Fractal Concepts Research Articles

Bio-inspired plasmonic leaf for enhanced light-matter interactions

Abstract The mathematical concept of fractals is widely applied to photonics as planar structures ranging from terahertz resonators, optical antennas, to photodetectors. Here, instead of a direct mathematical abstract, we design a plasmonic leaf with fractal geometry from the outline of a leaf from Wargrave Pink. The enhanced light-matter interactions are observed numerically from the improvement in both absorption and near-field intensification. To demonstrate the effect experimentally, a three-dimensional fractal structure is realised through direct laser writing, which significantly improves the photothermal conversion. By virtue of the self-similarity in geometry, the artificial leaf improves the absorption of a 10-nm-thick gold film with 14 × temperature increment compared to flat Au film. Not limited to the proof-of-concept photothermal experiment demonstrated here, the fractal structure with improved light-matter interactions can be utilised in a variety of applications ranging from non-linear harmonic generation, plasmonic-enhanced fluorescence, to hot electron generation for photocatalysis.

Optimising virtual networks over time by using Windows Multiplicative DEA model

Recently, the prediction of the most efficient configuration of a vast set of devices used for mounting an optimised cloud computing services and virtual networks environments have attracted growing attention. This paper proposes a paradigm shift in modelling transmission control protocol (TCP) behaviour over time in virtual networks by using data envelopment analysis (DEA) models. Firstly, it proves that self-similarity with long-range dependency is presented differently in every network device. This study implements a novel fractal dimension concept on virtual networks for prediction, where this key index informs if the transport layer forwards services with smooth or jagged behaviour over time. Another substantial contribution is proving that virtual network devices have a distinct fractal memory, TCP bandwidth performance, and fractal dimension over time, presenting themselves as important factor for forecasting of spatiotemporal data. Thus, a continuous stepwise fractal performance evaluation framework methodology is developed as an expert system for virtual network assessment and performs a fractal analysis as a knowledge representation. In addition, due to the limitations of classical DEA models, the windows multiplicative data envelopment analysis (WMDEA) model is used to dynamically assess the fractal time series from virtual network hypervisors. For knowledge acquisition, 50 different virtual network hypervisors were appraised as decision-making units (DMU). Finally, this expert system also acts as a math hypervisor capable of determining the correct fractal pattern to follow when delivering TCP services in an optimised virtual network.

Fractal analysis and mathematical models for the investigation of photothermal inactivation of Candida albicans using carbon nanotubes

Candida albicans is responsible for the majority of nosocomial infections affecting immunocompromised patients. Systemic antifungals may promote microbial resistance, which has led to the search for alternative treatments, such as photothermal therapy (PTT). PTT assumes that the interaction of electromagnetic radiation with a photothermal agent generates heat that can lead to the destruction of tumor cells and the death of microorganisms. Carbon nanotubes (CNTs) have the potential for applications in biomedical systems, including acting as controlled deliverers of drugs, biosensors and scaffolds for tissue engineering and regenerative medicine. Furthermore, the absorption of radiation by CNTs in the infrared region induces an increase in temperature, which makes CNTs candidates for photothermal agents. In this work, the photothermal inactivation of C. albicans was evaluated by multiple wall CNTs associated with laser radiation in the near-infrared region. The mechanisms that are involved in inactivation were evaluated through cell susceptibility studies and an analysis of microscopic images that are associated with mathematical models and fractal concepts. The results indicate that direct contact between the cells and CNTs without irradiation does not lead to cell death, whereas the laser-mediated process is effective in inactivation. The application of the laws of scale and fractal concepts indicate that in the control groups, there are two distinct regimes that are delimited by the mean diameter of the microorganisms, as described by the Eden model and by the quasi-Euclidean surface. For the irradiated groups, the surfaces present only one regime described by Kardar-Parisi-Zhang, KPZ. The analysis of the fractality of the system by mathematical models can help in the identification of new strategies for the inactivation of microorganisms.

The Transformation of using Fractal Forms between Islamic and Digital Architecture from a Sustainable Approach

Along eras and civilizations, nature as a sustainable reference always has a great role in inspiration of the architectural form by the diversity of its vocabulary, rules, and colors…etc. Consequently, fractals as the main natural elements appeared in the achitectural morphology, since the ancient eras especially in the Islamic architecture in a various levels of forms, details, and patterns. However, nowadays the new architectural theories deal with fractals concept in a different ways depending on the deep understanding of universal and cosmic nature, also the new digital techniques in architectural design and construction that enhanced the appearance of new fractal forms. In this context, the research discusses the transformations of fractal systems, concepts and applications in architecture at the resent architectural theories, focusing on the digital architecture and how it deals with fractal forms, aiming to root the new architectural theories and link it with Islamic heritage to achieve a new Arab architecture that respects and reveals the local heritage and adapt with the latest architectural theories and techniques.

Multicolor Symmetrical Fractal Pattern Generator

A fractal is a geometric figure that combines the several characteristics among others: its parts have the same form as the whole, fragmented, and formation by iteration. The concept of fractals has been spread over all fields of sciences, technology, and art. The properties of fractal sets are necessary to be investigated and favorable for the practical applications. This paper aims to provide an algorithm for creating multicolor symmetrical fractal pattern generator. Generator algorithm consists of base, iteration, coloration, and duplication. To help the reader better understand the algorithm, we will present some script using Matlab. We describe a mathematically based algorithm that can fill a spatial region with a sequence of randomly placed which may be transformed copies of one motif or several motifs. This flexible algorithm can be used to produce a variety of aesthetically pleasing fractal patterns, of which we show some examples. We hope that the existence of fractals can show that mathematics is not a dry and flat subject, but it is a beautiful subject and can produce works that have a high degree of art and intellectual value.

BEAUTY IN MATHEMATICAL EDUCATION: SYNERGETIC WORLDVIEW

Introduction. The most important concepts underlying beauty are the mathematical concepts of symmetry and fractality. These categories are fundamental for modern mathematics, science and culture in general. However, in mathematical education and pedagogical literature, the ratio of these main categories has not been considered yet. Of special interest is the fact that the concepts of fractals, fractality and fractal geometry and fractal graphics are not included in the vast majority of high school programmes, although they have become commonly used among mathematicians and graphic designers.The aims of the article were the following: to demonstrate intersectionality and correlations of the basic concepts of symmetry and fractals from the point of view of synergetics, to establish the relevance of studying these concepts in the course of mathematics for aesthetic education of students and development of their worldview.Methodology and research methods. A significant role in the study is given to post-non-classical methodology based on synergetic worldview. The author employed the provisions of trinitarian methodology: in addition to two binary oppositions, the third element is necessary to solve the problem of contradiction of theseoppositions and integration into one coherent whole as the onditions of their coexistence. In the course of the research, analysis and generalisation of pedagogical and methodical literature, methods of comparative, historical and logical types of analysis were used.Results and scientific novelty.For centuries, beauty has been understood as a stable order and symmetry. The synergetrics as a general scientific theory about self-organisation of complex systems allows us to give another interpretation of beauty – as a kind of attractor, the result of self-organisation of nature or theflight of human thought. In the most general view, symmetry can be considered as transformation of similarity, which is also the core of another concept – fractality. On the one hand, fractality can be considered as one of the manifestations of symmetry in the broad sense. On the other hand, symmetry can be considered as a manifestation of fractality with a finite number of iterations. Thus, the concepts of symmetry and fractality are closely interrelated. Symmetry and fractality are two opposites, mutually complementing each other, aesthetically and mathematically mutually passing into each other. Symmetry reveals the beauty of a sustainable order and fractality reflects the beauty of the result of self-organisation of the chaos of nature or the freedom of the human mind. Therefore, symmetry and fractals are the most important concepts for the disclosure of the beauty of the universe, which determines their importance for mathematical learning and for aesthetic education of students.Practical significance. Taking into account the fact that the concepts of symmetry and fractals are directly related to each other, they should be jointly-taught. This will contribute to the development concept of mathematics education: to increase motivation for mathematical studies, to develop cognitive interests and activities, to narrow the gap between education and research processes, to overcome the problems with aesthetic education of students.

Fractal Theory and its Implication for Acquisition, Processing and Interpretation (API) of Geophysical Investigation: A Review

Abstract It has been observed that the sources of geophysical anomalies, such as density, susceptibility, conductivity, reflectivity, etc. may be conveniently defined according to a scaling/fractal distribution. Acquisition, processing and interpretation (collectively known as API) are the three main steps in any geophysical investigation. The proper design of acquisition of data can enhance the signal to noise ratio of geophysical response. Fractal theory has been applied and found to be worthwhile in deciding a geophysical survey network. Similarly, the second step, geophysical data processing for interpolation of missing data is key to reduce spurious anomaly due to aliased-interpolated data, and fractal or multi-fractal concepts have been applied to improve data processing significantly. Finally, interpretation is the main step which depends on the nature of the source. This lead to the development of new interpretation methods such as scaling spectral method and modified centroid method to accommodate scaling/fractal distribution of source for scaling geology for improved results. In the present paper, in-depth review of the application of fractal/multi-fractal approach in geophysical API for better geophysical understanding, is presented. Also, the advancement in geophysical API for filling the gaps in our understanding of sub-surface source characterization with recent examples in seismology and heat flow is highlighted, which can be further helpful in conducting future research of scaling geology.

Методологические аспекты анализа традиционных форм этнического предпринимательства

The article presents proven methodological approaches of traditional ethnic entrepreneurship forms study. The author stands on the positions of primordialism and considers the of ethnic entrepreneurship phenomenon on the base of fractal and network organization concepts. Ethnic entrepreneurship manifestations in the urban and rural environment are proposed to be considered in an organic relationship, as a process, that allows ethnos adaptation potential realization at all levels of social organization of ethnic minorities (from “households” to large trading firms), on the one hand. And, to integrate the primary institutions of the ethnos (fractals) into the system of regional and interregional economic (market) ties without the threat of losing ethnic identity, on the other hand. Special attention is paid to the ethnic economy place in the structure of ethnic processes (migration, adaptation, integration, consolidation, assimilation). This methodological approach allows us to consider ethnic entrepreneurship as a traditional economy structural element and a sociocultural phenomenon in statics. So, it is in dynamics, - as a manifestation of a multicomponent, hierarchically coordinated and latent (in form) process. According to the author, the actual ethnic economy should be attributed to the ethnic processes group. This allows you to expand the subject field of research in the field of social and applied anthropology (ethnology) significantly; to reach the level of scientific forecasting of the mechanism of interaction and interdependence of ethno-political and socio-economic processes.

Scaling and fractality in fatigue crack growth: Implications to Paris’ law and Wöhler’s curve

Size effects on crack growth are of paramount importance on the fatigue problem. Paris’ and Wöhler’s fatigue curves exhibit scale effects, which can be explained in the framework of incomplete self-similarity and fractal geometry concepts. In particular, scaling laws are found for the main fatigue parameters, that is, the fatigue threshold ΔKth and the fatigue limit Δσfl. The fatigue threshold increases with the crack length, whereas the fatigue limit decreases with the specimen size. Eventually, the proposed models are positively compared to experimental data available in the literature.

Visibility graph analysis of speech evoked auditory brainstem response in persistent developmental stuttering

In this research, the concept of fractality based on nonlinear science and chaos theory is explored to study and evaluate the complexity of speech-evoked auditory brainstem response (s-ABR) time series in order to capture its intrinsic multiscale dynamics. The visibility graph of the s-ABR series is proposed as a quantitative method to differentiate subjects with persistent developmental stuttering (PDS) from the normal group. Differential complexities between normal and PDS subjects is quantified using Graph index complexity (GIC). The model is applied to 14 individuals with PDS and 15 normal subjects. The results reveal the promising ability of GIC for assessment of abnormal activation of brainstem level in PDS group. It is observed that all s-ABR series have visibility graphs with a power-law topology and fractality in the s-ABR series is dictated by a mechanism associated with long-term memory of the auditory system dynamics at the brainstem level.

Reproductive Homeostasis and Senescence in Drosophila melanogaster.

The homeostatic properties of reproduction in aging female Drosophila melanogaster are investigated. Classic studies based on cohort analysis suggest that homeostatic capacity declines gradually as daily oviposition rates decline in aging flies. Analysis at the level of individuals gives a very different picture: reproductive homeostasis remains relatively constant for most of adult life until a critical point when oviposition either ceases entirely or continues in dysregulated fashion. The collapse of homeostatic capacity is abrupt. Enhanced homeostasis is associated with increased lifetime fecundity and improved prospects for survival. The fractal concept of lacunarity can be used to parameterize the "roughness" of individual fecundity trajectories and is inversely related to homeostatic capacity.

Cracking in an expansive soil under freeze–thaw cycles

Expansive soils located in cold regions can easily endure the action of frost heaving and cyclic freezing–thawing. Cracking can also occur in expansive clayey soils under freeze–thaw cycles, of which little attention has been paid on this issue. In this study, laboratory experiment and cracking analysis were performed on an expansive soil. Crack patterns were quantitatively analyzed using the fractal concept. The relationships among crack pattern, water loss, number of freeze–thaw cycles, and fractal dimension were discussed. It was found that crack patterns on the surface exhibit a hierarchical network structure that is fractal at a statistical level. Cracks induced by freeze–thaw cycles are shorter, more irregularly oriented, and slowly evolves from an irregularly rectilinear pattern towards a polygonal or quasi–hexagonal one; water loss, closely related to specimen thickness, plays a significant role in the process of soil cracking; crack development under freeze-thaw cycles are not only attributed to capillary effect, but also to expansion and absorption effects.

A fractal based approach to evaluate the progression of esophageal squamous cell dysplasia

Abstract Esophageal squamous cell carcinoma (ESCC) is the most prevalent malignancy of the esophagus with a very poor prognosis. Nevertheless, squamous cell dysplasia (ESD) has been identified as the only histological precursors of ESCC. Since, tissue alterations are slight in the early stage of ESD, human diagnosis is subjective. Hence, this work presents a first computer-aided system to differentiate low-grade dysplasia (LGD) from normal esophageal mucosa according to Vienna grading system, which is the most widespread method for histological grading of esophagus tissues. We captured microscopic images of a well-oriented region of Normal and LGD biopsies to characterize the architectural and cytological properties of specimens based on the computational analysis. We produced two sets of enhanced images. Then, by considering the fractal concept, we defined a new scale-dependent function in the generalized fractal dimension formulation to include the special information of both preprocessed images together. Then, for each image, a pattern was computed from variations of tissue fractal geometry across the pathway of dysplasia development. We proposed features extracted from these patterns to classify deviations of tissue characteristics from the normal stage. This method successfully differentiated the two diagnosis classes with statistical significance and high performance (accuracy = 97.78% ± 0.05, p

Advanced Method for Quantifying Socket Roughness and Empirical Correlations

Abstract This article describes an advanced method for quantifying socket profiles and proposes empirical relations to estimate the maximum side resistance at the soft rock socket. At three boreholes in Busan, South Korea, socket profiles with a continuous 500-mm length were recorded by the Kyungsung Sonar Roughness Profiling System that was developed by the authors, and full-scale bidirectional pile-loading tests (BDPLTs) were performed to evaluate the mobilized side resistances for the soft rock sockets. Based on the reports of previous research, the concept of self-affine fractals was adopted to idealize the measured profiles, and the roughness ratio, which is a nondimensional factor, was proposed to quantitatively describe the asperity of the idealized profiles. From the results of the BDPLTs, the unit-side-load transfer curves, f-w curves, for the soft rock socket were plotted to determine the maximum unit-side resistances in accordance with the new mobilized displacement criterion predicated upon the reviews of 17 f-w curves for the soft rock socket obtained from BDPLTs conducted at other sites in South Korea. Furthermore, using these data, including the socket diameter, the unconfined compressive strength of the soft rock, and the roughness ratio, advanced empirical correlations considering socket roughness were studied and proposed in this article.

Substrate effect on the evolution of surface morphology of BaF2thin films: A study based on fractal concepts

Fractal concepts are used to explore how different substrates affect the morphology of deposited barium fluoride (BaF2) thin films. BaF2 thin films of thickness 20 nm are prepared by electron beam evaporation technique at room temperature on glass, silicon and aluminum substrates. The structural properties and surface morphologies are investigated using glancing angle X-ray diffraction (GAXRD) and atomic force microscopy (AFM), respectively. Crystallite size, calculated from the XRD peaks, using the Debye Scherrer’s formula, is found to be significantly less for the case of Si substrate than for Al and glass substrates. Higuchi’s algorithm is applied to extract the fractal dimension of the horizontal and vertical sections of AFM images of the films. From the fractal dimension, the values of Hurst exponent (H) are inferred. For Al and glass substrates we find that H>0.5, which shows that the height fluctuations at neighboring pixels are positively correlated. However, for Si substrate, it is found that H<0.5 indicating that the height fluctuations at neighboring pixels are negatively correlated.

On multifractals: A non-linear study of actigraphy data

The healthcare process generates a vast quantity of data. Exploitation and analysis of such material might lead discoveries and potentially improve the clinical activity. Over the last years, several studies focused on the non-linear features exhibited by physiological signals and medical data. In general, the complex features of these signals have been demonstrated in some studies, ranging from heartbeat time intervals to neuron spikes series. Some works employed fractal and multifractal concepts to the analysis of such medical data. Similar evaluations of movement patterns recorded by actigraphy devices on individuals with fibromyalgia might provide new information that can be potentially translated into the clinical practice. Chronic pain is a condition characterised by sleep disturbances and psychological disorders that is correlated with impairment and reduced physical activity. This work aimed to determine the characteristics of activity series from fractal geometry concepts, in addition to evaluate the possibility of identifying individuals with fibromyalgia. Activity level data were collected from 27 healthy subjects and 27 fibromyalgia patients, with the use of clock-like devices equipped with accelerometers, for about two weeks, all day long. The activity series were evaluated through fractal and multifractal methods. Hurst exponent analysis exhibited values according to other studies for both groups, however, it is not possible to distinguish between the two groups by such analysis. Activity time series also exhibited a multifractal pattern. A paired analysis of the spectra indices for the sleep and awake states revealed differences between healthy subjects and fibromyalgia patients. The individuals feature differences between awake and sleep states having statistically significant differences for αq−−α0 in healthy subjects only, suggesting that there are not differences between awake and sleep state for patients with fibromyalgia. The approach has proven to be an option on the characterisation of such kind of signals and was able to (indirectly) differ between both healthy and fibromyalgia groups. This outcome suggests changes in the physiologic mechanisms of movement control.

Coupling gas transport measurements and X-ray tomography scans for multiscale analysis in silty soils

Scale is a key issue in soil studies. The idea that a sample must be of adequate size to embody a specific physical property comes from the representative elementary volume (REV) concept that defines minimum sample size for consistent results. Other approaches to describe soil spatial heterogeneity rely on the fractal dimension concept (FDC), which assumes structure changes with scale continuously, and the discrete hierarchy concept (DHC), which assumes change occurs discretely. This study considered using gas transport measurements combined with X-ray computed micro-tomography (μCT) for multi-scale analyses. Specifically, 24 large (“L cores”, 628.3 cm3) core volume samples were collected from two farms and two soil depths (3–11 cm and 20–28 cm) in northeast Italy. Gas transport parameters, such as air-filled porosity, air permeability, and gas diffusivity, were measured on the original cores and on successively sub-sampled medium (“M cores”, 100.4 cm3) and small (“S cores”, 4.7 cm3) cores. X-ray μCT–derived porosity indices were calculated for the two smaller scales. Soil core sub-sampling resulted in reduced soil gas transport property measurements, especially in the deepest depth when related to large and continuous bio-pore decreases in root channels and wormholes. In small core volumes, the pore network was dominated by small isolated pores, which might obstruct gas diffusion at that scale. All three concepts named above could be reconciled with our data. The limited numbers of samples and observation scales hindered identifying which model described soil spatial heterogeneity best. Finally, our results suggested the importance of considering scale effects on soil physical properties and their measurement consistency.

Effect of changes in the fractal structure of a littoral zone in the course of lake succession on the abundance, body size sequence and biomass of beetles.

Dystrophic lakes undergo natural disharmonic succession, in the course of which an increasingly complex and diverse, mosaic-like pattern of habitats evolves. In the final seral stage, the most important role is played by a spreading Sphagnum mat, which gradually reduces the lake’s open water surface area. Long-term transformations in the primary structure of lakes cause changes in the structure of lake-dwelling fauna assemblages. Knowledge of the succession mechanisms in lake fauna is essential for proper lake management. The use of fractal concepts helps to explain the character of fauna in relation to other aspects of the changing complexity of habitats. Our 12-year-long study into the succession of water beetles has covered habitats of 40 selected lakes which are diverse in terms of the fractal dimension. The taxonomic diversity and density of lake beetles increase parallel to an increase in the fractal dimension. An in-depth analysis of the fractal structure proved to be helpful in explaining the directional changes in fauna induced by the natural succession of lakes. Negative correlations appear between the body size and abundance. An increase in the density of beetles within the higher dimension fractals is counterbalanced by a change in the size of individual organisms. As a result, the biomass is constant, regardless of the fractal dimension.

Fractal Analysis Application Outlook for Improving Process Monitoring and Machine Maintenance in Manufacturing 4.0

Industry 4.0 has been advertised for a decade as the next disruptive evolution for production. It relies on automation growth and particularly on data exchange using numerous sensors in order to develop faster production with tight monitoring. The huge amount of data generated by clouds of sensors during production is often used to feed machine learning systems in order to detect faults, monitor and find possible ways for improvement. However, the artificial intelligence within machine learning requires finding and selecting key features, such as average and root mean square. While current machine learning has already proven its use in diverse applications, its efficiency could be further improved by generating better characteristics such as fractal parameters. In this paper, fractal analysis concept is presented and its current and future applications in machining are discussed. This sensitive and robust technique is already extracting high performance key features that could fill in monitoring and prediction systems. On top of improving features selection and, thus, improving the overall performance of monitoring and predictive systems in machining, this could lead to a more rapid artificial intelligence implementation into manufacturing.

ATRIUM SPACES: REALIZATION OF FRACTAL MODELS OF LOCAL AND URBAN STRUCTURES

The paper refers to the functional purpose of the atrium in architecture and town planning. The authors interpret atrium spaces in the context of a fractal-cluster model of environment formation. The study includes theoretical and conceptual development of the formation of space as a multilayer structure, based on key elements (clusters). The authors investigate the elements of space that are formed by means of atria taking into account fractal concepts. The hypothesis of sustainable development of urban spaces is stated by introducing atriums as transit and unifying elements of the urban environment.