scholarly journals Normal forms for CR singular codimension-two Levi-flat submanifolds

2015 ◽  
Vol 275 (1) ◽  
pp. 115-165 ◽  
Author(s):  
Xianghong Gong ◽  
Jiří Lebl
Keyword(s):  
Normal Forms ◽  
Codimension Two ◽  
Flat Submanifolds

scholarly journals One-Flat submanifolds with codimension two

1969 ◽  
Vol 13 (1) ◽  
pp. 220-223
Author(s):  
Hiroshi Noguchi
Keyword(s):  
Codimension Two ◽  
Flat Submanifolds

scholarly journals Codimension Two Determinantal Varieties with Isolated Singularities

2014 ◽  
Vol 115 (2) ◽  
pp. 161 ◽  
Author(s):  
Maria Aparecida Soares Ruas ◽  
Miriam Da Silva Pereira
Keyword(s):  
Betti Number ◽  
Normal Forms ◽  
Milnor Number ◽  
Isolated Singularities ◽  
Generic Fiber ◽  
Linear Projection ◽  
Codimension Two ◽  
Determinantal Varieties ◽  
Surface Singularities ◽  
Generic Section

We study codimension two determinantal varieties with isolated singularities. These singularities admit a unique smoothing, thus we can define their Milnor number as the middle Betti number of their generic fiber. For surfaces in $\mathsf{C}^4$, we obtain a Lê-Greuel formula expressing the Milnor number of the surface in terms of the second polar multiplicity and the Milnor number of a generic section. We also relate the Milnor number with Ebeling and Gusein-Zade index of the $1$-form given by the differential of a generic linear projection defined on the surface. To illustrate the results, in the last section we compute the Milnor number of some normal forms from Frühbis-Krüger and Neumer [7] list of simple determinantal surface singularities.

A NOTE ON THE TRIPLE-ZERO LINEAR DEGENERACY: NORMAL FORMS, DYNAMICAL AND BIFURCATION BEHAVIORS OF AN UNFOLDING

2002 ◽  
Vol 12 (12) ◽  
pp. 2799-2820 ◽  
Author(s):  
E. FREIRE ◽  
E. GAMERO ◽  
A. J. RODRÍGUEZ-LUIS ◽  
A. ALGABA
Keyword(s):  
Periodic Orbits ◽  
Normal Forms ◽  
Zero Eigenvalue ◽  
Codimension Two ◽  
Local Bifurcations ◽  
Linear Degeneracy

This paper is devoted to the analysis of bifurcations in a three-parameter unfolding of a linear degeneracy corresponding to a triple-zero eigenvalue. We carry out the study of codimension-two local bifurcations of equilibria (Takens–Bogdanov and Hopf-zero) and show that they are nondegenerate. This allows to put in evidence the presence of several kinds of bifurcations of periodic orbits (secondary Hopf,…) and of global phenomena (homoclinic, heteroclinic). The results obtained are applied in the study of the Rössler equation.

Dynamical Systems with a Codimension-One Invariant Manifold: The Unfoldings and Its Bifurcations

2015 ◽  
Vol 25 (06) ◽  
pp. 1550091 ◽  
Author(s):  
Kie Van Ivanky Saputra
Keyword(s):  
Dynamical System ◽  
Invariant Manifold ◽  
Normal Forms ◽  
Volterra Model ◽  
Infection Model ◽  
Global Bifurcations ◽  
Codimension Two ◽  
Codimension One ◽  
Variation Of Parameters ◽  
Local And Global Bifurcations

We investigate a dynamical system having a special structure namely a codimension-one invariant manifold that is preserved under the variation of parameters. We derive conditions such that bifurcations of codimension-one and of codimension-two occur in the system. The normal forms of these bifurcations are derived explicitly. Both local and global bifurcations are analyzed and yield the transcritical bifurcation as the codimension-one bifurcation while the saddle-node–transcritical interaction and the Hopf–transcritical interactions as the codimension-two bifurcations. The unfolding of this degeneracy is also analyzed and reveal global bifurcations such as homoclinic and heteroclinic bifurcations. We apply our results to a modified Lotka–Volterra model and to an infection model in HIV diseases.

METHODS OF REDUCING OF LARGE BOOLEAN FORMULAS REPRESENTED IN A CONJUNCTIVE NORMAL FORM FOR DETERMINING THEIR SATISFIABILITY

2020 ◽  
Author(s):  
N.I. Gdansky ◽  
◽  
A.A. Denisov ◽  
Keyword(s):  
Normal Form ◽  
Normal Forms ◽  
Conjunctive Normal Form ◽  
Boolean Formulas

The article explores the satisfiability of conjunctive normal forms used in modeling systems.The problems of CNF preprocessing are considered.The analysis of particular methods for reducing this formulas, which have polynomial input complexity is given.

Regularization of Integral Operators in Singularly Perturbed Problems Using Normal Forms

Vestnik MEI ◽  
2019 ◽  
Vol 6 ◽  
pp. 131-137
Author(s):  
Abdukhafiz A. Bobodzhanova ◽  
◽  
Valeriy F. Safonov ◽  
Keyword(s):  
Normal Forms ◽  
Integral Operators ◽  
Singularly Perturbed ◽  
Singularly Perturbed Problems

Constructing symplectic manifolds

2017 ◽  
Author(s):  
Dusa McDuff ◽  
Dietmar Salamon
Keyword(s):  
Final Section ◽  
Symplectic Manifolds ◽  
Homotopy Equivalence ◽  
Codimension Two ◽  
Open Manifolds ◽  
Connected Sums ◽  
Symplectic Forms ◽  
Ambient Manifold ◽  
Blowing Up ◽  
Four Manifolds

This chapter examines various ways to construct symplectic manifolds and submanifolds. It begins by studying blowing up and down in both the complex and the symplectic contexts. The next section is devoted to a discussion of fibre connected sums and describes Gompf’s construction of symplectic four-manifolds with arbitrary fundamental group. The chapter also contains an exposition of Gromov’s telescope construction, which shows that for open manifolds the h-principle rules and the inclusion of the space of symplectic forms into the space of nondegenerate 2-forms is a homotopy equivalence. The final section outlines Donaldson’s construction of codimension two symplectic submanifolds and explains the associated decompositions of the ambient manifold.

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