Singularities of equidistants and global centre symmetry sets of Lagrangian submanifolds

We define the Global Centre Symmetry set (GCS) of a smooth closed m-dimensional submanifold M of R^n, $n \leq 2m$, which is an affinely invariant generalization of the centre of a k-sphere in R^{k+1}. The GCS includes both the centre symmetry set defined by Janeczko and the Wigner caustic defined by Berry. We develop a new method for studying generic singularities of the GCS which is suited to the case when M is lagrangian in R^{2m} with canonical symplectic form. The definition of the GCS, which slightly generalizes one by Giblin and Zakalyukin, is based on the notion of affine equidistants, so, we first study singularities of affine equidistants of Lagrangian submanifolds, classifying all the stable ones. Then, we classify the affine-Lagrangian stable singularities of the GCS of Lagrangian submanifolds and show that, already for smooth closed convex curves in R^2, many singularities of the GCS which are affine stable are not affine-Lagrangian stable.Comment: 26 pages, 2 figure

Development of the Magnetic Excitations of Charge-Stripe Ordered La(2-x)Sr(x)NiO(4) on Doping Towards Checkerboard Charge Order

The magnetic excitation spectrums of charge stripe ordered La(2-x)Sr(x)NiO(4) x = 0.45 and x = 0.4 were studied by inelastic neutron scattering. We found the magnetic excitation spectrum of x = 0.45 from the ordered Ni^2+ S = 1 spins to match that of checkerboard charge ordered La(1.5)Sr(0.5)NiO(4). The distinctive asymmetry in the magnetic excitations above 40 meV was observed for both doping levels, but an additional ferromagnetic mode was observed in x = 0.45 and not in the x = 0.4. We discuss the origin of crossover in the excitation spectrum between x = 0.45 and x = 0.4 with respect to discommensurations in the charge stripe structure.Comment: 4 Figures. To be appear in the J. Kor. Phys. Soc. as a proceedings paper from the ICM 2012 conferenc

Gamma-Ray Burst Pulse Correlations as Redshift Indicators

Correlations among pulse properties in the prompt emission of long GRBs can potentially be used as cosmological distance indicators to estimate redshifts of GRBs to which these pulses belong. We demonstrate application of this technique to a sample of GRBs for which redshifts are not known. We also study the scatter of predicted redshifts of pulses found within individual bursts. We explore the characteristics of this scatter in hopes of identifying systematic corrections and/or pulse subsets that can be used to increase the technique's reliability.Comment: 3 pages and 6 figures, to appear in the proceedings of the Sixth Huntsville Gamma-Ray Burst Symposium, edited by C.A. Meegan, N. Gehrels, and C. Kouvelioto

Level sets of functions and symmetry sets of smooth surface sections

We prove that the level sets of a real C^s function of two variables near a non-degenerate critical point are of class C^[s/2] and apply this to the study of planar sections of surfaces close to the singular section by the tangent plane at hyperbolic points or elliptic points, and in particular at umbilic points. We also analyse the cases coming from degenerate critical points, corresponding to elliptic cusps of Gauss on a surface, where the differentiability is now reduced to C^[s/4]. However in all our applications to symmetry sets of families of plane curves, we assume the C^infty smoothness.Comment: 15 pages, Latex, 6 grouped figures. The final version will appear in Mathematics of Surfaces. Lecture Notes in Computer Science (2005