Paweł Oświęcimka

Rafał Rak, Jarosław Kwapień, Paweł Oświęcimka et al.

Compared to the heavily studied surface drainage systems, the mountain ridge systems have been a subject of less attention even on the empirical level, despite the fact that their structure is richer. To reduce this deficiency, we analyze different mountain ranges by means of a network approach and grasp some essential features of the ridge branching structure. We also employ a fractal analysis as it is especially suitable for describing properties of rough objects and surfaces. As our approach differs from typical analyses that are carried out in geophysics, we believe that it can initialize a research direction that will allow to shed more light on the processes that are responsible for landscape formation and will contribute to the network theory by indicating a need for the construction of new models of the network growth as no existing model can properly describe the ridge formation. We also believe that certain features of our study can offer help in the cartographic generalization. Specifically, we study structure of the ridge networks based on the empirical elevation data collected by SRTM. We consider mountain ranges from different geological periods and geographical locations. For each mountain range, we construct a simple topographic network representation (the ridge junctions are nodes) and a ridge representation (the ridges are nodes and the junctions are edges) and calculate the parameters characterizing their topology. We observe that the topographic networks inherit the fractal structure of the mountain ranges but do not show any other complex features. In contrast, the ridge networks, while lacking the proper fractality, reveal the power-law degree distributions with the exponent $1.6\le β\le 1.7$. By taking into account the fact that the analyzed mountains differ in many properties, these values seem to be universal for the earthly mountainous terrain.

Stanisław Drożdż, Paweł Oświęcimka, Andrzej Kulig et al.

In natural language using short sentences is considered efficient for communication. However, a text composed exclusively of such sentences looks technical and reads boring. A text composed of long ones, on the other hand, demands significantly more effort for comprehension. Studying characteristics of the sentence length variability (SLV) in a large corpus of world-famous literary texts shows that an appealing and aesthetic optimum appears somewhere in between and involves selfsimilar, cascade-like alternation of various lengths sentences. A related quantitative observation is that the power spectra S(f) of thus characterized SLV universally develop a convincing `1/f^beta' scaling with the average exponent beta =~ 1/2, close to what has been identified before in musical compositions or in the brain waves. An overwhelming majority of the studied texts simply obeys such fractal attributes but especially spectacular in this respect are hypertext-like, "stream of consciousness" novels. In addition, they appear to develop structures characteristic of irreducibly interwoven sets of fractals called multifractals. Scaling of S(f) in the present context implies existence of the long-range correlations in texts and appearance of multifractality indicates that they carry even a nonlinear component. A distinct role of the full stops in inducing the long-range correlations in texts is evidenced by the fact that the above quantitative characteristics on the long-range correlations manifest themselves in variation of the full stops recurrence times along texts, thus in SLV, but to a much lesser degree in the recurrence times of the most frequent words. In this latter case the nonlinear correlations, thus multifractality, disappear even completely for all the texts considered. Treated as one extra word, the full stops at the same time appear to obey the Zipfian rank-frequency distribution, however.