Fawaz Alharbi - Academia.edu (original) (raw)

Papers by Fawaz Alharbi

International Journal of Mathematical Analysis, 2019

This article is distributed under the Creative Commons by-nc-nd Attribution License.

Topology and its Applications, 2012

Quasi boundary and quasi corner singularities of functions are discussed. They correspond to the ... more Quasi boundary and quasi corner singularities of functions are discussed. They correspond to the classifications of Lagrangian projections with a boundary or a corner. The geometry of bifurcation diagrams and caustics of simple quasi boundary and quasi corner singularities in R 3 and R 4 are described.

International Journal of Algebra, 2019

We obtain a list of all simple classes of singularities of map germs from R 2 to R 3 with respect... more We obtain a list of all simple classes of singularities of map germs from R 2 to R 3 with respect to the quasi equivalence relation. A comparison between quasi and Mond's classes and their relations with boundary (Arnold's) and corner classes are discussed.

Facta Universitatis, Series: Mathematics and Informatics, 2019

We obtain a list of all simple classes of singularities of curves (irreducible and reducible)... more We obtain a list of all simple classes of singularities of curves (irreducible and reducible) in real spaces of any dimension with respect to the quasi equivalence relation.

Mathematics

We obtain a list of simple classes of singularities of function germs with respect to the quasi m... more 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 . Our main results are Theorem on the classification of simple quasi classes, and Theorem on the classification of Lagrangian submanifolds with smooth varieties. The brief description of the main results is given in the next section.

Quasi-Projections of Surfaces with Boundaries

Journal of Mathematical Sciences, 2014

We classify simple singularities of projections to a plane of surfaces embedded into threespace a... more We classify simple singularities of projections to a plane of surfaces embedded into threespace and equipped with a boundary up to a special equivalence relation that is more rough than the standard one. Relations to other classifications are described as well.

Quasi corner singularities

Proceedings of the Steklov Institute of Mathematics, 2010

We consider the classification of germs of functions up to a nonstandard equivalence relation sim... more We consider the classification of germs of functions up to a nonstandard equivalence relation similar to the quasi boundary equivalence and quasi equivalence of projections recently introduced by the second author. In fact, it is more rough than the classification of functions with respect to the group of diffeomorphisms preserving a corner (that is, a union of a pair of

International Journal of Mathematical Analysis, 2019

This article is distributed under the Creative Commons by-nc-nd Attribution License.

Topology and its Applications, 2012

Quasi boundary and quasi corner singularities of functions are discussed. They correspond to the ... more Quasi boundary and quasi corner singularities of functions are discussed. They correspond to the classifications of Lagrangian projections with a boundary or a corner. The geometry of bifurcation diagrams and caustics of simple quasi boundary and quasi corner singularities in R 3 and R 4 are described.

International Journal of Algebra, 2019

We obtain a list of all simple classes of singularities of map germs from R 2 to R 3 with respect... more We obtain a list of all simple classes of singularities of map germs from R 2 to R 3 with respect to the quasi equivalence relation. A comparison between quasi and Mond's classes and their relations with boundary (Arnold's) and corner classes are discussed.

Facta Universitatis, Series: Mathematics and Informatics, 2019

We obtain a list of all simple classes of singularities of curves (irreducible and reducible)... more We obtain a list of all simple classes of singularities of curves (irreducible and reducible) in real spaces of any dimension with respect to the quasi equivalence relation.

Mathematics

We obtain a list of simple classes of singularities of function germs with respect to the quasi m... more 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 . Our main results are Theorem on the classification of simple quasi classes, and Theorem on the classification of Lagrangian submanifolds with smooth varieties. The brief description of the main results is given in the next section.

Quasi-Projections of Surfaces with Boundaries

Journal of Mathematical Sciences, 2014

We classify simple singularities of projections to a plane of surfaces embedded into threespace a... more We classify simple singularities of projections to a plane of surfaces embedded into threespace and equipped with a boundary up to a special equivalence relation that is more rough than the standard one. Relations to other classifications are described as well.

Quasi corner singularities

Proceedings of the Steklov Institute of Mathematics, 2010

We consider the classification of germs of functions up to a nonstandard equivalence relation sim... more We consider the classification of germs of functions up to a nonstandard equivalence relation similar to the quasi boundary equivalence and quasi equivalence of projections recently introduced by the second author. In fact, it is more rough than the classification of functions with respect to the group of diffeomorphisms preserving a corner (that is, a union of a pair of