Coarse-graining in the spring-block approach.

<p>The figure illustrates how the links are inherited when blocks juncture.</p

The two-dimensional spring-block system used for modeling quasi-static fragmentation phenomena.

<p>The blocks are interconnected by springs and they can slide on a two-dimensional substrate.</p

Results of the method for Transylvania using census data between 1850 and 2002.

<p>The picture on the left illustrates a rough partition, confirming that Banat is historically not belonging to Transylvania. The picture on the right shows a finer partition in four, revealing the main regions: north of Transylvania, south of Transylvania, Banat and the Szekler region.</p

Results of the spring-block approach for USA using a five year census data.

<p>On the left we illustrate a rough partition in 4 and on the right we illustrate a finer partition in 48 elements.</p

Voronoi tessellation for determining the neighboring relation of the blocks.

<p>The red lines are the segments of the Voronoi tessellation, the black ones are the edges of the Delaunay graph (triangulation).</p

Testing the spring-block system on a model situation.

<p>As shown in the figure on the left four domains are defined so that connectivity inside the domains are stronger than connectivity between the domains. The picture in the middle shows an intermediate simulation step, with the existing springs and blocks. The figure on the right shows the detected partition.</p

Main elements of the one-dimensional Burridge-Knopoff model.

<p>The blocks are connected to each other and to the upper plane by springs. The upper plane is dragged with a constant velocity. Since the blocks are allowed to slide, they will slide in avalanches following the motion of the upper surface.</p

Sketch of a Delaunay Triangulation.

<p>The Delaunay triangulation and its dual, the Voronoi tessellation for a random set of points. The blue lines are the segments of the Voronoi tessellation, the red ones are the edges of the Delaunay graph (triangulation).</p

Cluster analysis of the spatial distribution of heterochromatin and euchromatin after irradiation with 0.5 Gy and 3.5 Gy.

<p>The analysis was performed on non-irradiated samples, and on samples fixed 6 h and 24 h post irradiation, respectively. <b>Panel A</b> present the behaviour of heterochromatic regions irradiated with 0.5 Gy and 3.5 Gy. We observe a decrease in the ratio of the clustered points 6 h after irradiation for both doses. 24 h after irradiation these changes are reverted. This means that a few hours after irradiation the heterochromatin decondenses and after one day the decondensation is reverted. <b>Panel B</b> shows the cluster analysis result for euchromatin. These regions behave differently, compared to heterochromatin. For 0.5 Gy, the number of clustered points is increased after 6 h, and the trend continues at the 24 h timestamp. However, for 3.5 Gy, after the initial increase in the percentage of clustered points at 6 h, the ratio drops again in the 24 h measurements. The interpretation of the increase/decrease in the number of clustered points is similar in this case: when the ratio increases, the studied domains condense, when the ratio decreases, the domains relax. This means, that heterochromatin and euchromatin react to irradiation in an opposite way: This analysis, in accordance with the other results, indicates that while heterochromatin opens up soon after irradiation, euchromatin condensates. Given enough time, these processes are reverted in both cases.</p

Conditional Probabilities of Edge Lengths for Random Data.

<p>The figures illustrate how the <i>p</i>(<i>r</i>|<i>r</i>′) conditional probability looks for randomly generated point positions. Panel A. Conditional probability of points with coordinates generated according to a uniform distribution. Panel B. Conditional probability of points with coordinates generated according to a mixture of uniform distribution and clusters of Gaussian distributions. In the latter example an emphasized diagonal is observed which is the result of the Gaussian clusters. Tightly packed points tend to produce short edges, while points from the edges of the clusters mostly have longer edges.</p