Fractals and multifractals

Emerging wireless communication antennas presently need to operate at ISM. In this paper, a Cantor fractal monopole antenna using multi fractal structure is proposed and presented for ISM band applications. The antenna is designed in FR4 substrate with dielectric constant of 4.6. Simulation is performed with CST microwave studio. The bandwidth of the antenna is optimized and the gain of the proposed antenna is improved.

This article argues that any ecological finance theory devised to fit the Sustainable Development Goals (SDGs) needs a paradigm shift in the morphology of randomness underlying financial risk modelling, by integrating the characteristics of "nature" and sustainability into the modelling carried out. It extends the common diagnosis of the 2008 financial crisis with considerations on the morphology of randomness and the reasons why neoclassical finance theory is not sustainable from this perspective. It argues that the main problem with unsustainable neoclassical finance risk modelling is its underlying morphology of randomness that creates a dangerous risk culture. It presents Leibniz's principle of continuity and Quetelet's theory of average as cornerstones of classical risk culture in finance, acting as a mental model for financial experts and practitioners. It links the notion of sustainability with the morphology of randomness and presents a possible alternative approach to financial risk modelling defined by rough randomness. If morphology of randomness in nature is properly described by fractal and multifractal methods, hence ecological finance theory has to include fractal properties into financial risk models. The conclusion proposes a new agenda for future research.

Regarding the foundations of matter as energy fields in a fractal gearing structure can redefine the foundations of quantum physics in a unique new way, while remaining consistent with modern probabilisitic theories.  Further, working experimental models of fractal gear arrays have created effects that cannot be explained with current theories, and they are described here in detail.  This theory is the only way to explain the experimental phenomenon, and it is an explanation to phenomenon which I have been able to create in the lab for the first time in physics history.  Fractal gear theory is a deterministic model of quantum fields that can be used simultaneously with the current standard model and they both can borrow concepts from each other without breaking the logical consistency.  An example is the double-slit experiment; fractal gear theory states that a harmonic field structure is created by the geometry of the matter (the slits) and the electron or photon rides on it in an oscillatory manner so it does not matter if one or many particles enter simultaneously through the slits.  Many of the modern theories provide quantitative answers to the questions that arise from this new deterministic unified field theory. A prime example is that strings from string theory can be understood as a line of contact thru time within a fractal gear field array.  Physicists today have obfuscated theories with statistical data , whereas most of the real advances in physics have arisen from thought experiments and theoretical and geometric analysis. This theory was devised with very little mathematics and a whole lot of 3D geometry and analysis with almost zero experimental data or references beyond the foundations of modern quantum physics.  It is a grass roots endeavor to start from scratch and rebuild quantum physics with a correct picture that almost any mechanically inclined person can understand; a picture that has eluded physicists for over 100 years.  I believe that if a picture is worth 1000 words, then a geometry is worth 1000 formulas, but geometry is much easier to work with.

Fractal gear fields are a quantum foam theory and provides means for transmitting information and by its explanation of emergent space time, it is directly implied that information is able to travel faster than the speed of light within a bound domain.  The speed of transmission dissipates at a rate proportional to the total energy within the domain.  Fractal gear theory is a geometrical foundation for a pilot-wave theory in the spirit of DeBroglie -Bohm theory.  Further, the 3D geometric form of fractal gear fields shows clearly that all bell inequality measurements using photon polarization is biased and in error, due to angular variations in density of the field structure.

1. My paper "ENTROPY DYNAMICS ASSOCIATED WITH SELF-ORGANIZATION" (pp. 229-242).
ABSTRACT: A general model linking dynamics of informational entropy and the self-organization process in an open, steady-state, nonequilibrium system is proposed. Formulas for dynamics of the informational entropy flow and its rate are developed with respect to random process of influences exerted upon a system. It was revealed that the open system responds to a strong change of conditions by steep growth ofthe informational entropy flow up to a maximum value at the critical point associated with the self-organization process. An example of self-organization during elasto-plastic deformation of metal is considered.
2. My paper "REVEALING THE MULTIFRACTAL NATURE OF FAILURE SEQUENCE" (pp. 295-296).
ABSTRACT: The purpose of the paper is to introduce a new application of the muMfractals in the reliability engineering, risk analysis, historical chronology and others fields where we deal with series of events of various natures. The spectrum of such series is quite broad. In particular, we do not know whether some temporal pattern is hidden in an apparently disordered set of events. The multifractal theory is a good basis for revealing such an order and describing the event sequence in time. It can provide a deeper understanding the nature of the event flow.

Desde una búsqueda diferente de las fuentes históricas de la Colonia y con la idea de que el altépetl se destruye en una trayectoria compleja después de la Conquista española, los autores realizan un estudio de caso sobre la vieja cabecera de Itzmiquilpan en la que demuestran que la segregación de los barrios indígenas sujetos respecto a su antigua cabecera, heredera del gobierno indígena prehispánico, se configuró de tal manera que el tiempo transcurrido entre los eventos es una función del número de pueblos independizados, el tamaño de las fronteras que construyeron y la superficie del territorio enajenado, de forma tal que, incluso, la dimensión fractal de las fronteras y de la superficie independizada es una función logarítmica del tiempo, donde las grandes extensiones territoriales tienden a reducirse hasta una dimensión fractal que converge en un valor cercano al 1.3 y finalmente exploran la idea de que existe una autosimilitud espacial en la acumulación de los procesos, también como una función del tiempo.

Urban intervention processes in the historical cities demands a thorough comprehension of the urban
morphogenesis approaches in these complex structures. It took hundreds or even thousands of years
urban structures to emerge and evolve till the current morphology co-evolving with the human biofunctional
interrelations evolution. The focus of this research is on fractal geometry role’s in emerging
urban morphogenesis, by analyzing two main rules (Multifractality and Sensitive Adaptability)
capable of achieving comprehensive homogeneity at all Urban and Architecture levels. To understand
these new approach and to create a new epistemological framework for urban morphology,
evolutionary techniques could be used in order to have a set of rules for building different genotypes
of Architecture and urban forms which, by using computational processes, develop as output 3D
models of urban processes, more suitable with complex adaptive systems such as historical contexts.

Agro-meteorological quantities are often in the form of time series, and knowledge about their temporal scaling properties is fundamental for transferring locally measured fluctuations to larger scales and vice versa. However, the scaling analysis of these quantities is complicated due to the presence of localized trends and nonstationarities. The objective of this study was to characterise scaling properties (i.e. statistical self-similarity) of the chosen agro-meteorological quantities through multifractal detrended fluctuation analysis (MFDFA). For this purpose, MFDFA was performed with 11322 measured time series (31 yr) of daily air temperature, wind velocity, relative air humidity, global radiation and precipitation from stations located in Finland, Germany, Poland and Spain. The empirical singularity spectra indicated their multifractal structure. The richness of the studied multifractals was evaluated by the width of their spectrum, indicating considerable differences in dynamics and development. In log-log plots of the cumulative distributions of all meteorological parameters the linear functions prevailed for high values of the response, indicating that these distributions were consistent with power-law asymptotic behaviour. Additionally, we investigated the type of multifractality that underlies the q-dependence of the generalized Hurst exponent by analysing the corresponding shuffled and surrogate time series. For most of the studied meteorological parameters, the multifractality is due to different long-range correlations for small and large fluctuations. Only for precipitation does the multifractality result mainly from broad probability function. This feature may be especially valuable for assessing the effect of change in climate dynamics.

In this work a generic rugged fracture surface was modeled starting from the fractal theory. This fracture surface (or crack profile) was considered as being a self-affine fractal, with averaged dimension, H (Hurst’s exponent). A relationship between the real fractured area (rugged) and the projected area (it planes) was obtained. Surfaces of fractures and experimental cracks were analyzed and compared with the model. Was concluded that the Hurst dimension of ruggedness is capable to express the irregularities of fracture surfaces to different materials, characterizing its fractal morphology. Soon after, the geometry fractal was introduced in Elasto-Plastic Fracture Mechanics (EPFM) with the purpose of modeling the fracture and the stable crack propagation in brittle and ductile materials. The equation that defines the elasto-plastic energy release rate, J, [ANDERSON 1995] was modified to take in account the ruggedness of fracture surface, using the mathematical expression that relates the rugged surface with the projected surface. By means of this relationship it was obtained, an mathematical expression to J-R curves that depends of noch size, the minimum fracture size, the Griffith critical size and the fractal dimension of fracture, showing with this the advantageous consequences of this modification. Experimental testings of J-R curve, to different materials, were accomplished by the compliance method, whose results were shown in great agreement with the proposed model. As a consequence of theoretical fitting curves to the experimental measures of integral-J, it was obtained the energy, geometric and fractais parameters of the fracture such as, the effective energy of fracture surface, 2geff = 2ge + gp, and the minimum crack size, lo, and the ruggedness dimension, H, (exponent Hurst). In particular the parameter H, obtained by the fitting, was compared with that obtained by the ruggedness fractal characterization, H, to each fractured sample. It was shown with that, of an inambiguous way, the rising of the J-R curve has a strong correlation with the fractal dimension and with the ruggedness dimension (Hurst’s exponent) of the fractured surface. The proposition of this work was compared with others authors showing that it differs from those currently used in the literature, differing the results for therein presented. As consequence it was shown too a correction for the mathematical expressions proposed by MU [1988] and LUNG [1988], that they relate the fractal dimension with the elastic energy released rate, Go, and also for the mathematical expressions proposed by MECHOLSKY and PASSOJA [1989] that relate the fractal dimension with the fracture toughness, KIC. By last, as a consequent fracture of the fractal fracture scaling, was shown the existence of a new property equivalent to fracture toughness. Therefore the fractal theory, become possible a mathemathical revision of the Classical equations of Fracture Mechanics (FM) which was historically established using the Euclidian geometry. But now, was made a mathematical reformulation of CFM using the fractal geometry, in order to obtain relationships that include in a explicit way the ruggedness of fractured surface becoming the mathemathical description most realistic and authentic. Therefore this work tried to study a way for which the fractal theory can enrich and elucidate several aspects of the Fracture Mechanics. The results obtained in this work can solve some of the existent doubts in the literature when the fractal scaling is used in the formulation of physical greatness that depend on the ruggedness and the projected area of fracture.

The motivation for the continuous dimensionality of spacetime near the Fermi scale stems from two premises, namely, 1) dimensional regularization of perturbative quantum field theory (QFT), 2) the existence of non-trivial fixed points of Renormalization Group equations. Here we discuss a third reason, rooted in the behavior of non-equilibrium phase-space flows.

We perform the Multifractal Detrended Fluctuation Analysis (MF-DFA) method to
investigate the multifractal properties of the Moroccan All Shared Index (MASI) and the
Moroccan Most Active Shares Index (MADEX) from the Casablanca Stock Exchange
(CSE). By applying the MF-DFA method we first calculate the generalized Hurst
exponents and we then deduce the Rényi exponents as well as the singularity spectrum of
the MASI and MADEX indices. Furthermore, we perform the shuffling and the phase-
randomization techniques to detect the sources of the multifractality. We show that there
are two major sources of multifractality, the long-range temporal correlations and the fat-
tail distribution. We show notably that the first source contributes mainly to the
multifractality of MASI index while the two sources contribute almost equally to the
multifractality of the MADEX index. By comparing the multifractal behavior of the MASI
and MADEX indices we find finally that the first one exhibits a richer multifractal feature
than the second one. This permits us to conclude that the greater is the stock market the
more complex is the dynamics of the stock market index representing all of the market,
which is traduced by richer multifractal behavior of this index. This study leads to the
principal conclusion that the Casablanca Stock Exchange is characterized by a multifractal
behavior.

Gas movement in porous media is important for many environmental, engineering, ecological, and agricultural questions, including soil-atmosphere gas exchange, respiration of plants and micro-organisms, and various chemical reactions. The number, size, geometry, connectivity, and tortuosity of pores influence gas movement. The complexity of soil pore systems can be described by multi-fractal analysis at the microscopic scale, while soil gas diffusion coefficients (Ds/Do) are usually measured at the core scale. The goals of this study were therefore to 1) quantitatively describe air-macroporosity from high-resolution X-ray computed tomography (CT) scanning images of ten intact soil cores, 2) analyze the frequency distribution of this air-macroporosity among columns using multi-fractals, and 3) assess the relationships, based on Spearman’s correlation coefficients, between Ds/Do measured in laboratory and the multi-fractal parameter estimates describing pore structural variability. Dep...

Today's evolving cyber security threats demand new, modern, and cognitive computing approaches to network security systems. In the early years of the Internet, a simple packet inspection firewall was adequate to stop the then-contemporary attacks, such as Denial of Service (DoS), ports scans, and phishing. Since then, DoS has evolved to include Distributed Denial of Service (DDoS) attacks, especially against the Domain Name Service (DNS). DNS based DDoS amplification attacks cannot be stopped easily by traditional signature based detection mechanisms because the attack packets contain authentic data, and signature based detection systems look for specific attack-byte patterns. This paper proposes a chaos based complexity measure and a cognitive machine classification algorithm to detect DNS DDoS amplification attacks. In particular, this paper computes the Lyapunov exponent to measure the complexity of a flow of packets, and classifies the traffic as either normal or anomalous, ...

Additive manufacturing (AM) provides a unique platform for the rapid design and fabrication of complex structures. These can be used as is or be used as templates to be decorated with electrochemical deposited nanomaterials, which are then used as electrocatalytic sensing platforms. Novel methods are consequently needed to determine the area of the electrochemical deposited morphology present on the electrode surfaces. Additively manufactured graphene macroelectrodes (AM-GMs) were fabricated using a commercially available feedstock and molybdenum (di)oxide (MoO2) was successfully electrochemically deposited onto the electrode surface. The electrochemically deposited MoO2 was analysed using Scanning electron microscopy (SEM), Optical Interferometry, Raman Spectroscopy and Multifractal analysis (MFA). Although the electrochemical deposition of MoO2 nanowires were clearly visible using SEM, the use of MFA enabled the facile quantification of the MoO2 nanowires deposited at a variety of time points (20 – 300 s). For the first time MFA was utilised to generate quantitative data, derived from ƒ(α) curves, to determine the area of the electrochemical deposited MoO2 nanowires, and also coverage (%), density, dispersion and clustering. The AM-GMs which were subjected to 300 s (the maximum time period used in this study) of MoO2 electrodeposition demonstrated the greatest percentage area coverage (20.14 %). The use of such mathematical systems offers a cost-effective method to characterise and determine useful parameters of electrochemically deposited materials which can applied to a range of macroelectrodes decorated with nanomaterials.

Emergent gravity builds upon the formal connection between classical Thermodynamics and the field equations of General Relativity. Drawing from the thermodynamic interpretation of multifractals, we speculate here that gravitational dynamics follows from the multifractal geometry of spacetime far above the Fermi scale. Our brief report suggests that a) informational entropy may be linked to the Beckenstein-Hawking entropy and b), Newton's constant may be linked to the information dimension, a defining metric of multifractal scaling.

This thesis presents a fractal phase calculus that contributes to the advancement of computational emulation of human behavior. Previous disparities between theoretical foundations of computability and behavioral complexity are coordinated into alignment. Gödel's theorems, rooted in fractal properties of natural, developmental stages of evolving complexity, describe downwardly assimilated properties of human experience. Tarski recognized Gödel's theorems are not limited to mathematics, but the relationship between mathematics, computation, and human behavior could not be made without the recent identification of the fractal nature of measurable behavioral transition dynamics. One conclusion drawn here is that Gödel's theorems and Church-Turing computability should no longer be described just as properties of mathematics and logic, but of human behavior in general. This fractal phase calculus unifies these seemingly disparate domains. In doing so, the lack of integral architectonic models for unified semantic architectures and operational processes that emulate human behavior in computational environments is addressed with two novel integral architectonic models of qualia abstraction (architecture) and phase complexity (process). When synthesized as a recursive architectonic they account for the broadest possible range of human experience and behavior. Recursive architectonic properties universal to human experience are captured and made computable through the fractal phase calculus. In addition to presenting its axioms, the fractal phase calculus is demonstrated by applying it to the development of a molecule to show that these universal architectures and processes underpin not just humans, but entities of all scales that humans conceive. This research was motivated by the needed functions of recursive architectonic software to facilitate a new means for the knowledge creation process. This current contribution has implications for (a) disseminating information, (b) unifying dissemination feedback, (c) supporting the integration and synthesis of differentiation across scales and breadths of participation, (d) identifying uncharted knowledge frontiers, (e) modeling dynamic knowledge, (f) empowering communication, and (g) the creation of a globally participatory theory of everything.

Over the years, a number of measures have been defined for the purpose of determining the number of independent dimensions contained in a space. The most common dimensionality measures are the topological dimensionality and various kinds of fractal dimensionalities. While each of these dimensionality measures is useful in its own right, none of them accurately quantifies the effective number of independent directions passing through locations contained in a local region of a space. This article introduces a new dimensionality measure, called the connectivity dimensionality field, which is the true measure for the effective number of independent directions passing through locations in a space. In contrast to the fractal dimensionality, the connectivity dimensionality field is a topological property because its value at each material location is invariant to deformations of the space preserving connectivity. The connectivity dimensionality field is a fundamental concept that applies to many different kinds of discrete spaces, continuous spaces, and discrete-continuous dual spaces. A discrete space is a space in which positions cannot be varied differentially, and a continuous space is a space in which positions can be varied differentially. A discrete-continuous dual space has complementary discrete and continuous representations, and a process called discrete-continuous dual matching relates the discrete and continuous representations to each other. This article formally defines two basic types of discrete-continuous duality: (a) asymptotic and (b) strict. A rigorous method is provided for computing the connectivity dimensionality field in edge-vertex graphs, continuous spaces, and discrete-continuous dual spaces. Many examples are given to illustrate the key concepts. Single points, unbranched lines, and periodic lattices are examples of discrete-continuous dual spaces in which the connectivity dimensionality field is a constant nonnegative integer. In other types of discrete-continuous dual spaces, the connectivity dimensionality field contains inherent uncertainty. For the first time, a comprehensive theory is derived that predicts the inherent uncertainty associated with the connectivity dimensionality field in discrete-continuous dual spaces. The study of discrete-continuous dual spaces with variable connectivity dimensionality fields transcends variable-based mathematics.

We demonstrate that the Rorschach ink-blots are multifractals. Normal subjects submitted to their perception for two minutes have been studied recording their GSR neurological and psychophysiological signal during the observation of the ink blots. It has resulted that also the induced perceptive dynamics in the subject is a multifractal. In the framework of the theory of complexity we elaborate that the intrinsic complexity of the ink-blots induces in the subjects an emotive and cognitive condition of profound inner conflict that may be estimated by the generalized Hurst exponent of the subject as arising from the GSR recording. We expect that the results are important not only in the theory of complexity but also in the clinical study of cases of neurological, psychological and psychiatric interest.

The reduction of bacteria and biofilm formation is important when designing surfaces for use in industry. Molybdenum disulfide surfaces (MoS2SUR) were produced using MoS2 particle (MoS2PAR) sizes of 90 nm, 2 μm, and 6 μm containing MoS2PAR concentrations of 5%, 10%, 15%, and 20%. These were tested to determine the efficacy of the MoS2SUR to impede bacterial retention and biofilm formation of two different types of bacteria, Staphylococcus aureus and Pseudomonas aeruginosa. The MoS2SUR were characterized using Fourier transform infrared spectroscopy, ion-coupled plasma atomic emission spectroscopy, scanning electron microscopy, optical profilometry, and water contact angles. The MoS2SUR made with the smaller 90 nm MoS2PAR sizes demonstrated smaller topographical-shaped features. As the size of the incorporated MoS2PAR increased, the MoS2SUR demonstrated wider surface features, and they were less wettable. The increase in MoS2PAR concentration within the MoS2SUR groups did not affect the surface topography but did increase wettability. However, the increase in MoS2PAR size increased both the surface topography and wettability. The MoS2SUR with the smaller topographical-shaped features influenced the retention of the S. aureus bacteria. Increased MoS2SUR topography and wettability resulted in the greatest reduction in bacterial retention, and the bacteria became more heterogeneously dispersed and less clustered across the surfaces. The surfaces that exhibited decreased bacterial retention (largest particle sizes, largest features, greatest roughness, and most wettable) resulted in decreased biofilm formation. Cytotoxicity testing of the surface using cell viability demonstrated that the MoS2SUR were not toxic against HK-2 cells at MoS2PAR sizes of 90 nm and 2 μm. This work demonstrated that individual surface variables (MoS2SUR topographic shape and roughness, MoS2PAR size, and concentration) decreased bacterial loading on the surfaces, which then decreased biofilm formation. By optimizing MoS2SUR properties, it was possible to impede bacterial retention and subsequent biofilm formation.

The current experiment investigated the effect of visual accuracy feedback on the structure of variability of time interval estimates in the continuation tapping paradigm. Participants were asked to repeatedly estimate a 1-s interval for a prolonged period of time by tapping their index finger. In some conditions, participants received accuracy feedback after every estimate, whereas in other conditions, no feedback was given. Also, the likelihood of receiving visual feedback was manipulated by adjusting the tolerance band around the 1-s target interval so that feedback was displayed only if the temporal estimate deviated from the target interval by more than 50, 100, or 200 ms respectively. We analyzed the structure of variability of the inter-tap intervals with fractal and multifractal methods that allow for a quantification of complex long-range correlation patterns in the timing performance. Our results indicate that feedback changes the long-range correlation structure of time estimates: Increased amounts of feedback lead to a decrease in fractal long-range correlations, as well to a decrease in the magnitude of local fluctuations in the performance. The multifractal characteristics of the time estimates were not impacted by the presence of accuracy feedback. Nevertheless, most of the data sets show significant multifractal signatures. We interpret these findings as showing that feedback acts to constrain and possibly reorganize timing performance. Implications for mechanistic and complex systems-based theories of timing behavior are discussed.

Fractals and multifractals are well-known trademarks of nonlinear dynamics and classical chaos. The goal of this work is to tentatively uncover the unforeseen path from multifractals and selfsimilarity to the framework of effective field theory (EFT). An intriguing finding is that the partition function of multifractal geometry includes the signature of non-Euclidean metric. Our results also suggest that the scale-free structure of multifractal geometry lies behind the concept of covariance in EFT.

Hamiltonian time crystals (HTC) are time-dependent solutions of the equation of motion that develop in a minimum energy configuration. A subset of HTC's includes periodic trajectories that spontaneously break time-translation invariance and occur at a local minimum of the free energy. Recent studies suggest that HTC's may be realized as novel topological structures formed by knotted molecules. In this brief note we show that fractional dynamics of harmonic oscillators may replicate the properties of HTC's outside the realm of traditional condensed matter applications.

Urban intervention processes in the historical cities demands a thorough comprehension of the urban morphogenesis approaches in these complex structures. It took hundreds or even thousands of years urban structures to emerge and evolve till the current morphology co-evolving with the human biofunctional interrelations evolution. The focus of this research is on fractal geometry role’s in emerging urban morphogenesis, by analyzing two main rules (Multifractality and Sensitive Adaptability) capable of achieving comprehensive homogeneity at all Urban and Architecture levels. To understand these new approach and to create a new epistemological framework for urban morphology, evolutionary techniques could be used in order to have a set of rules for building different genotypes of Architecture and urban forms which, by using computational processes, develop as output 3D models of urban processes, more suitable with complex adaptive systems such as historical contexts. -The interrelation b...

Urban intervention processes in the historical cities demands a thorough comprehension of the urban morphogenesis approaches in these complex structures. It took hundreds or even thousands of years urban structures to emerge and evolve till the current morphology co-evolving with the human biofunctional interrelations evolution. The focus of this research is on fractal geometry role’s in emerging urban morphogenesis, by analyzing two main rules (Multifractality and Sensitive Adaptability) capable of achieving comprehensive homogeneity at all Urban and Architecture levels. To understand these new approach and to create a new epistemological framework for urban morphology, evolutionary techniques could be used in order to have a set of rules for building different genotypes of Architecture and urban forms which, by using computational processes, develop as output 3D models of urban processes, more suitable with complex adaptive systems such as historical contexts. -The interrelation b...

As paradigm of complex behavior, multifractals describe the underlying geometry of self-similar objects or processes. Building on the connection between entropy and multifractals, we show here that the generalized dimension of geodesic trajectories in General Relativity coincides with the four-dimensionality of classical spacetime.

Fractals are mathematical structures under exploration and development in virtually every academic discipline and beyond: “besides mathematics and science, fractals have direct applications in many fields, including music, literature, visual art, architecture, sculpture, dance, technology, business, finance, economics, psychology, and sociology” (Frame & Mandelbrot, 2002, p. 4). Since fractal geometry today is “the only available language for the study of roughness, a concept that is basic and related to our senses, but has been the last to give rise to a science,” (Ibid., p. 11) and ‘roughness’ or unevenness or irregularity is a natural, universal tenet of thought and matter, the science behind fractal geometry unifying it across disciplines holds much promise.

The rise of a multidisciplinary discourse containing measurable -- and predictable – patterns of irregularity is attributed to fractal geometry with the seriousness and playfulness of mathematics at its root. Because of these fundamentals, the newness of the study of fractal geometry lends itself to extreme unpredictability. Since formulas and algorithms create fractal images and patterns, much of the fun and exploration of fractals lies in ability to experience new mathematical conjectures in anticipation of new discoveries.

Such an exploratory method has been developing across the disciplines with diverse jargon but one common language uniting them all, that being this staunchly predictable yet wildly unpredictable mathematical integrity of the fractal, dependent upon the available formula or algorithm used to suggest its creation and level of complexity.

We have yet to discover or unveil a serious application of fractal geometries related to patterns of irregularity in consideration of the work of Irwin Panofsky and the positing of iconography in art history, e.g., or in philosophy with study of Michel Foucault’s architecture of knowledge or history of insanity, which appear to naturally lend themselves to such interrogations.

We explore the idea that Minkowski spacetime and the principle of locality reflect the asymptotic properties of self-organized criticality (SOC). Both properties arise from demanding that the scaling behavior of space and time coordinates follows the power-law distribution of Gaussian random walks.

Here we show that there is an approximate mapping between multifractal theory and the Born rule of Quantum Mechanics. The derivation is based on the fractal geometry of quantum mechanical paths, which replicates the geometry of unrestricted random walks in d > 2 Euclidean dimensions.

the ever decreasing area of forests has lead to environmental and economical challenges and has brought with it a renewed interest in developing methodologies that quantify the extent of deforestation and reforestation. in this study we analyzed the deforested areas of the Apuseni Mountains, which has been under economic pressure in recent years and resulted in widespread deforestation as a means of income. Deforested surface dynamics modeling was based on images contained in the Global forest Database, provided by the Department of Geographical Sciences at Maryland University between 2000 and 2014. The results of the image particle analysis and modelling were based on total Area (ha), count of patches and Average Size whereas deforested area distribution was based on the Local connected fractal Dimension, fractal fragmentation index and tug-of-War Lacunarity as indicators of forest fragmentation or heterogeneity. The major findings of the study indicated a reduction of the tree cover area by 3.8%, an increase in fragmentation of 17.7% and an increase in heterogeneity by 29%, while fractal connectivity decreased only by 0.1%. The fractal and particle analysis showed a clustering of forest loss areas with an average increase from 1.1 to 3.0 ha per loss site per year. in conclusion, the fractal and particle analysis provide a relevant methodological framework to further our understanding of the spatial effects of economic pressure on forestry. In view of its effect on climate, deforestation is athe threatening challenge for contemporary society. The major factors leading to the decline of forested areas include an expansion of agricultural land 1-4 , human settlements 5 , timber demand 6 , mining exploitations and population growth 7 leading to environmental and economic challenges in the long term. Therefore the aim of the local or regional authorities is to reduce deforestation and to regenerate the existing forests wherever possible due to their importance in supporting the biotic system, regional and global climate change, the preservation of atmospheric oxygen and carbon, conservation of biodiversity and benefits to local communities 6,8-12. Forest fragmentation is a major result of deforestation 13,14. It leads to habitat modification 15 , and subdivision of plant and animal populations. Thus, changes in species interactions occur 13,14 , leading to further tree mortality

We show that the monthly recorded history (1866–2014) of the Southern Oscillation Index (SOI), a descriptor of the El Niño Southern Oscillation (ENSO) phenomenon, can be correctly described as a dynamic system supporting a potential nonlinear predictability well beyond the spring barrier. Long-term predictability is strongly connected to a detailed knowledge about the topology of the attractor obtained by embedding the SOI index in a wavelet base state space. By utilizing the state orbits on the attractor, we show that the information contained in the SOI is sufficient to provide nonlinear attractor information, allowing the detection of predictability for longer than a year: 2, 3, and 4 years in advance throughout the record with an acceptable error. This is possible due to the fact that the lower-frequency variability of the SOI presents long-term positive autocorrelation. Thus, by using complementary methods, we confirm that the reconstructed attractor of the low-frequency part (lower than 1/year) of SOI time series cannot be attributed to stochastic influences. Furthermore, we establish its multifractality. As an example of the capabilities of the methodology, we investigate a few specific El Niño (1972–1973, 1982–1983, 1997–1998) and La Niña (1973–1974, 1988–1989 and 2010–2011) events. Our results indicate that each of these present several equivalent temporal structures over other eras of these 149 years (1866–2014). Accordingly, none of these cases, including extreme events, presents temporal singularity. We conclude that the methodology’s simplicity of implementation and ease of use makes it suitable for studying nonlinear predictability in any area where observations are similar to those describing the ENSO phenomenon.