English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

Let O be a closed geodesic polygon in S^2. Maps from O into S^2 are said tosatisfy tangent boundary conditions if the edges of O are mapped into thegeodesics which contain them. Taking O to be an octant of S^2, we compute theinfimum Dirichlet energy, Ecal(H), for continuous maps satisfying tangentboundary conditions of arbitrary homotopy type H. The expression for Ecal(H)involves a topological invariant - the spelling length - associated with the(nonabelian) fundamental group of the n-times punctured two-sphere, pi_1(S^2 -{s_1,..., s_n},*). The lower bound for Ecal(H) is obtained from combinatorialgroup theory arguments, while the upper bound is obtained by constructingexplicit representatives which, on all but an arbitrarily small subset of O,are alternatively locally conformal or anticonformal. For conformal andanticonformal classes (classes containing wholly conformal and anticonformalrepresentatives respectively), the expression for Ecal(H) reduces to a previousresult involving the degrees of a set of regular values s_1,..., s_n in thetarget S^2 space. These degrees may be viewed as invariants associated with theabelianization of \pi_1(S^2 - {s_1,..., s_n}, *). For nonconformal classes,however, Ecal(H) may be strictly greater than the abelian bound. This stemsfrom the fact that, for nonconformal maps, the number of preimages of certainregular values may necessarily be strictly greater than the absolute value oftheir degrees.  This work is motivated by the theoretical modelling of nematic liquidcrystals in confined polyhedral geometries. The results imply new lower andupper bounds for the Dirichlet energy (one-constant Oseen-Frank energy) ofreflection-symmetric tangent unit-vector fields in a rectangular prism.

Tangent unit-vector fields: nonabelian homotopy invariants and the Dirichlet energy