We construct a new invariant-the trunkenness-for volume-perserving vector fields on S^3 up to volume-preserving diffeomorphism. We prove that the trunkenness is independent from the helicity and that it is the limit of a knot invariant (called the trunk) computed on long pieces of orbits.
The pendulum, in the presence of linear dissipation and a constant torque, is a non-integrable, nonlinear differential equation. In this paper, using the idea of rotated vector fields, derives the relation between the applied force $beta$ and the periodic solution, and a conclusion that the critical value of $beta$ is a fixed one in the over damping situation. These results are of practical significance in the study of charge-density waves in physics.
A proof is given of the vector identity proposed by Gubarev, Stodolsky and Zakarov that relates the volume integral of the square of a 3-vector field to non-local integrals of the curl and divergence of the field. The identity is applied to the case of the magnetic vector potential and magnetic field of a rotating charged shell. The latter provides a straightforward exercise in the use of the addition theorem of spherical harmonics.
In this paper, we focus on the construction of high order volume preserving in- tegrators for divergence-free vector fields: the monomial basis, the exponential basis and tensor product of the monomial and the exponential basis. We first prove that the commutators of elementary divergence-free vector fields (EDFVF) of those three kinds are still divergence-free vector fields of the same kind. Assuming then there is only diagonal part of divergence-free vector field of the monomial basis, for those three kinds of divergence-free vector fields, we construct high order volume-preserving inte- grators using the multi-commutators for EDFVFs. Moreover, we consider the ordering of the EDFVFs and their commutators to reduce the error of the schemes, showing by numerical tests that the strategy in [9] works very well.
In 1985, Barnsley and Harrington defined a ``Mandelbrot Set $mathcal{M}$ for pairs of similarities --- this is the set of complex numbers $z$ with $0<|z|<1$ for which the limit set of the semigroup generated by the similarities $x mapsto zx$ and $x mapsto z(x-1)+1$ is connected. Equivalently, $mathcal{M}$ is the closure of the set of roots of polynomials with coefficients in $lbrace -1,0,1 rbrace$. Barnsley and Harrington already noted the (numerically apparent) existence of infinitely many small ``holes in $mathcal{M}$, and conjectured that these holes were genuine. These holes are very interesting, since they are ``exotic components of the space of (2 generator) Schottky semigroups. The existence of at least one hole was rigorously confirmed by Bandt in 2002, and he conjectured that the interior points are dense away from the real axis. We introduce the technique of traps to construct and certify interior points of $mathcal{M}$, and use them to prove Bandts Conjecture. Furthermore, our techniques let us certify the existence of infinitely many holes in $mathcal{M}$.