Generalized vector variational-like inequalities and nonsmooth vector optimization problems

In this article, we establish some relationships between a solution of generalized vector variational-like inequalities and an efficient solution or a weakly efficient solution to the nonsmooth vector optimization problem under the assumptions of pseudoinvexity or invariant pseudomonotonicity. Our results extend and improve the corresponding results in the literature.     

Impulsive control systems with commutative vector fields

We consider variational problems with control laws given by systems of ordinary differential equations whose vector fields depend linearly on the time derivativeu=(u ¹,...,u ^m ) of the controlu=(u ¹,...,u ^m ). The presence of the derivativeu, which is motivated by recent applications in Lagrangian mechanics, causes an impulsive dynamics: at any jump of the control, one expects a jump of the state.The main assumption of this paper is the commutativity of the vector fields that multiply theu . This hypothesis allows us to associate our impulsive systems and the corresponding adjoint systems to suitable nonimpulsive control systems, to which standard techniques can be applied. In particular, we prove a maximum principle, which extends Pontryagin's maximum principle to impulsive commutative systems.     

Integration of locally exponential Lie algebras of vector fields

A locally convex Lie algebra is said to be locally exponential if it belongs to some local Lie group in canonical coordinates. In this note we give criteria for locally exponential Lie algebras of vector fields on an infinite-dimensional manifold to integrate to global Lie group actions. Moreover, we show that all necessary conditions are satisfied if the manifold is finite-dimensional connected and σ-compact, which leads to a generalization of Palais’ Integrability Theorem.      

Deformations of vector fields and canonical coordinates on coadjoint orbits

We establish the concurrence of the existence of a canonical linear embedding, enabling us to pass to the Darboux coordinates on the coadjoint orbits, and the existence of a polarization of a certain linear functional. In a corollary to the main theorem we prove that every polarization is normal, which means that it satisfies the Pukansky condition.     

Projective vector fields on Finsler manifolds

In this paper, we give the equation that characterizes projective vector fields on a Finsler manifold by the local coordinate. Moreover, we obtain a feature of the projective fields on the compact Finsler manifold with non-positive flag curvature and the non-existence of projective vector fields on the compact Finsler manifold with negative flag curvature. Furthermore, we deduce some expectable, but non-trivial relationships between geometric vector fields such as projective, affine, conformal, homothetic and Killing vector fields on a Finsler manifold.     

Commutativity preserving linear maps and Lie automorphisms of strictly triangular matrix space

In this paper we classify linear maps preserving commutativity in both directions on the space N(F) of strictly upper triangular (n+1)×(n+1) matrices over a field F. We show that for n3 a linear map on N(F) preserves commutativity in both directions if and only if =^′+f where ^′ is a product of standard maps on N(F) and f is a linear map of N(F) into its center.     

Asymptotic stability at infinity for differentiable vector fields of the plane

Let be a differentiable (but not necessarily C¹) vector field, where σ>0 and . Denote by R(z) the real part of z∈C. If for some ?>0 and for all , no eigenvalue of DpX belongs to , then: (a) for all , there is a unique positive semi-trajectory of X starting at p; (b) it is associated to X, a well-defined number I(X) of the extended real line [−∞,∞) (called the index of X at infinity) such that for some constant vector v∈R² the following is satisfied: if I(X) is less than zero (respectively greater or equal to zero), then the point at infinity ∞ of the Riemann sphere R²∪{∞} is a repellor (respectively an attractor) of the vector field X+v.     

Homology of the Lie algebra of vector fields on the line with coefficients in symmetric powers of its adjoint representation

We compute the homology of the Lie algebra W ₁ of (polynomial) vector fields on the line with coefficients in symmetric powers of its adjoint representation. We also list the results obtained so far for the homology with coefficients in tensor powers and, in turn, use them for partially computing the homology of the Lie algebra of W ₁-valued currents on the line.     

Effect of magnetic field on the flow of a fourth order fluid

A study is made of the flow engendered in a semi-infinite expanse of an incompressible non-Newtonian fluid by an infinite rigid plate moving with an arbitrary velocity in its own plane. The fluid is considered to be fourth order and electrically conducting. A magnetic field is applied in the transverse direction to the flow. The nonlinear problem is solved for constant magnetic field analytically using reduction methods as well as numerically and expressions for the velocity field are obtained. Limiting cases of interest can be deduced by choosing suitable parametric values.     

On the geometry of tangent hyperquadric bundles: CR and pseudoharmonic vector fields

We study the natural almost CR structure on the total space of a subbundle of hyperquadrics of the tangent bundle T(M) over a semi-Riemannian manifold (M, g) and show that if the Reeb vector ξ of an almost contact Riemannian manifold is a CR map then the natural almost CR structure on M is strictly pseudoconvex and a posteriori ξ is pseudohermitian. If in addition ξ is geodesic then it is a harmonic vector field. As an other application, we study pseudoharmonic vector fields on a compact strictly pseudoconvex CR manifold M, i.e. unit (with respect to the Webster metric associated with a fixed contact form on M) vector fields X ε H(M) whose horizontal lift X↑ to the canonical circle bundle S¹ → C(M) → M is a critical point of the Dirichlet energy functional associated to the Fefferman metric (a Lorentz metric on C(M)). We show that the Euler–Lagrange equations satisfied by X^↑ project on a nonlinear system of subelliptic PDEs on M. Mathematics Subject Classifications (2000): 53C50, 53C25, 32V20     

On conformal vector fields on Randers manifolds

In this paper,we study conformal vector fields on a Randers manifold with certain curvature properties.In particular,we completely determine conformal vector fields on a Randers manifold of weakly isotropic flag curvature.     

Killing vector fields of constant length on Riemannian manifolds

We study the nontrivial Killing vector fields of constant length and the corresponding flows on complete smooth Riemannian manifolds. Various examples are constructed of the Killing vector fields of constant length generated by the isometric effective almost free but not free actions of S ¹ on the Riemannian manifolds close in some sense to symmetric spaces. The latter manifolds include “almost round” odd-dimensional spheres and unit vector bundles over Riemannian manifolds. We obtain some curvature constraints on the Riemannian manifolds admitting nontrivial Killing fields of constant length.     

Eigenvalues of vector fields,Bott's residue formula and integral invariants

Given a compatible vector field on a compact connected almost-complex manifold, we show in this article that the multiplicities of eigenvalues among the zero point set of this vector field have intimate relations. We highlight a special case of our result and reinterpret it as a vanishing-type result in the framework of the celebrated Atiyah–Bott–Singer localization formula. This new point of view, via the Chern–Weil theory and a strengthened version of Bott's residue formula, can lead to an obstruction to Killing real holomorphic vector fields on compact Hermitian manifolds in terms of a curvature integral.     

Bifurcations of limit cycles in equivariant quintic planar vector fields

In this paper, we obtain 23 limit cycles for a _Z3$Z_3$-equivariant near-Hamiltonian system of degree 5 which is the perturbation of a _Z6$Z_6$-equivariant quintic Hamiltonian system. The configuration of these limit cycles is new and different from the configuration obtained by H.S.Y. Chan, K.W. Chung and J. Li, where the unperturbed system is a _Z3$Z_3$-equivariant quintic Hamiltonian system. Our unperturbed system is different from the unperturbed systems studied by Y. Wu and M. Han. The limit cycles are obtained by Poincaré–Pontryagin theorem and Poincaré–Bendixson theorem.     

A converse of asymptotic formulae in simultaneous approximation

In the general setting of simultaneous approximation by sequences of linear shape preserving operators, this paper contains a sort of converse result of Voronovskaya-type asymptotic formulae. As a by-product a saturation result is derived. Applications to some very well-known approximation processes are also presented.     

Exact classification of nondegenerate devergence-free vector fields on surfaces of small genus

The paper contains the classification up to conjugation in the categoryC ^k,k=1,...,∞, of complete nondegenerate divergence-free vector fieldson a compact surface (possibly with boundary) almost all of whose trajectories are closed. The classification up to conjugation in given for arbitrary nondegenerate divergence-free vector fields on the surfacesS ², ,K ²,T ², , possibly with holes. Translated fromMatematicheskie Zametki, Vol. 65, No. 3, pp. 336–353 March, 1999.     

Exact smooth classification of hamiltonian vector fields on two-dimensional manifolds

A complete exact classification of Hamiltonian systems with Morse Hamiltonians on two-dimensional manifolds is given, i.e., the systems are classified up to diffeomorphisms mapping vector fields into vector fields. The classification imposes no restrictions on Morse functions. Translated fromMatematicheskie Zametki, Vol. 61, No. 2, pp. 179–200, February, 1997. Translated by O. V. Sipacheva     

On the Dulac's problem for piecewise analytic vector fields

This paper presents a version of Dulac's Problem for piecewise analytic vector fields that states conditions for the number of limit cycles around certain minimal sets. A suitable model-theoretic structure is introduced under which a qualitative investigation of the problem is settled.