Quasi corner singularities

Alharbi, F. M.; Zakalyukin, V. M.

doi:10.1134/s0081543810030016

Cited by 3 publications

(4 citation statements)

References 4 publications

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“…In the definition of the quasi border equivalence relation in [3], it was assumed that the border was stratified set and the stratification satisfied the Whitney Condition A. Therefore, in many cases, we will be able to establish similar results to that in [1][2][3] for semi-borders that satisfy such conditions. In particular, two function germs are pseudo semi-border equivalent, if there is a diffeomorphism acting as a change of variables, taking one germ to the other and satisfying the following condition: if one of these functions has a critical point at the semi-borders, then its image (or respectively the inverse image) also belongs to the semi-borders.…”

Section: Introductionmentioning

confidence: 68%

“…Moser's homotopy method is the main tool that will be used to establish quasi semi-border equivalence between function germs. In addition, we will be using the following technique, which is similar to Lemma 8.1 in [2] and Lemma 2.10 in [3]. It will be used to establish quasi semi-border equivalence between function germs.…”

Section: Basic Techniques Of the Classificationmentioning

confidence: 99%

“…Following the applications of quasi boundary, quasi corner, and quasi cusp equivalence relations considered in [1][2][3], respectively, we introduce the following. The concepts of the stability and simplicity of Lagrangian submanifolds with semi-borders can be defined analogously to the standard notions.…”

Section: Lagrangian Submanifolds With Smooth Varietiesmentioning

confidence: 99%

“…The aim of the relation is to preserve only the positions of critical points of functions defined on a space along a smooth hypersurface. Using Zakalyukin's idea, a similar quasi equivalence relation was introduced in which case the smooth hypersurface was replaced by an arbitrary one, called a border, in particular, when the border is a cylinder over a corner (a union of two transversal intersecting smooth hypersurfaces) or a cuspidal edge, as considered in [2,3], respectively. In the present paper, we study an even more general setting and consider an arbitrary algebraic variety of any dimension, which will be called a semi-border, and the respective relation will be named quasi semi-border equivalence.…”

Section: Introductionmentioning

confidence: 99%

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Quasi Semi-Border Singularities

Alharbi¹,

Alsaeed²

2019

Mathematics

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We obtain a list of simple classes of singularities of function germs with respect to the quasi m-boundary equivalence relation, with m ≥ 2 . The results obtained in this paper are a natural extension of Zakalyukin’s work on the new non-standard equivalent relation. In spite of the rather artificial nature of the definitions, the quasi relations have very natural applications in symplectic geometry. In particular, they are used to classify singularities of Lagrangian projections equipped with a submanifold. The main method that is used in the classification is the standard Moser’s homotopy technique. In addition, we adopt the version of Arnold’s spectral sequence method, which is described in Lemma 2. Our main results are Theorem 4 on the classification of simple quasi classes, and Theorem 5 on the classification of Lagrangian submanifolds with smooth varieties. The brief description of the main results is given in the next section.

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Section: Introductionmentioning

confidence: 68%

Section: Basic Techniques Of the Classificationmentioning

confidence: 99%

Section: Lagrangian Submanifolds With Smooth Varietiesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Quasi Semi-Border Singularities

Alharbi¹,

Alsaeed²

2019

Mathematics

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Bifurcation diagrams and caustics of simple quasi border singularities

Alharbi

2012

Topology and its Applications

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Vector fields on bifurcation diagrams of quasi singularities

Alharbi,

2024

MATH

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<p>We describe the generators of the vector fields tangent to the bifurcation diagrams and caustics of simple quasi boundary singularities. As an application, submersions on the pair $ (G, B) $, which consists of a cuspidal edge $ G $ in $ \mathbb{R}^3 $ that contains a distinguishing regular curve $ B $, are classified. This classification was used as a means to investigate the contact that a general cuspidal edge $ G $ equipped with a regular curve $ B\subset G $ has with planes. The singularities of the height functions on $ (G, B) $ are discussed and they are related to the curvatures and torsions of the distinguished curves on the cuspidal edge. In addition to this, the discriminants of the versal deformations of the submersions that were accomplished are described and they are related to the duality of the cuspidal edge.</p>

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Quasi corner singularities

Cited by 3 publications

References 4 publications

Quasi Semi-Border Singularities

Quasi Semi-Border Singularities

Bifurcation diagrams and caustics of simple quasi border singularities

Vector fields on bifurcation diagrams of quasi singularities

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