubes, Alan Weinstein, Eduard Zehnder (editors), The Floer Memorial Volume, Birkhäuser Verlag, page 99,
      There exist interesting and unexplored relations between symplectic geometry and the theory of critical points of holomorphic functions.
    • 1997, C. H. Cushman-de Vries (translator), Richard H. Cushman, Gijs M. Tuynman (translation editors), Jean-Marie Souriau, Structure of Dynamical Systems: A Symplectic View of Physics, Springer Science & Business Media (Birkhäuser).
    • 2003, Fabrizio Catanese, Gang Tian (editors), Symplectic 4-Manifolds and Algebraic Surfaces: Lectures given at the C.I.M.E Summer School, Springer, Lecture Notes in Mathematics No. 1938.
    • 2003, Yakov Eliashberg, Boris A. Khesin, François Lalonde (editors), Symplectic and Contact Topology: Interactions and Perspectives, American Mathematical Society.
    • 2003, Maung Min-Oo, The Dirac Operator in Geometry and Physics, Steen Markvorsen, Maung Min-Oo (editors), Global Riemannian Geometry: Curvature and Topology, Springer, page 72,
      In symplectic geometry, there is a notion of fibrations   with a symplectic manifold F as fiber, where the structure group is the group of (exact) Hamiltonian symplectomorphisms of the fiber. These are called symplectic fibrations. If the base manifold   is also symplectic, there is a weak coupling construction, originally due to Thurston, of defining a symplectic structure on the total space  .
  5. That moves in the same direction as a system of synchronized waves.

AntonymsEdit

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NounEdit

symplectic (plural symplectics)

  1. (mathematics) A symplectic bilinear form, manifold, geometry, etc.
    • 1967, Journal of Mathematics and Mechanics, Volume 16, Issue 1, Indiana University, page 339,
      The structure of stable symplectics on finite dimensional spaces has been studied by Krein [8], Gelfand & Lidskii [9], and Moser [10] in work of considerable practical importance.
  1. (ichthyology) A symplectic bone.
    • 1914, The Philippine Journal of Science, Volume 9, page 27,
      The symplectics (9) consist of a somewhat curved central triangular portion with the base upward, and anteriorly and posteriorly from this extends a wing-like process.
    • 1965, Agra University Journal of Research: Science, Volume 14, page 71,
      The symplectics (Fig. 8, sym) are thin slender bones placed vertically in between the quadrates and the hyomandibulars.
    • 1967, Tyson R. Roberts, Studies on the Osteology and Phylogeny of Characoid Fishes, page 59,
      In many teleosts, on the other hand, including the catfishes, the symplectics have been entirely lost.

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