Generalized distance-squared mappings of the plane into the plane

Abstract

Abstract. We define generalized distance-squared mappings, and we concentrate on the plane to plane case. We classify generalized distance-squared mappings of the plane into the plane in a recognizable way.

Preservation of Immersed or Injective Properties by Composing Generic Generalized Distance-Squared Mappings

“…In [7], a classification result on generalized distance-squared mappings of the plane into the plane is given. If the rank of A is two, a generalized distance-squared mapping having a generic central point is a mapping of which any singular point is a fold point except one cusp point.…”

“…In [6] and [7], the properties of generic generalized distance-squared mappings are investigated. Hence, it is natural to investigate the properties of compositions with generic generalized distance-squared mappings…”

STABILITY OF <i>C</i><b>^∞</b>CONVEX INTEGRANDS

“…For proper C ∞ mappings of special type, it is natural to ask a similar question: are generic proper mappings of special type stable? Such investigations, for instance, can be found in [20] for generic projections of submanifolds, in [6] for generic projections of stable mappings and in [8][9][10][11] for generic distance-squared mappings and their generalizations.…”

Spherical method for studying Wulff shapes and related topics

“…Question 9 may be considered as the density question of stable functions in a restricted function space. Researches arisen from similar motivations as Question 9 can be found in [46,12,26,27,28,29,30]. Thus, from the viewpoint of Singularity Theory, Question 9 may be regarded as a natural question.…”

“…". Such investigations, for instance, can be found in [46] for generic projections of submanifolds, in [12] for generic projections of stable mappings and in [26,27,28,29,30] for generic distance-squared mappings and their generalizations. Motivated by these researches, we posed Question 9 in Section 1.…”