Formation of singularities for equivariant 2+1 dimensional wave maps into the two-sphere

In this paper we report on numerical studies of the Cauchy problem for equivariant wave maps from 2+1 dimensional Minkowski spacetime into the two-sphere. Our results provide strong evidence for the conjecture that large energy initial data develop singularities in finite time and that singularity formation has the universal form of adiabatic shrinking of the degree-one harmonic map from $\mathbb{R}^2$ into $S^2$.Comment: 14 pages, 5 figures, final version to be published in Nonlinearit

Dispersion and collapse of wave maps

We study numerically the Cauchy problem for equivariant wave maps from 3+1 Minkowski spacetime into the 3-sphere. On the basis of numerical evidence combined with stability analysis of self-similar solutions we formulate two conjectures. The first conjecture states that singularities which are produced in the evolution of sufficiently large initial data are approached in a universal manner given by the profile of a stable self-similar solution. The second conjecture states that the codimension-one stable manifold of a self-similar solution with exactly one instability determines the threshold of singularity formation for a large class of initial data. Our results can be considered as a toy-model for some aspects of the critical behavior in formation of black holes.Comment: 14 pages, Latex, 9 eps figures included, typos correcte

Bifurcation and fine structure phenomena in critical collapse of a self-gravitating σ\sigma-field

Building on previous work on the critical behavior in gravitational collapse of the self-gravitating SU(2) $\sigma$-field and using high precision numerical methods we uncover a fine structure hidden in a narrow window of parameter space. We argue that this numerical finding has a natural explanation within a dynamical system framework of critical collapse

APPLICATION OF IMAGE ANALYSIS METHODS FOR ISOCENTER QUALITY ASSURANCE IN RADIOTHERAPY

One of the major procedures for testing the geometrical accuracy of devices used in radiotherapy treatments is the test of the geometrical position of the radiation isocenter. The importance of the test reflects the fact that geometrical position of the radiation isocenter generally affects the tumor targeting. At present the geometric accuracy is assessed with the Winston-Lutz test which checks the position of an image of a ball marker with the respect to the center of the radiation field as projected on a detector plane. Obviously, determination of coordinates of a single marker is not sufficient to fully account for a complicated geometry of a therapeutic device. The purpose of the study was to design a new image analysis tool to better determine the isocenter. The proposed automated procedure for determining isocenter position uses projection data acquired for a special cube phantom. The projection images of a phantom are acquired for various angles of rotation of the gantry. A procedure is proposed to extract some geometric characteristics of a therapeutic device from the projection images

APPLICATION OF IMAGE ANALYSIS METHODS FOR ISOCENTER QUALITY ASSURANCE IN RADIOTHERAPY

One of the major procedures for testing the geometrical accuracy of devices used in radiotherapy treatments is the test of the geometrical position of the radiation isocenter. The importance of the test reflects the fact that geometrical position of the radiation isocenter generally affects the tumor targeting. At present the geometric accuracy is assessed with the Winston-Lutz test which checks the position of an image of a ball marker with the respect to the center of the radiation field as projected on a detector plane. Obviously, determination of coordinates of a single marker is not sufficient to fully account for a complicated geometry of a therapeutic device. The purpose of the study was to design a new image analysis tool to better determine the isocenter. The proposed automated procedure for determining isocenter position uses projection data acquired for a special cube phantom. The projection images of a phantom are acquired for various angles of rotation of the gantry. A procedure is proposed to extract some geometric characteristics of a therapeutic device from the projection images