Parallel bundles in planar map geometries

Parallel lines are very important objects in Euclid plane geometry and its behaviors can be gotten by one's intuition. But in a planar map geometry, a kind of the Smarandache geometries, the sutation is complex since it may contains elliptic or hyperbolic points. This paper concentrates on the behavior of parallel bundles, a generazation of parallel lines in plane geometry and obtains characteristics for for parallel bundles.Comment: 16page

On the parallel lines for nondegenerate conics

Computation of parallel lines (envelopes) to parabolas, ellipses, and hyperbolas is of importance in structure engineering and theory of mechanisms. Homogeneous polynomials that implicitly define parallel lines for the given offset to a conic are found by computing Groebner bases for an elimination ideal of a suitably defined affine variety. Singularity of the lines is discussed and their singular points are explicitly found as functions of the offset and the parameters of the conic. Critical values of the offset are linked to the maximum curvature of each conic. Application to a finite element analysis is shown. Keywords: Affine variety, elimination ideal, Groebner basis, homogeneous polynomial, singularity, family of curves, envelope, pitch curve, undercutting, cam surfaceComment: 40 pages, 10 figures, TOC, 3 appendices, short version of this paper was presented at the 5th Annual Hawaii International Conference on Statistics, Mathematics and Related Fields, January 16 - 18, 2006, Honolulu Hawaii, US

Curve diffusion and straightening flows on parallel lines

In this paper, we study families of immersed curves $\gamma:(-1,1)\times[0,T)\rightarrow\mathbb{R}^2$ with free boundary supported on parallel lines $\{\eta_1, \eta_2\}:\mathbb{R}\rightarrow\mathbb{R}^2$ evolving by the curve diffusion flow and the curve straightening flow. The evolving curves are orthogonal to the boundary and satisfy a no-flux condition. We give estimates and monotonicity on the normalised oscillation of curvature, yielding global results for the flows.Comment: 35 pages, 3 figure

Analysis of double-parallel amplified recirculating optical-delay lines

A novel method of analysis of double-parallel amplified recirculating optical-delay lines (DPAROD) is presented. The location of the maxima and the minima of the transfer function for this configuration is calculated and experimentally demonstrated. The influence of different parameters, such as the coupling coefficients, gains, lengths of the fiber loops and fractional losses of the directional couplers, on the shape of the transfer function are analyzed. Different measurements have been taken to verify this model. The potential application of these interconnected delay loops as filters is a reason for developing this method.Publicad

Melting of Flux Lines in an Alternating Parallel Current

We use a Langevin equation to examine the dynamics and fluctuations of a flux line (FL) in the presence of an {\it alternating longitudinal current} $J_{\parallel}(\omega)$. The magnus and dissipative forces are equated to those resulting from line tension, confinement in a harmonic cage by neighboring FLs, parallel current, and noise. The resulting mean-square FL fluctuations are calculated {\it exactly}, and a Lindemann criterion is then used to obtain a nonequilibrium `phase diagram' as a function of the magnitude and frequency of $J_{\parallel}(\omega)$. For zero frequency, the melting temperature of the mixed phase (a lattice, or the putative "Bose" or "Bragg Glass") vanishes at a limiting current. However, for any finite frequency, there is a non-zero melting temperature.Comment: 5 pages, 1 figur

Monte-Carlo study of anisotropic scaling generated by disorder

We analyze the critical properties of the three-dimensional Ising model with linear parallel extended defects. Such a form of disorder produces two distinct correlation lengths, a parallel correlation length $\xi_\parallel$ in the direction along defects, and a perpendicular correlation length $\xi_\perp$ in the direction perpendicular to the lines. Both $\xi_\parallel$ and $\xi_\perp$ diverge algebraically in the vicinity of the critical point, but the corresponding critical exponents $\nu_\parallel$ and $\nu_\perp$ take different values. This property is specific for anisotropic scaling and the ratio $\nu_\parallel/\nu_\perp$ defines the anisotropy exponent $\theta$. Estimates of quantitative characteristics of the critical behaviour for such systems were only obtained up to now within the renormalization group approach. We report a study of the anisotropic scaling in this system via Monte Carlo simulation of the three-dimensional system with Ising spins and non-magnetic impurities arranged into randomly distributed parallel lines. Several independent estimates for the anisotropy exponent $\theta$ of the system are obtained, as well as an estimate of the susceptibility exponent $\gamma$. Our results corroborate the renormalization group predictions obtained earlier.Comment: 22 pages, 9 figure