Symmetry Detection of Rational Space Curves from their Curvature and Torsion

We present a novel, deterministic, and efficient method to detect whether a given rational space curve is symmetric. By using well-known differential invariants of space curves, namely the curvature and torsion, the method is significantly faster, simpler, and more general than an earlier method addressing a similar problem. To support this claim, we present an analysis of the arithmetic complexity of the algorithm and timings from an implementation in Sage.Comment: 25 page

Involutions of polynomially parametrized surfaces

We provide an algorithm for detecting the involutions leaving a surface defined by a polynomial parametrization invariant. As a consequence, the symmetry axes, symmetry planes and symmetry center of the surface, if any, can be determined directly from the parametrization, without computing or making use of the implicit representation. The algorithm is based on the fact, proven in the paper, that any involution of the surface comes from an involution of the parameter space (the real plane, in our case); therefore, by determining the latter, the former can be found. The algorithm has been implemented in the computer algebra system Maple 17. Evidence of its efficiency for moderate degrees, examples and a complexity analysis are also given

Detecting Similarity of Rational Plane Curves

A novel and deterministic algorithm is presented to detect whether two given rational plane curves are related by means of a similarity, which is a central question in Pattern Recognition. As a by-product it finds all such similarities, and the particular case of equal curves yields all symmetries. A complete theoretical description of the method is provided, and the method has been implemented and tested in the Sage system for curves of moderate degrees.Comment: 22 page

Dark matter and U(1)' symmetry for the right-handed neutrinos

We consider a U(1)' gauge symmetry acting on three generations of right-handed neutrinos. The U(1)' symmetry is broken at the TeV scale and its remnant discrete symmetry makes one of the right-handed neutrinos stable. As a natural consequence of the anomaly cancellation, the neutrino mass matrix consists of a combination of Type I (TeV scale) seesaw and radiative correction. The stable right-handed neutrino communicates with the Standard Model via s-channel exchange of the Higgs field and the U(1)' gauge boson, so that the observed relic density for dark matter is obtained in a wide range of the parameter space. The experimental signatures in collider and other experiments are briefly discussed.Comment: 16 pages, 4 figure

Lie symmetries of birational maps preserving genus 0 fibrations

Preprint.We prove that any planar birational integrable map, which preserves a fibration given by genus $0$ curves has a Lie symmetry and some associated invariant measures. The obtained results allow to study in a systematic way the global dynamics of these maps. Using this approach, the dynamics of several maps is described. In particular we are able to give, for particular examples, the explicit expression of the rotation number function, and the set of periods of the considered maps.Preprin

Stationary Rotating Strings as Relativistic Particle Mechanics

Stationary rotating strings can be viewed as geodesic motions in appropriate metrics on a two-dimensional space. We obtain all solutions describing stationary rotating strings in flat spacetime as an application. These rotating strings have infinite length with various wiggly shapes. Averaged value of the string energy, the angular momentum and the linear momentum along the string are discussed.Comment: 20pages, 7 figure