different between isomorphism vs symplectomorphism

English

Etymology

iso- +? -morphism

Pronunciation

Noun

isomorphism (plural isomorphisms)

1. Similarity of form
   • 1984 Brigitte Asbach-Schnitker, "Introduction", Mercury or The Secret and Swift Messenger, ?ISBN.

     The postulated isomorphism between words and things constitutes the characterizing feature of all philosophically based universal languages.

   1. (biology) the similarity in form of organisms, which may be due to convergent evolution or shared genetic background, e.g. an algae species in which the haploid and diploid life stages are indistinguishable based on morphology.
   2. (chemistry) the similarity in the crystal structures of similar chemical compounds
      • 1874 C. Rammelsberg, "Crystallographic and chemical relations of the natural sulphides, arsenides, and sulpharsenides", The Chemical News and Journal of Physical Science, page 197.

        The isomorphism of compounds does not prove the isomorphism of their respective constituents.

   3. (sociology) the similarity in the structure or processes of different organizations
2. A one-to-one correspondence
   1. (group algebra) A bijection f such that both f and its inverse f ^?1 are homomorphisms, that is, structure-preserving mappings.
   2. (computer science) a one-to-one correspondence between all the elements of two sets, e.g. the instances of two classes, or the records in two datasets
   3. (category theory) A morphism which has an inverse; the composition of the morphism and its inverse yields either one of two identity morphisms (depending on the order of composition).

Synonyms

• (in category theory): iso

Antonyms

• anisomorphism

Related terms

• isomorphic
• isomorphous

Translations

See also

• Isomorphism on Wikipedia.Wikipedia
• Isomorphism on Encyclopedia of Mathematics
• isomorphism on nLab
• Isomorphism on Wolfram MathWorld
• Graph Isomorphism on Wolfram MathWorld
• Natural Isomorphism on Wolfram MathWorld

English

Etymology

Blend of symplectic +? isomorphism.

Noun

symplectomorphism (plural symplectomorphisms)

1. (mathematics) An isomorphism of a symplectic manifold; a diffeomorphism which preserves symplectic structure.
   • 1977, Alan Weinstein, Lectures on Symplectic Manifolds, American Mathematical Society, page 34,

     In prequantizing symplectomorphisms of the type T^*g, the only special property which we used was the fact that they preserve canonical 1-forms.

   • 2001, A. Dzhamay, G. Wassermann (translators), V. I. Arnol'd, A. B. Givental' Symplectic Geometry, V. I. Arnol'd, S. P. Novikov (editors), Dynamical Systems IV: Symplectic Geometry and its Applications, Springer, 2nd Edition, page 39,

     Poincare's argument is based on the fact that the fixed points of a symplectomorphism of the annulus are precisely the critical points of the function ${\displaystyle \textstyle F(x,y)=\int {(fdv-gdu)}}$, where ${\displaystyle u=(X+x)/2}$, ${\displaystyle v=(Y+y)/2}$, true under the assumption that the Jacobian ${\displaystyle \partial (u,v)/\partial (x,y)}$ is different from zero.

   • 2008, Ana Cannas da Silva, Lectures on Symplectic Geometry, Springer, 2nd printing with corrections, page 63,

     The symplectomorphisms of a symplectic manifold ${\displaystyle (M,\omega )}$ form the group

     ${\displaystyle {\text{Sympl}}(M,\omega )=\lbrace f:M{\overset {\simeq }{\longrightarrow }}M\ \vert \ f^{*}\omega =\omega \rbrace }$.

Translations