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Nilpotent normal form for divergence-free vector fields and volume-preserving maps
We study the normal forms for incompressible flows and maps in the
neighborhood of an equilibrium or fixed point with a triple eigenvalue. We
prove that when a divergence free vector field in has nilpotent
linearization with maximal Jordan block then, to arbitrary degree, coordinates
can be chosen so that the nonlinear terms occur as a single function of two
variables in the third component. The analogue for volume-preserving
diffeomorphisms gives an optimal normal form in which the truncation of the
normal form at any degree gives an exactly volume-preserving map whose inverse
is also polynomial inverse with the same degree.Comment: laTeX, 20 pages, 1 figur
Nilpotent normal form for divergence-free vector fields and volume-preserving maps
We study the normal forms for incompressible flows and maps in the
neighborhood of an equilibrium or fixed point with a triple eigenvalue. We
prove that when a divergence free vector field in has nilpotent
linearization with maximal Jordan block then, to arbitrary degree, coordinates
can be chosen so that the nonlinear terms occur as a single function of two
variables in the third component. The analogue for volume-preserving
diffeomorphisms gives an optimal normal form in which the truncation of the
normal form at any degree gives an exactly volume-preserving map whose inverse
is also polynomial inverse with the same degree.Comment: laTeX, 20 pages, 1 figur
On the representations and -equivariant normal form for solenoidal Hopf-zero singularities
In this paper, we deal with the solenoidal conservative Lie algebra
associated to the classical normal form of Hopf-zero singular system. We
concentrate on the study of some representations and -equivariant
normal form for such singular differential equations. First, we list some of
the representations that this Lie algebra admits. The vector fields from this
Lie algebra could be expressed by the set of ordinary differential equations
where the first two of them are in the canonical form of a one-degree of
freedom Hamiltonian system and the third one depends upon the first two
variables. This representation is governed by the associated Poisson algebra to
one sub-family of this Lie algebra. Euler's form, vector potential, and Clebsch
representation are other representations of this Lie algebra that we list here.
We also study the non-potential property of vector fields with Hopf-zero
singularity from this Lie algebra. Finally, we examine the unique normal form
with non-zero cubic terms of this family in the presence of the symmetry group
. The theoretical results of normal form theory are illustrated
with the modified Chua's oscillator
Secants of Lagrangian Grassmannians
We study the dimensions of secant varieties of the Grassmannian of Lagrangian
subspaces in a symplectic vector space. We calculate these dimensions for third
and fourth secant varieties. Our result is obtained by providing a normal form
for four general points on such a Grassmannian and by explicitly calculating
the tangent spaces at these four points
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