Search results

mathematical field of differential topology, the Hopf fibration (also known as the Hopf bundle or Hopf map) describes a 3-sphere (a hypersphere in four-dimensional...

In mathematics, a Hopf algebra, named after Heinz Hopf, is a structure that is simultaneously an (unital associative) algebra and a (counital coassociative)...

the Hopf invariant is a homotopy invariant of certain maps between n-spheres. In 1931 Heinz Hopf used Clifford parallels to construct the Hopf map η :...

mathematics, the Poincaré–Hopf theorem (also known as the Poincaré–Hopf index formula, Poincaré–Hopf index theorem, or Hopf index theorem) is an important...

In the mathematical theory of bifurcations, a Hopf bifurcation is a critical point where, as a parameter changes, a system's stability switches and a periodic...

and was discovered by Heinz Hopf, who constructed a nontrivial map from S3 to S2, now known as the Hopf fibration. This map generates the homotopy group...

transformation, locally quadratic transformation, dilatation, σ-process, or Hopf map. The simplest case of a blowup is the blowup of a point in a plane. Most...

Heinz Hopf (19 November 1894 – 3 June 1971) was a German mathematician who worked on the fields of dynamical systems, topology and geometry. Hopf was born...

representation of a Hopf algebra is a representation of its underlying associative algebra. That is, a representation of a Hopf algebra H over a field...

path-connected H-space with finitely generated and free cohomology groups is a Hopf algebra. Also, one can define the Pontryagin product on the homology groups...

The Hopf theorem (named after Heinz Hopf) is a statement in differential topology, saying that the topological degree is the only homotopy invariant of...

geometrically how the icosahedron works as a map of a Hopf fibration of the entire 600-cell, and how the Hopf fibration is an expression of an isoclinic...

_{n}(S^{n})\cong \mathbb {Z} } . In the second example the Hopf map, η {\displaystyle \eta } , is mapped to its suspension Σ η {\displaystyle \Sigma \eta } ...

In mathematics, a braided Hopf algebra is a Hopf algebra in a braided monoidal category. The most common braided Hopf algebras are objects in a Yetter–Drinfeld...

the theory of Hopf algebras, a Hopf algebroid is a generalisation of weak Hopf algebras, certain skew Hopf algebras and commutative Hopf k-algebroids....

In complex geometry, a Hopf manifold (Hopf 1948) is obtained as a quotient of the complex vector space (with zero deleted) ( C n ∖ 0 ) {\displaystyle...

spirit, weak Hopf algebras are weak bialgebras together with a linear map S satisfying specific conditions; they are generalizations of Hopf algebras. These...

Hopf–Rinow theorem is a set of statements about the geodesic completeness of Riemannian manifolds. It is named after Heinz Hopf and his student Willi...

fibers of the Hopf map are circles in S3" (page 95). Lyons gives an elementary introduction to quaternions to elucidate the Hopf fibration as a mapping...

In complex geometry, a Hopf surface is a compact complex surface obtained as a quotient of the complex vector space (with zero deleted) C 2 ∖ { 0 } {\displaystyle...