Zbisław Tabor

Formation of singularities for equivariant (2+1)-dimensional wave maps into the 2-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 2-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 2 into S2.

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 behaviour in the formation of black holes.

The potential of multi-slice computed tomography based quantification of the structural anisotropy of vertebral trabecular bone

In the present paper we addressed the problem of whether the information about the structural anisotropy of trabecular bone can be retrieved from low-quality data, captured with clinical multi-raw spiral CT scanners. Two measures of quantifying structural anisotropy were tested - the current standard mean intercept length (MIL) and the gray-level structure tensor (GST). Thirty two vertebral bodies were μCT and CT scanned. The reference values of structural anisotropy were measured in μCT images and compared with the measures of structural anisotropy determined from low-quality CT data. MIL-based measures of structural anisotropy cannot be reliably determined from CT data. The assessment of the GST is significantly better than that of MIL, but the accuracy is not, in general, satisfactory. Based on the results of experiments with artificial data and the analysis of the real images, it can be concluded that a possible reason of the poor performance is anisotropic resolution of clinical CT scanners.

On the equivalence of two methods of determining fabric tensor

In this paper it is studied how three methods of quantifying structural anisotropy are related. Mean intercept length (MIL) method has been designed for the analysis of binary images. Autocorrelation function and the covariance matrix of the gray-level intensity gradient (GST method) are approaches designed for the analysis of gray-level data. It is shown here that the autocorrelation function and the MIL methods are not related in a general case. In contrast, an analytical proof is provided to show that MIL and GST methods are strictly equivalent. The standard definition of MIL is expressed in terms of a gradient field. Next it is shown that eigenvectors of the MIL fabric tensor are also eigenvectors of the GST fabric tensor and eigenvalues of the MIL fabric tensor can be determined if the eigenvectors of the GST fabric tensor are known. It follows from the study that the application of MIL in the assessing quality of trabecular bone can be replaced in all cases by the application of the GST method, which is more general (as defined for gray-level data), easier to implement and less computationally expensive.

On blowup for Yang-Mills fields

We study development of singularities for the spherically symmetric Yang-Mills equations in

Equivalence of mean intercept length and gradient fabric tensors – 3d study

d+1

Comparison of trabecular bone architecture in young and old bones

dimensional Minkowski spacetime for

Vacuum gravitational collapse in nine dimensions

d=4

A microcomputed tomography-based comparison of root canal filling quality following different instrumentation and obturation techniques.

(the critical dimension) and

A novel method of estimating structure model index from gray-level images

d=5