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Singularities of implicit differential systems and maximum principle

Stanisław Janeczko, Fernand Pelletier (2003)

Banach Center Publications

The integrability condition for the Lagrangian implicit differential systems of (TP,ω̇), introduced in [7], is applied for the specialized control theory systems. The Pontryagin maximum principle was reformulated in the framework of implicit differential systems and the corresponding necessary and sufficient conditions were proved. The beginning of the classification list of normal forms for Lagrangian implicit differential systems according to the symplectic equivalence is provided and the corresponding...

Singularities of non-degenerate n-ruled (n + 1)-manifolds in Euclidean space

Kentaro Saji (2004)

Banach Center Publications

The objective of this paper is to study singularities of n-ruled (n + 1)-manifolds in Euclidean space. They are one-parameter families of n-dimensional affine subspaces in Euclidean space. After defining a non-degenerate n-ruled (n + 1)-manifold we will give a necessary and sufficient condition for such a map germ to be right-left equivalent to the cross cap × interval. The behavior of a generic n-ruled (n + 1)-manifold is also discussed.

Spaces of polynomials with roots of bounded multiplicity

M. Guest, A. Kozlowski, K. Yamaguchi (1999)

Fundamenta Mathematicae

We describe an alternative approach to some results of Vassiliev ([Va1]) on spaces of polynomials, by applying the "scanning method" used by Segal ([Se2]) in his investigation of spaces of rational functions. We explain how these two approaches are related by the Smale-Hirsch Principle or the h-Principle of Gromov. We obtain several generalizations, which may be of interest in their own right.

Stratifications of polynomial spaces

Lev Birbrair (1998)

Publicacions Matemàtiques

In the paper we construct some stratifications of the space of monic polynomials in real and complex cases. These stratifications depend on properties of roots of the polynomials on some given semialgebraic subset of R or C. We prove differential triviality of these stratifications. In the real case the proof is based on properties of the action of the group of interval exchange transformations on the set of all monic polynomials of some given degree. Finally we compare stratifications corresponding...

Sur les singularités isolées d'intersections complètes quasi-homogènes

Marc Giusti (1977)

Annales de l'institut Fourier

Le résultat principal de cet article est une formule explicite donnant le nombre de Milnor d’une singularité isolée d’intersection complète quasi-homogène d’une courbe de C 3 en fonction des degrés et des poids. Ce calcul effectué par des méthodes topologiques repose sur le théorème suivant : la fibre de Milnor d’une singularité isolée d’intersection complète quasi-homogène ne dépend que des degrés et des poids à difféomorphisme près. Une autre conséquence de ce théorème est l’existence d’une morsification...

Systems of rays in the presence of distribution of hyperplanes

S. Janeczko (1995)

Banach Center Publications

Horizontal systems of rays arise in the study of integral curves of Hamiltonian systems v H on T*X, which are tangent to a given distribution V of hyperplanes on X. We investigate the local properties of systems of rays for general pairs (H,V) as well as for Hamiltonians H such that the corresponding Hamiltonian vector fields v H are horizontal with respect to V. As an example we explicitly calculate the space of horizontal geodesics and the corresponding systems of rays for the canonical distribution...

Currently displaying 101 – 120 of 148