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An Unexpected Renaissance Age
Published in Alessio Plebe, Pietro Perconti, The Future of the Artificial Mind, 2021
Alessio Plebe, Pietro Perconti
On a side very different from mathematical topology, there are intriguing attempts at explaining DL by drawing an analogy with theoretical physics. There is a technique, known as renormalization group, which has played a fundamental role in contemporary theoretical physics, overcoming the problem of series summing up to infinite probability in fundamental equations, like Dirac’s quantum electrodynamics (Stueckelberg and Petermann, 1953). The renormalization group allows one to relate changes of a physical system that appear at different scales, yet exhibit scale invariance properties. By applying renormalization group it was possible to resolve the critical divergence towards infinity of the series in the Dirac equation. The renormalization group is also the best tool for the analysis of critical phenomena, phase transitions at the boundaries between the ordinary discontinuous behavior between phases, and the continuum of phases observed at temperatures above a certain threshold. Physical systems approaching the critical point have a remarkable invariance in scale of some of their parameters, making the renormalization group very effective in connecting phenomena which occur at quite different length scales (Wilson and Kogut, 1974). What does the renormalization group have to do with artificial neural networks?
Chaos in Space
Published in Pier Luigi Gentili, Untangling Complex Systems, 2018
A fractal is a complex geometric shape. It maintains its sophisticated form, or the main features of its complex structure, under different levels of magnification, i.e., under changes of spatial scale. This peculiarity of fractals is defined as either “scale invariance” or “scale symmetry,” or more often, “self-similarity.” Several fractals can be built by following simple rules. A fractal that often appears in the strange attractors generated by chaotic dissipative dynamics is the Cantor set. For example, it is the fractal that can be detected in the Lorenz attractor. The set is named after the nineteenth-century mathematician Georg Cantor. However, it was created by Henry Smith, who was a nineteenth-century geometry professor at the Oxford University.2 To build a Cantor set, we start with a bounded interval C0. We remove an open interval from inside C0. C0 is divided into two subintervals, each containing more than one point. Iteratively, we remove an open interval from each remaining interval. In this way, the length of the remaining intervals at each step shrinks to zero as the construction of the set proceeds. After an infinite number of iterations, we obtain the Cantor set. The most often mentioned Cantor set is that obtained by removing the middle third of every interval (see Figure 11.3). In the end, it consists of an infinite number of infinitesimal pieces, separated by gaps of various lengths, and having a negligible length.3 Of course, it is impossible to print a fractal on a piece of paper.
Fractal Analysis
Published in Shyam S. Sablani, M. Shafiur Rahman, Ashim K. Datta, Arun S. Mujumdar, Handbook of Food and Bioprocess Modeling Techniques, 2006
Fractal analysis is mainly applied when other methods fail or become tedious to solve complex or chaotic problems. Many natural patterns are either irregular or fragmented to such an extreme degree that Euclidian or classical geometry could not describe their form (Mandelbrot 1977, 1987). Any shape can be characterized by whether or not it has a characteristic length (Takayasu 1990). For example, a sphere has a characteristic length defined as the diameter. Shapes with characteristic lengths have an important common property of smoothness of surface. A shape having no characteristic length is called self-similar. Self-similarity is also known as scale-invariance, because selfsimilar shapes do not change their shape under a change of observational scale. This important symmetry gives a clue to understanding complicated shapes, which have no characteristic length, such as the Koch curve or clouds (Takayasu 1990). The idea of a fractal is based on the lack of characteristic length or on self-similarity. The word fractal is a new term introduced by Mandelbrot (1977) to represent shapes or phenomena having no characteristic length. The origin of this word is the Latin adjective fractus meaning broken. The English words “fractional” and “fracture” are derived from this Latin word fractus, which means the state of broken pieces being gathered together irregularly.
A study of the association involving pore characteristics and compressive strength of cement pastes incorporating fine fly ash
Published in Mechanics of Advanced Materials and Structures, 2023
Min Bai, Kaiyue Cao, Yangbo Lu, Peng Zhao, Zhe Zhang, Hui Li
Fractals are used to quantitatively describe geometric shapes based on the self-similarity and scale invariance characteristics of fractal media. Materials derived from cement are well suited for this application due to their complex structure [35]. In general, fractals can be divided into two categories: regular and irregular. Pore structures in materials derived from cement belong to the irregular fractal category. Based on the mercury intrusion results, the fractal dimension of different pore sizes in materials derived from cement could be fitted through relating the measured intrusion volume to the material’s porosity and maximum pore size, represented by the letter D. When the pore surface is relatively smooth, the fractal dimension approaches 2; if the surface is rough and irregular, the fractal dimension approaches 3; if it is greater than 3, the complexity of the pore structure is high, and its distribution pattern cannot be described using images.
A laboratory prototype of automatic pavement crack sealing based on a modified 3D printer
Published in International Journal of Pavement Engineering, 2021
Jingwei Liu, Xu Yang, Xin Wang, Jian Wei Yam
Fractal thresholding algorithm was used in this research to differentiate pavement cracks from the pavement background in the images, because cracks can be effectively differentiated by fractals with high accuracy, fast computational speed, and least influence of noise (Zuo et al. 2008). Fractal theory is based on the concept of self-similarity and it can be used to differentiate cracks from pavement, which has the property of self-similarity. Scale invariance is the basic concept of self-similarity, which means that a small part of a crack has the property of self-similarity when it looks like the whole crack in an enlarged or shrunk form. The enhanced image is segmented into a matrix size of s×s, and the threshold value for each segment is then evaluated by the algorithm shown in Eq. (2): where, A is the surface area of fractals, ε is the distance of three-dimensional space, k is the threshold value, D can be evaluated by applying least-square fitting, and k can then be calculated.
Research on building energy consumption prediction model based on fractal theory
Published in Intelligent Buildings International, 2020
Junqi Yu, Sen Jiao, Yue Zhang, Xisheng Ding, Jiali Wang, Tong Ran
Building energy consumption has typical features of multivariate, nonlinear, strong coupling and multi-skilling complex. By observing the time series curve of the building energy consumption, it is found that this is a very typical nonlinear curve. Branch theory is an emerging nonlinear theory developed in recent years and has been applied in various subject areas. Branch theory is widely used because it can directly analyze some rules of its own changes from abstract complexion complex nonlinear things. Natural things generally have fractal characteristics. And fractal characteristics include self-similarity and scale invariance. After research, it is found that building energy consumption has fractal characteristics, which lays a theoretical foundation for the application of fractal theory to building energy consumption analysis.