Fractals: Form, Chance and Dimension
... In the box-counting method, the image's Euclidean space is divided into a grid of boxes with the size of r, with the initial box size being equal to the original image's size, in order to estimate the FDs. Then, as r gets smaller, the number of nonempty boxes (N(r)) is counted [17]. The gradient of a line in a diagram of The FD is called a dimension due to its measurement of how completely an object fills space. ...
... For the FD value as an integer, an ideal point has zero dimensions, an ideal line has one dimension, an ideal plane has two dimensions, and an ideal volume has three dimensions. As the trabecular bone structures in two dimensions are not perfectly straight lines and do not fill the two-dimensional space entirely, their FD values range from 1 to 2 [17]. For example, compared to the trabecular bone structures with higher FD values, such as 1.45, those with low FD values, such as 1.2, would have fewer trabecular structures and would not entirely fill the two-dimensional space. ...
... All techniques rely on the association between a measuring tool and the geographical distribution of the object. There are various techniques for calculating FD values, such as the Hausdorff dimension, the Minkowski-Bouligand dimension, the mass-radius method, and the box-counting method [17]. In the box-counting method, the image's Euclidean space is divided into a grid of boxes with the size of r, with the initial box size being equal to the original image's size, in order to estimate the FDs. ...
Abstract: (1) Background: Fractal analysis has been used as a mathematical method for studying the complexity of fractal structures such as trabecular bone that look similar at different scales. Bruxism is a disorder involving nonfunctional grinding and clenching of the teeth that leads to bone resorption and fractal dimension reduction. This study aimed to evaluate the trabecular pattern of the mandibular condyle, angle, and dental region in panoramic radiographs of individuals with and without bruxism using fractal analysis and a larger sample size. (2) Methods: A total of 365 panoramic radiographs belonging to two groups consisting of bruxism and non-bruxism individuals were assessed using fractal analysis. Fractal dimension (FD) values were calculated on each side for the three regions of interest (ROIs): the mandibular condylar head, mandibular angle, and interdental region between the second premolar and first molar. Statistical analysis was performed using binominal and chi-square tests, the Shapiro-Wilk test, and the Mann-Whitney test (α = 0.05). (3) Results: No significant differences were observed between the FD values of the ROIs in the two groups (p > 0.05). (4) Conclusions: No significant differences existed in the FD values of the ROIs in patients with and without bruxism. This result shows that fractal analysis of panoramic radiographs cannot be useful in detecting patients with bruxism.
... Tree architecture assessment dates back to the 1970s (Vinci and Richter, 1970). However, with the introduction of the concept of "fractal dimension" (Mandelbrot, 1977) measurement of structures of objects has changed. Tree crown assessment has also evolved from 2D fractal dimension assessment (Morse et al., 1985) of leaves, digitized 3D fractal analysis of crown shapes (Boudon et al., 2006), the ratio of crown size to volume (Zeide and Gresham, 1991;Zeide and Pfeifer, 1991) to the analysis of "box-dimension" through 3D information obtained from terrestrial LiDAR technology (Seidel, 2018). ...
... Until some time ago, measuring natural objects was done in one, two, or three dimensions of geometry by considering it to be a solid body (Mandelbrot, 1982). However, with the concept of fractal dimension, the way to measure structures especially of three-dimensional objects has changed since objects are considered as non-uniform, porous, and even empty structures with self-similar replicates (Mandelbrot, 1977). ...
... However, in trees Mandelbrot (1977) suggested that box-dimension value could range between 1 and 2. Seidel et al. (2019a) indicated that box-dimension of a tree could be less than 2.72 if it is to consider efficient light utilization avoiding within tree shading. In the same study, it is stated that box-dimension of 2.72 would be ineffective for light utilization due to maximized within tree shading as it has the highest size to the surface ratio which is like Menger sponge developed by Menger (1926). ...
Forest stand structure is built by the architecture of individual trees. Individual Tree architecture is affected by genetic and external environmental drivers. This study used high resolution three-dimensional (3D) information on the tree architecture to investigate the relationship between (i) tree’s seed dispersal strategy and (ii) the latitude of the species’ home range (genetical legacy) and the box-dimension of a tree. The latter is a measure of the structural complexity of tree architecture. The aim was to improve our understanding of the development of tree architecture based on fractal analysis. Using a ZEB HORIZON mobile laser scanner, the required 3D data to retrieve the box-dimension was captured. The results indicate that the mean latitude of a tree species’ home range had a significant effect (p<0.001) on the box-dimension of the tested 473 tree species in the Stutel-Arboretum.This indicates that, despite adaptation to the growing conditions in Stutel, the trees exhibited a genetical legacy of the evolutionary adaptation to the latitude of their origin. The coefficient of determination of the relationship was however rather low with R2=0.016. Furthermore, we found a significantly different box-dimension for tree species that rely on wind as a means of seed dispersal as opposed to those relying on animals.
From these results, we conclude that genetic factors contribute notably to the architecture of trees growing in the Arboretum. This architecture can be efficiently studied using the ZEB HORIZON mobile laser scanner.
KEYWORDS
Tree architecture, LiDAR, box-dimension, latitude, seed dispersal strategy, fractal analysis.
... En 1977, alors que jusque-là les mathématiques se limitaient à des formes lisses, le mathématicien polono-franco-américain Benoît Mandelbrot publie un livre suggérant que la plupart des formes dans la nature peuvent être décrites mathématiquement comme des fractales, un mot qu'il a lui-même inventé pour désigner des formes au contour irrégulier et fragmenté 82 (Mandelbrot, 1977). A ce moment-là, il affirme qu'on peut générer des fractales à partir de formes au contour lisse qu'on fractionne successivement en petits morceaux. ...
... Le film « Star Trek 2 : la colère de Khan » en est un bon exemple où une planète entière est générée à partir de la géométrie fractale ; c'est la première image de synthèse dans un long métrage. La caractéristique principale des fractales est l'auto-similarité, c'est-à-dire que le tout ressemble à la partie qui ressemble elle-même à une partie encore plus petite, le motif se répète ainsi par un processus itératif infini (Mandelbrot, 1977). Cette auto-similarité se retrouve en fait partout : arbres, montagnes, vaisseaux sanguins, surface de la lune… Le flocon de Koch en est un bel exemple ( Figure 45) (Koch, 1904). ...
Après une course d'endurance, le patron de récupération fonctionnelle est décrit comme biphasique, étant caractérisé par des déficits fonctionnels immédiats, suivis d'une récupération partielle à 2 h, avant de nouveaux déficits 1 à 2 jours plus tard ne s'atténuant que progressivement sur plusieurs jours. En raison du potentiel effet protecteur des hormones œstrogènes, notamment au niveau musculaire, les femmes pourraient mieux résister à la fatigue et récupérer plus rapidement. La littérature s’est néanmoins focalisée sur la récupération des hommes et principalement en phase aiguë. La phase retardée se caractérise pourtant par un phénomène inflammatoire lié à la régénérescence des microlésions musculaires causées par la course. Cette phase s’accompagne de courbatures musculaires qui disparaissent avant que la récupération ne soit complète, ce qui constitue un risque potentiellement accru de blessure à la reprise de la pratique. L’objectif principal de ce travail de thèse était d’établir et de comparer la cinétique de récupération structurale et fonctionnelle de coureurs féminins et masculins après une course d’endurance de 20 km avec dénivelé. Nos résultats soulignent l‘interaction entre le sexe et le test d’évaluation utilisé. Les femmes ont présenté plus de courbatures et d’altérations structurales (tant à l’échographie que par imagerie par résonnance magnétique) des muscles ischio-jambiers que les hommes. Par contre, leurs déficits fonctionnels étaient moindres et leur récupération plus précoce dans certains tests. Ce travail souligne la faiblesse des liens entre les altérations structurales et les déficits fonctionnels, autant que la richesse des ajustements neuromusculaires en situations dynamiques pluri-articulaires et musculaires. Les différences fonctionnelles observées entre les sexes semblent fortement influencées par l’organisation spécifique des synergies musculaires propres à chaque sexe.
... Термін "фрактал" запровадив у 1975 р. французький математик Бенуа Мандельброт [13], [14]. Польський математик Вацлав Серпінський ще раніше запропонував різні геометричні структури, [15] по суті фрактальні, які можна успішно використовувати в техніці, наприклад, при створенні антен [16]. ...
... Для збереження конфіденційності цю морфологічну модель можна подати як морфологічну множинуповну морфологічну матрицю [13,14] Для ручного ПЗ, який застосовується переважно в слюсарних лещатах (альтернатива 1.1), й електромеханічного ПЗ, який використовується в машинних лещатах (альтернатива 1.2), з розповсюдженим гвинтовим ПТ (альтернатива 2.1) запишемо кортежі морфологічної множини як морфологічні формули (сполучення альтернатив кожної ознаки) для ОЗ простої форми (альтернатива 7.1) X 1 -X 8 і складної форми типу "шарнірний палець кермової тяги автомобіля" (альтернатива 7.2) X 9 і X 10 (рис. 3): Жирним шрифтом позначено альтернативи, які відрізняються від варіанта X 1 . ...
Background. On metal-cutting machines, in mechanical assembly production and everyday life, vices of various designs are widelyused, which, as a rule, have a constant structure and one kinematic chain from the engine (or manual drive) to the clamping elements –jaws. The vast majority of them are adapted for clamping cylindrical objects (parts) and objects with plane-parallel sides. At the same time,in mechanical assembly production, as well as during repair work, it often becomes necessary to clamp spherical, wedge-shaped, conical,and irregularly shaped objects.Objective. The purpose of the paper is to propose a morphological set of universal vices taking into account the achievements inthe theory of fractals and alternatives to new features of the state, type of execution and mobility of clamping elements, and to come upwith morphological formulae of vices.Methods. A morphological model of the vices is being developed, which takes into account the achievements of the fractal theoryand makes it possible to predict new technical solutions for universal vices at the level of inventions.Results. The morphological formulae of universal vices from the morphological set which contains alternatives of new signs of acondition, a type of execution and mobility of clamping elements, and also prerequisites for using the fractal theory for clamping processmodelling of complex shape details are offered.Conclusions. Using the proposed approach, it is possible to predict new technical solutions of universal vices at the level of inven-tions for the manufacture, research and subsequent commercialization
... Historically, the breakage has also been described using the concept of fractal dimension. Fractal dimension, introduced by Mandelbrot and Aizenman (1979), is a way to quantify the complexity or irregularity of a geometric pattern or structure that exhibits self-similarity at different scales. Fractal dimension is a non-integer value, and it can be used to describe phenomena in nature, such as coastlines, clouds, mountains, and biological structures, that do not fit well with classical Euclidean geometry. ...
Comminution, which refers to the process of rock breakage, is an extremely energy-intensive activity in mineral processing. It consumes approximately 37% of the total energy used in mining operations. This process suffers from notable equipment-related inefficiencies. If opportunities for improving this process can be identified, these could lead to significant economic and environmental advantages. However, identifying such opportunities is challenging due to the inherent limitations of the current standard methods used to evaluate comminution circuits. Many of these standards are either based on empirical approaches or lack effectiveness in accurately determining equipment efficiency. Additionally, the existing methods for measurement are excessively laborious and require large sample sizes. These factors can compromise: the accurate characterisation of an orebody when conducting variability test programs; benchmarking studies to evaluate comminution circuit performance; or assessment of the suitability of new technologies. To address these issues, it is proposed that a measure of the practical minimum energy required for comminution, which offers the realistically achievable efficiency of comminution circuits, can provide the benchmark against which to compare current and future technologies and processes. Secondly, the method needs to be simple, relatively fast and have minimal sample requirements.
This thesis is focused on developing a novel ore characterisation method for the precise measurement of practical minimum comminution energy. A monolayer of particles is broken using a precision rolls crusher (PRC) – a rigidly mounted (double) roller mill instrumented with precise torque measurement hardware. The torque transducers are mounted on rolls shafts to measure the breakage response of rock particles in the form of precise torque and angular speed readings, which are then converted into energy. Significant efforts have been made to isolate the effect of the grinding environment on breakage and achieve energy measurement at the primary fracture of rock particles. The test iteratively crushes feed particles down to final product size in many controlled breakage stages. The generation of fines in each breakage stage is then plotted as a function of input energy. The inverse of the gradient of this plot provides the size specific energy (SSE) in kWh/t of material generated below a predefined marker size.
The PRC has been applied for benchmarking on-site comminution circuits and evaluating the performance of other testing methods that involve different modes of breakage, such as bulk grindability, bed breakage, and single particle breakage. The on-site circuits were found to be inefficient, using 3.2 to 3.8 times more energy than the PRC when using 106 µm marker. Similarly, lab-HPGR exhibited inefficiencies of 2.4 to 2.9 times, while the Instrumented Bond ball mill (IBM) showed even greater inefficiencies, ranging from 3.4 to 3.9 times. These results demonstrate that the PRC has potential to be used as a suitable choice for ore characterisation applied to quantifying the performance of industrial circuits. Additionally, there is potential for the PRC to evolve into a geometallurgical ore characterisation test – it is simple, fast, requires small sample mass (~300 g), and covers a wide feed particle size range (12 mm to 75 µm).
Moreover, the analysis of relative size specific energy ratio of full-scale comminution circuits and testing methods in relation to the PRC showed that common concept of comminution efficiency being an absolute number is a myth, with efficiency instead being dependent on the product size against which it is assessed.
Finally, fractal dimension analysis was employed to study breakage behaviour of different ore types in the PRC, revealing a transition in breakage mode as the gap decreased during progressive breakage experiment on the PRC. The transition went from less efficient mode of breakage (crushing/ crack branching) at coarser gaps to more efficient mode of breakage (bulk splitting) at finer gaps.
... This perspective is evident in the work of Milasinovic et al. (2010), who view a crisis as "unexpected, undesirable, unpredictable and unthinkable, often leading to uncertainty and disbelief. " Topper & Lagadec (2013) explicitly address the "fractal" nature of crises, drawing parallels with the application of Mandelbrot's principles to financial crises characterized by high complexity, volatility and discontinuity (Mandelbrot, 1995). They develop a theory of the "fractal nature" of crises, justified by properties such as scale, impossibility of prediction, interdependence of systems and instantaneous impact. ...
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... Indeed, theoretical (ideal) fractals are objects showing scale-invariant behavior [4], that, in the spatial domain, are 'self-similar' geometric items with features on an infinite number of scales and their hallmark is a power-law scaling. In a real scenario, as for natural (statistical) fractals, scale-invariance holds only for a limited range of scales related to a particular scaling range [5]. ...
Brain-related neuronal recordings, such as local field potential (LFP), electroencephalogram (EEG) and magnetoencephalogram (MEG), offer the opportunity to study the complexity of the human brain at different spatial and temporal scales. The complex properties of neuronal signals are intrinsically related to the concept of “scale-free” behavior and irregular dynamic, which cannot be fully described through standard linear methods, but can be measured by nonlinear indexes. A remarkable application of these analysis methods on electrophysiological recordings is the deep comprehension of the pathophysiology of neurodegenerative diseases, that has been shown to be associated to changes in brain activity complexity. In particular, a decrease of global complexity has been associated to Alzheimer’s disease, while a local increase of brain signals complexity characterizes Parkinson’s disease. Despite the recent proliferation of studies using fractal and entropy-based analysis, the application of these techniques is still far from clinical practice, due to the lack of an agreement about their correct estimation and a conclusive and shared interpretation. Along with the aim of helping towards the realization of a multidisciplinary audience to approach nonlinear methods based on the concepts of fractality and irregularity, this survey describes the implementation and proper employment of the mostly known and applied indexes in the context of Alzheimer’s and Parkinson’s diseases.
... We replace the initial high-temperature random configuration by an artificial sponge structure to probe its effect on the dynamics. As a prototypical sponge structure we use Menger sponges 30 , the three-dimensional generalization of Sierpinski carpets [see Fig. 4a]. The starting configuration [see Fig. 4b] of dimensions L 3 is created by repeating sponges of kth iteration of size (3 k ) 3 . ...
One key aspect of coarsening following a quench below the critical temperature is domain growth. For the non-conserved Ising model a power-law growth of domains of like spins with exponent α=1/2
is predicted. Including recent work, it was not possible to clearly observe this growth law in the special case of a zero-temperature quench in the three-dimensional model. Instead a slower growth with α<1/2
was reported. We attempt to clarify this discrepancy by running large-scale Monte Carlo simulations on simple-cubic lattices with linear lattice sizes up to L=2048
employing an efficient GPU implementation. Indeed, at late times we measure domain sizes compatible with the expected growth law—but surprisingly, at still later times domains even grow superdiffusively, i.e., with α>1/2
. We argue that this new problem is possibly caused by sponge-like structures emerging at early times.
... A higher value implies a higher complexity of time series, with a lower likelihood that the similarity will continue as the embedding dimension grows (or a grater probability that similarity changes). The Approximate Entropy (ApEn) and Sample Entropy (SampEn) algorithms are the most popular ones (Mandelbrot and Aizenman, 1979). ...
In a variety of diseases, altered respiratory modulation is often an early sign of autonomic dysfunction. Therefore, understanding and evaluating the effects of mechanical ventilation on the autonomic nervous system is vital. The effects of mechanical ventilation on autonomic balance have been assessed by heart
rate variability (HRV) using frequency domain and non-linear analysis including fractal complexity and entropy analysis in anesthetized mice. BALB/c mice (n=48) were divided into two groups: Spontaneous breathing and mechanical ventilation. The electrocardiograms were recorded. Four different types of analysis were employed: i. frequency domain analysis, ii. Poincaré plots, iii. Detrended Fluctuation
Analysis (DFA) and iv. Entropy analysis. An unpaired t-test was used for statistical analysis. In a ventilated group, very low frequency (VLF) and low frequency (LF) parameters were not changed, whereas the high-frequency parameter was decreased compared to spontaneous breathing mice. DFAα 1
was significantly increased due to mechanical ventilation but DFAα 2 was unchanged. In Poincaré plots analysis, the standard deviation 2 (SD2) / standard deviation 1 (SD1) ratio was increased, however, SD1 and SD2 were not significantly affected. Also, Approximate Entropy and Sample Entropy remained unchanged. HF parameter, DFAα1, and SD2/SD1 were affected by mechanical ventilation. Decreased HF and increased DFAα1, further support the notion that HRV is dominated by respiratory sinus arrhythmia at high frequencies, this may be due to decreased vagal tone caused by mechanical ventilation. These novel
results of HRV analysis are important considering the increased usage of HRV techniques day by day in animal models and other medical practices.
... In general, fractals have played a leading role in various fields of sciences and engineering such as biology [1], medicine [2], the science of materials [3][4][5], elasticity theory [6][7][8][9][10], quantum mechanics and molecular physics [11][12][13][14][15][16], astronomy [17,18], fluid mechanics [19][20][21], statistical physics [11,[22][23][24][25][26][27][28][29][30], plasma physics and magnetohydrodynamics (MHD) [31][32][33], and many others. In general, the notion of fractal dimension was introduced in aggregation, gelation, random walks, fracture of cracked metals, diffusion-limited aggregates, etc. [34][35][36]. It is noteworthy that fractal dimension is also used to measure the complexity of spatiotemporal objects in geometrical frames [37] and a complete spectrum of fractal dimensions such as the Renyi dimensions is therefore required [38][39][40]. ...
Abstract: In this study, we have used the concept of ‘product-like fractal measure’ introduced by Li and Ostoja-Starzewski in their formulation of anisotropic and elastic media to introduce Vlasov equation in fractal dimension. We have been motivated by this new concept since it is safely applied to several systems with length scales bounded by lower and upper cutoffs via the dimensional regularization technique. We have considered through this study collisionless plasma in fractal dimension and we have analyzed the impacts of the Vlasov equation in some points in plasma physics starting from the Hamiltonian dynamics. Several results have been obtained and discussed accordingly. The main outcomes of the present study concern first the facts that waves in cold plasma are associated with low fractal dimensions and hence by low complexity and that the effective damping is affected by the sign of the fractal dimension and differs from the conventional Landau damping in magnitude and in sign for negative dimension. Besides, for a very low fractal dimension, the Landau damping is negligible or removed. This result was observed in kinetic turbulent plasmas and may solve the paradox problem emerging in the application of critical balance to a kinetic turbulence cascade.
https://doi.org/10.1080/17455030.2022.2155321
... Оскільки під час аналізу експериментальних даний у якості методів монофрактального аналізу автори роботи зазвичай використовують методи УФА та ДинФА, а також класичний метод нормованого розмаху [9], то розповімо про застосування методу КФ, у першу чергу, для регуляризаційної [10], кліткової [11], варіаційної [12] та херстової [13] фрактальних розмірностей. ...
One of the main numerical characteristics used in numerous methods of fractal analysis is the corresponding fractal dimensions. The accuracy of estimating these dimensions in the vast majority of cases is quite small, which cannot satisfy, first of all, researchers-practitioners. The method of the corrective function is put forward, which makes it possible to compensate for the ever-existing nonlinearity of the dependence between the true value of the fractal dimension and its estimation, performed using the selected method of monofractal analysis of signals and processes for a known number of samples of the discrete data vector of the investigated signal. The main idea of the method is to build and apply a special correction function using a set of model fractal signals with previously known values of the fractal dimension. The mathematical bases of the new method are outlined. Features of the practical application of the corrective function method are considered on the example of the evaluation of regularization, boxing, variation and Hurst fractal dimensions. For them, the minimum values of the number of samples of the discrete data vector of the investigated signal, at which these dimensions can still be estimated, are defined. Using a set of model monofractal and multifractal signals on the example of the dynamical fractal analysis method, the effectiveness of the created method of the corrective function is shown. It is proven that due to the application of the correction function method, the maximum deviation of the estimated fractal dimension from the true known value for the specified dimensions is reduced from 25 – 55% to 5 – 7%.
... The fractal dimension is a measure of clustering AE information in time series [35][36][37][38]. To analyze the fracturing source of rock samples using fractal theory, we classified the commonly used fractal analysis statistical method of AE event time series into two categories: (1) Taking event frequency as the main body, we determined a sliding dynamic statistical event number with the fixed sliding step size as the boundary. ...
Nonlinear analysis of the rock mass failure process is a powerful tool for the study of instability and prevention of disaster in rock mass engineering. Nonlinear analysis has rapidly developed in recent decades. We propose a new method to determine the fractal characteristics of the failure process using time series analysis based on an iterative cycle division of the box dimension of fractal theory. We use AE technology to carry out time series analysis of fracture sources in the failure process of rock mass. We show that the disordered failure of rock mass exhibits notable fractal characteristics in the time series. The advantage of this method is that only the time and frequency of AE monitoring are used, resulting in a better identification and classification of damage, compared with traditional natural sequence frequency distribution. In addition, MATLAB numerical simulation analysis shows that these time series fractal dimensions can accurately characterize the failure characteristics of rock mass. In brief, we show the potential for future orderly time analyses of the complex and disorderly process of rock mass failure.
... We present lecture notes regarding a statistical dialect for population and community studies in which we use what James Gleick calls math beyond the derivative (Gleick 1987). The math genres that fit Gleick's description include Benoit Mandelbrot's (1961Mandelbrot's ( , 1977 fractal geometry and the Tesla Vortex algebra (Mathloger. 2022). ...
The paper presents the conceptual tools. Worked examples illustrate the analysis with remote-sensed data from the Cerrado (Savanna) Biome in the Norther Brazil Highlands.
... In the literature there have been many works on or related to the Hausdorff dimension of the graph of the Weierstrass type functions (see e.g. [8,5,9,10,11,12,18,13,14,16,17]). Recently, for the case φ is real analytic, thanks to the new theories in [11,18], Shen and the author [19] proved the following dichotomy for W φ λ,b : Either W φ λ,b is real analytic, or the Hausdorff dimension of its graph is equal to D 1 . ...
For a Lipschitz Z−periodic function φ : R → R 2 satisfied that R 2 \ {φ(x) : x ∈ R} is not connected, an integer b ≥ 2 and λ ∈ (c/b 1 2 , 1), we prove the following for the generalized Weierstrass-type function W(x) = ∞ n=0 λ n φ(b n x): the box dimension of its graph is equal to 3 + 2 log b λ, where c is a constant depending on φ.
... Starting in the mid 1970s and early 1980s, the turbulent approach underwent its own revolution: nonlinear physics and geophysics especially deterministic chaos [39][40][41] and fractals [42,43]. From the point of view of atmospheric dynamics, there were two key advances. ...
Since the first climate models in the 1970s, algorithms and computer speeds have increased by a factor of ≈1017 allowing the simulation of more and more processes at finer and finer resolutions. Yet, the spread of the members of the multi-model ensemble (MME) of the Climate Model Intercomparison Project (CMIP) used in last year’s 6th IPCC Assessment Report was larger than ever: model uncertainty, in the sense of MME uncertainty, has increased. Even if the holy grail is still kilometric scale models, bigger may not be better. Why model structures that live for ≈15 min only to average them over factors of several hundred thousand in order to produce decadal climate projections? In this commentary, I argue that alongside the development of “seamless” (unique) weather-climate models that chase ever smaller—and mostly irrelevant—details, the community should seriously invest in the development of stochastic macroweather models. Such models exploit the statistical laws that are obeyed at scales longer than the lifetimes of planetary scale structures, beyond the deterministic prediction limit (≈10 days). I argue that the conventional General Circulation Models and these new macroweather models are complementary in the same way that statistical mechanics and continuum mechanics are equally valid with the method of choice determined by the application. Candidates for stochastic macroweather models are now emerging, those based on the Fractional Energy Balance Equation (FEBE) are particularly promising. The FEBE is an update and generalization of the classical Budyko–Sellers energy balance models, it respects the symmetries of scaling and energy conservation and it already allows for both state-of-the-art monthly and seasonal, interannual temperature forecasts and multidecadal projections. I demonstrate this with 21st century FEBE climate projections for global mean temperatures. Overall, the projections agree with the CMIP5 and CMIP6 multi-model ensembles and the FEBE parametric uncertainty is about half of the MME structural uncertainty. Without the FEBE, uncertainties are so large that climate policies (mitigation) are largely decoupled from climate consequences (warming) allowing policy makers too much “wiggle room”. The lower FEBE uncertainties will help overcome the current “uncertainty crisis”. Both model types are complementary, a fact demonstrated by showing that CMIP global mean temperatures can be accurately projected using such stochastic macroweather models (validating both approaches). Unsurprisingly, they can therefore be combined to produce an optimum hybrid model in which the two model types are used as copredictors: when combined, the various uncertainties are reduced even further.
... Fractals are the graphical representation of Chaos theory (Mandelbrot 1977;Camacho Velázquez and Vásquez Cruz 2015). Fractal geometry describes structures whose curves, spirals and filaments repeat at different scales (principle of self-similarity) (Mandelbrot and Wheeler 1983) contrasting with traditional Euclidean geometry, which describes artificially created objects. ...
The fundamentals of Chaos theory allow the study of climatic conditions and long-term modifications produced by changes in their spatial and temporal scales. The aim of this work is to analyse the variability and changes produced in the annual cycles of temperature and precipitation in Northern Patagonia, Argentina, applying multifractal analysis as a practical mathematical tool of Chaos theory. Data from the NASA POWER Project (2021) was implemented as an alternative dataset for carrying out climatological studies in the area. Annual mean temperature and precipitation time-series data (1981–2019) were analysed at 72 grid points with 1° of spatial resolution. The Mann–Kendall test was used to calculate the trends through the annual cycles of the meteorological variables. Fractal dimension values were calculated using Multifractal Detrended Fluctuation Analysis. The Hurst exponent, complexity and asymmetry were the multifractal dimensions describing the persistence of time-series trends and climatic variability. The results showed changes in the annual cycles of both variables during the study period. The most significant finding was a large area in the centre and north of the study area, where the decrease in the rainfall regime was persistent. The Hurst exponent detected a sector in the Patagonian Andes mountain range where the temperature increase was constant. This work demonstrates that fractal geometry is useful to describe meteorological variability and obtain better short-, medium- and long-term forecasts.
... However, taking the coastline of Britain as an example, Mandelbrot showed that the self-similarity of curves can give rise to noninteger, i.e. fractional, dimensions larger than one. Later, he generalized this to higher-dimensional cases and introduced the term "fractal" for geometrical objects whose Hausdorff-Besicovitch dimension strictly exceeds their "topological" dimension (Mandelbrot 1977;Mandelbrot and Wheeler 1983). ...
The past 40 years have witnessed extensive research on fractal structure and scale-free dynamics in the brain. Although considerable progress has been made, a comprehensive picture has yet to emerge, and needs further linking to a mechanistic account of brain function. Here, we review these concepts, connecting observations across different levels of organization, from both a structural and functional perspective. We argue that, paradoxically, the level of cortical circuits is the least understood from a structural point of view and perhaps the best studied from a dynamical one. We further link observations about scale-freeness and fractality with evidence that the environment provides constraints that may explain the usefulness of fractal structure and scale-free dynamics in the brain. Moreover, we discuss evidence that behavior exhibits scale-free properties, likely emerging from similarly organized brain dynamics, enabling an organism to thrive in an environment that shares the same organizational principles. Finally, we review the sparse evidence for and try to speculate on the functional consequences of fractality and scale-freeness for brain computation. These properties may endow the brain with computational capabilities that transcend current models of neural computation and could hold the key to unraveling how the brain constructs percepts and generates behavior.
... Then, the results in Mandelbrot (1972) and Mandelbrot (1974) further emphasized the usefulness of scaling relations in the setting of turbulence modelling. For the first time in 1975, Benoit Mandelbrot defined the term fractal which was later developed in Mandelbrot (1977). ...
Water pipe failures result in real water losses in the form of water outflowing into the porous medium, such as the surrounding soil. Such an outflow may result in the creation of suffosion holes. The appropriate management of the water supply network may contribute to reducing the number of failures, but due to their random nature, it is not possible to completely eliminate them. Therefore, alternative solutions are being sought to reduce the effects of the failures. This article presents a fragment of the results from a broader scope of the research, which attempted to determine the outflow zone in relation to the fractal characteristics of water outflows. The research included the analysis of the actual geometric structures created by the water outflows, which were simplified into linear structures using isometric transformations. The structures were analyzed in terms of the parameters characterizing them, including their fractal dimensions. As a result, it was found that there was no relationship between the analyzed fractal parameters and the leakage area or hydraulic pressure in the water pipe. However, the influence of the number of points forming each linear structure on the analyzed parameters was shown. This allowed for the determination of further research aimed at estimating the size of the water outflow zone after the unsealing of an underground water supply pipe.
The mathematician Benoit Mandelbrot (1924–2010) created a new way to look at much of nature by inventing fractal geometry. He had a varied career in which he followed his instinct and interest. He left behind a unique legacy.
À partir d’une recherche-intervention déployée au cœur des cellules de crise d’une multinationale faisant face à la crise Covid-19, nous abordons un point peu abordé dans la littérature, celui de l’antinomie spécifique rencontrée par les multinationales. Ces dernières doivent en effet déployer une réponse cohérente tout en naviguant à travers les contextes variés de leurs entités locales. L’étude de la littérature nous permet de nous aligner avec les travaux ayant synthétisés les retours d’expérience des crises sanitaires, à ceci près qu’ils ne tiennent souvent pas compte des complexités rencontrées par les multinationales face à des réglementations, contextes et environnements variés. Dans le cadre d’un questionnaire partagé avec 19 cellules de crise activées d’une même multinationale, nous montrons que l’organisation homothétique de la crise favorise la flexibilité grâce à une influence ascendante des cellules de crise locales. En illustrant les bénéfices de la subsidiarité dans l’organisation des différentes cellules de crise, les résultats permettent de proposer aux multinationales de développer une organisation de gestion de crise adaptée pour faire face aux crises complexes et multicouches à venir.
Context
Landscape ecology is endowed with a wealth of accumulated insights into how spatial, ecological and social research can be fruitfully combined and synthesised. This has the potential to contribute significantly to how cultural landscapes are observed, analysed, conceptualised and explained.
Objectives
This article provides an overview of theories in landscape ecology relevant to the study of cultural landscapes. Based on a review of selected contributions formulated since the field's first inception, it is outlined how theory was developed within the field and how proven methods of theory generation can inspire further development.
Methods
A systematic review covering historical and contemporary theoretical contributions to landscape ecology was conducted. Theories were analysed to uncover by what methods they were formulated. On this basis, an overview of theories in landscape ecology relevant to the study of cultural landscapes was developed.
Results
A total of 32 theories were included in the review and described. Four pathways of theory development characteristic for the way knowledge is accumulated in landscape ecology were identified. These pathways exhibit modes of knowledge transfer between observations and actions taking place in concrete empirical contexts, knowledge which is transferable across contexts, as well as generally applicable concepts. An annotated overview of primary and secondary sources is provided. Contemporary literature building on the theories was identified, linking sources of conceptual inspiration to the current state of the art.
Conclusions
The review illustrates that a wealth of complementary theories exists in the field, creating a condition of theoretical multiplicity. Key theories and tendencies for theory development are outlined, and it is discussed how theoretical advancement in the study of cultural landscapes may be improved.
Herein theoretical ideas are presented to demonstrate the utility of CEs in uncovering what is often hidden in multiply interacting ON time series. We focus on what can be learned about ONs’ resilience using what we know about modified diffusion entropy analysis (MDEA) to reveal the influence of CEs on an ON’s ability to perform its healthy function. We emphasize processing empirical time series hosting invisible CEs that consists of a mix of CEs and non-CEs. These notions have recently been applied to heart rate variability (HRV) time series, and we review those applications to convince the reader of their wider range of utility in the form of CERT. The utility of CE time series is explored to determine the efficacy of a noninvasive type of rehabilitation for certain kinds of neurodegenerative diseases. We hypothesize how complexity matching among complex dynamic ONs can be used to define this kind of rehabilitation, one that mimics nature’s own strategy for healing injuries and recovering from disease based on the noninvasive driving of a morbid ON by a healthy ON of the same kind. The first question we need to answer is: How can invisible CEs be detected?
Operator algebra is introduced without being explicit about how we do things but hopefully conveys in words what we do not show with formal mathematics. The nomenclature “fractal” calculus replaces the historic misnomer “fractional” calculus and thereby restricts CERT to the non-integer calculus applied to phenomena that satisfy the fractal paradigm. The scaled PDF is shown to be the solution to a fractal kinetic equation (FKE) and to be necessary for obtaining the scaling by the fractal dimension observed in CE time series. The arguments for replacing integer-order rate equations with fractal-order rate equations, when the empirical evidence forces us to do so, are presented, as is the fractal probability calculus having IPL solutions. The latter solution satisfies the scaling behavior of simultaneously measured time series from the brain, heart, and lungs and provides a new kind of synchronization [complexity synchronization (CS)]. Two or more time series may not have their time series in synchrony and yet have their scaling indices in CS because the latter occurs at the level of the IPL index. This provides new insight into health, disease, and rehabilitation of ONs. We argue that a morbid ON can be rehabilitated to healthy functionality using a CS protocol.
This paper presents two applications of the general principle – the everything is a matter of degree – the principle that underlies fuzzy techniques. The first – qualitative – application helps explain the fact that while most earthquakes occur close to faults (borders between tectonic plates or terranes), earthquakes have also been observed in areas which are far away from the known faults. The second – more quantitative – application is to the problem of which of the collaborators should be listed as authors and which should be simply thanked in the paper. We argue that the best answer to this question is to explicitly state the degree of authorship – in contrast to the usual yes-no approach. We also show how to take into account that this degree can be estimated only with some uncertainty – i.e., that we need to deal with interval-valued degrees.
Safe and resistant infrastructure is an essential component of public safety. However, existing structures are vulnerable to damage resulting from excessive ground movement due to seismic activity or underground explosions. The aim of this paper, which is part of an extensive study, is to develop an isolation system based on periodic materials with H-fractal geometry in order to obstruct, absorb or completely modify the pattern of seismic energy before it reaches the foundations of structures. Fractal metamaterial structures have shown promise for increasing the frequency range prohibited for seismic protection. We report the anti-seismic properties of a seismic metamaterial model based on an H-shaped quasi-fractal cell. The fractal design, also known as seismic metamaterials, has an important impact on the band structures of seismic crystals. Using the fractal as a base unit, anti-seismic phononic crystals were developed, and their band-gap characteristics were shown to display unique features due to the increasing wave propagation path and hybridization between local resonances and Bragg scattering. The seismic–mechanical duality is supposed to provide flexible solutions capable of increasing/widening the band-gaps to improve the level of seismic protection.
The morphology of granular materials, such as sands, is of significant importance due to the effect of grain shape on their physical, mechanical, and hydraulic behavior. As technology has progressed from visual identification to modern computer-based techniques, numerous methods have been developed for quantifying grain shapes, many of which utilize digital image analysis and advances in computational techniques. A comprehensive understanding of available shape characterization methods is essential to make better use of these tools. This paper presents a state-of-the-art review of current methods for characterizing the morphology of granular materials, focusing particularly on digital image analysis techniques. It critically evaluates two essential aspects of shape characterization: the acquisition of particle shape information and shape measurement methods, discussing the strengths and limitations of each approach. Further, the application of grain shape characterization to analyze the effect of particle shape on the macro-scale behavior of sand is discussed. The review emphasizes the need to shift from classical shape characterizations developed by sedimentologists to objective-oriented shape characterizations that enable micro-to-macro correlations, taking into account the availability of robust tools and technologies.
Responses to sustainability challenges are not delivering results at the scale and speed called for by science, international agreements, and concerned citizens. Yet there is a tendency to underestimate the large-scale impacts of small-scale, local, and contextualized actions, and particularly the role of individuals in scaling transformations. Here, we explore a fractal approach to scaling sustainability transformations based on "universal values." Universal values are proposed as intrinsic characteristics that connect humans and nature in a coherent, acausal way. Drawing on the Three Spheres of Transformation framework, we consider how enacting universal values can generate fractal-like patterns of sustainability that repeat recursively across scales. Fractal approaches shift the focus from scaling through "things" (e.g., technologies, behaviors, projects) to scaling through a quality of agency based on values that apply to all. We discuss practical steps involved in fractal approaches to scaling transformations to sustainability, provide examples, and conclude with questions for future research.
Natural Language Processing (NLP) makes use of Artificial Intelligence algorithms to extract meaningful information from unstructured texts, i.e., content that lacks metadata and cannot easily be indexed or mapped onto standard database fields. It has several applications, from sentiment analysis and text summary to automatic language translation. In this work, we use NLP to figure out similar structural linguistic patterns among several different languages. We apply the word2vec algorithm that creates a vector representation for the words in a multidimensional space that maintains the meaning relationship between the words. From a large corpus we built this vectorial representation in a 100-dimensional space for English, and Estonian. Then, we calculated the fractal dimensions of the structure that represents each language. The structures are multi-fractals with two different dimensions that we use, in addition to the token-dictionary size rate of the languages, to represent the languages in a three-dimensional space. Finally, analyzing the distance among languages in this space, we conclude that the closeness there is tendentially related to the distance in the Phylogenetic tree that depicts the lines of evolutionary descent of the languages from a common ancestor.
Community governance quality is of interest to entire human beings across the globe, as communities are the very environment they are living in, hence deciding on people’s life quality. However, the approaches to an inhabitable community vary and the specific governance patterns of community often differ from country to country. Some communities exhibit strong favor for spontaneous order while other communities obviously result from certain deliberate plan. The list of cross-community difference can go on. No matter what kind of pattern a community has chosen willy-nilly, it is noticeable that the realistic governance pattern for a community to embrace is determined by many factors, such as residents’ mores, external institutions and historical tradition among others. To demonstrate the hypothesis, this paper employs a comparative case study of community governance respectively in USA and China (i.e., the city of Oswego, NY, USA and a few urban communities in China). First, a literature review is given on the significant approaches in China to improving community governance. Then, encouraged by the paradigm of social physics (that is a mindset about borrowing scientific principles to handle social problems), the authors choose fractal theory (especially the relevant concept of self-similarity) as an analysis framework to arrive at the conclusion that specific community governance pattern is usually not freely chosen but determined by the fundamental variables which the community happens to be embedded in. The method of analysis and its subsequent implications are expected to turn out useful in understanding and further bettering the community governance effect in future.
Our hypertechnological civilization, obsessed with attempting to control and predict every event and every aspect of our lives, has not yet understood that emergency is a connotative element of complexity and of the complex systems that we call "life". This means that emer-gency, like error, is an intrinsic part of our lives and can never be predicted, prevented or managed, much less eliminated. Seeking solutions by delegating carte blanche to technology, stak-ing all on know-how, speed and simulation, is the "great mistake" of today's digitalized society and of its educational institutions. In dealing with emergency, rather than rationalizing our inadequacies and those of our authorities and experts by using the age-old metaphor "black swan" (Taleb, 2007; 2012), students and teachers alike need to be empowered to inhabit complexity, to expect unpredictability, and to tackle emergency through creativity and self-organization, in order to be able to fully comprehend how emergency can become emergence.
The chapter discusses the results of a neural network learning algorithm for recognizing fractal images of the “Fractal dust” type (Kantor's set). This algorithm allows you to restore a fractal image with low quality. A generated set of matrices of the randomized system of iterative functions (RSIF), and fractals built on these matrices were used for training the neural network Mathematical formulas and schemes for calculating and constructing a fractal image and recognition are derived. This algorithm consists of finding RSIF for an existing fractal with different image quality. A learnt neural network makes it possible to compress an existing fractal image to a small set of coefficients. With their help you can reproduce an image of any format (resolution), as well as improve (distort) the quality of the image. The use of the algorithm in the future will make it possible to develop an automated library which can be integrated for graphic editors, data protection, information compression and improvement of display quality.KeywordsNeural networkFractalImage recognitionData compressionRandomized system of iterative functions (RSIF)
The presence of discontinuities in the rock mass is extremely crucial in many engineering applications of rock mechanics. The estimation of the joint roughness coefficient (JRC) utilising easily accessible camera-friendly devices and application software has recently gained prominence. The joint roughness coefficient of a jointed rough surface may be calculated using either contact methods (physically contacting the surface to get the profile) or non-contact methods (extraction of the profile without physically touching the surface using laser profilometry or 3D laser scanning and photogrammetry techniques). The 3D laser scanner is widely used for its accuracy in obtaining the surface topography of rock masses; however, the practice is often uneconomical due to the high equipment cost. In this study, the smartphone photogrammetry technology is used to extract the three-dimensional surface topography of a jointed rough surface in order to evaluate the JRC. For the comparative study, the surface profile was retrieved using a surface profilometer, 3D scanner and photogrammetry method. The JRC values for the surface profiles gathered through smartphone photogrammetry, 3D scanning and surface profilometer have been estimated using visual analysis, fractal dimension and statistical techniques. The study found that the surface profile generated using the photogrammetric approach utilising smartphone delivers more realistic profiling than the surface profilometer because of the lower least count and greater precision. The predicted JRC values for smartphone photogrammetry profiles agreed well with the JRC values obtained for 3D scanner and surface profilometer profiles.
In this paper, we introduce a class of novel C1-rational quartic spline zipper fractal interpolation functions (RQS ZFIFs) with variable scalings, where rational spline has a quartic polynomial in the numerator and a cubic polynomial in the denominator with two shape control parameters. We derive an upper bound for the uniform error of the proposed interpolant with a C3 data generating function, and it is shown that our fractal interpolant has O(h2) convergence and can be increased to O(h3) under certain conditions. We restrict the scaling functions and shape control parameters so that the proposed RQS ZFIF is positive, when the given data set is positive. Using this sufficient condition, some numerical examples of positive RQS ZFIFs are presented to support our theory.
In this paper, homotopy analysis method (HAM) is used to obtain the analytic solution for fragmentation population balance equation. Different sample problems are solved using HAM and their series solution is obtained. A detailed analysis of the series solution and the region of convergence of the solution is also studied. It is observed that the convergence region of the series solution can be adjusted with the help of certain parameters involved in HAM.KeywordsHomotopy analysis methodPopulation balance equationAnalytic approximationsFragmentationConvergence
One of the main tasks confronting contemporary educators, is to maintain optimism and hope and be pro-actively creative amidst the bureaucracy and perfunctory processes of the university. Despite expectations to conform to institutional prescriptions, pedagogical tactics can be invoked, which can operate to repurpose the strategies and established practices of the learning environment. To develop this argument, the chapter will adapt and develop a range of concepts as pedagogic possibilities, associated with Gilles Deleuze (The Fold), Deleuze and Guattari (the Rhizome), Benoit Mandelbrot (Fractality) and Michel de Certeau (tactics). In so doing, the chapter will invite educators to investigate and facilitate the possibility of fostering alternative and ultimately creative and hopeful learning approaches (also incorporating bell hooks), that incorporate the unpredictable emergence and heterogeneous ruptures and Folds of individual learners. Through the development and suggested application of these concepts, the chapter will argue that beyond established curriculum and pedagogy, untapped spaces of collaborative hope and possibility can be recognised and embraced. In challenging and short-circuiting boredom-inducing prescriptions and closed-system learning environments, a more open-system and ‘process’ of knowledge-fluidity and nonlinearity can emerge. The possibility of uncovering and adapting dynamic learning spaces, with hopeful pedagogical tactics, means that flexible and creative practices can incite inner-worlds of unpredictable rhythm and articulation to emerge.
This paper presents a novel method for the characterization of the sea state using a set of raw experimental Synthetic Aperture Radar (SAR) data in two dimensions, i.e., in the “fast time” and “slow time” directions (as explained in the text) and the so-called “Modified Fractal Signature” (MFS) method. That is, experimental SAR radar signatures in the above two dimensions (i.e., “raw data” in the time domain) were provided to our research group by the Norwegian Institute of Defence (FFI Institute), Norway, which we processed and analyzed using the MFS method in a novel way, as presented in detail in this paper. The numerical results obtained here show an easy categorization of the sea surface as “calm sea” or “turbulent sea”, thus establishing a very promising technique for the characterization of sea state in real time, as described in detail in the text.
Based on the rockery photos of ten classical gardens including world cultural heritage, this paper incorporates the combined analysis and comparison of constituent elements (rockeries, buildings and plants) of scenic surfaces, and probes into the fractal characteristics of landscaping through the theory and the fractal dimension (FD) value analysis software. Studies have shown that the quantitative evaluation data of visual complexity (FD, i.e., fractal dimension) can characterize the contour morphology of constituent elements of the scenic surface of rockeries. The relevant analysis results are as follows:
(1) FD can directly quantify the morphological contour of each element. Through the statistical analysis, it can effectively avoid the misjudgment of empirical cognition and subjective feeling. Therefore, FD value can be used as one of the effective indexes to evaluate the complexity and diversity of rockery and landscape elements.
(2) The change in the level of the FD value enables the intuitive analysis of the effects of the plant varieties and landscaping techniques on rockery morphological complexity.
(3) Higher FD value is not always better. Necessary morphological maintenance is required to avoid excessive FD value of plants.
The USDA Forest Service, in collaboration with the California Natural Resources Agency and other partners, is committed to increasing the “pace and scale” of forest treatments in California. Multiple federal and state initiatives in the last few years detail this commitment. The Forest Service developed the “Strategy for Shared Stewardship” (2018), a program to work with land management partners to co-manage fire risk across broad landscapes. The State of California issued a “Wildfire and Forest Resilience Action Plan” (January 2021) designed to strategically accelerate efforts to restore the health and resilience of California forests through a joint State-Forest
Service framework to enhance stewardship in California. In all cases, land managers need support to plan and implement treatments to address restoration at a landscape scale.
An essential component of these initiatives is the spatial data representing landscape conditions and new analytical tools for planning management investments. Pacific Southwest Research Station (PSW) scientists and staff from Region 5 Information Management, Mapping and Remote Sensing (MARS) Team, joined forces to
develop and/or collect and assemble existing sources of spatial data. This project, referred to as the ACCEL project (for accelerating pace and scale of treatments), combines the expertise and experience of research and management to build this library of data on landscape conditions.
Scientific Philosophy and Principles in Medicine is an accessible treatise on the philosophy that guides medical practice. It lays the foundation of a multidisciplinary framework behind the development of the medical profession. The book presents 10 chapters that cover issues that are frequently encountered by medical professionals in their career: philosophical and linguistic principles of rational thought, scientific, crisp and fuzzy logic, diagnostic aspects, the history of medicine, epistemological concepts, approximate reasoning, principles of medical wisdom, numerical and graphical diagnostics, and the collaboration of researchers involved in the fields of engineering and medicine.
The author of the book brings several years of teaching experience and medical practice into this reference with the goal of integrating principles of scientific philosophy and logic into medical education. Readers will understand the process of devising rational diagnostic and treatment approaches that support human health as a generative process that seeks to solve problems through creativity, rather than a classical process of following medical protocols.
This book is intended as a basic reference for medical students, teachers, and general readers interested in the application of logic, philosophy and scientific principles in medicine.
This article is devoted to sets having the Moran structure. The main attention is given to topological, metric, and fractal properties of certain sets whose elements have restrictions on using digits or combinations of digits in own representations.
In this paper, we study the long memory behavior of the hourly cryptocurrency returns during the COVID-19 pandemic period. Initially, we apply different tests against the spurious long memory, with the results indicating the presence of true long memory for most cryptocurrencies. Yet, using the multivariate test, the series are found to be contaminated by level shifts or smooth trends. Then, we adopt the wavelet-based multivariate long memory approach suggested by Achard and Gannaz (2016) to model their long memory connectivity. The findings indicate a change in persistence for all series during the sample period. The fractal connectivity clustering indicates a similarity among Ethereum (ETH) and Litecoin (LTC), Monero (XMR), Bitcoin (BTC), and EOC token (EOS), while Stellar (XLM) is clustered away from the remaining series, indicating the absence of any interdependence with other crypto returns. Overall, shocks arising from COVID-19 crisis have led to changes in long-run correlation structure.
Zeolite Y and Ti-containing zeolite Y (1%, 2% and 5% TiO2) were synthesized by a hy-drothermal seed-assisted method. In order to evidence the evolution of morphology, structure, and fractal dimensions during the zeolitization process at certain time intervals, a small volume from the reaction medium was isolated and frozen by lyophilization. The obtained samples were characterized by scanning electron microscopy (SEM), wide-angle X-ray diffraction (XRD), and small-angle X-ray scattering (SAXS). The fractal dimension values of the isolated samples, calculated from SAXS data, evidenced a transition from small particles with a smooth surface (2.021) to compact structures represented by zeolite crystallites with rough surfaces (2.498) and specific organization for zeolite Y. The formation of new structures during hydrothermal treatment, the increase in crystallite size and roughness due to the continuous growth were suggested by variation of fractal dimensions values, SEM microscopy images and X-ray diffractograms. The incorporation of titanium in low concentration into the zeolite Y framework led to the obtaining of low fractal dimensions of 2.034-2.275 (smooth surfaces and compact structures). On the other hand, higher titanium concentration (2%) led to an increase in fractal dimensions indicating structures with rougher surfaces and well-defined self-similarity properties. A mechanism for zeolite synthesis was proposed by correlation of the results obtained through morphological, structural, and fractal analysis.
Bu çalışmada, granüler zeminlerin fiziksel özellikleri ile şekilsel özellikleri arasındaki ilişkiye dikkat çekilerek, granüler zeminlerin şekilsel özellikleri (şekil, köşelilik, boyut, yüzey pürüzlülüğü, fraktal boyut, v.b.) detaylı olarak anlatılmıştır. Ayrıca görüntü analiz yöntemi hakkında özet bilgi verilerek, granüler zeminlerin şekilsel özelliklerinin belirlenmesi örneklendirilmiştir. Sonuçta şekilsel özelliklerin, daneli zeminlerin bazı fiziksel-mühendislik özelliklerine etkileri araştırılmış ve görüntü analiz yönteminin zeminlerin fiziksel-mühendislik özelliklerinin tespitinde kullanılabileceği ortaya konulmuştur.
In the review the spatial organization of cells, cell ensembles, tissues and metazoan body are considered using the concepts of fractal geometry, topology and dynamic chaos theory. We investigated both the scenario of transition from chaos into order during self-organization of cells in vitro and the reverse scenario of transition from order to chaos in the fractal morphogenesis of metazoan cell systems. Chaotic features in animal morphology were identified and quantified. Fractal morphogenesis was studied using epithelial branching channels of gastrovascular system in the scyphomeduse Aurelia aurita and tracheal gill system in the mayfly larvae Siphlonurus immanis and Parameletus chelifer, as well as structures of colonial interna in rhizocephalan crustaceans Peltogasterella gracilis and Polyascus polygenea. It was shown that completely identical fractal patterns do not occur even within a single animal body with radial or bilateral symmetric, functionally equivalent repetitive modules. Fractal dimension was used to quantify the spatial complexity of neuron morphology in central nervous system of the fishes Pholidapus dybowskii, Oncorhyhchus keta and Oncorhyhchus masou. During ontogenesis of Oncorhyhchus masou the values of fractal dimension and linear morphometric indicators were rising in studied neuron groups. Probably biological morphogenesis with chaotic fractal regime had an advantage in evolution, providing morphofunctional variability, plasticity and adaptability to unpredictable environmental changes.
The purpose of this chapter is to enhance the capacity for recognizing, comprehending and learning how to inhabit complexity, in particular social complexity, on the part of both experts and laymen, by describing the unique characteristics of complex systems, and above all by clarifying the crucial differences between complex and complicated systems, so often confused even among prominent scientists and researchers. Equally important is the purpose of changing the trajectory of those experts, educators, economists and political leaders who persist in the erroneous conviction that in the near future it will be possible to obtain a thorough understanding of virtually all phenomena, controlling and managing their complexity, and eliminating error, doubt and unpredictability from our societies and our lives.Design/Methodology/Approach—Beginning with a brief historical description of the scientific awareness gradually acquired on the implications of complexity, hierarchical systems and the capacities for self-organization and emergence inherent to all biological, physical, human and social complex adaptive systems (CAS), and the inadequacy of defining reality through mathematical formulas or sets of rules, as had previously been used, we will provide: (1) a working definition of social complexity and “hypercomplexity”; (2) epistemological methods for teaching and training students to undertake a systemic approach, engage in systems thinking and understand the full implications of the “observer/participant” and of qualitative factors; (3) techniques for including error, uncertainty and unpredictability, conflict and debate in education, training hybrid figures capable of intersecting inter/multi/transdisciplinary fields of knowledge.Findings—We will identify the illusions of the hypertechnological, hyperconnected civilization and its ongoing anthropological transformation, including (1) the “tyranny of concreteness” and “the great mistake”: the belief that all problems can be solved by delegating solutions solely to technology, and that complexity can be measured, managed and predicted through data, algorithms, formulas and statistics; (2) the fracture between the sciences and the humanities and between the natural and the artificial represented by “false dichotomies”; (3) the illusions of social control and elimination of error; 4) the vision of an ordered, regular society occasionally interrupted by “black swans”, without recognizing that emergency, error, uncertainty and unpredictability are intrinsic to all complex adaptive systems, which follow an irreversible arrow of time.Originality—(1) Bringing forth evidence that managing complexity is impossible; at best, we can only cope with or inhabit complexity; (2) Rethinking education and training radically and not simply as adaptations and extensions of educational processes to the technological changes; technology is a part of culture and can never be separate from it. (3) Introducing the concept of the “overturn”: today biological evolution is being determined by cultural evolution.Social and Environmental Implications—Profound and systemic change can only be triggered from grassroots communities and individual actions, and can never be imposed top-down by intelligentsia, elites or governments. This implicates the fundamental importance of educational processes teaching systemic and critical thinking. Otherwise, any innovation will simply become a “would-be” innovation. It is furthermore essential to understand that innovating means destabilizing, at least temporarily.Research limitations—An intrinsic limitation to the approach described in this chapter is based on qualitative methods and factors, which are, in the opinion of this writer, impossible to measure, classify, or in general translate into quantitative data, as each person and/or community is unique and in continual interaction with others. Although this can be considered a criticality, it is useful for avoiding the standardization of the findings obtained.KeywordsErrorUncertaintyUnpredictabilityThe great mistakeFalse dichotomiesComplex adaptive systems
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