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A survey on polygonal portraits of manifolds

Kobayashi M.

Demonstratio Mathematica

2010

Vol. 43, nr 2

433-445

Planar portraits are geometric representations of smooth manifolds defined by their generic maps into the plane. A simple subclass called the polygonal portraits is introduced, their realisations, and relations of their shapes to the topology of source manifolds are discussed. Generalisations and analogies of the results to other planar portraits are also mentioned. A list of manifolds which possibly admit polygonal portraits is given, up to diffeomorphism and up to homotopy spheres. This article is intended to give a summary on our research on the topic, and hence precise proofs will be given in other papers.

Properties of non-linear waves with singularities along a cusp

Kołakowski H.

Bulletin of the Polish Academy of Sciences. Mathematics

2000

Vol. 48, no 2

193-201

The paper deals with local solutions to the equation ([...]u = f (x, y, z, u, Du) for which the cusp [x^z = y^3] is characteristic (Du = grad u). A regularity theorem in Gevrey class is proved.

Linear waves with singularities along a cusp

Kołakowski H.

Biuletyn Wojskowej Akademii Technicznej

2000

Vol. 49, nr 1

33-40

Examples of solutions to a partial differential equation with singularities exactly on cusp {x ² = y ³} ⊂ R³ are given.

W pracy podano przykłady rowiązań pewnego równania cząstkowego z osobliwościami na {x ² = y ³} ⊂ R³.