Double power laws, fractals and self-similarity

Power law (PL) distributions have been largely reported in the modeling of distinct real phenomena and have been associated with fractal structures and self-similar systems. In this paper, we analyze real data that follows a PL and a double PL behavior and verify the relation between the PL coefficient and the capacity dimension of known fractals. It is to be proved a method that translates PLs coefficients into capacity dimension of fractals of any real data.

Double power laws, fractals and self-similarity

Power law (PL) distributions have been largely reported in the modeling of distinct real phenomena and have been associated with fractal structures and self-similar systems. In this paper, we analyze real data that follows a PL and a double PL behavior and verify the relation between the PL coefficient and the capacity dimension of known fractals. It is to be proved a method that translates PLs coefficients into capacity dimension of fractals of any real data.

Power Law Behavior and Self-similarity in Modern Industrial Accidents

Advances in technology have produced more and more intricate industrial systems, such as nuclear power plants, chemical centers and petroleum platforms. Such complex plants exhibit multiple interactions among smaller units and human operators, rising potentially disastrous failure, which can propagate across subsystem boundaries. This paper analyzes industrial accident data-series in the perspective of statistical physics and dynamical systems. Global data is collected from the Emergency Events Database (EM-DAT) during the time period from year 1903 up to 2012. The statistical distributions of the number of fatalities caused by industrial accidents reveal Power Law (PL) behavior. We analyze the evolution of the PL parameters over time and observe a remarkable increment in the PL exponent during the last years. PL behavior allows prediction by extrapolation over a wide range of scales. In a complementary line of thought, we compare the data using appropriate indices and use different visualization techniques to correlate and to extract relationships among industrial accident events. This study contributes to better understand the complexity of modern industrial accidents and their ruling principles.

Can power laws help us understand gene and proteome information?

Proteins are biochemical entities consisting of one or more blocks typically folded in a 3D pattern. Each block (a polypeptide) is a single linear sequence of amino acids that are biochemically bonded together. The amino acid sequence in a protein is defined by the sequence of a gene or several genes encoded in the DNA-based genetic code. This genetic code typically uses twenty amino acids, but in certain organisms the genetic code can also include two other amino acids. After linking the amino acids during protein synthesis, each amino acid becomes a residue in a protein, which is then chemically modified, ultimately changing and defining the protein function. In this study, the authors analyze the amino acid sequence using alignment-free methods, aiming to identify structural patterns in sets of proteins and in the proteome, without any other previous assumptions. The paper starts by analyzing amino acid sequence data by means of histograms using fixed length amino acid words (tuples). After creating the initial relative frequency histograms, they are transformed and processed in order to generate quantitative results for information extraction and graphical visualization. Selected samples from two reference datasets are used, and results reveal that the proposed method is able to generate relevant outputs in accordance with current scientific knowledge in domains like protein sequence/proteome analysis.

Self-similarity principle: the reduced description of randomness

A new general fitting method based on the Self-Similar (SS) organization of random sequences is presented. The proposed analytical function helps to fit the response of many complex systems when their recorded data form a self-similar curve. The verified SS principle opens new possibilities for the fitting of economical, meteorological and other complex data when the mathematical model is absent but the reduced description in terms of some universal set of the fitting parameters is necessary. This fitting function is verified on economical (price of a commodity versus time) and weather (the Earth’s mean temperature surface data versus time) and for these nontrivial cases it becomes possible to receive a very good fit of initial data set. The general conditions of application of this fitting method describing the response of many complex systems and the forecast possibilities are discussed.

A review of power laws in real life phenomena

Power law distributions, also known as heavy tail distributions, model distinct real life phenomena in the areas of biology, demography, computer science, economics, information theory, language, and astronomy, amongst others. In this paper, it is presented a review of the literature having in mind applications and possible explanations for the use of power laws in real phenomena. We also unravel some controversies around power laws.

A straightforward method to obtain the cohesive laws of bonded joints under mode I loading

A simple procedure to measure the cohesive laws of bonded joints under mode I loading using the double cantilever beam test is proposed. The method only requires recording the applied load–displacement data and measuring the crack opening displacement at its tip in the course of the experimental test. The strain energy release rate is obtained by a procedure involving the Timoshenko beam theory, the specimen’s compliance and the crack equivalent concept. Following the proposed approach the influence of the fracture process zone is taken into account which is fundamental for an accurate estimation of the failure process details. The cohesive law is obtained by differentiation of the strain energy release rate as a function of the crack opening displacement. The model was validated numerically considering three representative cohesive laws. Numerical simulations using finite element analysis including cohesive zone modeling were performed. The good agreement between the inputted and resulting laws for all the cases considered validates the model. An experimental confirmation was also performed by comparing the numerical and experimental load–displacement curves. The numerical load–displacement curves were obtained by adjusting typical cohesive laws to the ones measured experimentally following the proposed approach and using finite element analysis including cohesive zone modeling. Once again, good agreement was obtained in the comparisons thus demonstrating the good performance of the proposed methodology.

Multidimensional Scaling Visualization using Parametric Similarity Indices

In this paper, we apply multidimensional scaling (MDS) and parametric similarity indices (PSI) in the analysis of complex systems (CS). Each CS is viewed as a dynamical system, exhibiting an output time-series to be interpreted as a manifestation of its behavior. We start by adopting a sliding window to sample the original data into several consecutive time periods. Second, we define a given PSI for tracking pieces of data. We then compare the windows for different values of the parameter, and we generate the corresponding MDS maps of ‘points’. Third, we use Procrustes analysis to linearly transform the MDS charts for maximum superposition and to build a global MDS map of “shapes”. This final plot captures the time evolution of the phenomena and is sensitive to the PSI adopted. The generalized correlation, the Minkowski distance and four entropy-based indices are tested. The proposed approach is applied to the Dow Jones Industrial Average stock market index and the Europe Brent Spot Price FOB time-series.

Fractais - Aplicações em Engenharia

Num universo despovoado de formas geométricas perfeitas, onde proliferam superfícies irregulares, difíceis de representar e de medir, a geometria fractal revelou-se um instrumento poderoso no tratamento de fenómenos naturais, até agora considerados erráticos, imprevisíveis e aleatórios. Contudo, nem tudo na natureza é fractal, o que significa que a geometria euclidiana continua a ser útil e necessária, o que torna estas geometrias complementares. Este trabalho centra-se no estudo da geometria fractal e na sua aplicação a diversas áreas científicas, nomeadamente, à engenharia. São abordadas noções de auto-similaridade (exata, aproximada), formas, dimensão, área, perímetro, volume, números complexos, semelhança de figuras, sucessão e iterações relacionadas com as figuras fractais. Apresentam-se exemplos de aplicação da geometria fractal em diversas áreas do saber, tais como física, biologia, geologia, medicina, arquitetura, pintura, engenharia eletrotécnica, mercados financeiros, entre outras. Conclui-se que os fractais são uma ferramenta importante para a compreensão de fenómenos nas mais diversas áreas da ciência. A importância do estudo desta nova geometria, é avassaladora graças à sua profunda relação com a natureza e ao avançado desenvolvimento tecnológico dos computadores.