1,204 research outputs found
Detecting Symmetries of Rational Plane and Space Curves
This paper addresses the problem of determining the symmetries of a plane or
space curve defined by a rational parametrization. We provide effective methods
to compute the involution and rotation symmetries for the planar case. As for
space curves, our method finds the involutions in all cases, and all the
rotation symmetries in the particular case of Pythagorean-hodograph curves. Our
algorithms solve these problems without converting to implicit form. Instead,
we make use of a relationship between two proper parametrizations of the same
curve, which leads to algorithms that involve only univariate polynomials.
These algorithms have been implemented and tested in the Sage system.Comment: 19 page
On the state space geometry of the Kuramoto-Sivashinsky flow in a periodic domain
The continuous and discrete symmetries of the Kuramoto-Sivashinsky system
restricted to a spatially periodic domain play a prominent role in shaping the
invariant sets of its chaotic dynamics. The continuous spatial translation
symmetry leads to relative equilibrium (traveling wave) and relative periodic
orbit (modulated traveling wave) solutions. The discrete symmetries lead to
existence of equilibrium and periodic orbit solutions, induce decomposition of
state space into invariant subspaces, and enforce certain structurally stable
heteroclinic connections between equilibria. We show, on the example of a
particular small-cell Kuramoto-Sivashinsky system, how the geometry of its
dynamical state space is organized by a rigid `cage' built by heteroclinic
connections between equilibria, and demonstrate the preponderance of unstable
relative periodic orbits and their likely role as the skeleton underpinning
spatiotemporal turbulence in systems with continuous symmetries. We also offer
novel visualizations of the high-dimensional Kuramoto-Sivashinsky state space
flow through projections onto low-dimensional, PDE representation independent,
dynamically invariant intrinsic coordinate frames, as well as in terms of the
physical, symmetry invariant energy transfer rates.Comment: 31 pages, 17 figures; added references, corrected typos. Due to file
size restrictions some figures in this preprint are of low quality. A high
quality copy may be obtained from
http://www.cns.gatech.edu/~predrag/papers/preprints.html#rp
Affine equivalences of surfaces of translation and minimal surfaces, and applications to symmetry detection and design
We introduce a characterization for affine equivalence of two surfaces of translation defined by either rational or meromorphic generators. In turn, this induces a similar characterization for minimal surfaces. In the rational case, our results provide algorithms for detecting affine equivalence of these surfaces, and therefore, in particular, the symmetries of a surface of translation or a minimal surface of the considered types. Additionally, we apply our results to designing surfaces of translation and minimal surfaces with symmetries, and to computing the symmetries of the higher-order Enneper surfaces.publishedVersio
Knot commensurability and the Berge conjecture
We investigate commensurability classes of hyperbolic knot complements in the
generic case of knots without hidden symmetries. We show that such knot
complements which are commensurable are cyclically commensurable, and that
there are at most hyperbolic knot complements in a cyclic commensurability
class. Moreover if two hyperbolic knots have cyclically commensurable
complements, then they are fibered with the same genus and are chiral. A
characterisation of cyclic commensurability classes of complements of periodic
knots is also given. In the non-periodic case, we reduce the characterisation
of cyclic commensurability classes to a generalization of the Berge conjecture.Comment: v3: This version is reorganized with minor errors fixed. Proposition
4.1, Corollary 4.2, and Proposition 5.8 were added. Question 7.2 was upgraded
to Theorem 7.2. 30 pages, 1 figur
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