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Boundary asymptotics of the relative Bergman kernel metric for hyperelliptic curves

100%

Dong R. X.

Complex Manifolds

2017

tom 4

nr 1

7-15

We survey variations of the Bergman kernel and their asymptotic behaviors at degeneration. For a Legendre family of elliptic curves, the curvature form of the relative Bergman kernel metric is equal to the Poincaré metric on ℂ \ {0,1}. The cases of other elliptic curves are either the same or trivial. Two proofs depending on elliptic functions’ special properties and Abelian differentials’ Taylor expansions are discussed, respectively. For a holomorphic family of hyperelliptic nodal or cuspidal curves and their Jacobians, we announce our results on the Bergman kernel asymptotics near various singularities. For genus-two curves particularly, asymptotic formulas with precise coefficients involving the complex structure information are written down explicitly.

Linear waves with singularities along a cusp

100%

Kołakowski H.

Biuletyn Wojskowej Akademii Technicznej

2000

tom 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.

A survey on polygonal portraits of manifolds

88%

Kobayashi M.

Demonstratio Mathematica

2010

tom 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

75%

Kołakowski H.

Bulletin of the Polish Academy of Sciences. Mathematics

2000

tom 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.

The structure of the cusps of valves in the human foetal great saphenous vein

75%

Czarniawska-Grzesinska M. , Bruska M.

Folia Morphologica

2003

tom 62

nr 3

275-276

The study was performed on 110 great saphenous veins in human foetuses of both sexes aged 9 to 37 weeks. The earliest well-shaped valves were observed in foetuses aged 13 weeks. In these foetuses the number of valves varies from 2 to 7. Consecutive microscopic sections revealed that the developing valves at their origin present thickening of the endothelium which is continuous into the cusps of the valves. The bicuspid cusps are crescent-shaped and both surfaces are lined by endothelium.

Notes on the morphology of the tricuspid valve in the adult human heart

63%

Skwarek M. , Grzybiak M. , Kosinski A. , Hreczecha J.

Folia Morphologica

2004

tom 63

nr 3

Rapid progress in the field of interventional cardiology has caused research in the field of morphometry of the heart to be in constant demand [7–10, 12]. In this study, performed on a group of 75 adult human hearts, the authors have attempted to assess the form and number of the main and accessory cusps in the tricuspid valve. We have classified particular forms into 8 groups, depending on the number of cusps and we have divided the cusps into 3 main groups, depending on the support of the chordae tendineae.