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 Tools annual

Global: Zendy Free Plan

Global: Zendy Tools monthly

Global: Zendy Plus monthly

We consider several problems involving points and planes in three dimensions. Our main results are: (i) The maximum number of faces boundingm distinct cells in an arrangement ofn planes isO(m2/3n logn +n2); we can calculatem such cells specified by a point in each, in worst-case timeO(m2/3n log3n+n2 logn). (ii) The maximum number of incidences betweenn planes andm vertices of their arrangement isO(m2/3n logn+n2), but this number is onlyO(m3/5−δn4/5+2δ+m+n logm), for anyδ>0, for any collection of points no three of which are collinear. (iii) For an arbitrary collection ofm points, we can calculate the number of incidences between them andn planes by a randomized algorithm whose expected time complexity isO((m3/4−δn3/4+3δ+m) log2n+n logn logm) for anyδ>0. (iv) Givenm points andn planes, we can find the plane lying immediately below each point in randomized expected timeO([m3/4−δn3/4+3δ+m] log2n+n logn logm) for anyδ>0. (v) The maximum number of facets (i.e., (d−1)-dimensional faces) boundingm distinct cells in an arrangement ofn hyperplanes ind dimensions,d>3, isO(m2/3nd/3 logn+nd−1). This is also an upper bound for the number of incidences betweenn hyperplanes ind dimensions andm vertices of their arrangement. The combinatorial bounds in (i) and (v) and the general bound in (ii) are almost tight.

The complexity of many cells in arrangements of planes and related problems