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Register a new userThe trunkenness of a volume-preserving vector field
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Authors
Pierre Dehornoy
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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.
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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.
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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 $mathcal{M}$.
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