What is fractal, and why fractals should matter to the petroleum geologist. [The use of fractal geometry to determine the occurrence and movement of hydrocarbons

DOE Office of Scientific and Technical Information (OSTI.GOV)

Mandelbrot, B.B.

1991-03-01

The following statements are obviously quite wrong: oil fields are circular; they are the same size and are distributed uniformly throughout the world; soil is of uniform porosity and permeability; after water has been pumped into a field it seeps through as an underground sphere. The preceding statements are so grossly incorrect that they do not even provide useful first approximations that one could improve upon by adding so-called corrective terms. For example, one gains little by starting with the notion of a uniform distribution of oil fields and then assuming it is perturbed by small Gaussian scatter. The flowmore » of water in a porous medium often fingers out in a pattern so diffuse that a sphere is not a useful point of departure in describing it. In summary, even the simplest data underlying petroleum geology exhibit very gross irregularity and unevenness. Fractal geometry is the proper geometry of manageable irregularity, fragmentation, and unevenness. It is the only workable alternative between the excessive order of the Euclidean geometry and unmanageable disorder. The main features of fractal geometry will be described and several techniques will be pointed out that show promise for the petroleum geologist.« less

Fractals in the Classroom

ERIC Educational Resources Information Center

Fraboni, Michael; Moller, Trisha

2008-01-01

Fractal geometry offers teachers great flexibility: It can be adapted to the level of the audience or to time constraints. Although easily explained, fractal geometry leads to rich and interesting mathematical complexities. In this article, the authors describe fractal geometry, explain the process of iteration, and provide a sample exercise.…

Introducing geometry concept based on history of Islamic geometry

NASA Astrophysics Data System (ADS)

Maarif, S.; Wahyudin; Raditya, A.; Perbowo, K. S.

2018-01-01

Geometry is one of the areas of mathematics interesting to discuss. Geometry also has a long history in mathematical developments. Therefore, it is important integrated historical development of geometry in the classroom to increase’ knowledge of how mathematicians earlier finding and constructing a geometric concept. Introduction geometrical concept can be started by introducing the Muslim mathematician who invented these concepts so that students can understand in detail how a concept of geometry can be found. However, the history of mathematics development, especially history of Islamic geometry today is less popular in the world of education in Indonesia. There are several concepts discovered by Muslim mathematicians that should be appreciated by the students in learning geometry. Great ideas of mathematicians Muslim can be used as study materials to supplement religious character values taught by Muslim mathematicians. Additionally, by integrating the history of geometry in teaching geometry are expected to improve motivation and geometrical understanding concept.

Fractal Electronic Circuits Assembled From Nanoclusters

NASA Astrophysics Data System (ADS)

Fairbanks, M. S.; McCarthy, D.; Taylor, R. P.; Brown, S. A.

2009-07-01

Many patterns in nature can be described using fractal geometry. The effect of this fractal character is an array of properties that can include high internal connectivity, high dispersivity, and enhanced surface area to volume ratios. These properties are often desirable in applications and, consequently, fractal geometry is increasingly employed in technologies ranging from antenna to storm barriers. In this paper, we explore the application of fractal geometry to electrical circuits, inspired by the pervasive fractal structure of neurons in the brain. We show that, under appropriate growth conditions, nanoclusters of Sb form into islands on atomically flat substrates via a process close to diffusion-limited aggregation (DLA), establishing fractal islands that will form the basis of our fractal circuits. We perform fractal analysis of the islands to determine the spatial scaling properties (characterized by the fractal dimension, D) of the proposed circuits and demonstrate how varying growth conditions can affect D. We discuss fabrication approaches for establishing electrical contact to the fractal islands. Finally, we present fractal circuit simulations, which show that the fractal character of the circuit translates into novel, non-linear conduction properties determined by the circuit's D value.

Fractals in the neurosciences, Part II: clinical applications and future perspectives.

PubMed

Di Ieva, Antonio; Esteban, Francisco J; Grizzi, Fabio; Klonowski, Wlodzimierz; Martín-Landrove, Miguel

2015-02-01

It has been ascertained that the human brain is a complex system studied at multiple scales, from neurons and microcircuits to macronetworks. The brain is characterized by a hierarchical organization that gives rise to its highly topological and functional complexity. Over the last decades, fractal geometry has been shown as a universal tool for the analysis and quantification of the geometric complexity of natural objects, including the brain. The fractal dimension has been identified as a quantitative parameter for the evaluation of the roughness of neural structures, the estimation of time series, and the description of patterns, thus able to discriminate different states of the brain in its entire physiopathological spectrum. Fractal-based computational analyses have been applied to the neurosciences, particularly in the field of clinical neurosciences including neuroimaging and neuroradiology, neurology and neurosurgery, psychiatry and psychology, and neuro-oncology and neuropathology. After a review of the basic concepts of fractal analysis and its main applications to the basic neurosciences in part I of this series, here, we review the main applications of fractals to the clinical neurosciences for a holistic approach towards a fractal geometry model of the brain. © The Author(s) 2013.

Fractal dimension and nonlinear dynamical processes

NASA Astrophysics Data System (ADS)

McCarty, Robert C.; Lindley, John P.

1993-11-01

Mandelbrot, Falconer and others have demonstrated the existence of dimensionally invariant geometrical properties of non-linear dynamical processes known as fractals. Barnsley defines fractal geometry as an extension of classical geometry. Such an extension, however, is not mathematically trivial Of specific interest to those engaged in signal processing is the potential use of fractal geometry to facilitate the analysis of non-linear signal processes often referred to as non-linear time series. Fractal geometry has been used in the modeling of non- linear time series represented by radar signals in the presence of ground clutter or interference generated by spatially distributed reflections around the target or a radar system. It was recognized by Mandelbrot that the fractal geometries represented by man-made objects had different dimensions than the geometries of the familiar objects that abound in nature such as leaves, clouds, ferns, trees, etc. The invariant dimensional property of non-linear processes suggests that in the case of acoustic signals (active or passive) generated within a dispersive medium such as the ocean environment, there exists much rich structure that will aid in the detection and classification of various objects, man-made or natural, within the medium.

Modeling of frost crystal growth over a flat plate using artificial neural networks and fractal geometries

NASA Astrophysics Data System (ADS)

Tahavvor, Ali Reza

2017-03-01

In the present study artificial neural network and fractal geometry are used to predict frost thickness and density on a cold flat plate having constant surface temperature under forced convection for different ambient conditions. These methods are very applicable in this area because phase changes such as melting and solidification are simulated by conventional methods but frost formation is a most complicated phase change phenomenon consists of coupled heat and mass transfer. Therefore conventional mathematical techniques cannot capture the effects of all parameters on its growth and development because this process influenced by many factors and it is a time dependent process. Therefore, in this work soft computing method such as artificial neural network and fractal geometry are used to do this manner. The databases for modeling are generated from the experimental measurements. First, multilayer perceptron network is used and it is found that the back-propagation algorithm with Levenberg-Marquardt learning rule is the best choice to estimate frost growth properties due to accurate and faster training procedure. Second, fractal geometry based on the Von-Koch curve is used to model frost growth procedure especially in frost thickness and density. Comparison is performed between experimental measurements and soft computing methods. Results show that soft computing methods can be used more efficiently to determine frost properties over a flat plate. Based on the developed models, wide range of frost formation over flat plates can be determined for various conditions.

Buried mine detection using fractal geometry analysis to the LWIR successive line scan data image

NASA Astrophysics Data System (ADS)

Araki, Kan

2012-06-01

We have engaged in research on buried mine/IED detection by remote sensing method using LWIR camera. A IR image of a ground, containing buried objects can be assumed as a superimposed pattern including thermal scattering which may depend on the ground surface roughness, vegetation canopy, and effect of the sun light, and radiation due to various heat interaction caused by differences in specific heat, size, and buried depth of the objects and local temperature of their surrounding environment. In this cumbersome environment, we introduce fractal geometry for analyzing from an IR image. Clutter patterns due to these complex elements have oftentimes low ordered fractal dimension of Hausdorff Dimension. On the other hand, the target patterns have its tendency of obtaining higher ordered fractal dimension in terms of Information Dimension. Random Shuffle Surrogate method or Fourier Transform Surrogate method is used to evaluate fractional statistics by applying shuffle of time sequence data or phase of spectrum. Fractal interpolation to each line scan was also applied to improve the signal processing performance in order to evade zero division and enhance information of data. Some results of target extraction by using relationship between low and high ordered fractal dimension are to be presented.

Definition of fractal topography to essential understanding of scale-invariance

NASA Astrophysics Data System (ADS)

Jin, Yi; Wu, Ying; Li, Hui; Zhao, Mengyu; Pan, Jienan

2017-04-01

Fractal behavior is scale-invariant and widely characterized by fractal dimension. However, the cor-respondence between them is that fractal behavior uniquely determines a fractal dimension while a fractal dimension can be related to many possible fractal behaviors. Therefore, fractal behavior is independent of the fractal generator and its geometries, spatial pattern, and statistical properties in addition to scale. To mathematically describe fractal behavior, we propose a novel concept of fractal topography defined by two scale-invariant parameters, scaling lacunarity (P) and scaling coverage (F). The scaling lacunarity is defined as the scale ratio between two successive fractal generators, whereas the scaling coverage is defined as the number ratio between them. Consequently, a strictly scale-invariant definition for self-similar fractals can be derived as D = log F /log P. To reflect the direction-dependence of fractal behaviors, we introduce another parameter Hxy, a general Hurst exponent, which is analytically expressed by Hxy = log Px/log Py where Px and Py are the scaling lacunarities in the x and y directions, respectively. Thus, a unified definition of fractal dimension is proposed for arbitrary self-similar and self-affine fractals by averaging the fractal dimensions of all directions in a d-dimensional space, which . Our definitions provide a theoretical, mechanistic basis for understanding the essentials of the scale-invariant property that reduces the complexity of modeling fractals.

Application of Fractal Geometry in Evaluation of Effective Stimulated Reservoir Volume in Shale Gas Reservoirs

NASA Astrophysics Data System (ADS)

Sheng, Guanglong; Su, Yuliang; Wang, Wendong; Javadpour, Farzam; Tang, Meirong

According to hydraulic-fracturing practices conducted in shale reservoirs, effective stimulated reservoir volume (ESRV) significantly affects the production of hydraulic fractured well. Therefore, estimating ESRV is an important prerequisite for confirming the success of hydraulic fracturing and predicting the production of hydraulic fracturing wells in shale reservoirs. However, ESRV calculation remains a longstanding challenge in hydraulic-fracturing operation. In considering fractal characteristics of the fracture network in stimulated reservoir volume (SRV), this paper introduces a fractal random-fracture-network algorithm for converting the microseismic data into fractal geometry. Five key parameters, including bifurcation direction, generating length (d), deviation angle (α), iteration times (N) and generating rules, are proposed to quantitatively characterize fracture geometry. Furthermore, we introduce an orthogonal-fractures coupled dual-porosity-media representation elementary volume (REV) flow model to predict the volumetric flux of gas in shale reservoirs. On the basis of the migration of adsorbed gas in porous kerogen of REV with different fracture spaces, an ESRV criterion for shale reservoirs with SRV is proposed. Eventually, combining the ESRV criterion and fractal characteristic of a fracture network, we propose a new approach for evaluating ESRV in shale reservoirs. The approach has been used in the Eagle Ford shale gas reservoir, and results show that the fracture space has a measurable influence on migration of adsorbed gas. The fracture network can contribute to enhancement of the absorbed gas recovery ratio when the fracture space is less than 0.2 m. ESRV is evaluated in this paper, and results indicate that the ESRV accounts for 27.87% of the total SRV in shale gas reservoirs. This work is important and timely for evaluating fracturing effect and predicting production of hydraulic fracturing wells in shale reservoirs.

Fractal nematic colloids

NASA Astrophysics Data System (ADS)

Hashemi, S. M.; Jagodič, U.; Mozaffari, M. R.; Ejtehadi, M. R.; Muševič, I.; Ravnik, M.

2017-01-01

Fractals are remarkable examples of self-similarity where a structure or dynamic pattern is repeated over multiple spatial or time scales. However, little is known about how fractal stimuli such as fractal surfaces interact with their local environment if it exhibits order. Here we show geometry-induced formation of fractal defect states in Koch nematic colloids, exhibiting fractal self-similarity better than 90% over three orders of magnitude in the length scales, from micrometers to nanometres. We produce polymer Koch-shaped hollow colloidal prisms of three successive fractal iterations by direct laser writing, and characterize their coupling with the nematic by polarization microscopy and numerical modelling. Explicit generation of topological defect pairs is found, with the number of defects following exponential-law dependence and reaching few 100 already at fractal iteration four. This work demonstrates a route for generation of fractal topological defect states in responsive soft matter.

Fractal nematic colloids

PubMed Central

Hashemi, S. M.; Jagodič, U.; Mozaffari, M. R.; Ejtehadi, M. R.; Muševič, I.; Ravnik, M.

2017-01-01

Fractals are remarkable examples of self-similarity where a structure or dynamic pattern is repeated over multiple spatial or time scales. However, little is known about how fractal stimuli such as fractal surfaces interact with their local environment if it exhibits order. Here we show geometry-induced formation of fractal defect states in Koch nematic colloids, exhibiting fractal self-similarity better than 90% over three orders of magnitude in the length scales, from micrometers to nanometres. We produce polymer Koch-shaped hollow colloidal prisms of three successive fractal iterations by direct laser writing, and characterize their coupling with the nematic by polarization microscopy and numerical modelling. Explicit generation of topological defect pairs is found, with the number of defects following exponential-law dependence and reaching few 100 already at fractal iteration four. This work demonstrates a route for generation of fractal topological defect states in responsive soft matter. PMID:28117325

Sierpinski triangles as a tool to introduce fractal geometry to children and their parents

NASA Astrophysics Data System (ADS)

Gires, Auguste; Schertzer, Daniel

2017-04-01

There are currently two somehow contradictory trends in the public debates involving scientific issues. On the one hand there is a need to address topics of increasing complexity, while on the other hand simple(istic) solutions are suggested by numerous people (including high level ones). Meanwhile there seems to be growing defiance towards science findings. Such problems are faced in numerous fields including geosciences where famous examples are the debates dealing with climate change, or water / air contamination. Such unfortunate trends means that the input of scientists in the society and public debates is strongly required. Although it not actually their job, scientists should get involved as a citizens. They should try to explain the complexity of the issues at stake, and take the necessary time to achieve this; not all problems can be explained with the help of a 140 characters tweet! Rather than hiding the uncertainties, they should try to explain this notion often not well understood, and admit the current limitations of knowledge. In the meantime it would be positive if this dialogue could help children and their parents to get familiarized with science and scientists, show that science is not obscure and actually present in everyday life. Scientists obviously also have the hope of fostering a desire for understanding, enhancing scientific culture and even promoting careers in this field. Fractals and fractal geometry are actually a rather good tool to achieve this. Indeed through numerous iterations of a simple process, one can easily obtain a rather complex shape, exhibiting some of the features observed in the nature. Fractal shapes are scale invariant, i.e. the more you zoom in, the more details you see; a portion of the shape is similar to the full one. This paper aims at presenting a series of activities presenting fractals to young people developed primarily around the famous Sierpinski triangles. Two types of activities were carefully designed

The fractal geometry of Hartree-Fock

NASA Astrophysics Data System (ADS)

Theel, Friethjof; Karamatskou, Antonia; Santra, Robin

2017-12-01

The Hartree-Fock method is an important approximation for the ground-state electronic wave function of atoms and molecules so that its usage is widespread in computational chemistry and physics. The Hartree-Fock method is an iterative procedure in which the electronic wave functions of the occupied orbitals are determined. The set of functions found in one step builds the basis for the next iteration step. In this work, we interpret the Hartree-Fock method as a dynamical system since dynamical systems are iterations where iteration steps represent the time development of the system, as encountered in the theory of fractals. The focus is put on the convergence behavior of the dynamical system as a function of a suitable control parameter. In our case, a complex parameter λ controls the strength of the electron-electron interaction. An investigation of the convergence behavior depending on the parameter λ is performed for helium, neon, and argon. We observe fractal structures in the complex λ-plane, which resemble the well-known Mandelbrot set, determine their fractal dimension, and find that with increasing nuclear charge, the fragmentation increases as well.

Determination of Uniaxial Compressive Strength of Ankara Agglomerate Considering Fractal Geometry of Blocks

NASA Astrophysics Data System (ADS)

Coskun, Aycan; Sonmez, Harun; Ercin Kasapoglu, K.; Ozge Dinc, S.; Celal Tunusluoglu, M.

2010-05-01

dimensions. To determine fractal dimensions of more than hundred andesite blocks in cores, a computer program namely FRACRUN were developed. Fractal geometry has been used as practical and popular tool to define particularly irregular shaped bodies in literature since the theory of fractal was developed by Mandelbrot (1967) (Hyslip and Vallejo, 1997; Kruhl and Nega, 1996; Bagde etal., 2002; Gulbin and Evangulova, 2003; Pardini, 2003; Kolay and Kayabali, 2006; Hamdi, 2008; Zorlu, 2009 and Sezer, 2009). Although there are some methods to determine fractal dimensions, square grid-cell count method for 2D and segment count method for 1D were followed in the algorithm of FRACRUN. FRACRUN has capable of determine fractal dimensions of many closed polygons on a single surface. In the study, a database composed of uniaxial compressive strength, volumetric block proportion, fractal dimensions and number of blocks for each core was established. Finally, prediction models were developed by regression analyses and compared with the empirical equations proposed by Sonmez et al. (2006). Acknowledgement This study is a product of ongoing project supported by TUBITAK (The Scientific and Technological Research Council of Turkey - Project No: 108Y002). References Bagde, M.N., Raina, A.K., Chakraborty, A.K., Jethwa, J.L., 2002. Rock mass characterization by fractal dimension. Engineering Geology 63, 141-155. Gokceoglu, C., 2002. A fuzzy triangular chart to predict the uniaxial compressive strength of the Ankara agglomerates from their petrographic composition. Engineering Geology, 66 (1-2), 39-51. Gulbin, Y.L., Evangulova, E.B., 2003. Morphometry of quartz aggregates in granites: fractal images referring to nucleation and growth processes. Mathematical Geology 35 (7), 819-833 Hamdi, E., 2008. A fractal description of simulated 3D discontinuity networks. Rock Mechanics and Rock Engineering 41, 587-599. Hyslip, J.P., Vallejo, L.E., 1997. Fractals analysis of the roughness and size distribution

Fractal morphometry of cell complexity.

PubMed

Losa, Gabriele A

2002-01-01

Irregularity and self-similarity under scale changes are the main attributes of the morphological complexity of both normal and abnormal cells and tissues. In other words, the shape of a self-similar object does not change when the scale of measurement changes, because each part of it looks similar to the original object. However, the size and geometrical parameters of an irregular object do differ when it is examined at increasing resolution, which reveals more details. Significant progress has been made over the past three decades in understanding how irregular shapes and structures in the physical and biological sciences can be analysed. Dominant influences have been the discovery of a new practical geometry of Nature, now known as fractal geometry, and the continuous improvements in computation capabilities. Unlike conventional Euclidean geometry, which was developed to describe regular and ideal geometrical shapes which are practically unknown in nature, fractal geometry can be used to measure the fractal dimension, contour length, surface area and other dimension parameters of almost all irregular and complex biological tissues. We have used selected examples to illustrate the application of the fractal principle to measuring irregular and complex membrane ultrastructures of cells at specific functional and pathological stage.

Electro-chemical manifestation of nanoplasmonics in fractal media

NASA Astrophysics Data System (ADS)

Baskin, Emmanuel; Iomin, Alexander

2013-06-01

Electrodynamics of composite materials with fractal geometry is studied in the framework of fractional calculus. This consideration establishes a link between fractal geometry of the media and fractional integrodifferentiation. The photoconductivity in the vicinity of the electrode-electrolyte fractal interface is studied. The methods of fractional calculus are employed to obtain an analytical expression for the giant local enhancement of the optical electric field inside the fractal composite structure at the condition of the surface plasmon excitation. This approach makes it possible to explain experimental data on photoconductivity in the nano-electrochemistry.

Broadband High Efficiency Fractal-Like and Diverse Geometry Silicon Nanowire Arrays for Photovoltaic Applications

NASA Astrophysics Data System (ADS)

AL-Zoubi, Omar H.

Solar energy has many advantages over conventional sources of energy. It is abundant, clean and sustainable. One way to convert solar energy directly into electrical energy is by using the photovoltaic solar cells (PVSC). Despite PVSC are becoming economically competitive, they still have high cost and low light to electricity conversion efficiency. Therefore, increasing the efficiency and reducing the cost are key elements for producing economically more competitive PVSC that would have significant impact on energy market and saving environment. A significant percentage of the PVSC cost is due to the materials cost. For that, thin films PVSC have been proposed which offer the benefits of the low amount of material and fabrication costs. Regrettably, thin film PVSC show poor light to electricity conversion efficiency because of many factors especially the high optical losses. To enhance conversion efficiency, numerous techniques have been proposed to reduce the optical losses and to enhance the absorption of light in thin film PVSC. One promising technique is the nanowire (NW) arrays in general and the silicon nanowire (SiNW) arrays in particular. The purpose of this research is to introduce vertically aligned SiNW arrays with enhanced and broadband absorption covering the entire solar spectrum while simultaneously reducing the amount of material used. To this end, we apply new concept for designing SiNW arrays based on employing diversity of physical dimensions, especially radial diversity within certain lattice configurations. In order to study the interaction of light with SiNW arrays and compute their optical properties, electromagnetic numerical modeling is used. A commercial numerical electromagnetic solver software package, high frequency structure simulation (HFSS), is utilized to model the SiNW arrays and to study their optical properties. We studied different geometries factors that affect the optical properties of SiNW arrays. Based on this study, we

Langevin Equation on Fractal Curves

NASA Astrophysics Data System (ADS)

Satin, Seema; Gangal, A. D.

2016-07-01

We analyze random motion of a particle on a fractal curve, using Langevin approach. This involves defining a new velocity in terms of mass of the fractal curve, as defined in recent work. The geometry of the fractal curve, plays an important role in this analysis. A Langevin equation with a particular model of noise is proposed and solved using techniques of the Fα-Calculus.

Investigation of the Fractal Geometry of Tundra Lake Patterns using Historical Topographic Maps and Satellite Imagery.

NASA Astrophysics Data System (ADS)

Kariyawasam, T.; Essa, A.; Gong, M.; Sudakov, I.

2017-12-01

Greenhouse gas emissions from tundra lakes are a significant positive feedback to the atmosphere in a changing climate as a pronounced growth of the numbers of tundra lake patterns has been observed in the Arctic region. Detailed knowledge of spatial dynamics of lake patterns in a changing arctic tundra landscape and their geometrical properties is therefore potentially valuable, in order to understand and accurately model the sources of greenhouse gas emissions from boreal permafrost. Our goal is to use a collection of historical topographic maps and satellite imagery of tundra lakes to conduct computational image analyses for examining spatial dynamics of Tundra lake patterns. Our approach is based upon analyzing area-perimeter data of thousands of tundra lakes to compute the fractal dimension to study the tundra lake pattern geometry, which have been used to classify pollen grains by textual patterning (Mander, 2016), vegetation in dryland ecosystems (Mander, 2017) and melt pond patterns (Hohenegger, 2012). By analyzing area - perimeter data for over 900 lakes we find that for both historical topographic maps and current satellite imagery, the fractal dimension D is stable at 1.6 for Tundra lakes with area less than about 100km2. For Tundra lake sizes bigger than 100 km2 fractal dimension takes values close to 2 and less than one indicative of structural changes in Tundra lake pattern geometry. Furthermore the current study did not reveal any percolation transition above some critical threshold in Tundra lake evolution. The results of the study will provide scientists with new data on these aspects of tundra lakes to help characterize the geomorphology of spatial patterns in arctic tundra lakes.

Fractals: To Know, to Do, to Simulate.

ERIC Educational Resources Information Center

Talanquer, Vicente; Irazoque, Glinda

1993-01-01

Discusses the development of fractal theory and suggests fractal aggregates as an attractive alternative for introducing fractal concepts. Describes methods for producing metallic fractals and a computer simulation for drawing fractals. (MVL)

Fractal characterization of fracture surfaces in concrete

USGS Publications Warehouse

Saouma, V.E.; Barton, C.C.; Gamaleldin, N.A.

1990-01-01

Fractal geometry is used to characterize the roughness of cracked concrete surfaces through a specially built profilometer, and the fractal dimension is subsequently correlated to the fracture toughness and direction of crack propagation. Preliminary results indicate that the fracture surface is indeed fractal over two orders of magnitudes with a dimension of approximately 1.20. ?? 1990.

A fractal model of effective stress of porous media and the analysis of influence factors

NASA Astrophysics Data System (ADS)

Li, Wei; Zhao, Huan; Li, Siqi; Sun, Wenfeng; Wang, Lei; Li, Bing

2018-03-01

The basic concept of effective stress describes the characteristics of fluid and solid interaction in porous media. In this paper, based on the theory of fractal geometry, a fractal model was built to analyze the relationship between the microstructure and the effective stress of porous media. From the microscopic point of view, the influence of effective stress on pore structure of porous media was demonstrated. Theoretical analysis and experimental results show that: (i) the fractal model of effective stress can be used to describe the relationship between effective stress and the microstructure of porous media; (ii) a linear increase in the effective stress leads to exponential increases in fractal dimension, porosity and pore number of the porous media, and causes a decreasing trend in the average pore radius.

Deterministic chaos and fractal complexity in the dynamics of cardiovascular behavior: perspectives on a new frontier.

PubMed

Sharma, Vijay

2009-09-10

Physiological systems such as the cardiovascular system are capable of five kinds of behavior: equilibrium, periodicity, quasi-periodicity, deterministic chaos and random behavior. Systems adopt one or more these behaviors depending on the function they have evolved to perform. The emerging mathematical concepts of fractal mathematics and chaos theory are extending our ability to study physiological behavior. Fractal geometry is observed in the physical structure of pathways, networks and macroscopic structures such the vasculature and the His-Purkinje network of the heart. Fractal structure is also observed in processes in time, such as heart rate variability. Chaos theory describes the underlying dynamics of the system, and chaotic behavior is also observed at many levels, from effector molecules in the cell to heart function and blood pressure. This review discusses the role of fractal structure and chaos in the cardiovascular system at the level of the heart and blood vessels, and at the cellular level. Key functional consequences of these phenomena are highlighted, and a perspective provided on the possible evolutionary origins of chaotic behavior and fractal structure. The discussion is non-mathematical with an emphasis on the key underlying concepts.

Deterministic Chaos and Fractal Complexity in the Dynamics of Cardiovascular Behavior: Perspectives on a New Frontier

PubMed Central

Sharma, Vijay

2009-01-01

Physiological systems such as the cardiovascular system are capable of five kinds of behavior: equilibrium, periodicity, quasi-periodicity, deterministic chaos and random behavior. Systems adopt one or more these behaviors depending on the function they have evolved to perform. The emerging mathematical concepts of fractal mathematics and chaos theory are extending our ability to study physiological behavior. Fractal geometry is observed in the physical structure of pathways, networks and macroscopic structures such the vasculature and the His-Purkinje network of the heart. Fractal structure is also observed in processes in time, such as heart rate variability. Chaos theory describes the underlying dynamics of the system, and chaotic behavior is also observed at many levels, from effector molecules in the cell to heart function and blood pressure. This review discusses the role of fractal structure and chaos in the cardiovascular system at the level of the heart and blood vessels, and at the cellular level. Key functional consequences of these phenomena are highlighted, and a perspective provided on the possible evolutionary origins of chaotic behavior and fractal structure. The discussion is non-mathematical with an emphasis on the key underlying concepts. PMID:19812706

Materials and fractal designs for 3D multifunctional integumentary membranes with capabilities in cardiac electrotherapy.

PubMed

Xu, Lizhi; Gutbrod, Sarah R; Ma, Yinji; Petrossians, Artin; Liu, Yuhao; Webb, R Chad; Fan, Jonathan A; Yang, Zijian; Xu, Renxiao; Whalen, John J; Weiland, James D; Huang, Yonggang; Efimov, Igor R; Rogers, John A

2015-03-11

Advanced materials and fractal design concepts form the basis of a 3D conformal electronic platform with unique capabilities in cardiac electrotherapies. Fractal geometries, advanced electrode materials, and thin, elastomeric membranes yield a class of device capable of integration with the entire 3D surface of the heart, with unique operational capabilities in low power defibrillation. Co-integrated collections of sensors allow simultaneous monitoring of physiological responses. Animal experiments on Langendorff-perfused rabbit hearts demonstrate the key features of these systems. © 2015 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim.

Fundamental Fractal Antenna Design Process

NASA Astrophysics Data System (ADS)

Zhu, L. P.; Kim, T. C.; Kakas, G. D.

2017-12-01

Antenna designers are always looking to come up with new ideas to push the envelope for new antennas, using a smaller volume while striving for higher bandwidth, wider bandwidth, and antenna gain. One proposed method of increasing bandwidth or shrinking antenna size is via the use of fractal geometry, which gives rise to fractal antennas. Fractals are those fun shapes that if one zooms in or zoom out, the structure is always the same. Design a new type of antenna based on fractal antenna design by utilize the Design of Experiment (DOE) will be shown in fractal antenna design process. Investigate conformal fractal antenna design for patterns, dimensions, and size, of the antenna but maintaining or improving the antenna performance. Research shows an antenna designer how to create basic requirements of the fractal antenna through a step by step process, and provides how to optimize the antenna design with the model prediction, lab measurement, and actual results from the compact range measurement on the antenna patterns.

Self-stabilized Fractality of Sea-coasts Through Damped Erosion

NASA Astrophysics Data System (ADS)

Sapoval, B.; Baldassari, A.; Gabrielli, A.

2004-05-01

Coastline morphology is of current interest in geophysical research and coastline erosion has important economic consequences. At the same time, although the geometry of seacoasts is often used as an introductory archetype of fractal morphology in nature there has been no explanation about which physical mechanism could justify that empirical observation. The present work propose a minimal, but robust, model of evolution of rocky coasts towards fractality. The model describes how a stationary fractal geometry arises spontaneously from the mutual self-stabilization of a rocky coast morphology and sea eroding power. If, on one hand, erosion generally increases the geometrical irregularity of the coast, on the other hand this increase creates a stronger damping of the sea and a consequent diminution of its eroding power. The increased damping argument relies on the studies of fractal acoustical cavities, which have shown that viscous damping is augmented on a longer, irregular, surface. A minimal two-dimensional model of erosion is introduced which leads to the through a complex dynamics of the earth-sea interface, to the appearance of a stationary fractal seacoast with dimension close to 4/3. Fractal geometry plays here the role of a morphological attractor directly related to percolation geometry. The model reproduces at least qualitatively some of the features of real coasts using only simple ingredients: the randomness of the lithology and the decrease of the erosion power of the sea. B. Sapoval, Fractals (Aditech, Paris, 1989). B. Sapoval, O. Haeberlé, and S.Russ, J. Acoust. Soc. Am., 2014 (1997). B. Hébert B., B. Sapoval, and S.Russ, J. Acoust. Soc. Am., 1567 (1999).

Fractal design concepts for stretchable electronics.

PubMed

Fan, Jonathan A; Yeo, Woon-Hong; Su, Yewang; Hattori, Yoshiaki; Lee, Woosik; Jung, Sung-Young; Zhang, Yihui; Liu, Zhuangjian; Cheng, Huanyu; Falgout, Leo; Bajema, Mike; Coleman, Todd; Gregoire, Dan; Larsen, Ryan J; Huang, Yonggang; Rogers, John A

2014-01-01

Stretchable electronics provide a foundation for applications that exceed the scope of conventional wafer and circuit board technologies due to their unique capacity to integrate with soft materials and curvilinear surfaces. The range of possibilities is predicated on the development of device architectures that simultaneously offer advanced electronic function and compliant mechanics. Here we report that thin films of hard electronic materials patterned in deterministic fractal motifs and bonded to elastomers enable unusual mechanics with important implications in stretchable device design. In particular, we demonstrate the utility of Peano, Greek cross, Vicsek and other fractal constructs to yield space-filling structures of electronic materials, including monocrystalline silicon, for electrophysiological sensors, precision monitors and actuators, and radio frequency antennas. These devices support conformal mounting on the skin and have unique properties such as invisibility under magnetic resonance imaging. The results suggest that fractal-based layouts represent important strategies for hard-soft materials integration.

Fractal design concepts for stretchable electronics

NASA Astrophysics Data System (ADS)

Fan, Jonathan A.; Yeo, Woon-Hong; Su, Yewang; Hattori, Yoshiaki; Lee, Woosik; Jung, Sung-Young; Zhang, Yihui; Liu, Zhuangjian; Cheng, Huanyu; Falgout, Leo; Bajema, Mike; Coleman, Todd; Gregoire, Dan; Larsen, Ryan J.; Huang, Yonggang; Rogers, John A.

2014-02-01

Stretchable electronics provide a foundation for applications that exceed the scope of conventional wafer and circuit board technologies due to their unique capacity to integrate with soft materials and curvilinear surfaces. The range of possibilities is predicated on the development of device architectures that simultaneously offer advanced electronic function and compliant mechanics. Here we report that thin films of hard electronic materials patterned in deterministic fractal motifs and bonded to elastomers enable unusual mechanics with important implications in stretchable device design. In particular, we demonstrate the utility of Peano, Greek cross, Vicsek and other fractal constructs to yield space-filling structures of electronic materials, including monocrystalline silicon, for electrophysiological sensors, precision monitors and actuators, and radio frequency antennas. These devices support conformal mounting on the skin and have unique properties such as invisibility under magnetic resonance imaging. The results suggest that fractal-based layouts represent important strategies for hard-soft materials integration.

Human physiological benefits of viewing nature: EEG responses to exact and statistical fractal patterns.

PubMed

Hagerhall, C M; Laike, T; Küller, M; Marcheschi, E; Boydston, C; Taylor, R P

2015-01-01

Psychological and physiological benefits of viewing nature have been extensively studied for some time. More recently it has been suggested that some of these positive effects can be explained by nature's fractal properties. Virtually all studies on human responses to fractals have used stimuli that represent the specific form of fractal geometry found in nature, i.e. statistical fractals, as opposed to fractal patterns which repeat exactly at different scales. This raises the question of whether human responses like preference and relaxation are being driven by fractal geometry in general or by the specific form of fractal geometry found in nature. In this study we consider both types of fractals (statistical and exact) and morph one type into the other. Based on the Koch curve, nine visual stimuli were produced in which curves of three different fractal dimensions evolve gradually from an exact to a statistical fractal. The patterns were shown for one minute each to thirty-five subjects while qEEG was continuously recorded. The results showed that the responses to statistical and exact fractals differ, and that the natural form of the fractal is important for inducing alpha responses, an indicator of a wakefully relaxed state and internalized attention.

Investigation into How 8th Grade Students Define Fractals

ERIC Educational Resources Information Center

Karakus, Fatih

2015-01-01

The analysis of 8th grade students' concept definitions and concept images can provide information about their mental schema of fractals. There is limited research on students' understanding and definitions of fractals. Therefore, this study aimed to investigate the elementary students' definitions of fractals based on concept image and concept…

ABC of multi-fractal spacetimes and fractional sea turtles

NASA Astrophysics Data System (ADS)

Calcagni, Gianluca

2016-04-01

We clarify what it means to have a spacetime fractal geometry in quantum gravity and show that its properties differ from those of usual fractals. A weak and a strong definition of multi-scale and multi-fractal spacetimes are given together with a sketch of the landscape of multi-scale theories of gravitation. Then, in the context of the fractional theory with q-derivatives, we explore the consequences of living in a multi-fractal spacetime. To illustrate the behavior of a non-relativistic body, we take the entertaining example of a sea turtle. We show that, when only the time direction is fractal, sea turtles swim at a faster speed than in an ordinary world, while they swim at a slower speed if only the spatial directions are fractal. The latter type of geometry is the one most commonly found in quantum gravity. For time-like fractals, relativistic objects can exceed the speed of light, but strongly so only if their size is smaller than the range of particle-physics interactions. We also find new results about log-oscillating measures, the measure presentation and their role in physical observations and in future extensions to nowhere-differentiable stochastic spacetimes.

Effect of Fractal Dimension on the Strain Behavior of Particulate Media

NASA Astrophysics Data System (ADS)

Altun, Selim; Sezer, Alper; Goktepe, A. Burak

2016-12-01

In this study, the influence of several fractal identifiers of granular materials on dynamic behavior of a flexible pavement structure as a particulate stratum is considered. Using experimental results and numerical methods as well, 15 different grain-shaped sands obtained from 5 different sources were analyzed as pavement base course materials. Image analyses were carried out by use of a stereomicroscope on 15 different samples to obtain quantitative particle shape information. Furthermore, triaxial compression tests were conducted to determine stress-strain and shear strength parameters of sands. Additionally, the dynamic response of the particulate media to standard traffic loads was computed using finite element modeling (FEM) technique. Using area-perimeter, line divider and box counting methods, over a hundred grains for each sand type were subjected to fractal analysis. Relationships among fractal dimension descriptors and dynamic strain levels were established for assessment of importance of shape descriptors of sands at various scales on the dynamic behavior. In this context, the advantage of fractal geometry concept to describe irregular and fractured shapes was used to characterize the sands used as base course materials. Results indicated that fractal identifiers can be preferred to analyze the effect of shape properties of sands on dynamic behavior of pavement base layers.

Developing the Concept of a Parabola in Taxicab Geometry

ERIC Educational Resources Information Center

Ada, Tuba; Kurtulus, Aytaç; Yanik, H. Bahadir

2015-01-01

The aim of this study was to observe the development process of the concept of a parabola in Taxicab geometry. The study was carried out in two stages. First, some activities related to Euclidean geometry and Taxicab geometry were designed based on concept development and real-life applications, and they were administered to a ninth-grade student.…

Electromagnetism on anisotropic fractal media

NASA Astrophysics Data System (ADS)

Ostoja-Starzewski, Martin

2013-04-01

Basic equations of electromagnetic fields in anisotropic fractal media are obtained using a dimensional regularization approach. First, a formulation based on product measures is shown to satisfy the four basic identities of the vector calculus. This allows a generalization of the Green-Gauss and Stokes theorems as well as the charge conservation equation on anisotropic fractals. Then, pursuing the conceptual approach, we derive the Faraday and Ampère laws for such fractal media, which, along with two auxiliary null-divergence conditions, effectively give the modified Maxwell equations. Proceeding on a separate track, we employ a variational principle for electromagnetic fields, appropriately adapted to fractal media, so as to independently derive the same forms of these two laws. It is next found that the parabolic (for a conducting medium) and the hyperbolic (for a dielectric medium) equations involve modified gradient operators, while the Poynting vector has the same form as in the non-fractal case. Finally, Maxwell's electromagnetic stress tensor is reformulated for fractal systems. In all the cases, the derived equations for fractal media depend explicitly on fractal dimensions in three different directions and reduce to conventional forms for continuous media with Euclidean geometries upon setting these each of dimensions equal to unity.

Fractals in physiology and medicine

NASA Technical Reports Server (NTRS)

Goldberger, Ary L.; West, Bruce J.

1987-01-01

The paper demonstrates how the nonlinear concepts of fractals, as applied in physiology and medicine, can provide an insight into the organization of such complex structures as the tracheobronchial tree and heart, as well as into the dynamics of healthy physiological variability. Particular attention is given to the characteristics of computer-generated fractal lungs and heart and to fractal pathologies in these organs. It is shown that alterations in fractal scaling may underlie a number of pathophysiological disturbances, including sudden cardiac death syndromes.

Evolving random fractal Cantor superlattices for the infrared using a genetic algorithm

PubMed Central

Bossard, Jeremy A.; Lin, Lan; Werner, Douglas H.

2016-01-01

Ordered and chaotic superlattices have been identified in Nature that give rise to a variety of colours reflected by the skin of various organisms. In particular, organisms such as silvery fish possess superlattices that reflect a broad range of light from the visible to the UV. Such superlattices have previously been identified as ‘chaotic’, but we propose that apparent ‘chaotic’ natural structures, which have been previously modelled as completely random structures, should have an underlying fractal geometry. Fractal geometry, often described as the geometry of Nature, can be used to mimic structures found in Nature, but deterministic fractals produce structures that are too ‘perfect’ to appear natural. Introducing variability into fractals produces structures that appear more natural. We suggest that the ‘chaotic’ (purely random) superlattices identified in Nature are more accurately modelled by multi-generator fractals. Furthermore, we introduce fractal random Cantor bars as a candidate for generating both ordered and ‘chaotic’ superlattices, such as the ones found in silvery fish. A genetic algorithm is used to evolve optimal fractal random Cantor bars with multiple generators targeting several desired optical functions in the mid-infrared and the near-infrared. We present optimized superlattices demonstrating broadband reflection as well as single and multiple pass bands in the near-infrared regime. PMID:26763335

Fractal and Multifractal Models Applied to Porous Media - Editorial

USDA-ARS?s Scientific Manuscript database

Given the current high level of interest in the use of fractal geometry to characterize natural porous media, a special issue of the Vadose Zone Journal was organized in order to expose established fractal analysis techniques and cutting-edge new developments to a wider Earth science audience. The ...

Fractal characteristic in the wearing of cutting tool

NASA Astrophysics Data System (ADS)

Mei, Anhua; Wang, Jinghui

1995-11-01

This paper studies the cutting tool wear with fractal geometry. The wearing image of the flank has been collected by machine vision which consists of CCD camera and personal computer. After being processed by means of preserving smoothing, binary making and edge extracting, the clear boundary enclosing the worn area has been obtained. The fractal dimension of the worn surface is calculated by the methods called `Slit Island' and `Profile'. The experiments and calciating give the conclusion that the worn surface is enclosed by a irregular boundary curve with some fractal dimension and characteristics of self-similarity. Furthermore, the relation between the cutting velocity and the fractal dimension of the worn region has been submitted. This paper presents a series of methods for processing and analyzing the fractal information in the blank wear, which can be applied to research the projective relation between the fractal structure and the wear state, and establish the fractal model of the cutting tool wear.

Mixing and the fractal geometry of piecewise isometries.

PubMed

Park, Paul P; Lynn, Thomas F; Umbanhowar, Paul B; Ottino, Julio M; Lueptow, Richard M

2017-04-01

Mathematical concepts often have applicability in areas that may have surprised their original developers. This is the case with piecewise isometries (PWIs), which transform an object by cutting it into pieces that are then rearranged to reconstruct the original object, and which also provide a paradigm to study mixing via cutting and shuffling in physical sciences and engineering. Every PWI is characterized by a geometric structure called the exceptional set, E, whose complement comprises nonmixing regions in the domain. Varying the parameters that define the PWI changes both the structure of E as well as the degree of mixing the PWI produces, which begs the question of how to determine which parameters produce the best mixing. Motivated by mixing of yield stress materials, for example granular media, in physical systems, we use numerical simulations of PWIs on a hemispherical shell and examine how the fat fractal properties of E relate to the degree of mixing for any particular PWI. We present numerical evidence that the fractional coverage of E negatively correlates with the intensity of segregation, a standard measure for the degree of mixing, which suggests that fundamental properties of E such as fractional coverage can be used to predict the effectiveness of a particular PWI as a mixing mechanism.

Applications of fractals in ecology.

PubMed

Sugihara, G; M May, R

1990-03-01

Fractal models describe the geometry of a wide variety of natural objects such as coastlines, island chains, coral reefs, satellite ocean-color images and patches of vegetation. Cast in the form of modified diffusion models, they can mimic natural and artificial landscapes having different types of complexity of shape. This article provides a brief introduction to fractals and reports on how they can be used by ecologists to answer a variety of basic questions, about scale, measurement and hierarchy in, ecological systems. Copyright © 1990. Published by Elsevier Ltd.

Plasmon confinement in fractal quantum systems

NASA Astrophysics Data System (ADS)

Westerhout, Tom; van Veen, Edo; Katsnelson, Mikhail I.; Yuan, Shengjun

2018-05-01

Recent progress in the fabrication of materials has made it possible to create arbitrary nonperiodic two-dimensional structures in the quantum plasmon regime. This paves the way for exploring the quantum plasmonic properties of electron gases in complex geometries. In this work we study systems with a fractal dimension. We calculate the full dielectric functions of two prototypical fractals with different ramification numbers, namely the Sierpinski carpet and gasket. We show that the Sierpinski carpet has a dispersion comparable to a square lattice, but the Sierpinski gasket features highly localized plasmon modes with a flat dispersion. This strong plasmon confinement in finitely ramified fractals can provide a novel setting for manipulating light at the quantum level.

Fractal analysis as a potential tool for surface morphology of thin films

NASA Astrophysics Data System (ADS)

Soumya, S.; Swapna, M. S.; Raj, Vimal; Mahadevan Pillai, V. P.; Sankararaman, S.

2017-12-01

Fractal geometry developed by Mandelbrot has emerged as a potential tool for analyzing complex systems in the diversified fields of science, social science, and technology. Self-similar objects having the same details in different scales are referred to as fractals and are analyzed using the mathematics of non-Euclidean geometry. The present work is an attempt to correlate fractal dimension for surface characterization by Atomic Force Microscopy (AFM). Taking the AFM images of zinc sulphide (ZnS) thin films prepared by pulsed laser deposition (PLD) technique, under different annealing temperatures, the effect of annealing temperature and surface roughness on fractal dimension is studied. The annealing temperature and surface roughness show a strong correlation with fractal dimension. From the regression equation set, the surface roughness at a given annealing temperature can be calculated from the fractal dimension. The AFM images are processed using Photoshop and fractal dimension is calculated by box-counting method. The fractal dimension decreases from 1.986 to 1.633 while the surface roughness increases from 1.110 to 3.427, for a change of annealing temperature 30 ° C to 600 ° C. The images are also analyzed by power spectrum method to find the fractal dimension. The study reveals that the box-counting method gives better results compared to the power spectrum method.

An Approach to Study Elastic Vibrations of Fractal Cylinders

NASA Astrophysics Data System (ADS)

Steinberg, Lev; Zepeda, Mario

2016-11-01

This paper presents our study of dynamics of fractal solids. Concepts of fractal continuum and time had been used in definitions of a fractal body deformation and motion, formulation of conservation of mass, balance of momentum, and constitutive relationships. A linearized model, which was written in terms of fractal time and spatial derivatives, has been employed to study the elastic vibrations of fractal circular cylinders. Fractal differential equations of torsional, longitudinal and transverse fractal wave equations have been obtained and solution properties such as size and time dependence have been revealed.

Fractal characterization of a fractured chalk reservoir - The Laegerdorf case

DOE Office of Scientific and Technical Information (OSTI.GOV)

Stoelum, H.H.; Koestler, A.G.; Feder, J.

1991-03-01

What is the matrix block size distribution of a fractured reservoir In order to answer this question and assess the potential of fractal geometry as a method of characterization of fracture networks, a pilot study has been done of the fractured chalk quarry in Laegerdorf. The fractures seen on the quarry walls were traced in the field for a total area of {approximately}200 {times} 45 m. The digitized pictures have been analyzed by a standard box-counting method. This analysis gave a fractal dimension of similarity varying from 1.33 for fractured areas between faults, to 1.43 for the fault zone, andmore » 1.53 for the highly deformed fault gouge. The amplitude showed a similar trend. The fractal dimension for the whole system of fractures is {approximately}1.55. In other words, fracture networks in chalk have a nonlinear, fractal geometry, and so matrix block size is a scaling property of chalk reservoirs. In terms of rock mechanics, the authors interpret the variation of the fractal dimension as follows: A small fractal dimension and amplitude are associated with brittle deformation in the elastic regime, while a large fractal dimension and amplitude are associated with predominantly ductile, strain softening deformation in the plastic regime. The interaction between the two regimes of deformation in the rock body is a key element of successful characterization and may be approached by seeing the rock as a non-Newtonian viscoelastic medium. The fractal dimension for the whole is close to a material independent limit that constrains the development of fractures.« less

Fractals, Coherence and Brain Dynamics

NASA Astrophysics Data System (ADS)

Vitiello, Giuseppe

2010-11-01

I show that the self-similarity property of deterministic fractals provides a direct connection with the space of the entire analytical functions. Fractals are thus described in terms of coherent states in the Fock-Bargmann representation. Conversely, my discussion also provides insights on the geometrical properties of coherent states: it allows to recognize, in some specific sense, fractal properties of coherent states. In particular, the relation is exhibited between fractals and q-deformed coherent states. The connection with the squeezed coherent states is also displayed. In this connection, the non-commutative geometry arising from the fractal relation with squeezed coherent states is discussed and the fractal spectral properties are identified. I also briefly discuss the description of neuro-phenomenological data in terms of squeezed coherent states provided by the dissipative model of brain and consider the fact that laboratory observations have shown evidence that self-similarity characterizes the brain background activity. This suggests that a connection can be established between brain dynamics and the fractal self-similarity properties on the basis of the relation discussed in this report between fractals and squeezed coherent states. Finally, I do not consider in this paper the so-called random fractals, namely those fractals obtained by randomization processes introduced in their iterative generation. Since self-similarity is still a characterizing property in many of such random fractals, my conjecture is that also in such cases there must exist a connection with the coherent state algebraic structure. In condensed matter physics, in many cases the generation by the microscopic dynamics of some kind of coherent states is involved in the process of the emergence of mesoscopic/macroscopic patterns. The discussion presented in this paper suggests that also fractal generation may provide an example of emergence of global features, namely long range

Fractal dimension analysis of complexity in Ligeti piano pieces

NASA Astrophysics Data System (ADS)

Bader, Rolf

2005-04-01

Fractal correlation dimensional analysis has been performed with whole solo piano pieces by Gyrgy Ligeti at every 50ms interval of the pieces. The resulting curves of development of complexity represented by the fractal dimension showed up a very reasonable correlation with the perceptional density of events during these pieces. The seventh piece of Ligeti's ``Musica ricercata'' was used as a test case. Here, each new part of the piece was followed by an increase of the fractal dimension because of the increase of information at the part changes. The second piece ``Galamb borong,'' number seven of the piano Etudes was used, because Ligeti wrote these Etudes after studying fractal geometry. Although the piece is not fractal in the strict mathematical sense, the overall structure of the psychoacoustic event-density as well as the detailed event development is represented by the fractal dimension plot.

Pond fractals in a tidal flat.

PubMed

Cael, B B; Lambert, Bennett; Bisson, Kelsey

2015-11-01

Studies over the past decade have reported power-law distributions for the areas of terrestrial lakes and Arctic melt ponds, as well as fractal relationships between their areas and coastlines. Here we report similar fractal structure of ponds in a tidal flat, thereby extending the spatial and temporal scales on which such phenomena have been observed in geophysical systems. Images taken during low tide of a tidal flat in Damariscotta, Maine, reveal a well-resolved power-law distribution of pond sizes over three orders of magnitude with a consistent fractal area-perimeter relationship. The data are consistent with the predictions of percolation theory for unscreened perimeters and scale-free cluster size distributions and are robust to alterations of the image processing procedure. The small spatial and temporal scales of these data suggest this easily observable system may serve as a useful model for investigating the evolution of pond geometries, while emphasizing the generality of fractal behavior in geophysical surfaces.

Pond fractals in a tidal flat

NASA Astrophysics Data System (ADS)

Cael, B. B.; Lambert, Bennett; Bisson, Kelsey

2015-11-01

Studies over the past decade have reported power-law distributions for the areas of terrestrial lakes and Arctic melt ponds, as well as fractal relationships between their areas and coastlines. Here we report similar fractal structure of ponds in a tidal flat, thereby extending the spatial and temporal scales on which such phenomena have been observed in geophysical systems. Images taken during low tide of a tidal flat in Damariscotta, Maine, reveal a well-resolved power-law distribution of pond sizes over three orders of magnitude with a consistent fractal area-perimeter relationship. The data are consistent with the predictions of percolation theory for unscreened perimeters and scale-free cluster size distributions and are robust to alterations of the image processing procedure. The small spatial and temporal scales of these data suggest this easily observable system may serve as a useful model for investigating the evolution of pond geometries, while emphasizing the generality of fractal behavior in geophysical surfaces.

Fractals and Transformations.

ERIC Educational Resources Information Center

Bannon, Thomas J.

1991-01-01

Discussed are several different transformations based on the generation of fractals including self-similar designs, the chaos game, the koch curve, and the Sierpinski Triangle. Three computer programs which illustrate these concepts are provided. (CW)

Spatial analysis of cities using Renyi entropy and fractal parameters

NASA Astrophysics Data System (ADS)

Chen, Yanguang; Feng, Jian

2017-12-01

The spatial distributions of cities fall into two groups: one is the simple distribution with characteristic scale (e.g. exponential distribution), and the other is the complex distribution without characteristic scale (e.g. power-law distribution). The latter belongs to scale-free distributions, which can be modeled with fractal geometry. However, fractal dimension is not suitable for the former distribution. In contrast, spatial entropy can be used to measure any types of urban distributions. This paper is devoted to generalizing multifractal parameters by means of dual relation between Euclidean and fractal geometries. The main method is mathematical derivation and empirical analysis, and the theoretical foundation is the discovery that the normalized fractal dimension is equal to the normalized entropy. Based on this finding, a set of useful spatial indexes termed dummy multifractal parameters are defined for geographical analysis. These indexes can be employed to describe both the simple distributions and complex distributions. The dummy multifractal indexes are applied to the population density distribution of Hangzhou city, China. The calculation results reveal the feature of spatio-temporal evolution of Hangzhou's urban morphology. This study indicates that fractal dimension and spatial entropy can be combined to produce a new methodology for spatial analysis of city development.

A new approach of sensorial evaluation of cooked cereal foods: fractal analysis of rheological data

NASA Astrophysics Data System (ADS)

Scher, J.; Hardy, J.

2002-11-01

An analytical method based on a fractal geometry concept was developed through the relationship between structure-texture of solid-like crackers, flat bread and Bretzels. An universal testing machine was used to determine indentation tests. The graphs were irregularly shaped so that usual interpretation was made not possible. Nevertheless, the irregular shape, or “roughness" displays auto-similarity properties which can be interpreted in terms of apparent fractal dimension texture (D_T). A trained panel able to quantify the “hardness", “porous structure" and “crispness" descriptors carried out sensorial characterisation of products. High correlation between sensorial hardness and resistance to indentation, on one hand, and between crispness and D_T on the other hand was found. Modelling mathematics methods for complex systems allow useful contribution to Food Science.

Characterisation of human non-proliferative diabetic retinopathy using the fractal analysis

PubMed Central

Ţălu, Ştefan; Călugăru, Dan Mihai; Lupaşcu, Carmen Alina

2015-01-01

AIM To investigate and quantify changes in the branching patterns of the retina vascular network in diabetes using the fractal analysis method. METHODS This was a clinic-based prospective study of 172 participants managed at the Ophthalmological Clinic of Cluj-Napoca, Romania, between January 2012 and December 2013. A set of 172 segmented and skeletonized human retinal images, corresponding to both normal (24 images) and pathological (148 images) states of the retina were examined. An automatic unsupervised method for retinal vessel segmentation was applied before fractal analysis. The fractal analyses of the retinal digital images were performed using the fractal analysis software ImageJ. Statistical analyses were performed for these groups using Microsoft Office Excel 2003 and GraphPad InStat software. RESULTS It was found that subtle changes in the vascular network geometry of the human retina are influenced by diabetic retinopathy (DR) and can be estimated using the fractal geometry. The average of fractal dimensions D for the normal images (segmented and skeletonized versions) is slightly lower than the corresponding values of mild non-proliferative DR (NPDR) images (segmented and skeletonized versions). The average of fractal dimensions D for the normal images (segmented and skeletonized versions) is higher than the corresponding values of moderate NPDR images (segmented and skeletonized versions). The lowest values were found for the corresponding values of severe NPDR images (segmented and skeletonized versions). CONCLUSION The fractal analysis of fundus photographs may be used for a more complete undeTrstanding of the early and basic pathophysiological mechanisms of diabetes. The architecture of the retinal microvasculature in diabetes can be quantitative quantified by means of the fractal dimension. Microvascular abnormalities on retinal imaging may elucidate early mechanistic pathways for microvascular complications and distinguish patients with DR from

Characterisation of human non-proliferative diabetic retinopathy using the fractal analysis.

PubMed

Ţălu, Ştefan; Călugăru, Dan Mihai; Lupaşcu, Carmen Alina

2015-01-01

To investigate and quantify changes in the branching patterns of the retina vascular network in diabetes using the fractal analysis method. This was a clinic-based prospective study of 172 participants managed at the Ophthalmological Clinic of Cluj-Napoca, Romania, between January 2012 and December 2013. A set of 172 segmented and skeletonized human retinal images, corresponding to both normal (24 images) and pathological (148 images) states of the retina were examined. An automatic unsupervised method for retinal vessel segmentation was applied before fractal analysis. The fractal analyses of the retinal digital images were performed using the fractal analysis software ImageJ. Statistical analyses were performed for these groups using Microsoft Office Excel 2003 and GraphPad InStat software. It was found that subtle changes in the vascular network geometry of the human retina are influenced by diabetic retinopathy (DR) and can be estimated using the fractal geometry. The average of fractal dimensions D for the normal images (segmented and skeletonized versions) is slightly lower than the corresponding values of mild non-proliferative DR (NPDR) images (segmented and skeletonized versions). The average of fractal dimensions D for the normal images (segmented and skeletonized versions) is higher than the corresponding values of moderate NPDR images (segmented and skeletonized versions). The lowest values were found for the corresponding values of severe NPDR images (segmented and skeletonized versions). The fractal analysis of fundus photographs may be used for a more complete undeTrstanding of the early and basic pathophysiological mechanisms of diabetes. The architecture of the retinal microvasculature in diabetes can be quantitative quantified by means of the fractal dimension. Microvascular abnormalities on retinal imaging may elucidate early mechanistic pathways for microvascular complications and distinguish patients with DR from healthy individuals.

A new numerical approximation of the fractal ordinary differential equation

NASA Astrophysics Data System (ADS)

Atangana, Abdon; Jain, Sonal

2018-02-01

The concept of fractal medium is present in several real-world problems, for instance, in the geological formation that constitutes the well-known subsurface water called aquifers. However, attention has not been quite devoted to modeling for instance, the flow of a fluid within these media. We deem it important to remind the reader that the concept of fractal derivative is not to represent the fractal sharps but to describe the movement of the fluid within these media. Since this class of ordinary differential equations is highly complex to solve analytically, we present a novel numerical scheme that allows to solve fractal ordinary differential equations. Error analysis of the method is also presented. Application of the method and numerical approximation are presented for fractal order differential equation. The stability and the convergence of the numerical schemes are investigated in detail. Also some exact solutions of fractal order differential equations are presented and finally some numerical simulations are presented.

Enhanced Graphene Photodetector with Fractal Metasurface.

PubMed

Fang, Jieran; Wang, Di; DeVault, Clayton T; Chung, Ting-Fung; Chen, Yong P; Boltasseva, Alexandra; Shalaev, Vladimir M; Kildishev, Alexander V

2017-01-11

Graphene has been demonstrated to be a promising photodetection material because of its ultrabroadband optical absorption, compatibility with CMOS technology, and dynamic tunability in optical and electrical properties. However, being a single atomic layer thick, graphene has intrinsically small optical absorption, which hinders its incorporation with modern photodetecting systems. In this work, we propose a gold snowflake-like fractal metasurface design to realize broadband and polarization-insensitive plasmonic enhancement in graphene photodetector. We experimentally obtain an enhanced photovoltage from the fractal metasurface that is an order of magnitude greater than that generated at a plain gold-graphene edge and such an enhancement in the photovoltage sustains over the entire visible spectrum. We also observed a relatively constant photoresponse with respect to polarization angles of incident light, as a result of the combination of two orthogonally oriented concentric hexagonal fractal geometries in one metasurface.

Using Disney's "Frozen" to Motivate Mathematics: Bringing Fractals into the Classroom

ERIC Educational Resources Information Center

Piatek-Jimenez, Katrina; Phelps, Christine M.

2016-01-01

The movie "Frozen" took the world by storm and this global popularity of the movie and its music can be harnessed by teachers of mathematics. This article builds on the "frozen fractal" lyric from "Let It Go" to incorporate fractal geometry into primary mathematics classrooms.

Self-Similarity of Plasmon Edge Modes on Koch Fractal Antennas.

PubMed

Bellido, Edson P; Bernasconi, Gabriel D; Rossouw, David; Butet, Jérémy; Martin, Olivier J F; Botton, Gianluigi A

2017-11-28

We investigate the plasmonic behavior of Koch snowflake fractal geometries and their possible application as broadband optical antennas. Lithographically defined planar silver Koch fractal antennas were fabricated and characterized with high spatial and spectral resolution using electron energy loss spectroscopy. The experimental data are supported by numerical calculations carried out with a surface integral equation method. Multiple surface plasmon edge modes supported by the fractal structures have been imaged and analyzed. Furthermore, by isolating and reproducing self-similar features in long silver strip antennas, the edge modes present in the Koch snowflake fractals are identified. We demonstrate that the fractal response can be obtained by the sum of basic self-similar segments called characteristic edge units. Interestingly, the plasmon edge modes follow a fractal-scaling rule that depends on these self-similar segments formed in the structure after a fractal iteration. As the size of a fractal structure is reduced, coupling of the modes in the characteristic edge units becomes relevant, and the symmetry of the fractal affects the formation of hybrid modes. This analysis can be utilized not only to understand the edge modes in other planar structures but also in the design and fabrication of fractal structures for nanophotonic applications.

Fractal pharmacokinetics of the drug mibefradil in the liver

NASA Astrophysics Data System (ADS)

Fuite, J.; Marsh, R.; Tuszyński, J.

2002-08-01

We explore the ramifications of the fractal geometry of the key organ for drug elimination, the liver, on pharmacokinetic data analysis. A formalism is developed for the use of a combination of well-stirred Euclidean and fractal compartments in the body. Perturbation analysis is carried out to obtain analytical solutions for the drug concentration time evolution. These results are then fitted to experimental data collected from clinically instrumented dogs [see, A. Skerjanec et al., J. Pharm. Sci. 85, 189 (1995)] using the drug mibefradil. The thus obtained spectral fractal dimension has a range of values that is consistent with the value found in independently performed ultrasound experiments on the liver.

The fractal heart — embracing mathematics in the cardiology clinic

PubMed Central

Captur, Gabriella; Karperien, Audrey L.; Hughes, Alun D.; Francis, Darrel P.; Moon, James C.

2017-01-01

For clinicians grappling with quantifying the complex spatial and temporal patterns of cardiac structure and function (such as myocardial trabeculae, coronary microvascular anatomy, tissue perfusion, myocyte histology, electrical conduction, heart rate, and blood-pressure variability), fractal analysis is a powerful, but still underused, mathematical tool. In this Perspectives article, we explain some fundamental principles of fractal geometry and place it in a familiar medical setting. We summarize studies in the cardiovascular sciences in which fractal methods have successfully been used to investigate disease mechanisms, and suggest potential future clinical roles in cardiac imaging and time series measurements. We believe that clinical researchers can deploy innovative fractal solutions to common cardiac problems that might ultimately translate into advancements for patient care. PMID:27708281

Fractal Image Filters for Specialized Image Recognition Tasks

DTIC Science & Technology

2010-02-11

butter sets of fractal geometers, such as Sierpinski triangles, twin- dragons , Koch curves, Cantor sets, fractal ferns, and so on. The geometries and...K is a centrally symmetric convex body in R m, then the function ‖x‖K defines a norm on R m. Moreover, the set K is the unit ball with respect to the...positive numbers r and R such that K contains a ball of radius r and is contained in a ball of radius R, the following proposition is clear

Using 3D Geometric Models to Teach Spatial Geometry Concepts.

ERIC Educational Resources Information Center

Bertoline, Gary R.

1991-01-01

An explanation of 3-D Computer Aided Design (CAD) usage to teach spatial geometry concepts using nontraditional techniques is presented. The software packages CADKEY and AutoCAD are described as well as their usefulness in solving space geometry problems. (KR)

Implementation for wideband applications using UWB fractal patch antenna

NASA Astrophysics Data System (ADS)

Kumar, D. Naresh

2018-04-01

This paper defines in detail about the diverse fractal patch antenna. Microstrip patch antennas has evolved in the field of research and development extending its impact across wide range of applications. A combination of patch antenna with fractal patterns has become a tryout to outspread it further. Because of its low profile nature patch antennas have added to a lot of prominence. Apart from have this property it can also be renovated further for wide bandwidth (2929 MHz) applications, as it exhibits self-analogous property. This antenna is premeditated on a patch using Sierpinski(4.040 GHz, 6.566 GHz) and Koch fractal geometries respectively. The antenna is designed using HFSS software.

Relationship between the anomalous diffusion and the fractal dimension of the environment

NASA Astrophysics Data System (ADS)

Zhokh, Alexey; Trypolskyi, Andrey; Strizhak, Peter

2018-03-01

In this letter, we provide an experimental study highlighting a relation between the anomalous diffusion and the fractal dimension of the environment using the methanol anomalous transport through the porous solid pellets with various pores geometries and different chemical compositions. The anomalous diffusion exponent was derived from the non-integer order of the time-fractional diffusion equation that describes the methanol anomalous transport through the solid media. The surface fractal dimension was estimated from the nitrogen adsorption isotherms using the Frenkel-Halsey-Hill method. Our study shows that decreasing the fractal dimension leads to increasing the anomalous diffusion exponent, whereas the anomalous diffusion constant is independent on the fractal dimension. We show that the obtained results are in a good agreement with the anomalous diffusion model on a fractal mesh.

Decoding the Margins: What Can the Fractal Geometry of Basaltic Flow Margins Tell Us?

NASA Astrophysics Data System (ADS)

Schaefer, E. I.; Hamilton, C.; Neish, C.; Beard, S. P.; Bramson, A. M.; Sori, M.; Rader, E. L.

2016-12-01

Studying lava flows on other planetary bodies is essential to characterizing eruption styles and constraining the bodies' thermal evolution. Although planetary basaltic flows are common, many key features are not resolvable in orbital imagery. We are thus developing a technique to characterize basaltic flow type, sub-meter roughness, and sediment mantling from these data. We will present the results from upcoming fieldwork at Craters of the Moon National Monument and Preserve with FINESSE (August) and at Hawai'i Volcanoes National Park (September). We build on earlier work that showed that basaltic flow margins are approximately fractal [Bruno et al., 1992; Gaonac'h et al., 1992] and that their fractal dimensions (D) have distinct `a`ā and pāhoehoe ranges under simple conditions [Bruno et al., 1994]. Using a differential GPS rover, we have recently shown that the margin of Iceland's 2014 Holuhraun flow exhibits near-perfect (R2=0.9998) fractality for ≥24 km across dm to km scales [Schaefer et al., 2016]. This finding suggests that a fractal-based technique has significant potential to characterize flows at sub-resolution scales. We are simultaneously seeking to understand how margin fractality can be modified. A preliminary result for an `a'ā flow in Hawaii's Ka'ū Desert suggests that although aeolian mantling obscures the original flow margin, the apparent margin (i.e., sediment-lava interface) remains fractal [Schaefer et al., 2015]. Further, the apparent margin's D is likely significantly modified from that of the original margin. Other factors that we are exploring include erosion, transitional flow types, and topographic confinement. We will also rigorously test the intriguing possibility that margin D correlates with the sub-meter Hurst exponent H of the flow surface, a common metric of roughness scaling [e.g., Shepard et al., 2001]. This hypothesis is based on geometric arguments [Turcotte, 1997] and is qualitatively consistent with all results so far.

Experimental investigation of the flow field and power consumption characteristics of regular and fractal blade impellers in a dynamic mixer

NASA Astrophysics Data System (ADS)

Steiros, K.; Bruce, P. J. K.; Buxton, O. R. H.; Vassilicos, J. C.

2015-11-01

Experiments have been performed in an octagonal un-baffled water tank, stirred by three radial turbines with different geometry impellers: (1) regular rectangular blades; (2) single-iteration fractal blades; (3) two-iteration fractal blades. Shaft torque was monitored and the power number calculated for each case. Both impellers with fractal geometry blades exhibited a decrease of turbine power number compared to the regular one (15% decrease for single-iteration and 19% for two iterations). Phase locked PIV in the discharge region of the blades revealed that the vortices emanating from the regular blades are more coherent, have higher kinetic energy, and advect faster towards the tank's walls where they are dissipated, compared to their fractal counterparts. This suggests a strong link between vortex production and behaviour and the energy input for the different impellers. Planar PIV measurements in the bulk of the tank showed an increase of turbulence intensity of over 20% for the fractal geometry blades, suggesting higher mixing efficiency. Experiments with pressure measurements on the different geometry blade surfaces are ongoing to investigate the distribution of forces, and calculate hydrodynamic centres of pressure. The authors would like to acknowledge the financial support given by European Union FP7 Marie Curie MULTISOLVE project (Grant Agreement No. 317269).

Fractal Risk Assessment of ISS Propulsion Module in Meteoroid and Orbital Debris Environments

NASA Technical Reports Server (NTRS)

Mog, Robert A.

2001-01-01

A unique and innovative risk assessment of the International Space Station (ISS) Propulsion Module is conducted using fractal modeling of the Module's response to the meteoroid and orbital debris environments. Both the environment models and structural failure modes due to the resultant hypervelocity impact phenomenology, as well as Module geometry, are investigated for fractal applicability. The fractal risk assessment methodology could produce a greatly simplified alternative to current methodologies, such as BUMPER analyses, while maintaining or increasing the number of complex scenarios that can be assessed. As a minimum, this innovative fractal approach will provide an independent assessment of existing methodologies in a unique way.

Towards thermomechanics of fractal media

NASA Astrophysics Data System (ADS)

Ostoja-Starzewski, Martin

2007-11-01

Hans Ziegler’s thermomechanics [1,2,3], established half a century ago, is extended to fractal media on the basis of a recently introduced continuum mechanics due to Tarasov [14,15]. Employing the concept of internal (kinematic) variables and internal stresses, as well as the quasiconservative and dissipative stresses, a field form of the second law of thermodynamics is derived. In contradistinction to the conventional Clausius Duhem inequality, it involves generalized rates of strain and internal variables. Upon introducing a dissipation function and postulating the thermodynamic orthogonality on any lengthscale, constitutive laws of elastic-dissipative fractal media naturally involving generalized derivatives of strain and stress can then be derived. This is illustrated on a model viscoelastic material. Also generalized to fractal bodies is the Hill condition necessary for homogenization of their constitutive responses.

Using Fractal And Morphological Criteria For Automatic Classification Of Lung Diseases

NASA Astrophysics Data System (ADS)

Vehel, Jacques Levy

1989-11-01

Medical Images are difficult to analyze by means of classical image processing tools because they are very complex and irregular. Such shapes are obtained for instance in Nuclear Medecine with the spatial distribution of activity for organs such as lungs, liver, and heart. We have tried to apply two different theories to these signals: - Fractal Geometry deals with the analysis of complex irregular shapes which cannot well be described by the classical Euclidean geometry. - Integral Geometry treats sets globally and allows to introduce robust measures. We have computed three parameters on three kinds of Lung's SPECT images: normal, pulmonary embolism and chronic desease: - The commonly used fractal dimension (FD), that gives a measurement of the irregularity of the 3D shape. - The generalized lacunarity dimension (GLD), defined as the variance of the ratio of the local activity by the mean activity, which is only sensitive to the distribution and the size of gaps in the surface. - The Favard length that gives an approximation of the surface of a 3-D shape. The results show that each slice of the lung, considered as a 3D surface, is fractal and that the fractal dimension is the same for each slice and for the three kind of lungs; as for the lacunarity and Favard length, they are clearly different for normal lungs, pulmonary embolisms and chronic diseases. These results indicate that automatic classification of Lung's SPECT can be achieved, and that a quantitative measurement of the evolution of the disease could be made.

Improvements in RF Shimming in High Field MRI Using High Permittivity Materials With Low Order Pre-Fractal Geometries.

PubMed

Schmidt, Rita; Webb, Andrew

2016-08-01

Ultra-high field MRI is an area of great interest for clinical research and basic science due to the increased signal-to-noise, spatial resolution and magnetic-susceptibility-based contrast. However, the fact that the electromagnetic wavelength in tissue is comparable to the relevant body dimensions means that the uniformity of the excitation field is much poorer than at lower field strengths. In addition to techniques such as transmit arrays, one simple but effective method to counteract this effect is to use high permittivity "pads". Very high permittivities enable thinner, flexible pads to be used, but the limiting factor is wavelength effects within the pads themselves, which can lead to image artifacts. So far, all studies have used simple continuous rectangular/circular pad geometries. In this work we investigate how the wavelength effects can be partially mitigated utilizing shaped pad with holes. Several arrangements have been simulated, including low order pre-fractal geometries, which maintain the overall coverage of the pad, but can provide better image homogeneity in the region of interest or higher sensitivity depending on the setup. Experimental data in the form of in vivo human images at 7T were acquired to validate the simulation results.

Navigation performance in virtual environments varies with fractal dimension of landscape.

PubMed

Juliani, Arthur W; Bies, Alexander J; Boydston, Cooper R; Taylor, Richard P; Sereno, Margaret E

2016-09-01

Fractal geometry has been used to describe natural and built environments, but has yet to be studied in navigational research. In order to establish a relationship between the fractal dimension (D) of a natural environment and humans' ability to navigate such spaces, we conducted two experiments using virtual environments that simulate the fractal properties of nature. In Experiment 1, participants completed a goal-driven search task either with or without a map in landscapes that varied in D. In Experiment 2, participants completed a map-reading and location-judgment task in separate sets of fractal landscapes. In both experiments, task performance was highest at the low-to-mid range of D, which was previously reported as most preferred and discriminable in studies of fractal aesthetics and discrimination, respectively, supporting a theory of visual fluency. The applicability of these findings to architecture, urban planning, and the general design of constructed spaces is discussed.

Space-Filling Supercapacitor Carpets: Highly scalable fractal architecture for energy storage

NASA Astrophysics Data System (ADS)

Tiliakos, Athanasios; Trefilov, Alexandra M. I.; Tanasǎ, Eugenia; Balan, Adriana; Stamatin, Ioan

2018-04-01

Revamping ground-breaking ideas from fractal geometry, we propose an alternative micro-supercapacitor configuration realized by laser-induced graphene (LIG) foams produced via laser pyrolysis of inexpensive commercial polymers. The Space-Filling Supercapacitor Carpet (SFSC) architecture introduces the concept of nested electrodes based on the pre-fractal Peano space-filling curve, arranged in a symmetrical equilateral setup that incorporates multiple parallel capacitor cells sharing common electrodes for maximum efficiency and optimal length-to-area distribution. We elucidate on the theoretical foundations of the SFSC architecture, and we introduce innovations (high-resolution vector-mode printing) in the LIG method that allow for the realization of flexible and scalable devices based on low iterations of the Peano algorithm. SFSCs exhibit distributed capacitance properties, leading to capacitance, energy, and power ratings proportional to the number of nested electrodes (up to 4.3 mF, 0.4 μWh, and 0.2 mW for the largest tested model of low iteration using aqueous electrolytes), with competitively high energy and power densities. This can pave the road for full scalability in energy storage, reaching beyond the scale of micro-supercapacitors for incorporating into larger and more demanding applications.

Reducing the Mismatch of Geometry Concept Definitions and Concept Images Held by Pre-Service Teachers

ERIC Educational Resources Information Center

Cunningham, Robert F.; Roberts, Allison

2010-01-01

Twenty-three female elementary pre-service teachers were assessed on their ability to answer questions involving geometry concepts. Despite being given the definitions of the altitude of a triangle and the diagonal of a polygon on the pretest, limited understanding of these concepts was evident. A treatment using both graphic organizer and concept…

Surface Fractal Analysis for Estimating the Fracture Energy Absorption of Nanoparticle Reinforced Composites

PubMed Central

Pramanik, Brahmananda; Tadepalli, Tezeswi; Mantena, P. Raju

2012-01-01

In this study, the fractal dimensions of failure surfaces of vinyl ester based nanocomposites are estimated using two classical methods, Vertical Section Method (VSM) and Slit Island Method (SIM), based on the processing of 3D digital microscopic images. Self-affine fractal geometry has been observed in the experimentally obtained failure surfaces of graphite platelet reinforced nanocomposites subjected to quasi-static uniaxial tensile and low velocity punch-shear loading. Fracture energy and fracture toughness are estimated analytically from the surface fractal dimensionality. Sensitivity studies show an exponential dependency of fracture energy and fracture toughness on the fractal dimensionality. Contribution of fracture energy to the total energy absorption of these nanoparticle reinforced composites is demonstrated. For the graphite platelet reinforced nanocomposites investigated, surface fractal analysis has depicted the probable ductile or brittle fracture propagation mechanism, depending upon the rate of loading. PMID:28817017

Increasing Communication in Geometry by Using a Personal Math Concept Chart

ERIC Educational Resources Information Center

Friedman, Rhonda; Kazerouni, Gety; Lax, Stacey; Weisdorf, Elli

2011-01-01

The action research team developed a "Personal Math Concept Chart". This chart required students to describe the mathematical concepts that they were studying in the Geometry strand of Mathematics using their own images and words. In this study, students were encouraged to express their own understanding of geometric concepts in order to…

Analysis of Geographical Distribution Patterns in Plants Using Fractals

NASA Astrophysics Data System (ADS)

Bari, A.; Ayad, G.; Padulosi, S.; Hodgkin, T.; Martin, A.; Gonzalez-Andujar, J. L.; Brown, A. H. D.

Geographical distribution patterns in plants have been observed since primeval times and have been used by plant explorers to trace the origin of plants species. These patterns embody the effects of fundamental law-like processes. Diversity in plants has also been found to be proportionate with the area, and this scaling behavior is also known as fractal behavior. In the present study, we use fractal geometry to analyze the distribution patterns of wild taxa of cowpea with the objective to locate where their diversity would be the highest to aid in the planning of targeted explorations and conservation measures.

Students' Conceptions of Congruency through the Use of Dynamic Geometry Software

ERIC Educational Resources Information Center

Gonzalez, Gloriana; Herbst, Patricio G.

2009-01-01

This paper describes students' interactions with dynamic diagrams in the context of an American geometry class. Students used the dragging tool and the measuring tool in Cabri Geometry to make mathematical conjectures. The analysis, using the cK[cent sign] model of conceptions, suggests that incorporating technology in mathematics classrooms…

Noteworthy fractal features and transport properties of Cantor tartans

NASA Astrophysics Data System (ADS)

Balankin, Alexander S.; Golmankhaneh, Alireza K.; Patiño-Ortiz, Julián; Patiño-Ortiz, Miguel

2018-06-01

This Letter is focused on the impact of fractal topology on the transport processes governed by different kinds of random walks on Cantor tartans. We establish that the spectral dimension of the infinitely ramified Cantor tartan ds is equal to its fractal (self-similarity) dimension D. Consequently, the random walk on the Cantor tartan leads to a normal diffusion. On the other hand, the fractal geometry of Cantor tartans allows for a natural definition of power-law distributions of the waiting times and step lengths of random walkers. These distributions are Lévy stable if D > 1.5. Accordingly, we found that the random walk with rests leads to sub-diffusion, whereas the Lévy walk leads to ballistic diffusion. The Lévy walk with rests leads to super-diffusion, if D >√{ 3 }, or sub-diffusion, if 1.5 < D <√{ 3 }.

Fractal Measure and Microscopic Modeling of Osseointegration.

PubMed

Santos, Leonardo Cavalcanti Bezerra; Carvalho, Alessandra Albuquerque; Leão, Jair Carneiro; Neto, Paulo Jose; Stosic, Tatijana; Stosic, Borko

2015-01-01

In this study, the process of osseointegration on titanium implant surfaces with different physicochemical treatments subjected to a simulated corporal fluid submersion was evaluated using the concept of fractal dimension. It was found that different treatments led to rather different calcium phosphate crystal growth patterns, with fractal dimension ranging from 1.68 to 1.93. The observed crystal patterns may be explained by a general deposition, diffusion, and aggregation growth mechanism, where diffusing particle sticking probability plays a fundamental role.

Launching the chaotic realm of iso-fractals: A short remark

DOE Office of Scientific and Technical Information (OSTI.GOV)

O'Schmidt, Nathan; Katebi, Reza; Corda, Christian

In this brief note, we introduce the new, emerging sub-discipline of iso-fractals by highlighting and discussing the preliminary results of recent works. First, we note the abundance of fractal, chaotic, non-linear, and self-similar structures in nature while emphasizing the importance of studying such systems because fractal geometry is the language of chaos. Second, we outline the iso-fractal generalization of the Mandelbrot set to exemplify the newly generated Mandelbrot iso-sets. Third, we present the cutting-edge notion of dynamic iso-spaces and explain how a mathematical space can be iso-topically lifted with iso-unit functions that (continuously or discretely) change; in the discrete casemore » examples, we mention that iteratively generated sequences like Fibonacci’s numbers and (the complex moduli of) Mandelbrot’s numbers can supply a deterministic chain of iso-units to construct an ordered series of (magnified and/or de-magnified) iso-spaces that are locally iso-morphic. Fourth, we consider the initiation of iso-fractals with Inopin’s holographic ring (IHR) topology and fractional statistics for 2D and 3D iso-spaces. In total, the reviewed iso-fractal results are a significant improvement over traditional fractals because the application of Santilli’s iso-mathematics arms us an extra degree of freedom for attacking problems in chaos. Finally, we conclude by proposing some questions and ideas for future research work.« less

The role of the circadian system in fractal neurophysiological control

PubMed Central

Pittman-Polletta, Benjamin R.; Scheer, Frank A.J.L.; Butler, Matthew P.; Shea, Steven A.; Hu, Kun

2013-01-01

Many neurophysiological variables such as heart rate, motor activity, and neural activity are known to exhibit intrinsic fractal fluctuations - similar temporal fluctuation patterns at different time scales. These fractal patterns contain information about health, as many pathological conditions are accompanied by their alteration or absence. In physical systems, such fluctuations are characteristic of critical states on the border between randomness and order, frequently arising from nonlinear feedback interactions between mechanisms operating on multiple scales. Thus, the existence of fractal fluctuations in physiology challenges traditional conceptions of health and disease, suggesting that high levels of integrity and adaptability are marked by complex variability, not constancy, and are properties of a neurophysiological network, not individual components. Despite the subject's theoretical and clinical interest, the neurophysiological mechanisms underlying fractal regulation remain largely unknown. The recent discovery that the circadian pacemaker (suprachiasmatic nucleus) plays a crucial role in generating fractal patterns in motor activity and heart rate sheds an entirely new light on both fractal control networks and the function of this master circadian clock, and builds a bridge between the fields of circadian biology and fractal physiology. In this review, we sketch the emerging picture of the developing interdisciplinary field of fractal neurophysiology by examining the circadian system’s role in fractal regulation. PMID:23573942

Fabrication of Multscale Fractal-Like Structures by Controlling Fluid Interface Instability

PubMed Central

Islam, Tanveer ul; Gandhi, Prasanna S.

2016-01-01

Nature, in quest for the best designs has shaped its vital systems into fractal geometries. Effectual way of spontaneous fabrication of scalable, ordered fractal-like structures by controlling Saffman-Taylor instability in a lifted Hele-Shaw cell is deployed here. In lifted Hele-Shaw cell uncontrolled penetration of low-viscosity fluid into its high-viscosity counterpart is known to develop irregular, non-repeatable, normally short-lived, branched patterns. We propose and characterize experimentally anisotropies in a form of spatially distributed pits on the cell plates to control initiation and further penetration of non-splitting fingers. The proposed control over shielding mechanism yields recipes for fabrication of families of ordered fractal-like patterns of multiple generations. As an example, we demonstrate and characterize fabrication of a Cayley tree fractal-like pattern. The patterns, in addition, are retained permanently by employing UV/thermally curable fluids. The proposed technique thus establishes solid foundation for bio-mimicking natural structures spanning multiple-scales for scientific and engineering use. PMID:27849003

Fabrication of Multscale Fractal-Like Structures by Controlling Fluid Interface Instability

NASA Astrophysics Data System (ADS)

Islam, Tanveer Ul; Gandhi, Prasanna S.

2016-11-01

Nature, in quest for the best designs has shaped its vital systems into fractal geometries. Effectual way of spontaneous fabrication of scalable, ordered fractal-like structures by controlling Saffman-Taylor instability in a lifted Hele-Shaw cell is deployed here. In lifted Hele-Shaw cell uncontrolled penetration of low-viscosity fluid into its high-viscosity counterpart is known to develop irregular, non-repeatable, normally short-lived, branched patterns. We propose and characterize experimentally anisotropies in a form of spatially distributed pits on the cell plates to control initiation and further penetration of non-splitting fingers. The proposed control over shielding mechanism yields recipes for fabrication of families of ordered fractal-like patterns of multiple generations. As an example, we demonstrate and characterize fabrication of a Cayley tree fractal-like pattern. The patterns, in addition, are retained permanently by employing UV/thermally curable fluids. The proposed technique thus establishes solid foundation for bio-mimicking natural structures spanning multiple-scales for scientific and engineering use.

Microtopographic Inspection and Fractal Analysis of Skin Neoplasia

NASA Astrophysics Data System (ADS)

Costa, Manuel F. M.; Hipolito, Alberto Valencia; Gutierrez, Gustavo Fidel; Chanona, Jorge; Gallegos, Eva Ramón

2008-04-01

Early detection of skin cancer is fundamental to a successful treatment. Changes in the shape, including the relief, of skin lesions are an indicator of a possible malignity. Optical microtopographic inspection of skin lesions can be used to identify diagnostic patterns of benign and malign skin' lesions. Statistical parameters like the mean roughness (Ra) may allow the discrimination between different types of lesions and degree of malignity. Fractal analysis of bi-dimensional and 3D images of skin lesions can validate or complement that assessment by calculation of its fractal dimensions (FD). On the study herein reported the microtopographic inspection of the skin lesions were performed using the optical triangulation based microtopographer developed at the Physics Department of the University of Minho, MICROTOP.03.MFC. The patients that participated in this research work were men and women older than 15 years with the clinical and histopathology diagnoses of: melanoma, basocellular carcinoma, epidermoide carcinoma, actinic keratosis, keratoacantosis and benign nevus. Latex impressions of the lesions were taken and microtopographically analyzed. Characteristic information for each type of studied lesion was obtained. For melanoma it was observed that on the average these tumors present an increased roughness of around 67 percent compared to the roughness of the healthy skin. This feature allows the distinction from other tumors as basocellular carcinoma (were the roughness increase was in the average of 49 percent) and benign lesions as the epidermoide cyst (37 percent) or the seborrhea keratosis (4 percent). Tumor size and roughness are directly proportional to the grade of malignality. The characterization of the fractal geometry of 2D (histological slides) and 3D images of skin lesions was performed by obtaining its FD evaluated by means of the Box counting method. Results obtained showed that the average fractal dimension of histological slide images (FDh

Music and fractals

NASA Astrophysics Data System (ADS)

Wuorinen, Charles

2015-03-01

Any of the arts may produce exemplars that have fractal characteristics. There may be fractal painting, fractal poetry, and the like. But these will always be specific instances, not necessarily displaying intrinsic properties of the art-medium itself. Only music, I believe, of all the arts possesses an intrinsically fractal character, so that its very nature is fractally determined. Thus, it is reasonable to assert that any instance of music is fractal...

Fractal Bread.

ERIC Educational Resources Information Center

Esbenshade, Donald H., Jr.

1991-01-01

Develops the idea of fractals through a laboratory activity that calculates the fractal dimension of ordinary white bread. Extends use of the fractal dimension to compare other complex structures as other breads and sponges. (MDH)

Using fractal analysis of thermal signatures for thyroid disease evaluation

NASA Astrophysics Data System (ADS)

Gavriloaia, Gheorghe; Sofron, Emil; Gavriloaia, Mariuca-Roxana; Ghemigean, Adina-Mariana

2010-11-01

The skin is the largest organ of the body and it protects against heat, light, injury and infection. Skin temperature is an important parameter for diagnosing diseases. Thermal analysis is non-invasive, painless, and relatively inexpensive, showing a great potential research. Since the thyroid regulates metabolic rate it is intimately connected to body temperature, more than, any modification of its function generates a specific thermal image on the neck skin. The shapes of thermal signatures are often irregular in size and shape. Euclidean geometry is not able to evaluate their shape for different thyroid diseases, and fractal geometry is used in this paper. Different thyroid diseases generate different shapes, and their complexity are evaluated by specific mathematical approaches, fractal analysis, in order to the evaluate selfsimilarity and lacunarity. Two kinds of thyroid diseases, hyperthyroidism and papillary cancer are analyzed in this paper. The results are encouraging and show the ability to continue research for thermal signature to be used in early diagnosis of thyroid diseases.

[Recent progress of research and applications of fractal and its theories in medicine].

PubMed

Cai, Congbo; Wang, Ping

2014-10-01

Fractal, a mathematics concept, is used to describe an image of self-similarity and scale invariance. Some organisms have been discovered with the fractal characteristics, such as cerebral cortex surface, retinal vessel structure, cardiovascular network, and trabecular bone, etc. It has been preliminarily confirmed that the three-dimensional structure of cells cultured in vitro could be significantly enhanced by bionic fractal surface. Moreover, fractal theory in clinical research will help early diagnosis and treatment of diseases, reducing the patient's pain and suffering. The development process of diseases in the human body can be expressed by the fractal theories parameter. It is of considerable significance to retrospectively review the preparation and application of fractal surface and its diagnostic value in medicine. This paper gives an application of fractal and its theories in the medical science, based on the research achievements in our laboratory.

The Application of Fractal and Multifractal Theory in Hydraulic-Flow-Unit Characterization and Permeability Estimation

NASA Astrophysics Data System (ADS)

Chen, X.; Yao, G.; Cai, J.

2017-12-01

Pore structure characteristics are important factors in influencing the fluid transport behavior of porous media, such as pore-throat ratio, pore connectivity and size distribution, moreover, wettability. To accurately characterize the diversity of pore structure among HFUs, five samples selected from different HFUs (porosities are approximately equal, however permeability varies widely) were chosen to conduct micro-computerized tomography test to acquire direct 3D images of pore geometries and to perform mercury injection experiments to obtain the pore volume-radii distribution. To characterize complex and high nonlinear pore structure of all samples, three classic fractal geometry models were applied. Results showed that each HFU has similar box-counting fractal dimension and generalized fractal dimension in the number-area model, but there are significant differences in multifractal spectrums. In the radius-volume model, there are three obvious linear segments, corresponding to three fractal dimension values, and the middle one is proved as the actual fractal dimension according to the maximum radius. In the number-radius model, the spherical-pore size distribution extracted by maximum ball algorithm exist a decrease in the number of small pores compared with the fractal power rate rather than the traditional linear law. Among the three models, only multifractal analysis can classify the HFUs accurately. Additionally, due to the tightness and low-permeability in reservoir rocks, connate water film existing in the inner surface of pore channels commonly forms bound water. The conventional model which is known as Yu-Cheng's model has been proved to be typically not applicable. Considering the effect of irreducible water saturation, an improved fractal permeability model was also deduced theoretically. The comparison results showed that the improved model can be applied to calculate permeability directly and accurately in such unconventional rocks.

Building Students' Understanding of Quadratic Equation Concept Using Naïve Geometry

ERIC Educational Resources Information Center

Fachrudin, Achmad Dhany; Putri, Ratu Ilma Indra; Darmawijoyo

2014-01-01

The purpose of this research is to know how Naïve Geometry method can support students' understanding about the concept of solving quadratic equations. In this article we will discuss one activities of the four activities we developed. This activity focused on how students linking the Naïve Geometry method with the solving of the quadratic…

Fractal-Like Materials Design with Optimized Radiative Properties for High-Efficiency Solar Energy Conversion

DOE Office of Scientific and Technical Information (OSTI.GOV)

Ho, Clifford K.; Ortega, Jesus D.; Christian, Joshua Mark

Novel designs to increase light trapping and thermal efficiency of concentrating solar receivers at multiple length scales have been conceived, designed, and tested. The fractal-like geometries and features are introduced at both macro (meters) and meso (millimeters to centimeters) scales. Advantages include increased solar absorptance, reduced thermal emittance, and increased thermal efficiency. Radial and linear structures at the meso (tube shape and geometry) and macro (total receiver geometry and configuration) scales redirect reflected solar radiation toward the interior of the receiver for increased absorptance. Hotter regions within the interior of the receiver can reduce thermal emittance due to reduced localmore » view factors to the environment, and higher concentration ratios can be employed with similar surface irradiances to reduce the effective optical aperture, footprint, and thermal losses. Coupled optical/fluid/thermal models have been developed to evaluate the performance of these designs relative to conventional designs. Modeling results showed that fractal-like structures and geometries can increase the effective solar absorptance by 5 – 20% and the thermal efficiency by several percentage points at both the meso and macro scales, depending on factors such as intrinsic absorptance. Meso-scale prototypes were fabricated using additive manufacturing techniques, and a macro-scale bladed receiver design was fabricated using Inconel 625 tubes. On-sun tests were performed using the solar furnace and solar tower at the National Solar Thermal Test facility. The test results demonstrated enhanced solar absorptance and thermal efficiency of the fractal-like designs.« less

Classical evolution of fractal measures on the lattice

NASA Astrophysics Data System (ADS)

Antoniou, N. G.; Diakonos, F. K.; Saridakis, E. N.; Tsolias, G. A.

2007-04-01

We consider the classical evolution of a lattice of nonlinear coupled oscillators for a special case of initial conditions resembling the equilibrium state of a macroscopic thermal system at the critical point. The displacements of the oscillators define initially a fractal measure on the lattice associated with the scaling properties of the order parameter fluctuations in the corresponding critical system. Assuming a sudden symmetry breaking (quench), leading to a change in the equilibrium position of each oscillator, we investigate in some detail the deformation of the initial fractal geometry as time evolves. In particular, we show that traces of the critical fractal measure can be sustained for large times, and we extract the properties of the chain that determine the associated time scales. Our analysis applies generally to critical systems for which, after a slow developing phase where equilibrium conditions are justified, a rapid evolution, induced by a sudden symmetry breaking, emerges on time scales much shorter than the corresponding relaxation or observation time. In particular, it can be used in the fireball evolution in a heavy-ion collision experiment, where the QCD critical point emerges, or in the study of evolving fractals of astrophysical and cosmological scales, and may lead to determination of the initial critical properties of the Universe through observations in the symmetry-broken phase.

Can fractal objects operate as efficient inline mixers?

NASA Astrophysics Data System (ADS)

Laizet, Sylvain; Vassilicos, John; Turbulence, Mixing; Flow Control Group Team

2011-11-01

Recently, Hurst & Vassilicos, PoF 2007, Seoud & Vassilicos, PoF 2007, Mazellier & Vassilicos, PoF, 2010 used different multiscale grids to generate turbulence in a wind tunnel and have shown that complex multiscale boundary/initial conditions can drastically influence the behaviour of a turbulent flow, but that the detailled specific nature of the multiscale geometry matters too. Multiscale (fractal) objects can be designed to be immersed in any fluid flow where there is a need to control and design the turbulence generated by the object. Different types of multiscale objects can be designed as different types of energy-efficient mixers with varying degrees of high turbulent intensities, small pressure drop and downstream distance from the grid where the turbulence is most vigorous. Here, we present a 3D DNS study of the stirring and mixing of a passive scalar by turbulence generated with either a fractal square grid or a regular grid in the presence of a mean scalar gradient. The results show that: (1) there is a linear increase for the passive scalar variance for both grids, (2) the passive scalar variance is ten times bigger for the fractal grid, (3) the passive scalar flux is constant after the production region for both grids, (4) the passive scalar flux is enhanced by an order of magnitude for the fractal grid. We acknowledge support from EPSRC, UK.

Nontrivial paths and periodic orbits of the T-fractal billiard table

NASA Astrophysics Data System (ADS)

Lapidus, Michel L.; Miller, Robyn L.; Niemeyer, Robert G.

2016-07-01

We introduce and prove numerous new results about the orbits of the T-fractal billiard. Specifically, in section 3, we give a variety of sufficient conditions for the existence of a sequence of compatible periodic orbits. In section 4, we examine the limiting behavior of particular sequences of compatible periodic orbits. Additionally, sufficient conditions for the existence of particular nontrivial paths are given in section 4. The proofs of two results of Lapidus and Niemeyer (2013 The current state of fractal billiards Fractal Geometry and Dynamical Systems in Pure and Applied Mathematics II: Fractals in Applied Mathematics (Contemporary Mathematics vol 601) ed D Carfi et al (Providence, RI: American Mathematical Society) pp 251-88 (e-print: arXiv:math.DS.1210.0282v2, 2013) appear here for the first time, as well. In section 5, an orbit with an irrational initial direction reaches an elusive point in a way that yields a nontrivial path of finite length, yet, by our convention, constitutes a singular orbit of the fractal billiard table. The existence of such an orbit seems to indicate that the classification of orbits may not be so straightforward. A discussion of our results and directions for future research is then given in section 6.

A New Fractal Model of Chromosome and DNA Processes

NASA Astrophysics Data System (ADS)

Bouallegue, K.

Dynamic chromosome structure remains unknown. Can fractals and chaos be used as new tools to model, identify and generate a structure of chromosomes?Fractals and chaos offer a rich environment for exploring and modeling the complexity of nature. In a sense, fractal geometry is used to describe, model, and analyze the complex forms found in nature. Fractals have also been widely not only in biology but also in medicine. To this effect, a fractal is considered an object that displays self-similarity under magnification and can be constructed using a simple motif (an image repeated on ever-reduced scales).It is worth noting that the problem of identifying a chromosome has become a challenge to find out which one of the models it belongs to. Nevertheless, the several different models (a hierarchical coiling, a folded fiber, and radial loop) have been proposed for mitotic chromosome but have not reached a dynamic model yet.This paper is an attempt to solve topological problems involved in the model of chromosome and DNA processes. By combining the fractal Julia process and the numerical dynamical system, we have finally found out four main points. First, we have developed not only a model of chromosome but also a model of mitosis and one of meiosis. Equally important, we have identified the centromere position through the numerical model captured below. More importantly, in this paper, we have discovered the processes of the cell divisions of both mitosis and meiosis. All in all, the results show that this work could have a strong impact on the welfare of humanity and can lead to a cure of genetic diseases.

From Fractal Trees to Deltaic Networks

NASA Astrophysics Data System (ADS)

Cazanacli, D.; Wolinsky, M. A.; Sylvester, Z.; Cantelli, A.; Paola, C.

2013-12-01

Geometric networks that capture many aspects of natural deltas can be constructed from simple concepts from graph theory and normal probability distributions. Fractal trees with symmetrical geometries are the result of replicating two simple geometric elements, line segments whose lengths decrease and bifurcation angles that are commonly held constant. Branches could also have a thickness, which in the case of natural distributary systems is the equivalent of channel width. In river- or wave-dominated natural deltas, the channel width is a function of discharge. When normal variations around the mean values for length, bifurcating angles, and discharge are applied, along with either pruning of 'clashing' branches or merging (equivalent to channel confluence), fractal trees start resembling natural deltaic networks, except that the resulting channels are unnaturally straight. Introducing a bifurcation probability fewer, naturally curved channels are obtained. If there is no bifurcation, the direction of each new segment depends on the direction the previous segment upstream (correlated random walk) and, to a lesser extent, on a general direction of growth (directional bias). When bifurcation occurs, the resulting two directions also depend on the bifurcation angle and the discharge split proportions, with the dominant branch following the direction of the upstream parent channel closely. The bifurcation probability controls the channel density and, in conjunction with the variability of the directional angles, the overall curvature of the channels. The growth of the network in effect is associated with net delta progradation. The overall shape and shape evolution of the delta depend mainly on the bifurcation angle average size and angle variability coupled with the degree of dominant direction dependency (bias). The proposed algorithm demonstrates how, based on only a few simple rules, a wide variety of channel networks resembling natural deltas, can be replicated

A stress sensitivity model for the permeability of porous media based on bi-dispersed fractal theory

NASA Astrophysics Data System (ADS)

Tan, X.-H.; Liu, C.-Y.; Li, X.-P.; Wang, H.-Q.; Deng, H.

A stress sensitivity model for the permeability of porous media based on bidispersed fractal theory is established, considering the change of the flow path, the fractal geometry approach and the mechanics of porous media. It is noted that the two fractal parameters of the porous media construction perform differently when the stress changes. The tortuosity fractal dimension of solid cluster DcTσ become bigger with an increase of stress. However, the pore fractal dimension of solid cluster Dcfσ and capillary bundle Dpfσ remains the same with an increase of stress. The definition of normalized permeability is introduced for the analyzation of the impacts of stress sensitivity on permeability. The normalized permeability is related to solid cluster tortuosity dimension, pore fractal dimension, solid cluster maximum diameter, Young’s modulus and Poisson’s ratio. Every parameter has clear physical meaning without the use of empirical constants. Predictions of permeability of the model is accordant with the obtained experimental data. Thus, the proposed model can precisely depict the flow of fluid in porous media under stress.

Electrochemical Evaluations of Fractal Microelectrodes for Energy Efficient Neurostimulation.

PubMed

Park, Hyunsu; Takmakov, Pavel; Lee, Hyowon

2018-03-12

Advancements in microfabrication has enabled manufacturing of microscopic neurostimulation electrodes with smaller footprint than ever possible. The smaller electrodes can potentially reduce tissue damage and allow better spatial resolution for neural stimulation. Although electrodes of any shape can easily be fabricated, substantial effort have been focused on identification and characterization of new materials and surface morphology for efficient charge injection, while maintaining simple circular or rectangular Euclidean electrode geometries. In this work we provide a systematic electrochemical evaluation of charge injection capacities of serpentine and fractal-shaped platinum microelectrodes and compare their performance with traditional circular microelectrodes. Our findings indicate that the increase in electrode perimeter leads to an increase in maximum charge injection capacity. Furthermore, we found that the electrode geometry can have even more significant impact on electrode performance than having a larger perimeter for a given surface area. The fractal-shaped microelectrodes, despite having smaller perimeter than other designs, demonstrated superior charge injection capacity. Our results suggest that electrode design can significantly affect both Faradaic and non-Faradaic electrochemical processes, which may be optimized to enable a more energy efficient design for neurostimulation.

A conservation law, entropy principle and quantization of fractal dimensions in hadron interactions

NASA Astrophysics Data System (ADS)

Zborovský, I.

2018-04-01

Fractal self-similarity of hadron interactions demonstrated by the z-scaling of inclusive spectra is studied. The scaling regularity reflects fractal structure of the colliding hadrons (or nuclei) and takes into account general features of fragmentation processes expressed by fractal dimensions. The self-similarity variable z is a function of the momentum fractions x1 and x2 of the colliding objects carried by the interacting hadron constituents and depends on the momentum fractions ya and yb of the scattered and recoil constituents carried by the inclusive particle and its recoil counterpart, respectively. Based on entropy principle, new properties of the z-scaling concept are found. They are conservation of fractal cumulativity in hadron interactions and quantization of fractal dimensions characterizing hadron structure and fragmentation processes at a constituent level.

Fractal Based Triple Band High Gain Monopole Antenna

NASA Astrophysics Data System (ADS)

Pandey, Shashi Kant; Pandey, Ganga Prasad; Sarun, P. M.

2017-10-01

A novel triple-band microstrip fed planar monopole antenna is proposed and investigated. A fractal antenna is created by iterating a narrow pulse (NP) generator model at upper side of modified ground plane, which has a rhombic patch, for enhancing the bandwidth and gain. Three iterations are carried out to study the effects of fractal geometry on the antenna performance. The proposed antenna can operate over three frequency ranges viz, 3.34-4.8 GHz, 5.5-10.6 GHz and 13-14.96 GHz suitable for WLAN 5.2/5.8 GHz, WiMAX 3.5/5.5 GHz and X band applications respectively. Simulated and measured results are in good agreements with each others. Results show that antenna provides wide/ultra wide bandwidths, monopole like radiation patterns and very high antenna gains over the operating frequency bands.

Unification of two fractal families

NASA Astrophysics Data System (ADS)

Liu, Ying

1995-06-01

Barnsley and Hurd classify the fractal images into two families: iterated function system fractals (IFS fractals) and fractal transform fractals, or local iterated function system fractals (LIFS fractals). We will call IFS fractals, class 2 fractals and LIFS fractals, class 3 fractals. In this paper, we will unify these two approaches plus another family of fractals, the class 5 fractals. The basic idea is given as follows: a dynamical system can be represented by a digraph, the nodes in a digraph can be divided into two parts: transient states and persistent states. For bilevel images, a persistent node is a black pixel. A transient node is a white pixel. For images with more than two gray levels, a stochastic digraph is used. A transient node is a pixel with the intensity of 0. The intensity of a persistent node is determined by a relative frequency. In this way, the two families of fractals can be generated in a similar way. In this paper, we will first present a classification of dynamical systems and introduce the transformation based on digraphs, then we will unify the two approaches for fractal binary images. We will compare the decoding algorithms of the two families. Finally, we will generalize the discussion to continuous-tone images.

Fractal Physiology and the Fractional Calculus: A Perspective

PubMed Central

West, Bruce J.

2010-01-01

fractal statistical process. Control of physiologic complexity is one of the goals of medicine, in particular, understanding and controlling physiological networks in order to ensure their proper operation. We emphasize the difference between homeostatic and allometric control mechanisms. Homeostatic control has a negative feedback character, which is both local and rapid. Allometric control, on the other hand, is a relatively new concept that takes into account long-time memory, correlations that are inverse power law in time, as well as long-range interactions in complex phenomena as manifest by inverse power-law distributions in the network variable. We hypothesize that allometric control maintains the fractal character of erratic physiologic time series to enhance the robustness of physiological networks. Moreover, allometric control can often be described using the fractional calculus to capture the dynamics of complex physiologic networks. PMID:21423355

Chaos, Fractals and Their Applications

NASA Astrophysics Data System (ADS)

Thompson, J. Michael T.

2016-12-01

This paper gives an up-to-date account of chaos and fractals, in a popular pictorial style for the general scientific reader. A brief historical account covers the development of the subject from Newton’s laws of motion to the astronomy of Poincaré and the weather forecasting of Lorenz. Emphasis is given to the important underlying concepts, embracing the fractal properties of coastlines and the logistics of population dynamics. A wide variety of applications include: NASA’s discovery and use of zero-fuel chaotic “superhighways” between the planets; erratic chaotic solutions generated by Euler’s method in mathematics; atomic force microscopy; spontaneous pattern formation in chemical and biological systems; impact mechanics in offshore engineering and the chatter of cutting tools; controlling chaotic heartbeats. Reference is made to a number of interactive simulations and movies accessible on the web.

Fractal density modeling of crustal heterogeneity from the KTB deep hole

NASA Astrophysics Data System (ADS)

Chen, Guoxiong; Cheng, Qiuming

2017-03-01

Fractal or multifractal concepts have significantly enlightened our understanding of crustal heterogeneity. Much attention has focused on 1/f scaling natures of physicochemical heterogeneity of Earth crust from fractal increment perspective. In this study, fractal density model from fractal clustering point of view is used to characterize the scaling behaviors of heterogeneous sources recorded at German Continental Deep Drilling Program (KTB) main hole, and of special contribution is the local and global multifractal analysis revisited by using Haar wavelet transform (HWT). Fractal density modeling of mass accumulation generalizes the unit of rock density from integer (e.g., g/cm3) to real numbers (e.g., g/cmα), so that crustal heterogeneities with respect to source accumulation are quantified by singularity strength of fractal density in α-dimensional space. From that perspective, we found that the bulk densities of metamorphic rocks exhibit fractal properties but have a weak multifractality, decreasing with the depth. The multiscaling natures of chemical logs also have been evidenced, and the observed distinct fractal laws for mineral contents are related to their different geochemical behaviors within complex lithological context. Accordingly, scaling distributions of mineral contents have been recognized as a main contributor to the multifractal natures of heterogeneous density for low-porosity crystalline rocks. This finally allows us to use de Wijs cascade process to explain the mechanism of fractal density. In practice, the proposed local singularity analysis based on HWT is suggested as an attractive high-pass filtering to amplify weak signatures of well logs as well as to delineate microlithological changes.

Fractal Interfaces for Stimulating and Recording Neural Implants

NASA Astrophysics Data System (ADS)

Watterson, William James

From investigating movement in an insect to deciphering cognition in a human brain to treating Parkinson's disease, hearing loss, or even blindness, electronic implants are an essential tool for understanding the brain and treating neural diseases. Currently, the stimulating and recording resolution of these implants remains low. For instance, they can record all the neuron activity associated with movement in an insect, but are quite far from recording, at an individual neuron resolution, the large volumes of brain tissue associated with cognition. Likewise, there is remarkable success in the cochlear implant restoring hearing due to the relatively simple anatomy of the auditory nerves, but are failing to restore vision to the blind due to poor signal fidelity and transmission in stimulating the more complex anatomy of the visual nerves. The critically important research needed to improve the resolution of these implants is to optimize the neuron-electrode interface. This thesis explores geometrical and material modifications to both stimulating and recording electrodes which can improve the neuron-electrode interface. First, we introduce a fractal electrode geometry which radically improves the restored visual acuity achieved by retinal implants and leads to safe, long-term operation of the implant. Next, we demonstrate excellent neuron survival and neurite outgrowth on carbon nanotube electrodes, thus providing a safe biomaterial which forms a strong connection between the electrode and neurons. Additional preliminary evidence suggests carbon nanotubes patterned into a fractal geometry will provide further benefits in improving the electrode-neuron interface. Finally, we propose a novel implant based off field effect transistor technology which utilizes an interconnecting fractal network of semiconducting carbon nanotubes to record from thousands of neurons simutaneously at an individual neuron resolution. Taken together, these improvements have the potential to

Theory and discretization of ideal magnetohydrodynamic equilibria with fractal pressure profiles

NASA Astrophysics Data System (ADS)

Kraus, B. F.; Hudson, S. R.

2017-09-01

In three-dimensional ideal magnetohydrodynamics, closed flux surfaces cannot maintain both rational rotational-transform and pressure gradients, as these features together produce unphysical, infinite currents. A proposed set of equilibria nullifies these currents by flattening the pressure on sufficiently wide intervals around each rational surface. Such rational surfaces exist at every scale, which characterizes the pressure profile as self-similar and thus fractal. The pressure profile is approximated numerically by considering a finite number of rational regions and analyzed mathematically by classifying the irrational numbers that support gradients into subsets. Applying these results to a given rotational-transform profile in cylindrical geometry, we find magnetic field and current density profiles compatible with the fractal pressure.

Interactions between a fractal tree-like object and hydrodynamic turbulence: flow structure and characteristic mixing length

NASA Astrophysics Data System (ADS)

Meneveau, C. V.; Bai, K.; Katz, J.

2011-12-01

The vegetation canopy has a significant impact on various physical and biological processes such as forest microclimate, rainfall evaporation distribution and climate change. Most scaled laboratory experimental studies have used canopy element models that consist of rigid vertical strips or cylindrical rods that can be typically represented through only one or a few characteristic length scales, for example the diameter and height for cylindrical rods. However, most natural canopies and vegetation are highly multi-scale with branches and sub-branches, covering a wide range of length scales. Fractals provide a convenient idealization of multi-scale objects, since their multi-scale properties can be described in simple ways (Mandelbrot 1982). While fractal aspects of turbulence have been studied in several works in the past decades, research on turbulence generated by fractal objects started more recently. We present an experimental study of boundary layer flow over fractal tree-like objects. Detailed Particle-Image-Velocimetry (PIV) measurements are carried out in the near-wake of a fractal-like tree. The tree is a pre-fractal with five generations, with three branches and a scale reduction factor 1/2 at each generation. Its similarity fractal dimension (Mandelbrot 1982) is D ~ 1.58. Detailed mean velocity and turbulence stress profiles are documented, as well as their downstream development. We then turn attention to the turbulence mixing properties of the flow, specifically to the question whether a mixing length-scale can be identified in this flow, and if so, how it relates to the geometric length-scales in the pre-fractal object. Scatter plots of mean velocity gradient (shear) and Reynolds shear stress exhibit good linear relation at all locations in the flow. Therefore, in the transverse direction of the wake evolution, the Boussinesq eddy viscosity concept is appropriate to describe the mixing. We find that the measured mixing length increases with increasing

Fractal astronomy.

NASA Astrophysics Data System (ADS)

Beech, M.

1989-02-01

The author discusses some of the more recent research on fractal astronomy and results presented in several astronomical studies. First, the large-scale structure of the universe is considered, while in another section one drops in scale to examine some of the smallest bodies in our solar system; the comets and meteoroids. The final section presents some thoughts on what influence the fractal ideology might have on astronomy, focusing particularly on the question recently raised by Kadanoff, "Fractals: where's the physics?"

Fractal mechanisms in the electrophysiology of the heart

NASA Technical Reports Server (NTRS)

Goldberger, A. L.

1992-01-01

The mathematical concept of fractals provides insights into complex anatomic branching structures that lack a characteristic (single) length scale, and certain complex physiologic processes, such as heart rate regulation, that lack a single time scale. Heart rate control is perturbed by alterations in neuro-autonomic function in a number of important clinical syndromes, including sudden cardiac death, congestive failure, cocaine intoxication, fetal distress, space sickness and physiologic aging. These conditions are associated with a loss of the normal fractal complexity of interbeat interval dynamics. Such changes, which may not be detectable using conventional statistics, can be quantified using new methods derived from "chaos theory.".

Learning Geometry through Dynamic Geometry Software

ERIC Educational Resources Information Center

Forsythe, Sue

2007-01-01

In this article, the author investigates effective teaching and learning of geometrical concepts using dynamic geometry software (DGS). Based from her students' reactions to her project, the author found that her students' understanding of the concepts was better than if they had learned geometry through paper-based tasks. However, mixing computer…

Fractal analysis of Xylella fastidiosa biofilm formation

NASA Astrophysics Data System (ADS)

Moreau, A. L. D.; Lorite, G. S.; Rodrigues, C. M.; Souza, A. A.; Cotta, M. A.

2009-07-01

We have investigated the growth process of Xylella fastidiosa biofilms inoculated on a glass. The size and the distance between biofilms were analyzed by optical images; a fractal analysis was carried out using scaling concepts and atomic force microscopy images. We observed that different biofilms show similar fractal characteristics, although morphological variations can be identified for different biofilm stages. Two types of structural patterns are suggested from the observed fractal dimensions Df. In the initial and final stages of biofilm formation, Df is 2.73±0.06 and 2.68±0.06, respectively, while in the maturation stage, Df=2.57±0.08. These values suggest that the biofilm growth can be understood as an Eden model in the former case, while diffusion-limited aggregation (DLA) seems to dominate the maturation stage. Changes in the correlation length parallel to the surface were also observed; these results were correlated with the biofilm matrix formation, which can hinder nutrient diffusion and thus create conditions to drive DLA growth.

Using Origami Boxes to Explore Concepts of Geometry and Calculus

ERIC Educational Resources Information Center

Wares, Arsalan

2011-01-01

The purpose of this classroom note is to provide an example of how a simple origami box can be used to explore important concepts of geometry and calculus. This article describes how an origami box can be folded, then it goes on to describe how its volume and surface area can be calculated. Finally, it describes how the box could be folded to…

Fractals for Geoengineering

NASA Astrophysics Data System (ADS)

Oleshko, Klaudia; de Jesús Correa López, María; Romero, Alejandro; Ramírez, Victor; Pérez, Olga

2016-04-01

The effectiveness of fractal toolbox to capture the scaling or fractal probability distribution, and simply fractal statistics of main hydrocarbon reservoir attributes, was highlighted by Mandelbrot (1995) and confirmed by several researchers (Zhao et al., 2015). Notwithstanding, after more than twenty years, it's still common the opinion that fractals are not useful for the petroleum engineers and especially for Geoengineering (Corbett, 2012). In spite of this negative background, we have successfully applied the fractal and multifractal techniques to our project entitled "Petroleum Reservoir as a Fractal Reactor" (2013 up to now). The distinguishable feature of Fractal Reservoir is the irregular shapes and rough pore/solid distributions (Siler, 2007), observed across a broad range of scales (from SEM to seismic). At the beginning, we have accomplished the detailed analysis of Nelson and Kibler (2003) Catalog of Porosity and Permeability, created for the core plugs of siliciclastic rocks (around ten thousand data were compared). We enriched this Catalog by more than two thousand data extracted from the last ten years publications on PoroPerm (Corbett, 2012) in carbonates deposits, as well as by our own data from one of the PEMEX, Mexico, oil fields. The strong power law scaling behavior was documented for the major part of these data from the geological deposits of contrasting genesis. Based on these results and taking into account the basic principles and models of the Physics of Fractals, introduced by Per Back and Kan Chen (1989), we have developed new software (Muukíl Kaab), useful to process the multiscale geological and geophysical information and to integrate the static geological and petrophysical reservoir models to dynamic ones. The new type of fractal numerical model with dynamical power law relations among the shapes and sizes of mesh' cells was designed and calibrated in the studied area. The statistically sound power law relations were established

Theory and discretization of ideal magnetohydrodynamic equilibria with fractal pressure profiles

DOE PAGES

Kraus, B. F.; Hudson, S. R.

2017-09-29

In three-dimensional ideal magnetohydrodynamics, closed flux surfaces cannot maintain both rational rotational-transform and pressure gradients, as these features together produce unphysical, infinite currents. A proposed set of equilibria nullifies these currents by flattening the pressure on sufficiently wide intervals around each rational surface. Such rational surfaces exist at every scale, which characterizes the pressure profile as self-similar and thus fractal. The pressure profile is approximated numerically by considering a finite number of rational regions and analyzed mathematically by classifying the irrational numbers that support gradients into subsets. As a result, applying these results to a given rotational-transform profile in cylindricalmore » geometry, we find magnetic field and current density profiles compatible with the fractal pressure.« less

Theory and discretization of ideal magnetohydrodynamic equilibria with fractal pressure profiles

DOE Office of Scientific and Technical Information (OSTI.GOV)

Kraus, B. F.; Hudson, S. R.

In three-dimensional ideal magnetohydrodynamics, closed flux surfaces cannot maintain both rational rotational-transform and pressure gradients, as these features together produce unphysical, infinite currents. A proposed set of equilibria nullifies these currents by flattening the pressure on sufficiently wide intervals around each rational surface. Such rational surfaces exist at every scale, which characterizes the pressure profile as self-similar and thus fractal. The pressure profile is approximated numerically by considering a finite number of rational regions and analyzed mathematically by classifying the irrational numbers that support gradients into subsets. As a result, applying these results to a given rotational-transform profile in cylindricalmore » geometry, we find magnetic field and current density profiles compatible with the fractal pressure.« less

On the design and optimisation of new fractal antenna using PSO

NASA Astrophysics Data System (ADS)

Rani, Shweta; Singh, A. P.

2013-10-01

An optimisation technique for newly shaped fractal structure using particle swarm optimisation with curve fitting is presented in this article. The aim of particle swarm optimisation is to find the geometry of the antenna for the required user-defined frequency. To assess the effectiveness of the presented method, a set of representative numerical simulations have been done and the results are compared with the measurements from experimental prototypes built according to the design specifications coming from the optimisation procedure. The proposed fractal antenna resonates at the 5.8 GHz industrial, scientific and medical band which is suitable for wireless telemedicine applications. The antenna characteristics have been studied using extensive numerical simulations and are experimentally verified. The antenna exhibits well-defined radiation patterns over the band.

Large Engine Technology Program. Task 22: Variable Geometry Concepts for Rich-Quench-Lean Combustors

NASA Technical Reports Server (NTRS)

Tacina, Robert R. (Technical Monitor); Cohen, J. M.; Padget, F. C.; Kwoka, D.; Wang, Q.; Lohmann, R. P.

2005-01-01

The objective of the task reported herein was to define, evaluate, and optimize variable geometry concepts suitable for use with a Rich-Quench-Lean (RQL) combustor. The specific intent was to identify approaches that would satisfy High Speed Civil Transport (HSCT) cycle operational requirements with regard to fuel-air ratio turndown capability, ignition, and stability margin without compromising the stringent emissions, performance, and reliability goals that this combustor would have to achieve. Four potential configurations were identified and three of these were refined and tested in a high-pressure modular RQL combustor rig. The tools used in the evolution of these concepts included models built with rapid fabrication techniques that were tested for airflow characteristics to confirm sizing and airflow management capability, spray patternation, and atomization characterization tests of these models and studies that were supported by Computational Fluid Dynamics analyses. Combustion tests were performed with each of the concepts at supersonic cruise conditions and at other critical conditions in the flight envelope, including the transition points of the variable geometry system, to identify performance, emissions, and operability impacts. Based upon the cold flow characterization, emissions results, acoustic behavior observed during the tests and consideration of mechanical, reliability, and implementation issues, the tri-swirler configuration was selected as the best variable geometry concept for incorporation in the RQL combustor evolution efforts for the HSCT.

Fractals, malware, and data models

NASA Astrophysics Data System (ADS)

Jaenisch, Holger M.; Potter, Andrew N.; Williams, Deborah; Handley, James W.

2012-06-01

We examine the hypothesis that the decision boundary between malware and non-malware is fractal. We introduce a novel encoding method derived from text mining for converting disassembled programs first into opstrings and then filter these into a reduced opcode alphabet. These opcodes are enumerated and encoded into real floating point number format and used for characterizing frequency of occurrence and distribution properties of malware functions to compare with non-malware functions. We use the concept of invariant moments to characterize the highly non-Gaussian structure of the opcode distributions. We then derive Data Model based classifiers from identified features and interpolate and extrapolate the parameter sample space for the derived Data Models. This is done to examine the nature of the parameter space classification boundary between families of malware and the general non-malware category. Preliminary results strongly support the fractal boundary hypothesis, and a summary of our methods and results are presented here.

The fourth dimension of life: fractal geometry and allometric scaling of organisms.

PubMed

West, G B; Brown, J H; Enquist, B J

1999-06-04

Fractal-like networks effectively endow life with an additional fourth spatial dimension. This is the origin of quarter-power scaling that is so pervasive in biology. Organisms have evolved hierarchical branching networks that terminate in size-invariant units, such as capillaries, leaves, mitochondria, and oxidase molecules. Natural selection has tended to maximize both metabolic capacity, by maximizing the scaling of exchange surface areas, and internal efficiency, by minimizing the scaling of transport distances and times. These design principles are independent of detailed dynamics and explicit models and should apply to virtually all organisms.

Fractal pharmacokinetics.

PubMed

Pereira, Luis M

2010-06-01

Pharmacokinetics (PK) has been traditionally dealt with under the homogeneity assumption. However, biological systems are nowadays comprehensively understood as being inherently fractal. Specifically, the microenvironments where drug molecules interact with membrane interfaces, metabolic enzymes or pharmacological receptors, are unanimously recognized as unstirred, space-restricted, heterogeneous and geometrically fractal. Therefore, classical Fickean diffusion and the notion of the compartment as a homogeneous kinetic space must be revisited. Diffusion in fractal spaces has been studied for a long time making use of fractional calculus and expanding on the notion of dimension. Combining this new paradigm with the need to describe and explain experimental data results in defining time-dependent rate constants with a characteristic fractal exponent. Under the one-compartment simplification this strategy is straightforward. However, precisely due to the heterogeneity of the underlying biology, often at least a two-compartment model is required to address macroscopic data such as drug concentrations. This simple modelling step-up implies significant analytical and numerical complications. However, a few methods are available that make possible the original desideratum. In fact, exploring the full range of parametric possibilities and looking at different drugs and respective biological concentrations, it may be concluded that all PK modelling approaches are indeed particular cases of the fractal PK theory.

Reply to "Comment on 'Hydrodynamics of fractal continuum flow' and 'Map of fluid flow in fractal porous medium into fractal continuum flow'".

PubMed

Balankin, Alexander S; Elizarraraz, Benjamin Espinoza

2013-11-01

The aim of this Reply is to elucidate the difference between the fractal continuum models used in the preceding Comment and the models of fractal continuum flow which were put forward in our previous articles [Phys. Rev. E 85, 025302(R) (2012); 85, 056314 (2012)]. In this way, some drawbacks of the former models are highlighted. Specifically, inconsistencies in the definitions of the fractal derivative, the Jacobian of transformation, the displacement vector, and angular momentum are revealed. The proper forms of the Reynolds' transport theorem and angular momentum principle for the fractal continuum are reaffirmed in a more illustrative manner. Consequently, we emphasize that in the absence of any internal angular momentum, body couples, and couple stresses, the Cauchy stress tensor in the fractal continuum should be symmetric. Furthermore, we stress that the approach based on the Cartesian product measured and used in the preceding Comment cannot be employed to study the path-connected fractals, such as a flow in a fractally permeable medium. Thus, all statements of our previous works remain unchallenged.

Scattering from fractals

NASA Astrophysics Data System (ADS)

Hurd, Alan J.

The realization that structures in Nature often can be described by Mandelbrot's fractals has led to a revolution in many areas of physics. The interaction of waves with fractal systems has, understandably, become intensely studied since scattering is the method of choice to probe delicate fractal structures such as chainlike particle aggregates. Not all of these waves are electromagnetic. Neutron scattering, for example, is an important complementary tool to structural studies by X-ray and light scattering. Since the phenomenology of small-angle neutron scattering (SANS), as it is applied to fractal systems, is identical to that of small-angle X-ray scattering (SAXS), it falls within the scope of this paper.

Lévy processes on a generalized fractal comb

NASA Astrophysics Data System (ADS)

Sandev, Trifce; Iomin, Alexander; Méndez, Vicenç

2016-09-01

Comb geometry, constituted of a backbone and fingers, is one of the most simple paradigm of a two-dimensional structure, where anomalous diffusion can be realized in the framework of Markov processes. However, the intrinsic properties of the structure can destroy this Markovian transport. These effects can be described by the memory and spatial kernels. In particular, the fractal structure of the fingers, which is controlled by the spatial kernel in both the real and the Fourier spaces, leads to the Lévy processes (Lévy flights) and superdiffusion. This generalization of the fractional diffusion is described by the Riesz space fractional derivative. In the framework of this generalized fractal comb model, Lévy processes are considered, and exact solutions for the probability distribution functions are obtained in terms of the Fox H-function for a variety of the memory kernels, and the rate of the superdiffusive spreading is studied by calculating the fractional moments. For a special form of the memory kernels, we also observed a competition between long rests and long jumps. Finally, we considered the fractal structure of the fingers controlled by a Weierstrass function, which leads to the power-law kernel in the Fourier space. This is a special case, when the second moment exists for superdiffusion in this competition between long rests and long jumps.

Concept of Fractal Dimension use of Multifractal Cloud Liquid Models Based on Real Data as Input to Monte Carlo Radiation Models

NASA Technical Reports Server (NTRS)

Wiscombe, W.

1999-01-01

The purpose of this paper is discuss the concept of fractal dimension; multifractal statistics as an extension of this; the use of simple multifractal statistics (power spectrum, structure function) to characterize cloud liquid water data; and to understand the use of multifractal cloud liquid water models based on real data as input to Monte Carlo radiation models of shortwave radiation transfer in 3D clouds, and the consequences of this in two areas: the design of aircraft field programs to measure cloud absorptance; and the explanation of the famous "Landsat scale break" in measured radiance.

Fractals in geology and geophysics

NASA Technical Reports Server (NTRS)

Turcotte, Donald L.

1989-01-01

The definition of a fractal distribution is that the number of objects N with a characteristic size greater than r scales with the relation N of about r exp -D. The frequency-size distributions for islands, earthquakes, fragments, ore deposits, and oil fields often satisfy this relation. This application illustrates a fundamental aspect of fractal distributions, scale invariance. The requirement of an object to define a scale in photograhs of many geological features is one indication of the wide applicability of scale invariance to geological problems; scale invariance can lead to fractal clustering. Geophysical spectra can also be related to fractals; these are self-affine fractals rather than self-similar fractals. Examples include the earth's topography and geoid.

Beyond maximum entropy: Fractal Pixon-based image reconstruction

NASA Technical Reports Server (NTRS)

Puetter, Richard C.; Pina, R. K.

1994-01-01

We have developed a new Bayesian image reconstruction method that has been shown to be superior to the best implementations of other competing methods, including Goodness-of-Fit methods such as Least-Squares fitting and Lucy-Richardson reconstruction, as well as Maximum Entropy (ME) methods such as those embodied in the MEMSYS algorithms. Our new method is based on the concept of the pixon, the fundamental, indivisible unit of picture information. Use of the pixon concept provides an improved image model, resulting in an image prior which is superior to that of standard ME. Our past work has shown how uniform information content pixons can be used to develop a 'Super-ME' method in which entropy is maximized exactly. Recently, however, we have developed a superior pixon basis for the image, the Fractal Pixon Basis (FPB). Unlike the Uniform Pixon Basis (UPB) of our 'Super-ME' method, the FPB basis is selected by employing fractal dimensional concepts to assess the inherent structure in the image. The Fractal Pixon Basis results in the best image reconstructions to date, superior to both UPB and the best ME reconstructions. In this paper, we review the theory of the UPB and FPB pixon and apply our methodology to the reconstruction of far-infrared imaging of the galaxy M51. The results of our reconstruction are compared to published reconstructions of the same data using the Lucy-Richardson algorithm, the Maximum Correlation Method developed at IPAC, and the MEMSYS ME algorithms. The results show that our reconstructed image has a spatial resolution a factor of two better than best previous methods (and a factor of 20 finer than the width of the point response function), and detects sources two orders of magnitude fainter than other methods.

Stiffness Indices and Fractal Dimension relationship in Arterial Pressure and Diameter Time Series in-Vitro

NASA Astrophysics Data System (ADS)

Cymberknop, L.; Legnani, W.; Pessana, F.; Bia, D.; Zócalo, Y.; Armentano, R. L.

2011-12-01

The advent of vascular diseases, such as hypertension and atherosclerosis, is associated to significant alterations in the physical properties of arterial vessels. Evaluation of arterial biomechanical behaviour is related to the assessment of three representative indices: arterial compliance, arterial distensibility and arterial stiffness index. Elasticity is the most important mechanical property of the arterial wall, whose natures is strictly non-linear. Intervention of elastin and collagen fibres, passive constituent elements of the arterial wall, is related to the applied wall stress level. Concerning this, appropriate tools are required to analyse the temporal dynamics of the signals involved, in order to characterize the whole phenomenon. Fractal geometry can be mentioned as one of those techniques. The aim of this study consisted on arterial pressure and diameter signals processing, by means of nonlinear techniques based on fractal geometry. Time series morphology was related to different arterial stiffness states, generated by means of blood flow variations, during experiences performed in vitro.

Miniature thermal matches: from nanoheaters to reactive fractals

NASA Astrophysics Data System (ADS)

Rebholz, Claus; Emre Gunduz, Ibrahim; Ando, Teiichi; Doumanidis, Charalabos C.

2015-04-01

Fine thermal actuation by miniature heat sources enables applications from electronics fabrication to tumor cauterization. This paper introduces the concept of nanoheaters, i.e., reactive bimetallic material dots (0D), ignited electrically to exothermically release precise heat amounts where and when needed. This concept is extended to nanoheater wires (1D) and foils (2D), as well as bulk nanoheaters (3D) manufactured via ball milling and ultrasonic consolidation of nickel and aluminum powders. The fractal structure of such powders and consolidates, with self-similar, multiscale Apollonian or lamellar packaging, is discovered to hold the key for their ignition sensitivity: nanoscale structures ignite first, to produce enough heat and raise the temperature of submicron formations, which then ignite microscale regions and so on; while inert areas quench and arrest the self-propagating exothermic reaction. Therefore, such engineered fractal reactive heaters lend themselves to affordable, high-throughput manufacture and controllable, safe, efficient, supplyless in situ thermal release. This can be transformative for innovations from self-healing composites and self-heating packages to underwater construction and mining.

Special issue of selected papers from the second UK-Japan bilateral Workshop and First ERCOFTAC Workshop on Turbulent Flows Generated/Designed in Multiscale/Fractal Ways, London, March 2012

NASA Astrophysics Data System (ADS)

Laizet, Sylvain; Sakai, Yasuhiko; Christos Vassilicos, J.

2013-12-01

This special issue of Fluid Dynamics Research includes nine papers which are based on nine of the presentations at the Second UK-Japan bilateral Workshop and First ERCOFTAC Workshop on 'Turbulent flows generated/designed in multiscale/fractal ways: fundamentals and applications' held from 26 to 27 March 2012 at Imperial College London, UK. The research area of fractal-generated turbulent flows started with a chapter published in 2001 in one of the conference proceedings which came out of the 1999 Isaac Newton Institute 6 month Programme on Turbulence in Cambridge (UK). However, the first results which formed the basis of much of the work reported in this special issue started appearing from 2007 onwards and progress since then could perhaps be described as not insignificant. Research in this area has resulted in the following six notable advances: (a) the definition of two new length-scales characterizing grid-generated turbulence; (b) enhanced and energy-efficient stirring and scalar transfer by fractal grid and fractal openings/flanges with applications, in particular, to improved turbulence generation for combustion; (c) the non-equilibrium turbulent dissipation law; (d) non-equilibrium axisymmetric wake laws; (e) insights into the dependence of drag forces and vortex shedding on the fractal geometry of fractal objects and simulation methods for the calculation of drag of fractal trees; and (f) the invention and successful proof of concept of fractal spoilers and fractal fences. The present special issue contains papers directly related to these advances and can be seen as a reflection of the current research in the field of fractal-generated turbulent flows and their differences and commonalities with other turbulent flows. The financial support from the Japan Society for the Promotion of Science has been decisive for the organization and success of this workshop. We are also grateful to ERCOFTAC who put in place the EU-wide Special Interest Group on multiscale

Fractality of sensations and the brain health: the theory linking neurodegenerative disorder with distortion of spatial and temporal scale-invariance and fractal complexity of the visible world

PubMed Central

Zueva, Marina V.

2015-01-01

The theory that ties normal functioning and pathology of the brain and visual system with the spatial–temporal structure of the visual and other sensory stimuli is described for the first time in the present study. The deficit of fractal complexity of environmental influences can lead to the distortion of fractal complexity in the visual pathways of the brain and abnormalities of development or aging. The use of fractal light stimuli and fractal stimuli of other modalities can help to restore the functions of the brain, particularly in the elderly and in patients with neurodegenerative disorders or amblyopia. Non-linear dynamics of these physiological processes have a strong base of evidence, which is seen in the impaired fractal regulation of rhythmic activity in aged and diseased brains. From birth to old age, we live in a non-linear world, in which objects and processes with the properties of fractality and non-linearity surround us. Against this background, the evolution of man took place and all periods of life unfolded. Works of art created by man may also have fractal properties. The positive influence of music on cognitive functions is well-known. Insufficiency of sensory experience is believed to play a crucial role in the pathogenesis of amblyopia and age-dependent diseases. The brain is very plastic in its early development, and the plasticity decreases throughout life. However, several studies showed the possibility to reactivate the adult’s neuroplasticity in a variety of ways. We propose that a non-linear structure of sensory information on many spatial and temporal scales is crucial to the brain health and fractal regulation of physiological rhythms. Theoretical substantiation of the author’s theory is presented. Possible applications and the future research that can experimentally confirm or refute the theoretical concept are considered. PMID:26236232

Fractal geometry as a new approach for proving nanosimilarity: a reflection note.

PubMed

Demetzos, Costas; Pippa, Natassa

2015-04-10

Nanosimilars are considered as new medicinal outcomes combining the generic drugs and the nanocarrier as an innovative excipient, in order to evaluate them as final products. They belong to the grey area - concerning the evaluation process - between generic drugs and biosimilar medicinal products. Generic drugs are well documented and a huge number of them are in market, replacing effectively the off-patent drugs. The scientific approach for releasing them to the market is based on bioequivalence studies, which are well documented and accepted by the regulatory agencies. On the other hand, the structural complexity of biological/biotechnology-derived products demands a new approach for the approval process taking into consideration that bioequivalence studies are not considered as sufficient as in generic drugs, and new clinical trials are needed to support their approval process of the product to the market. In proportion, due to technological complexity of nanomedicines, the approaches for proving the statistical identity or the similarity for generic and biosimilar products, respectively, with those of prototypes, are not considered as effective for nanosimilar products. The aim of this note is to propose a complementary approach which can provide realistic evidences concerning the nanosimilarity, based on fractal analysis. This approach is well fit with the structural complexity of nanomedicines and smooths the difficulties for proving the similarity between off-patent and nanosimilar products. Fractal analysis could be considered as the approach that completely characterizes the physicochemical/morphological characteristics of nanosimilar products and could be proposed as a start point for a deep discussion on nanosimilarity. Copyright © 2015 Elsevier B.V. All rights reserved.

Fractal analysis of MRI data for the characterization of patients with schizophrenia and bipolar disorder.

PubMed

Squarcina, Letizia; De Luca, Alberto; Bellani, Marcella; Brambilla, Paolo; Turkheimer, Federico E; Bertoldo, Alessandra

2015-02-21

Fractal geometry can be used to analyze shape and patterns in brain images. With this study we use fractals to analyze T1 data of patients affected by schizophrenia or bipolar disorder, with the aim of distinguishing between healthy and pathological brains using the complexity of brain structure, in particular of grey matter, as a marker of disease. 39 healthy volunteers, 25 subjects affected by schizophrenia and 11 patients affected by bipolar disorder underwent an MRI session. We evaluated fractal dimension of the brain cortex and its substructures, calculated with an algorithm based on the box-count algorithm. We modified this algorithm, with the aim of avoiding the segmentation processing step and using all the information stored in the image grey levels. Moreover, to increase sensitivity to local structural changes, we computed a value of fractal dimension for each slice of the brain or of the particular structure. To have reference values in comparing healthy subjects with patients, we built a template by averaging fractal dimension values of the healthy volunteers data. Standard deviation was evaluated and used to create a confidence interval. We also performed a slice by slice t-test to assess the difference at slice level between the three groups. Consistent average fractal dimension values were found across all the structures in healthy controls, while in the pathological groups we found consistent differences, indicating a change in brain and structures complexity induced by these disorders.

Fractal analysis of MRI data for the characterization of patients with schizophrenia and bipolar disorder

NASA Astrophysics Data System (ADS)

Squarcina, Letizia; De Luca, Alberto; Bellani, Marcella; Brambilla, Paolo; Turkheimer, Federico E.; Bertoldo, Alessandra

2015-02-01

Fractal geometry can be used to analyze shape and patterns in brain images. With this study we use fractals to analyze T1 data of patients affected by schizophrenia or bipolar disorder, with the aim of distinguishing between healthy and pathological brains using the complexity of brain structure, in particular of grey matter, as a marker of disease. 39 healthy volunteers, 25 subjects affected by schizophrenia and 11 patients affected by bipolar disorder underwent an MRI session. We evaluated fractal dimension of the brain cortex and its substructures, calculated with an algorithm based on the box-count algorithm. We modified this algorithm, with the aim of avoiding the segmentation processing step and using all the information stored in the image grey levels. Moreover, to increase sensitivity to local structural changes, we computed a value of fractal dimension for each slice of the brain or of the particular structure. To have reference values in comparing healthy subjects with patients, we built a template by averaging fractal dimension values of the healthy volunteers data. Standard deviation was evaluated and used to create a confidence interval. We also performed a slice by slice t-test to assess the difference at slice level between the three groups. Consistent average fractal dimension values were found across all the structures in healthy controls, while in the pathological groups we found consistent differences, indicating a change in brain and structures complexity induced by these disorders.

A modified abstraction of Sierpiński fractals towards enhanced sensitivity of a cross-coupled bow-tie nanostructure

NASA Astrophysics Data System (ADS)

Hasan, Dihan; Lee, Chengkuo

2018-06-01

We experimentally demonstrate a modified abstraction of a fractal geometry (up to order M = 2), namely the Sierpiński fractal, with intrinsic self-similarity for a multitude of infrared sensing applications. The modification particularly strengthens the dipolar resonance and enables optical magnetism at longer wavelengths on a relatively miniaturized footprint. In contrast to the conventional resonant sensing, we harness the broadband electric field enhancement of the modified fractal patterns originating from the lightning rod effect in the non-resonant regime. We demonstrate strong enhancement of molecular absorption at mid-IR by the fractal patterns in the non-resonant regime even under extreme thermal broadening. Finally, we extend the work towards the functional study of the molecular fingerprint of ultra-thin film (∼5 nm) on a non-complementary metamaterial platform in the non-resonant regime. With the help of the solid state chemical dewetting of the monolayer, we also successfully demonstrate a new type of cross-coupling mediated sensitivity of the multispectral and mutually coupled fractal patterns. The research clearly indicates the usefulness of broadband electric field enhancement by the second order fractal pattern for on chip, complete profiling of mid-IR fingerprints of biological elements, i.e. cell, and protein monolayer on a limited footprint and under versatile morphological states.

The Impact of The Fractal Paradigm on Geography

NASA Astrophysics Data System (ADS)

De Cola, L.

2001-12-01

Being itself somewhat fractal, Benoit Mandelbrot's magnum opus THE FRACTAL GEOMETRY OF NATURE may be deconstructed in many ways, including geometrically, systematically, and epistemologically. Viewed as a work of geography it may be used to organize the major topics of interest to scientists preoccupied with the understanding of real-world space in astronomy, geology, meteorology, hydrology, and biology. We shall use it to highlight such recent geographic accomplishments as automated feature detection, understanding urban growth, and modeling the spread of disease in space and time. However, several key challenges remain unsolved, among them: 1. It is still not possible to move continuously from one map scale to another so that objects change their dimension smoothly. I.e. as a viewer zooms in on a map the zero-dimensional location of a city should gradually become a 2-dimensional polygon, then a network of 1-dimensional streets, then 3-dimensional buildings, etc. 2. Spatial autocorrelation continues to be regarded more as an econometric challenge than as a problem of scaling. Similarities of values among closely-spaced observation is not so much a problem to be overcome as a source of information about spatial structure. 3. Although the fractal paradigm is a powerful model for data analysis, its ideas and techniques need to be brought to bear on the problems of understanding such hierarchies as ecosystems (the flow networks of energy and matter), taxonomies (biological classification), and knowledge (hierarchies of bureaucratic information, networks of linked data, etc).

Spatial Heterogeneity, Scale, Data Character and Sustainable Transport in the Big Data Era

NASA Astrophysics Data System (ADS)

Jiang, Bin

2018-04-01

In light of the emergence of big data, I have advocated and argued for a paradigm shift from Tobler's law to scaling law, from Euclidean geometry to fractal geometry, from Gaussian statistics to Paretian statistics, and - more importantly - from Descartes' mechanistic thinking to Alexander's organic thinking. Fractal geometry falls under the third definition of fractal - that is, a set or pattern is fractal if the scaling of far more small things than large ones recurs multiple times (Jiang and Yin 2014) - rather than under the second definition of fractal, which requires a power law between scales and details (Mandelbrot 1982). The new fractal geometry is more towards living geometry that "follows the rules, constraints, and contingent conditions that are, inevitably, encountered in the real world" (Alexander et al. 2012, p. 395), not only for understanding complexity, but also for creating complex or living structure (Alexander 2002-2005). This editorial attempts to clarify why the paradigm shift is essential and to elaborate on several concepts, including spatial heterogeneity (scaling law), scale (or the fourth meaning of scale), data character (in contrast to data quality), and sustainable transport in the big data era.

Comparison of two fractal interpolation methods

NASA Astrophysics Data System (ADS)

Fu, Yang; Zheng, Zeyu; Xiao, Rui; Shi, Haibo

2017-03-01

As a tool for studying complex shapes and structures in nature, fractal theory plays a critical role in revealing the organizational structure of the complex phenomenon. Numerous fractal interpolation methods have been proposed over the past few decades, but they differ substantially in the form features and statistical properties. In this study, we simulated one- and two-dimensional fractal surfaces by using the midpoint displacement method and the Weierstrass-Mandelbrot fractal function method, and observed great differences between the two methods in the statistical characteristics and autocorrelation features. From the aspect of form features, the simulations of the midpoint displacement method showed a relatively flat surface which appears to have peaks with different height as the fractal dimension increases. While the simulations of the Weierstrass-Mandelbrot fractal function method showed a rough surface which appears to have dense and highly similar peaks as the fractal dimension increases. From the aspect of statistical properties, the peak heights from the Weierstrass-Mandelbrot simulations are greater than those of the middle point displacement method with the same fractal dimension, and the variances are approximately two times larger. When the fractal dimension equals to 1.2, 1.4, 1.6, and 1.8, the skewness is positive with the midpoint displacement method and the peaks are all convex, but for the Weierstrass-Mandelbrot fractal function method the skewness is both positive and negative with values fluctuating in the vicinity of zero. The kurtosis is less than one with the midpoint displacement method, and generally less than that of the Weierstrass-Mandelbrot fractal function method. The autocorrelation analysis indicated that the simulation of the midpoint displacement method is not periodic with prominent randomness, which is suitable for simulating aperiodic surface. While the simulation of the Weierstrass-Mandelbrot fractal function method has

Turbulence Enhancement by Fractal Square Grids: Effects of the Number of Fractal Scales

NASA Astrophysics Data System (ADS)

Omilion, Alexis; Ibrahim, Mounir; Zhang, Wei

2017-11-01

Fractal square grids offer a unique solution for passive flow control as they can produce wakes with a distinct turbulence intensity peak and a prolonged turbulence decay region at the expense of only minimal pressure drop. While previous studies have solidified this characteristic of fractal square grids, how the number of scales (or fractal iterations N) affect turbulence production and decay of the induced wake is still not well understood. The focus of this research is to determine the relationship between the fractal iteration N and the turbulence produced in the wake flow using well-controlled water-tunnel experiments. Particle Image Velocimetry (PIV) is used to measure the instantaneous velocity fields downstream of four different fractal grids with increasing number of scales (N = 1, 2, 3, and 4) and a conventional single-scale grid. By comparing the turbulent scales and statistics of the wake, we are able to determine how each iteration affects the peak turbulence intensity and the production/decay of turbulence from the grid. In light of the ability of these fractal grids to increase turbulence intensity with low pressure drop, this work can potentially benefit a wide variety of applications where energy efficient mixing or convective heat transfer is a key process.

Towards a physics on fractals: Differential vector calculus in three-dimensional continuum with fractal metric

NASA Astrophysics Data System (ADS)

Balankin, Alexander S.; Bory-Reyes, Juan; Shapiro, Michael

2016-02-01

One way to deal with physical problems on nowhere differentiable fractals is the mapping of these problems into the corresponding problems for continuum with a proper fractal metric. On this way different definitions of the fractal metric were suggested to account for the essential fractal features. In this work we develop the metric differential vector calculus in a three-dimensional continuum with a non-Euclidean metric. The metric differential forms and Laplacian are introduced, fundamental identities for metric differential operators are established and integral theorems are proved by employing the metric version of the quaternionic analysis for the Moisil-Teodoresco operator, which has been introduced and partially developed in this paper. The relations between the metric and conventional operators are revealed. It should be emphasized that the metric vector calculus developed in this work provides a comprehensive mathematical formalism for the continuum with any suitable definition of fractal metric. This offers a novel tool to study physics on fractals.

Order-fractal transitions in abstract paintings

DOE Office of Scientific and Technical Information (OSTI.GOV)

Calleja, E.M. de la, E-mail: elsama79@gmail.com; Cervantes, F.; Calleja, J. de la

2016-08-15

In this study, we determined the degree of order for 22 Jackson Pollock paintings using the Hausdorff–Besicovitch fractal dimension. Based on the maximum value of each multi-fractal spectrum, the artworks were classified according to the year in which they were painted. It has been reported that Pollock’s paintings are fractal and that this feature was more evident in his later works. However, our results show that the fractal dimension of these paintings ranges among values close to two. We characterize this behavior as a fractal-order transition. Based on the study of disorder-order transition in physical systems, we interpreted the fractal-ordermore » transition via the dark paint strokes in Pollock’s paintings as structured lines that follow a power law measured by the fractal dimension. We determined self-similarity in specific paintings, thereby demonstrating an important dependence on the scale of observations. We also characterized the fractal spectrum for the painting entitled Teri’s Find. We obtained similar spectra for Teri’s Find and Number 5, thereby suggesting that the fractal dimension cannot be rejected completely as a quantitative parameter for authenticating these artworks. -- Highlights: •We determined the degree of order in Jackson Pollock paintings using the Hausdorff–Besicovitch dimension. •We detected a fractal-order transition from Pollock’s paintings between 1947 and 1951. •We suggest that Jackson Pollock could have painted Teri’s Find.« less

Entrainment to a real time fractal visual stimulus modulates fractal gait dynamics.

PubMed

Rhea, Christopher K; Kiefer, Adam W; D'Andrea, Susan E; Warren, William H; Aaron, Roy K

2014-08-01

Fractal patterns characterize healthy biological systems and are considered to reflect the ability of the system to adapt to varying environmental conditions. Previous research has shown that fractal patterns in gait are altered following natural aging or disease, and this has potential negative consequences for gait adaptability that can lead to increased risk of injury. However, the flexibility of a healthy neurological system to exhibit different fractal patterns in gait has yet to be explored, and this is a necessary step toward understanding human locomotor control. Fifteen participants walked for 15min on a treadmill, either in the absence of a visual stimulus or while they attempted to couple the timing of their gait with a visual metronome that exhibited a persistent fractal pattern (contained long-range correlations) or a random pattern (contained no long-range correlations). The stride-to-stride intervals of the participants were recorded via analog foot pressure switches and submitted to detrended fluctuation analysis (DFA) to determine if the fractal patterns during the visual metronome conditions differed from the baseline (no metronome) condition. DFA α in the baseline condition was 0.77±0.09. The fractal patterns in the stride-to-stride intervals were significantly altered when walking to the fractal metronome (DFA α=0.87±0.06) and to the random metronome (DFA α=0.61±0.10) (both p<.05 when compared to the baseline condition), indicating that a global change in gait dynamics was observed. A variety of strategies were identified at the local level with a cross-correlation analysis, indicating that local behavior did not account for the consistent global changes. Collectively, the results show that a gait dynamics can be shifted in a prescribed manner using a visual stimulus and the shift appears to be a global phenomenon. Copyright © 2014 Elsevier B.V. All rights reserved.

International trade network: fractal properties and globalization puzzle.

PubMed

Karpiarz, Mariusz; Fronczak, Piotr; Fronczak, Agata

2014-12-12

Globalization is one of the central concepts of our age. The common perception of the process is that, due to declining communication and transport costs, distance becomes less and less important. However, the distance coefficient in the gravity model of trade, which grows in time, indicates that the role of distance increases rather than decreases. This, in essence, captures the notion of the globalization puzzle. Here, we show that the fractality of the international trade system (ITS) provides a simple solution for the puzzle. We argue that the distance coefficient corresponds to the fractal dimension of ITS. We provide two independent methods, the box counting method and spatial choice model, which confirm this statement. Our results allow us to conclude that the previous approaches to solving the puzzle misinterpreted the meaning of the distance coefficient in the gravity model of trade.

International Trade Network: Fractal Properties and Globalization Puzzle

NASA Astrophysics Data System (ADS)

Karpiarz, Mariusz; Fronczak, Piotr; Fronczak, Agata

2014-12-01

Globalization is one of the central concepts of our age. The common perception of the process is that, due to declining communication and transport costs, distance becomes less and less important. However, the distance coefficient in the gravity model of trade, which grows in time, indicates that the role of distance increases rather than decreases. This, in essence, captures the notion of the globalization puzzle. Here, we show that the fractality of the international trade system (ITS) provides a simple solution for the puzzle. We argue that the distance coefficient corresponds to the fractal dimension of ITS. We provide two independent methods, the box counting method and spatial choice model, which confirm this statement. Our results allow us to conclude that the previous approaches to solving the puzzle misinterpreted the meaning of the distance coefficient in the gravity model of trade.

Fractal symmetry of protein interior: what have we learned?

PubMed

Banerji, Anirban; Ghosh, Indira

2011-08-01

The application of fractal dimension-based constructs to probe the protein interior dates back to the development of the concept of fractal dimension itself. Numerous approaches have been tried and tested over a course of (almost) 30 years with the aim of elucidating the various facets of symmetry of self-similarity prevalent in the protein interior. In the last 5 years especially, there has been a startling upsurge of research that innovatively stretches the limits of fractal-based studies to present an array of unexpected results on the biophysical properties of protein interior. In this article, we introduce readers to the fundamentals of fractals, reviewing the commonality (and the lack of it) between these approaches before exploring the patterns in the results that they produced. Clustering the approaches in major schools of protein self-similarity studies, we describe the evolution of fractal dimension-based methodologies. The genealogy of approaches (and results) presented here portrays a clear picture of the contemporary state of fractal-based studies in the context of the protein interior. To underline the utility of fractal dimension-based measures further, we have performed a correlation dimension analysis on all of the available non-redundant protein structures, both at the level of an individual protein and at the level of structural domains. In this investigation, we were able to separately quantify the self-similar symmetries in spatial correlation patterns amongst peptide-dipole units, charged amino acids, residues with the π-electron cloud and hydrophobic amino acids. The results revealed that electrostatic environments in the interiors of proteins belonging to 'α/α toroid' (all-α class) and 'PLP-dependent transferase-like' domains (α/β class) are highly conducive. In contrast, the interiors of 'zinc finger design' ('designed proteins') and 'knottins' ('small proteins') were identified as folds with the least conducive electrostatic

Using the Cambridge Structural Database to Teach Molecular Geometry Concepts in Organic Chemistry

ERIC Educational Resources Information Center

Wackerly, Jay Wm.; Janowicz, Philip A.; Ritchey, Joshua A.; Caruso, Mary M.; Elliott, Erin L.; Moore, Jeffrey S.

2009-01-01

This article reports a set of two homework assignments that can be used in a second-year undergraduate organic chemistry class. These assignments were designed to help reinforce concepts of molecular geometry and to give students the opportunity to use a technological database and data mining to analyze experimentally determined chemical…

Contour fractal analysis of grains

NASA Astrophysics Data System (ADS)

Guida, Giulia; Casini, Francesca; Viggiani, Giulia MB

2017-06-01

Fractal analysis has been shown to be useful in image processing to characterise the shape and the grey-scale complexity in different applications spanning from electronic to medical engineering (e.g. [1]). Fractal analysis consists of several methods to assign a dimension and other fractal characteristics to a dataset describing geometric objects. Limited studies have been conducted on the application of fractal analysis to the classification of the shape characteristics of soil grains. The main objective of the work described in this paper is to obtain, from the results of systematic fractal analysis of artificial simple shapes, the characterization of the particle morphology at different scales. The long term objective of the research is to link the microscopic features of granular media with the mechanical behaviour observed in the laboratory and in situ.

Fractal reaction kinetics.

PubMed

Kopelman, R

1988-09-23

Classical reaction kinetics has been found to be unsatisfactory when the reactants are spatially constrained on the microscopic level by either walls, phase boundaries, or force fields. Recently discovered theories of heterogeneous reaction kinetics have dramatic consequences, such as fractal orders for elementary reactions, self-ordering and self-unmixing of reactants, and rate coefficients with temporal "memories." The new theories were needed to explain the results of experiments and supercomputer simulations of reactions that were confined to low dimensions or fractal dimensions or both. Among the practical examples of "fractal-like kinetics" are chemical reactions in pores of membranes, excitation trapping in molecular aggregates, exciton fusion in composite materials, and charge recombination in colloids and clouds.

Pitfalls in Fractal Time Series Analysis: fMRI BOLD as an Exemplary Case

PubMed Central

Eke, Andras; Herman, Peter; Sanganahalli, Basavaraju G.; Hyder, Fahmeed; Mukli, Peter; Nagy, Zoltan

2012-01-01

This article will be positioned on our previous work demonstrating the importance of adhering to a carefully selected set of criteria when choosing the suitable method from those available ensuring its adequate performance when applied to real temporal signals, such as fMRI BOLD, to evaluate one important facet of their behavior, fractality. Earlier, we have reviewed on a range of monofractal tools and evaluated their performance. Given the advance in the fractal field, in this article we will discuss the most widely used implementations of multifractal analyses, too. Our recommended flowchart for the fractal characterization of spontaneous, low frequency fluctuations in fMRI BOLD will be used as the framework for this article to make certain that it will provide a hands-on experience for the reader in handling the perplexed issues of fractal analysis. The reason why this particular signal modality and its fractal analysis has been chosen was due to its high impact on today’s neuroscience given it had powerfully emerged as a new way of interpreting the complex functioning of the brain (see “intrinsic activity”). The reader will first be presented with the basic concepts of mono and multifractal time series analyses, followed by some of the most relevant implementations, characterization by numerical approaches. The notion of the dichotomy of fractional Gaussian noise and fractional Brownian motion signal classes and their impact on fractal time series analyses will be thoroughly discussed as the central theme of our application strategy. Sources of pitfalls and way how to avoid them will be identified followed by a demonstration on fractal studies of fMRI BOLD taken from the literature and that of our own in an attempt to consolidate the best practice in fractal analysis of empirical fMRI BOLD signals mapped throughout the brain as an exemplary case of potentially wide interest. PMID:23227008

Fractal dynamics in physiology: Alterations with disease and aging

PubMed Central

Goldberger, Ary L.; Amaral, Luis A. N.; Hausdorff, Jeffrey M.; Ivanov, Plamen Ch.; Peng, C.-K.; Stanley, H. Eugene

2002-01-01

According to classical concepts of physiologic control, healthy systems are self-regulated to reduce variability and maintain physiologic constancy. Contrary to the predictions of homeostasis, however, the output of a wide variety of systems, such as the normal human heartbeat, fluctuates in a complex manner, even under resting conditions. Scaling techniques adapted from statistical physics reveal the presence of long-range, power-law correlations, as part of multifractal cascades operating over a wide range of time scales. These scaling properties suggest that the nonlinear regulatory systems are operating far from equilibrium, and that maintaining constancy is not the goal of physiologic control. In contrast, for subjects at high risk of sudden death (including those with heart failure), fractal organization, along with certain nonlinear interactions, breaks down. Application of fractal analysis may provide new approaches to assessing cardiac risk and forecasting sudden cardiac death, as well as to monitoring the aging process. Similar approaches show promise in assessing other regulatory systems, such as human gait control in health and disease. Elucidating the fractal and nonlinear mechanisms involved in physiologic control and complex signaling networks is emerging as a major challenge in the postgenomic era. PMID:11875196

The Common Evolution of Geometry and Architecture from a Geodetic Point of View

NASA Astrophysics Data System (ADS)

Bellone, T.; Fiermonte, F.; Mussio, L.

2017-05-01

Throughout history the link between geometry and architecture has been strong and while architects have used mathematics to construct their buildings, geometry has always been the essential tool allowing them to choose spatial shapes which are aesthetically appropriate. Sometimes it is geometry which drives architectural choices, but at other times it is architectural innovation which facilitates the emergence of new ideas in geometry. Among the best known types of geometry (Euclidean, projective, analytical, Topology, descriptive, fractal,…) those most frequently employed in architectural design are: - Euclidean Geometry - Projective Geometry - The non-Euclidean geometries. Entire architectural periods are linked to specific types of geometry. Euclidean geometry, for example, was the basis for architectural styles from Antiquity through to the Romanesque period. Perspective and Projective geometry, for their part, were important from the Gothic period through the Renaissance and into the Baroque and Neo-classical eras, while non-Euclidean geometries characterize modern architecture.

Chaos, Fractals, and Polynomials.

ERIC Educational Resources Information Center

Tylee, J. Louis; Tylee, Thomas B.

1996-01-01

Discusses chaos theory; linear algebraic equations and the numerical solution of polynomials, including the use of the Newton-Raphson technique to find polynomial roots; fractals; search region and coordinate systems; convergence; and generating color fractals on a computer. (LRW)

Fractal-Based Image Analysis In Radiological Applications

NASA Astrophysics Data System (ADS)

Dellepiane, S.; Serpico, S. B.; Vernazza, G.; Viviani, R.

1987-10-01

We present some preliminary results of a study aimed to assess the actual effectiveness of fractal theory and to define its limitations in the area of medical image analysis for texture description, in particular, in radiological applications. A general analysis to select appropriate parameters (mask size, tolerance on fractal dimension estimation, etc.) has been performed on synthetically generated images of known fractal dimensions. Moreover, we analyzed some radiological images of human organs in which pathological areas can be observed. Input images were subdivided into blocks of 6x6 pixels; then, for each block, the fractal dimension was computed in order to create fractal images whose intensity was related to the D value, i.e., texture behaviour. Results revealed that the fractal images could point out the differences between normal and pathological tissues. By applying histogram-splitting segmentation to the fractal images, pathological areas were isolated. Two different techniques (i.e., the method developed by Pentland and the "blanket" method) were employed to obtain fractal dimension values, and the results were compared; in both cases, the appropriateness of the fractal description of the original images was verified.

Optimized multilayered wideband absorbers with graded fractal FSS

NASA Astrophysics Data System (ADS)

Vinoy, K. J.; Jose, K. A.; Varadan, Vijay K.; Varadan, Vasundara V.

2001-08-01

Various approaches have been followed for the reduction of radar cross section (RCS), especially of aircraft and missiles. In this paper we present the use of multiple layers of FSS-like fractal geometries printed on dielectric substrates for the same goal. The experimental results shown here indicate 15 dB reduction in the reflection of a flat surface, by the use of this configuration with low loss dielectrics. An extensive optimization scheme is required for extending the angle coverage as well as the bandwidth of the absorber. A brief investigation of such a scheme involving genetic algorithm for this purpose is also presented here.

Dual-band and polarization-insensitive terahertz absorber based on fractal Koch curves

NASA Astrophysics Data System (ADS)

Ma, Yan-Bing; Zhang, Huai-Wu; Li, Yuan-Xun; Wang, Yi-Cheng; Lai, Wei-En; Li, Jie

2014-05-01

We report the design, fabrication, and characterization of a dual-band and polarization-insensitive metamaterial absorber (MA), which consists of periodically arranged fractal Koch curves acting as the top resonator array and a metallic ground plane separated by a dielectric spacer. Compared with conventional MAs, a more compact size and multi-frequency operation are achieved by using fractal geometry as the unit cell of the MA. Both the effective medium theory and the multi-reflection interference theory are employed to investigate the underlying physical mechanism of the proposed terahertz MA, and results indicate that the latter theory is not suitable for explaining the absorption mechanism in our investigated structure. Two absorption peaks are observed at 0.226 THz and 0.622 THz with absorptivities of 91.3% and 95.6% respectively and good agreements between the full-wave simulation and experimental results are achieved.

Emergence of fractal scaling in complex networks

NASA Astrophysics Data System (ADS)

Wei, Zong-Wen; Wang, Bing-Hong

2016-09-01

Some real-world networks are shown to be fractal or self-similar. It is widespread that such a phenomenon originates from the repulsion between hubs or disassortativity. Here we show that this common belief fails to capture the causality. Our key insight to address it is to pinpoint links critical to fractality. Those links with small edge betweenness centrality (BC) constitute a special architecture called fractal reference system, which gives birth to the fractal structure of those reported networks. In contrast, a small amount of links with high BC enable small-world effects, hiding the intrinsic fractality. With enough of such links removed, fractal scaling spontaneously arises from nonfractal networks. Our results provide a multiple-scale view on the structure and dynamics and place fractality as a generic organizing principle of complex networks on a firmer ground.

Benoit B. Mandelbrot (1924-2010)

NASA Astrophysics Data System (ADS)

Barton, Christopher C.; Lovejoy, Shaun; Schertzer, Daniel J.; Turcotte, Donald L.

2012-01-01

Benoit B. Mandelbrot, who advanced the concept of power law scaling as a fundamental property of a broad range of natural processes and patterns in geophysics, economics, mathematics, and virtually all of science, died on 14 October 2010 in Cambridge, Mass., at the age of 85. Mandelbrot, known as the "father of fractal geometry," was a mathematician who developed the scaling concepts of self-similarity and self-affinity and found examples in spatial, temporal, and size patterns across a broad spectrum of disciplines. He coined the term "fractal" (from the Latin noun "fractus," meaning fragmented) for shapes and patterns that exhibit self-similarity, meaning that they are statistically scale independent. Such shapes are characterized by fractional power law exponents, between the integer (Euclidean) dimensions. He is best known through his books, including Les Objets Fractals: Forme, Hasard et Dimension; Fractals: Form, Chance and Dimension; The Fractal Geometry of Nature; andMultifractals and 1/f Noise: Wild Self-Affinity in Physics [Mandelbrot, 1975, Mandelbrot 1977, Mandelbrot 1982, Mandelbrot 1999].

Some problems in fractal differential equations

NASA Astrophysics Data System (ADS)

Su, Weiyi

2016-06-01

Based upon the fractal calculus on local fields, or p-type calculus, or Gibbs-Butzer calculus ([1],[2]), we suggest a constructive idea for "fractal differential equations", beginning from some special examples to a general theory. However, this is just an original idea, it needs lots of later work to support. In [3], we show example "two dimension wave equations with fractal boundaries", and in this note, other examples, as well as an idea to construct fractal differential equations are shown.

Trabecular Bone Mechanical Properties and Fractal Dimension

NASA Technical Reports Server (NTRS)

Hogan, Harry A.

1996-01-01

Countermeasures for reducing bone loss and muscle atrophy due to extended exposure to the microgravity environment of space are continuing to be developed and improved. An important component of this effort is finite element modeling of the lower extremity and spinal column. These models will permit analysis and evaluation specific to each individual and thereby provide more efficient and effective exercise protocols. Inflight countermeasures and post-flight rehabilitation can then be customized and targeted on a case-by-case basis. Recent Summer Faculty Fellowship participants have focused upon finite element mesh generation, muscle force estimation, and fractal calculations of trabecular bone microstructure. Methods have been developed for generating the three-dimensional geometry of the femur from serial section magnetic resonance images (MRI). The use of MRI as an imaging modality avoids excessive exposure to radiation associated with X-ray based methods. These images can also detect trabecular bone microstructure and architecture. The goal of the current research is to determine the degree to which the fractal dimension of trabecular architecture can be used to predict the mechanical properties of trabecular bone tissue. The elastic modulus and the ultimate strength (or strain) can then be estimated from non-invasive, non-radiating imaging and incorporated into the finite element models to more accurately represent the bone tissue of each individual of interest. Trabecular bone specimens from the proximal tibia are being studied in this first phase of the work. Detailed protocols and procedures have been developed for carrying test specimens through all of the steps of a multi-faceted test program. The test program begins with MRI and X-ray imaging of the whole bones before excising a smaller workpiece from the proximal tibia region. High resolution MRI scans are then made and the piece further cut into slabs (roughly 1 cm thick). The slabs are X-rayed again

Thermodynamics of photons on fractals.

PubMed

Akkermans, Eric; Dunne, Gerald V; Teplyaev, Alexander

2010-12-03

A thermodynamical treatment of a massless scalar field (a photon) confined to a fractal spatial manifold leads to an equation of state relating pressure to internal energy, PV(s) = U/d(s), where d(s) is the spectral dimension and V(s) defines the "spectral volume." For regular manifolds, V(s) coincides with the usual geometric spatial volume, but on a fractal this is not necessarily the case. This is further evidence that on a fractal, momentum space can have a different dimension than position space. Our analysis also provides a natural definition of the vacuum (Casimir) energy of a fractal. We suggest ways that these unusual properties might be probed experimentally.

An "ASYMPTOTIC FRACTAL" Approach to the Morphology of Malignant Cell Nuclei

NASA Astrophysics Data System (ADS)

Landini, Gabriel; Rippin, John W.

To investigate quantitatively nuclear membrane irregularity, 672 nuclei from 10 cases of oral cancer (squamous cell carcinoma) and normal cells from oral mucosa were studied in transmission electron micrographs. The nuclei were photographed at ×1400 magnification and transferred to computer memory (1 pixel = 35 nm). The perimeter of the profiles was analysed using the "yardstick method" of fractal dimension estimation, and the log-log plot of ruler size vs. boundary length demonstrated that there exists a significant effect of resolution on length measurement. However, this effect seems to disappear at higher resolutions. As this observation is compatible with the concept of asymptotic fractal, we estimated the parameters c, L and Bm from the asymptotic fractal formula Br = Bm {1 + (r / L)c}-1 , where Br is the boundary length measured with a ruler of size r, Bm is the maximum boundary for r → 0, L is a constant, and c = asymptotic fractal dimension minus topological dimension (D - Dt) for r → ∞. Analyses of variance showed c to be significantly higher in the normal than malignant cases (P < 0.001), but log(L) and Bm to be significantly higher in the malignant cases (P < 0.001). A multivariate linear discrimination analysis on c, log(L) and Bm re-classified 76.6% of the cells correctly (84.8% of the normal and 67.5% of the tumor). Furthermore, this shows that asymptotic fractal analysis applied to nuclear profiles has great potential for shape quantification in diagnosis of oral cancer.

Fractal structures and fractal functions as disease indicators

USGS Publications Warehouse

Escos, J.M; Alados, C.L.; Emlen, J.M.

1995-01-01

Developmental instability is an early indicator of stress, and has been used to monitor the impacts of human disturbance on natural ecosystems. Here we investigate the use of different measures of developmental instability on two species, green peppers (Capsicum annuum), a plant, and Spanish ibex (Capra pyrenaica), an animal. For green peppers we compared the variance in allometric relationship between control plants, and a treatment group infected with the tomato spotted wilt virus. The results show that infected plants have a greater variance about the allometric regression line than the control plants. We also observed a reduction in complexity of branch structure in green pepper with a viral infection. Box-counting fractal dimension of branch architecture declined under stress infection. We also tested the reduction in complexity of behavioral patterns under stress situations in Spanish ibex (Capra pyrenaica). Fractal dimension of head-lift frequency distribution measures predator detection efficiency. This dimension decreased under stressful conditions, such as advanced pregnancy and parasitic infection. Feeding distribution activities reflect food searching efficiency. Power spectral analysis proves to be the most powerful tool for character- izing fractal behavior, revealing a reduction in complexity of time distribution activity under parasitic infection.

Analysis of the fractal dimension of volcano geomorphology through Synthetic Aperture Radar (SAR) amplitude images acquired in C and X band.

NASA Astrophysics Data System (ADS)

Pepe, S.; Di Martino, G.; Iodice, A.; Manzo, M.; Pepe, A.; Riccio, D.; Ruello, G.; Sansosti, E.; Tizzani, P.; Zinno, I.

2012-04-01

In the last two decades several aspects relevant to volcanic activity have been analyzed in terms of fractal parameters that effectively describe natural objects geometry. More specifically, these researches have been aimed at the identification of (1) the power laws that governed the magma fragmentation processes, (2) the energy of explosive eruptions, and (3) the distribution of the associated earthquakes. In this paper, the study of volcano morphology via satellite images is dealt with; in particular, we use the complete forward model developed by some of the authors (Di Martino et al., 2012) that links the stochastic characterization of amplitude Synthetic Aperture Radar (SAR) images to the fractal dimension of the imaged surfaces, modelled via fractional Brownian motion (fBm) processes. Based on the inversion of such a model, a SAR image post-processing has been implemented (Di Martino et al., 2010), that allows retrieving the fractal dimension of the observed surfaces, dictating the distribution of the roughness over different spatial scales. The fractal dimension of volcanic structures has been related to the specific nature of materials and to the effects of active geodynamic processes. Hence, the possibility to estimate the fractal dimension from a single amplitude-only SAR image is of fundamental importance for the characterization of volcano structures and, moreover, can be very helpful for monitoring and crisis management activities in case of eruptions and other similar natural hazards. The implemented SAR image processing performs the extraction of the point-by-point fractal dimension of the scene observed by the sensor, providing - as an output product - the map of the fractal dimension of the area of interest. In this work, such an analysis is performed on Cosmo-SkyMed, ERS-1/2 and ENVISAT images relevant to active stratovolcanoes in different geodynamic contexts, such as Mt. Somma-Vesuvio, Mt. Etna, Vulcano and Stromboli in Southern Italy, Shinmoe

Holographic Characterization of Colloidal Fractal Aggregates

NASA Astrophysics Data System (ADS)

Wang, Chen; Cheong, Fook Chiong; Ruffner, David B.; Zhong, Xiao; Ward, Michael D.; Grier, David G.

In-line holographic microscopy images of micrometer-scale fractal aggregates can be interpreted with the Lorenz-Mie theory of light scattering and an effective-sphere model to obtain each aggregate's size and the population-averaged fractal dimension. We demonstrate this technique experimentally using model fractal clusters of polystyrene nanoparticles and fractal protein aggregates composed of bovine serum albumin and bovine pancreas insulin. This technique can characterize several thousand aggregates in ten minutes and naturally distinguishes aggregates from contaminants such as silicone oil droplets. Work supported by the SBIR program of the NSF.

Feature extraction algorithm for space targets based on fractal theory

NASA Astrophysics Data System (ADS)

Tian, Balin; Yuan, Jianping; Yue, Xiaokui; Ning, Xin

2007-11-01

In order to offer a potential for extending the life of satellites and reducing the launch and operating costs, satellite servicing including conducting repairs, upgrading and refueling spacecraft on-orbit become much more frequently. Future space operations can be more economically and reliably executed using machine vision systems, which can meet real time and tracking reliability requirements for image tracking of space surveillance system. Machine vision was applied to the research of relative pose for spacecrafts, the feature extraction algorithm was the basis of relative pose. In this paper fractal geometry based edge extraction algorithm which can be used in determining and tracking the relative pose of an observed satellite during proximity operations in machine vision system was presented. The method gets the gray-level image distributed by fractal dimension used the Differential Box-Counting (DBC) approach of the fractal theory to restrain the noise. After this, we detect the consecutive edge using Mathematical Morphology. The validity of the proposed method is examined by processing and analyzing images of space targets. The edge extraction method not only extracts the outline of the target, but also keeps the inner details. Meanwhile, edge extraction is only processed in moving area to reduce computation greatly. Simulation results compared edge detection using the method which presented by us with other detection methods. The results indicate that the presented algorithm is a valid method to solve the problems of relative pose for spacecrafts.

Fractal analysis of time varying data

DOEpatents

Vo-Dinh, Tuan; Sadana, Ajit

2002-01-01

Characteristics of time varying data, such as an electrical signal, are analyzed by converting the data from a temporal domain into a spatial domain pattern. Fractal analysis is performed on the spatial domain pattern, thereby producing a fractal dimension D.sub.F. The fractal dimension indicates the regularity of the time varying data.

Fractal electrodynamics via non-integer dimensional space approach

NASA Astrophysics Data System (ADS)

Tarasov, Vasily E.

2015-09-01

Using the recently suggested vector calculus for non-integer dimensional space, we consider electrodynamics problems in isotropic case. This calculus allows us to describe fractal media in the framework of continuum models with non-integer dimensional space. We consider electric and magnetic fields of fractal media with charges and currents in the framework of continuum models with non-integer dimensional spaces. An application of the fractal Gauss's law, the fractal Ampere's circuital law, the fractal Poisson equation for electric potential, and equation for fractal stream of charges are suggested. Lorentz invariance and speed of light in fractal electrodynamics are discussed. An expression for effective refractive index of non-integer dimensional space is suggested.

Experimental Study and Fractal Analysis on the Anisotropic Performance of Explosively Welded Interfaces of 304 Stainless Steel/245 Carbon Steel

NASA Astrophysics Data System (ADS)

Fu, Yanshu; Qiu, Yaohui; Li, Yulong

2018-03-01

The mechanical anisotropy of an explosive welding composite plate made of 304 stainless steel/245 steel was studied through shear experiments performed on explosively welded wavy interfaces along several orientation angles. The results indicated that the strength and the fracture energy of samples significantly varied with the orientation angles. The fracture surfaces of all samples were observed using a scanning electron microscope and through three-dimensional structure microscopy. The periodic features of all the fracture surfaces were clearly shown in different fracture modes. The fractal dimension of the fracture surfaces was calculated based on the fractal geometry by the box-counting method in MATLAB. The cohesive element model was used to analyze the fracture energy according to the physical dependence of the fractal dimension on thermodynamic entropy and interface separation energy. The fracture energy was an exponential function of the fractal dimension value, which was in good agreement with the experimental results. All results were validated for effective use in the application of anisotropy analysis to the welded interface and structural optimization of explosively welded composite plates.

Experimental Study and Fractal Analysis on the Anisotropic Performance of Explosively Welded Interfaces of 304 Stainless Steel/245 Carbon Steel

NASA Astrophysics Data System (ADS)

Fu, Yanshu; Qiu, Yaohui; Li, Yulong

2018-05-01

The mechanical anisotropy of an explosive welding composite plate made of 304 stainless steel/245 steel was studied through shear experiments performed on explosively welded wavy interfaces along several orientation angles. The results indicated that the strength and the fracture energy of samples significantly varied with the orientation angles. The fracture surfaces of all samples were observed using a scanning electron microscope and through three-dimensional structure microscopy. The periodic features of all the fracture surfaces were clearly shown in different fracture modes. The fractal dimension of the fracture surfaces was calculated based on the fractal geometry by the box-counting method in MATLAB. The cohesive element model was used to analyze the fracture energy according to the physical dependence of the fractal dimension on thermodynamic entropy and interface separation energy. The fracture energy was an exponential function of the fractal dimension value, which was in good agreement with the experimental results. All results were validated for effective use in the application of anisotropy analysis to the welded interface and structural optimization of explosively welded composite plates.

A fractal model of the Universe

NASA Astrophysics Data System (ADS)

Gottlieb, Ioan

The book represents a revisioned, extended, completed and translated version of the book "Superposed Universes. A scientific novel and a SF story" (1995). The book contains a hypothesis by the author concerning the complexity of the Nature. An introduction to the theories of numbers, manyfolds and topology is given. The possible connection with the theory of evolution of the Universe is discussed. The book contains also in the last chapter a SF story based on the hypothesis presented. A connection with fractals theory is given. A part of his earlier studies (1955-1956) were subsequently published without citation by Ali Kyrala (Phys. Rev. vol.117, No.5, march 1, 1960). The book contains as an important appendix the early papers (some of which are published in the coauthoprship with his scientific advisors): 1) T.T. Vescan, A. Weiszmann and I.Gottlieb, Contributii la studiul problemelor geometrice ale teoriei relativitatii restranse. Academia R.P.R. Baza Timisoara. Lucrarile consfatuirii de geometrie diferentiala din 9-12 iunie 1955. In this paper the authors show a new method of the calculation of the metrics. 2) Jean Gottlieb, L'hyphotese d'un modele de la structure de la matiere, Revista Matematica y Fisica Teorica, Serie A, Volumen XY, No.1, y.2, 1964 3) I. Gottlieb, Some hypotheses on space, time and gravitation, Studies in Gravitation Theory, CIP Press, Bucharest, 1988, pp.227-234 as well as some recent papers (published in the coauthorship with his disciples): 4)M. Agop, Gottlieb speace-time. A fractal axiomatic model of the Universe. in Particles and Fields, Editors: M.Agop and P.D. Ioannou, Athens University Press, 2005, pp. 59-141 5) I. Gottlieb, M.Agop and V.Enache, Games with Cantor's dust. Chaos, Solitons and Fractals, vol.40 (2009) pp. 940-945 6) I. Gottlieb, My picture over the World, Bull. of the Polytechnic Institute of Iasi. Tom LVI)LX, Fasc. 1, 2010, pp. 1-18. The book contains also a dedication to father Vasile Gottlieb and wife Cleopatra

The Effect of Various Media Scaffolding on Increasing Understanding of Students' Geometry Concepts

ERIC Educational Resources Information Center

Sutiarso, Sugeng; Coesamin, M.; Nurhanurawati

2018-01-01

This study is a quasi-experimental research with pretest-posttest control group design, which aims to determine (1) the tendency of students in using various media scaffolding based on gender, and (2) effect of media scaffolding on increasing understanding of students' geometry concepts. Media scaffolding used this study is chart, props, and…

Roughness Perception of Haptically Displayed Fractal Surfaces

NASA Technical Reports Server (NTRS)

Costa, Michael A.; Cutkosky, Mark R.; Lau, Sonie (Technical Monitor)

2000-01-01

Surface profiles were generated by a fractal algorithm and haptically rendered on a force feedback joystick, Subjects were asked to use the joystick to explore pairs of surfaces and report to the experimenter which of the surfaces they felt was rougher. Surfaces were characterized by their root mean square (RMS) amplitude and their fractal dimension. The most important factor affecting the perceived roughness of the fractal surfaces was the RMS amplitude of the surface. When comparing surfaces of fractal dimension 1.2-1.35 it was found that the fractal dimension was negatively correlated with perceived roughness.

A Double-Minded Fractal

ERIC Educational Resources Information Center

Simoson, Andrew J.

2009-01-01

This article presents a fun activity of generating a double-minded fractal image for a linear algebra class once the idea of rotation and scaling matrices are introduced. In particular the fractal flip-flops between two words, depending on the level at which the image is viewed. (Contains 5 figures.)

Paradigms of Complexity: Fractals and Structures in the Sciences

NASA Astrophysics Data System (ADS)

Novak, Miroslav M.

The Table of Contents for the book is as follows: * Preface * The Origin of Complexity (invited talk) * On the Existence of Spatially Uniform Scaling Laws in the Climate System * Multispectral Backscattering: A Fractal-Structure Probe * Small-Angle Multiple Scattering on a Fractal System of Point Scatterers * Symmetric Fractals Generated by Cellular Automata * Bispectra and Phase Correlations for Chaotic Dynamical Systems * Self-Organized Criticality Models of Neural Development * Altered Fractal and Irregular Heart Rate Behavior in Sick Fetuses * Extract Multiple Scaling in Long-Term Heart Rate Variability * A Semi-Continous Box Counting Method for Fractal Dimension Measurement of Short Single Dimension Temporal Signals - Preliminary Study * A Fractional Brownian Motion Model of Cracking * Self-Affine Scaling Studies on Fractography * Coarsening of Fractal Interfaces * A Fractal Model of Ocean Surface Superdiffusion * Stochastic Subsurface Flow and Transport in Fractal Fractal Conductivity Fields * Rendering Through Iterated Function Systems * The σ-Hull - The Hull Where Fractals Live - Calculating a Hull Bounded by Log Spirals to Solve the Inverse IFS-Problem by the Detected Orbits * On the Multifractal Properties of Passively Convected Scalar Fields * New Statistical Textural Transforms for Non-Stationary Signals: Application to Generalized Mutlifractal Analysis * Laplacian Growth of Parallel Needles: Their Mullins-Sekerka Instability * Entropy Dynamics Associated with Self-Organization * Fractal Properties in Economics (invited talk) * Fractal Approach to the Regional Seismic Event Discrimination Problem * Fractal and Topological Complexity of Radioactive Contamination * Pattern Selection: Nonsingular Saffman-Taylor Finger and Its Dynamic Evolution with Zero Surface Tension * A Family of Complex Wavelets for the Characterization of Singularities * Stabilization of Chaotic Amplitude Fluctuations in Multimode, Intracavity-Doubled Solid-State Lasers * Chaotic

Fractal dimension of turbulent black holes

NASA Astrophysics Data System (ADS)

Westernacher-Schneider, John Ryan

2017-11-01

We present measurements of the fractal dimension of a turbulent asymptotically anti-de Sitter black brane reconstructed from simulated boundary fluid data at the perfect fluid order using the fluid-gravity duality. We argue that the boundary fluid energy spectrum scaling as E (k )˜k-2 is a more natural setting for the fluid-gravity duality than the Kraichnan-Kolmogorov scaling of E (k )˜k-5 /3, but we obtain fractal dimensions D for spatial sections of the horizon H ∩Σ in both cases: D =2.584 (1 ) and D =2.645 (4 ), respectively. These results are consistent with the upper bound of D =3 , thereby resolving the tension with the recent claim in Adams et al. [Phys. Rev. Lett. 112, 151602 (2014), 10.1103/PhysRevLett.112.151602] that D =3 +1 /3 . We offer a critical examination of the calculation which led to their result, and show that their proposed definition of the fractal dimension performs poorly as a fractal dimension estimator on one-dimensional curves with known fractal dimension. Finally, we describe how to define and in principle calculate the fractal dimension of spatial sections of the horizon H ∩Σ in a covariant manner, and we speculate on assigning a "bootstrapped" value of fractal dimension to the entire horizon H when it is in a statistically quasisteady turbulent state.

Focusing behavior of the fractal vector optical fields designed by fractal lattice growth model.

PubMed

Gao, Xu-Zhen; Pan, Yue; Zhao, Meng-Dan; Zhang, Guan-Lin; Zhang, Yu; Tu, Chenghou; Li, Yongnan; Wang, Hui-Tian

2018-01-22

We introduce a general fractal lattice growth model, significantly expanding the application scope of the fractal in the realm of optics. This model can be applied to construct various kinds of fractal "lattices" and then to achieve the design of a great diversity of fractal vector optical fields (F-VOFs) combinating with various "bases". We also experimentally generate the F-VOFs and explore their universal focusing behaviors. Multiple focal spots can be flexibly enginnered, and the optical tweezers experiment validates the simulated tight focusing fields, which means that this model allows the diversity of the focal patterns to flexibly trap and manipulate micrometer-sized particles. Furthermore, the recovery performance of the F-VOFs is also studied when the input fields and spatial frequency spectrum are obstructed, and the results confirm the robustness of the F-VOFs in both focusing and imaging processes, which is very useful in information transmission.

Fractal Analysis of Rock Joint Profiles

NASA Astrophysics Data System (ADS)

Audy, Ondřej; Ficker, Tomáš

2017-10-01

Surface reliefs of rock joints are analyzed in geotechnics when shear strength of rocky slopes is estimated. The rock joint profiles actually are self-affine fractal curves and computations of their fractal dimensions require special methods. Many papers devoted to the fractal properties of these profiles were published in the past but only a few of those papers employed a convenient computational method that would have guaranteed a sound value of that dimension. As a consequence, anomalously low dimensions were presented. This contribution deals with two computational modifications that lead to sound fractal dimensions of the self-affine rock joint profiles. These are the modified box-counting method and the modified yard-stick method sometimes called the compass method. Both these methods are frequently applied to self-similar fractal curves but the self-affine profile curves due to their self-affine nature require modified computational procedures implemented in computer programs.

Aesthetic Responses to Exact Fractals Driven by Physical Complexity

PubMed Central

Bies, Alexander J.; Blanc-Goldhammer, Daryn R.; Boydston, Cooper R.; Taylor, Richard P.; Sereno, Margaret E.

2016-01-01

Fractals are physically complex due to their repetition of patterns at multiple size scales. Whereas the statistical characteristics of the patterns repeat for fractals found in natural objects, computers can generate patterns that repeat exactly. Are these exact fractals processed differently, visually and aesthetically, than their statistical counterparts? We investigated the human aesthetic response to the complexity of exact fractals by manipulating fractal dimensionality, symmetry, recursion, and the number of segments in the generator. Across two studies, a variety of fractal patterns were visually presented to human participants to determine the typical response to exact fractals. In the first study, we found that preference ratings for exact midpoint displacement fractals can be described by a linear trend with preference increasing as fractal dimension increases. For the majority of individuals, preference increased with dimension. We replicated these results for other exact fractal patterns in a second study. In the second study, we also tested the effects of symmetry and recursion by presenting asymmetric dragon fractals, symmetric dragon fractals, and Sierpinski carpets and Koch snowflakes, which have radial and mirror symmetry. We found a strong interaction among recursion, symmetry and fractal dimension. Specifically, at low levels of recursion, the presence of symmetry was enough to drive high preference ratings for patterns with moderate to high levels of fractal dimension. Most individuals required a much higher level of recursion to recover this level of preference in a pattern that lacked mirror or radial symmetry, while others were less discriminating. This suggests that exact fractals are processed differently than their statistical counterparts. We propose a set of four factors that influence complexity and preference judgments in fractals that may extend to other patterns: fractal dimension, recursion, symmetry and the number of segments in a

Fractal dynamics of earthquakes

DOE Office of Scientific and Technical Information (OSTI.GOV)

Bak, P.; Chen, K.

1995-05-01

Many objects in nature, from mountain landscapes to electrical breakdown and turbulence, have a self-similar fractal spatial structure. It seems obvious that to understand the origin of self-similar structures, one must understand the nature of the dynamical processes that created them: temporal and spatial properties must necessarily be completely interwoven. This is particularly true for earthquakes, which have a variety of fractal aspects. The distribution of energy released during earthquakes is given by the Gutenberg-Richter power law. The distribution of epicenters appears to be fractal with dimension D {approx} 1--1.3. The number of after shocks decay as a function ofmore » time according to the Omori power law. There have been several attempts to explain the Gutenberg-Richter law by starting from a fractal distribution of faults or stresses. But this is a hen-and-egg approach: to explain the Gutenberg-Richter law, one assumes the existence of another power-law--the fractal distribution. The authors present results of a simple stick slip model of earthquakes, which evolves to a self-organized critical state. Emphasis is on demonstrating that empirical power laws for earthquakes indicate that the Earth`s crust is at the critical state, with no typical time, space, or energy scale. Of course the model is tremendously oversimplified; however in analogy with equilibrium phenomena they do not expect criticality to depend on details of the model (universality).« less

Fractal Clustering and Knowledge-driven Validation Assessment for Gene Expression Profiling.

PubMed

Wang, Lu-Yong; Balasubramanian, Ammaiappan; Chakraborty, Amit; Comaniciu, Dorin

2005-01-01

DNA microarray experiments generate a substantial amount of information about the global gene expression. Gene expression profiles can be represented as points in multi-dimensional space. It is essential to identify relevant groups of genes in biomedical research. Clustering is helpful in pattern recognition in gene expression profiles. A number of clustering techniques have been introduced. However, these traditional methods mainly utilize shape-based assumption or some distance metric to cluster the points in multi-dimension linear Euclidean space. Their results shows poor consistence with the functional annotation of genes in previous validation study. From a novel different perspective, we propose fractal clustering method to cluster genes using intrinsic (fractal) dimension from modern geometry. This method clusters points in such a way that points in the same clusters are more self-affine among themselves than to the points in other clusters. We assess this method using annotation-based validation assessment for gene clusters. It shows that this method is superior in identifying functional related gene groups than other traditional methods.

Multispectral image fusion based on fractal features

NASA Astrophysics Data System (ADS)

Tian, Jie; Chen, Jie; Zhang, Chunhua

2004-01-01

composition of source pyramid images. So this fusion scheme is a multi-resolution analysis. The wavelet decomposition of image can be actually considered as special pyramid decomposition. According to wavelet decomposition theories, the approximation of image (formula available in paper) at resolution 2j+1 equal to its orthogonal projection in space , that is, where Ajf is the low-frequency approximation of image f(x, y) at resolution 2j and , , represent the vertical, horizontal and diagonal wavelet coefficients respectively at resolution 2j. These coefficients describe the high-frequency information of image at direction of vertical, horizontal and diagonal respectively. Ajf, , and are independent and can be considered as images. In this paper J is set to be 1, so the source image is decomposed to produce the son-images Af, D1f, D2f and D3f. To solve the problem of detecting artifacts, the concepts of vertical fractal dimension FD1, horizontal fractal dimension FD2 and diagonal fractal dimension FD3 are proposed in this paper. The vertical fractal dimension FD1 corresponds to the vertical wavelet coefficients image after the wavelet decomposition of source image, the horizontal fractal dimension FD2 corresponds to the horizontal wavelet coefficients and the diagonal fractal dimension FD3 the diagonal one. These definitions enrich the illustration of source images. Therefore they are helpful to classify the targets. Then the detection of artifacts in the decomposed images is a problem of pattern recognition in 4-D space. The combination of FD0, FD1, FD2 and FD3 make a vector of (FD0, FD1, FD2, FD3), which can be considered as a united feature vector of the studied image. All the parts of the images are classified in the 4-D pattern space created by the vector of (FD0, FD1, FD2, FD3) so that the area that contains man-made objects could be detected. This detection can be considered as a coarse recognition, and then the significant areas in each son-images are signed so that

Band structures in fractal grading porous phononic crystals

NASA Astrophysics Data System (ADS)

Wang, Kai; Liu, Ying; Liang, Tianshu; Wang, Bin

2018-05-01

In this paper, a new grading porous structure is introduced based on a Sierpinski triangle routine, and wave propagation in this fractal grading porous phononic crystal is investigated. The influences of fractal hierarchy and porosity on the band structures in fractal graidng porous phononic crystals are clarified. Vibration modes of unit cell at absolute band gap edges are given to manifest formation mechanism of absolute band gaps. The results show that absolute band gaps are easy to form in fractal structures comparatively to the normal ones with the same porosity. Structures with higher fractal hierarchies benefit multiple wider absolute band gaps. This work provides useful guidance in design of fractal porous phononic crystals.

Fractal Signals & Space-Time Cartoons

NASA Astrophysics Data System (ADS)

Oetama, H. C. Jakob; Maksoed, W. H.

2016-03-01

In ``Theory of Scale Relativity'', 1991- L. Nottale states whereas ``scale relativity is a geometrical & fractal space-time theory''. It took in comparisons to ``a unified, wavelet based framework for efficiently synthetizing, analyzing ∖7 processing several broad classes of fractal signals''-Gregory W. Wornell:``Signal Processing with Fractals'', 1995. Furthers, in Fig 1.1. a simple waveform from statistically scale-invariant random process [ibid.,h 3 ]. Accompanying RLE Technical Report 566 ``Synthesis, Analysis & Processing of Fractal Signals'' as well as from Wornell, Oct 1991 herewith intended to deducts =a Δt + (1 - β Δ t) ...in Petersen, et.al: ``Scale invariant properties of public debt growth'',2010 h. 38006p2 to [1/{1- (2 α (λ) /3 π) ln (λ/r)}depicts in Laurent Nottale,1991, h 24. Acknowledgment devotes to theLates HE. Mr. BrigadierGeneral-TNI[rtd].Prof. Ir. HANDOJO.

Active tectonics on Deception Island (West-Antarctica): A new approach by using the fractal anisotropy of lineaments, fault slip measurements and the caldera collapse shape

USGS Publications Warehouse

Pérez-López, R.; Giner-Robles, J.L.; Martínez-Díaz, J.J.; Rodríguez-Pascua, M.A.; Bejar, M.; Paredes, C.; González-Casado, J.M.

2007-01-01

The tectonic field on Deception Island (South Shetlands, West Antarctica) is determined from structural and fractal analyses. Three different analyses are applied to the study of the strain and stress fields in the area: (1) field measurements of faults (strain analysis), (2) fractal geometry of the spatial distribution of lineaments and (3) the caldera shape (stress analyses). In this work, the identified strain field is extensional with the maximum horizontal shortening trending NE-SW and NW-SE. The fractal technique applied to the spatial distribution of lineaments indicates a stress field with SHMAX oriented NE-SW. The elliptical caldera of Deception Island, determined from field mapping, satellite imagery, vents and fissure eruptions, has an elongate shape and a stress field with SHMAX trending NE-SW.

The Calculation of Fractal Dimension in the Presence of Non-Fractal Clutter

NASA Technical Reports Server (NTRS)

Herren, Kenneth A.; Gregory, Don A.

1999-01-01

The area of information processing has grown dramatically over the last 50 years. In the areas of image processing and information storage the technology requirements have far outpaced the ability of the community to meet demands. The need for faster recognition algorithms and more efficient storage of large quantities of data has forced the user to accept less than lossless retrieval of that data for analysis. In addition to clutter that is not the object of interest in the data set, often the throughput requirements forces the user to accept "noisy" data and to tolerate the clutter inherent in that data. It has been shown that some of this clutter, both the intentional clutter (clouds, trees, etc) as well as the noise introduced on the data by processing requirements can be modeled as fractal or fractal-like. Traditional methods using Fourier deconvolution on these sources of noise in frequency space leads to loss of signal and can, in many cases, completely eliminate the target of interest. The parameters that characterize fractal-like noise (predominately the fractal dimension) have been investigated and a technique to reduce or eliminate noise from real scenes has been developed. Examples of clutter reduced images are presented.

Nuclear patterns of human breast cancer cells during apoptosis: characterisation by fractal dimension and co-occurrence matrix statistics.

PubMed

Losa, Gabriele A; Castelli, Christian

2005-11-01

An analytical strategy combining fractal geometry and grey-level co-occurrence matrix (GLCM) statistics was devised to investigate ultrastructural changes in oestrogen-insensitive SK-BR3 human breast cancer cells undergoing apoptosis in vitro. Apoptosis was induced by 1 microM calcimycin (A23187 Ca(2+) ionophore) and assessed by measuring conventional cellular parameters during the culture period. SK-BR3 cells entered the early stage of apoptosis within 24 h of treatment with calcimycin, which induced detectable changes in nuclear components, as documented by increased values of most GLCM parameters and by the general reduction of the fractal dimensions. In these affected cells, morphonuclear traits were accompanied by the reduction of distinct gangliosides and loss of unidentifiable glycolipid molecules at the cell surface. All these changes were shown to be involved in apoptosis before the detection of conventional markers, which were only measurable during the active phases of apoptotic cell death. In overtly apoptotic cells treated with 1 microM calcimycin for 72 h, most nuclear components underwent dramatic ultrastructural changes, including marginalisation and condensation of chromatin, as reflected in a significant reduction of their fractal dimensions. Hence, both fractal and GLCM analyses confirm that the morphological reorganisation of nuclei, attributable to a loss of structural complexity, occurs early in apoptosis.

Fractal-based wideband invisibility cloak

NASA Astrophysics Data System (ADS)

Cohen, Nathan; Okoro, Obinna; Earle, Dan; Salkind, Phil; Unger, Barry; Yen, Sean; McHugh, Daniel; Polterzycki, Stefan; Shelman-Cohen, A. J.

2015-03-01

A wideband invisibility cloak (IC) at microwave frequencies is described. Using fractal resonators in closely spaced (sub wavelength) arrays as a minimal number of cylindrical layers (rings), the IC demonstrates that it is physically possible to attain a `see through' cloaking device with: (a) wideband coverage; (b) simple and attainable fabrication; (c) high fidelity emulation of the free path; (d) minimal side scattering; (d) a near absence of shadowing in the scattering. Although not a practical device, this fractal-enabled technology demonstrator opens up new opportunities for diverted-image (DI) technology and use of fractals in wideband optical, infrared, and microwave applications.

Exterior dimension of fat fractals

NASA Technical Reports Server (NTRS)

Grebogi, C.; Mcdonald, S. W.; Ott, E.; Yorke, J. A.

1985-01-01

Geometric scaling properties of fat fractal sets (fractals with finite volume) are discussed and characterized via the introduction of a new dimension-like quantity which is called the exterior dimension. In addition, it is shown that the exterior dimension is related to the 'uncertainty exponent' previously used in studies of fractal basin boundaries, and it is shown how this connection can be exploited to determine the exterior dimension. Three illustrative applications are described, two in nonlinear dynamics and one dealing with blood flow in the body. Possible relevance to porous materials and ballistic driven aggregation is also noted.

Power dissipation in fractal AC circuits

NASA Astrophysics Data System (ADS)

Chen, Joe P.; Rogers, Luke G.; Anderson, Loren; Andrews, Ulysses; Brzoska, Antoni; Coffey, Aubrey; Davis, Hannah; Fisher, Lee; Hansalik, Madeline; Loew, Stephen; Teplyaev, Alexander

2017-08-01

We extend Feynman’s analysis of an infinite ladder circuit to fractal circuits, providing examples in which fractal circuits constructed with purely imaginary impedances can have characteristic impedances with positive real part. Using (weak) self-similarity of our fractal structures, we provide algorithms for studying the equilibrium distribution of energy on these circuits. This extends the analysis of self-similar resistance networks introduced by Fukushima, Kigami, Kusuoka, and more recently studied by Strichartz et al.

The fractal dimension of cell membrane correlates with its capacitance: A new fractal single-shell model

PubMed Central

Wang, Xujing; Becker, Frederick F.; Gascoyne, Peter R. C.

2010-01-01

The scale-invariant property of the cytoplasmic membrane of biological cells is examined by applying the Minkowski–Bouligand method to digitized scanning electron microscopy images of the cell surface. The membrane is found to exhibit fractal behavior, and the derived fractal dimension gives a good description of its morphological complexity. Furthermore, we found that this fractal dimension correlates well with the specific membrane dielectric capacitance derived from the electrorotation measurements. Based on these findings, we propose a new fractal single-shell model to describe the dielectrics of mammalian cells, and compare it with the conventional single-shell model (SSM). We found that while both models fit with experimental data well, the new model is able to eliminate the discrepancy between the measured dielectric property of cells and that predicted by the SSM. PMID:21198103

Poiseuille equation for steady flow of fractal fluid

NASA Astrophysics Data System (ADS)

Tarasov, Vasily E.

2016-07-01

Fractal fluid is considered in the framework of continuous models with noninteger dimensional spaces (NIDS). A recently proposed vector calculus in NIDS is used to get a description of fractal fluid flow in pipes with circular cross-sections. The Navier-Stokes equations of fractal incompressible viscous fluids are used to derive a generalization of the Poiseuille equation of steady flow of fractal media in pipe.

Broadband Focusing Acoustic Lens Based on Fractal Metamaterials

PubMed Central

Song, Gang Yong; Huang, Bei; Dong, Hui Yuan; Cheng, Qiang; Cui, Tie Jun

2016-01-01

Acoustic metamaterials are artificial structures which can manipulate sound waves through their unconventional effective properties. Different from the locally resonant elements proposed in earlier studies, we propose an alternate route to realize acoustic metamaterials with both low loss and large refractive indices. We describe a new kind of acoustic metamaterial element with the fractal geometry. Due to the self-similar properties of the proposed structure, broadband acoustic responses may arise within a broad frequency range, making it a good candidate for a number of applications, such as super-resolution imaging and acoustic tunneling. A flat acoustic lens is designed and experimentally verified using this approach, showing excellent focusing abilities from 2 kHz and 5 kHz in the measured results. PMID:27782216

Fractal Feature of Particle-Size Distribution in the Rhizospheres and Bulk Soils during Natural Recovery on the Loess Plateau, China

PubMed Central

Song, Zilin; Zhang, Chao; Liu, Guobin; Qu, Dong; Xue, Sha

2015-01-01

The application of fractal geometry to describe soil structure is an increasingly useful tool for better understanding the performance of soil systems. Only a few studies, however, have focused on the structure of rhizospheric zones, where energy flow and nutrient recycling most frequently occur. We used fractal dimensions to investigate the characteristics of particle-size distribution (PSD) in the rhizospheres and bulk soils of six croplands abandoned for 1, 5, 10, 15, 20, and 30 years on the Loess Plateau of China and evaluated the changes over successional time. The PSDs of the rhizospheres and the fractal dimensions between rhizosphere soil and bulk soils during the natural succession differed significantly due to the influence of plant roots. The rhizospheres had higher sand (0.05–1.00 mm) contents, lower silt (<0.002 mm) contents, and lower fractal dimensions than the bulk soils during the early and intermediate successional stages (1–15 years). The fractal dimensions of the rhizosphere soil and bulk soil ranged from 2.102 to 2.441 and from 2.214 to 2.459, respectively, during the 30-year restoration. Rhizospheric clay and silt contents and fractal dimension tended to be higher and sand content tended to be lower as abandonment age increased, but the bulk soils had the opposite trend. Linear regression analysis indicated that the fractal dimensions of both the rhizospheres and bulk soils were significantly linearly correlated with clay, sand, organic-carbon, and total-nitrogen contents, with R 2 ranging from 0.526 to 0.752 (P<0.001). In conclusion, PSD differed significantly between the rhizosphere soil and bulk soil. The fractal dimension was a sensitive and useful index for quantifying changes in the properties of the different soil zones. This study will greatly aid the application of the fractal method for describing soil structure and nutrient status and the understanding of the performance of rhizospheric zones during ecological restoration. PMID

Fractal Interrelationships in Field and Seismic Data

DOE Office of Scientific and Technical Information (OSTI.GOV)

Wilson, T.H.; Dominic, Jovita; Halverson, Joel

1997-10-01

Size scaling interrelationships are evaluated in this study using a fractal model. Fractal models of several geologic variables are examined and include fracture patterns, reflection travel times, structural relief, drainage, topographic relief and active fault patterns. The fractal properties of structural relief inferred from seismic data and structural cross sections provide a quantitative means to characterize and compare complex structural patterns. Studies were conducted using seismic data from the Granny Creek oil field in the Appalachian Plateau. Previous studies of the field reveal that subtle detached structures present on the limb of a larger structure are associated with enhanced productionmore » from the field. Vertical increases of fractal dimension across the zone of detachment provide a measure of the extent to which detachment has occurred. The increases of fractal dimension are greatest in the more productive areas of the field. A result with equally important ramifications is that fracture systems do not appear to be intrinsically fractal as is often suggested in the literature. While examples of nearly identical patterns can be found at different scales supporting the idea of self-similarity, these examples are often taken from different areas and from different lithologies. Examination of fracture systems at different scales in the Valley and Ridge Province suggest that their distribution become increasingly sparse with scale reduction, and therefore are dissimilar or non-fractal. Box counting data in all cases failed to yield a fractal regime. The results obtained from this analysis bring into question the general applicability of reservoir simulations employing fractal models of fracture distribution. The same conclusions were obtained from the analysis of 1D fracture patterns such as those that might appear in a horizontal well.« less

Not just fractal surfaces, but surface fractal aggregates: Derivation of the expression for the structure factor and its applications

NASA Astrophysics Data System (ADS)

Besselink, R.; Stawski, T. M.; Van Driessche, A. E. S.; Benning, L. G.

2016-12-01

Densely packed surface fractal aggregates form in systems with high local volume fractions of particles with very short diffusion lengths, which effectively means that particles have little space to move. However, there are no prior mathematical models, which would describe scattering from such surface fractal aggregates and which would allow the subdivision between inter- and intraparticle interferences of such aggregates. Here, we show that by including a form factor function of the primary particles building the aggregate, a finite size of the surface fractal interfacial sub-surfaces can be derived from a structure factor term. This formalism allows us to define both a finite specific surface area for fractal aggregates and the fraction of particle interfacial sub-surfaces at the perimeter of an aggregate. The derived surface fractal model is validated by comparing it with an ab initio approach that involves the generation of a "brick-in-a-wall" von Koch type contour fractals. Moreover, we show that this approach explains observed scattering intensities from in situ experiments that followed gypsum (CaSO4 ṡ 2H2O) precipitation from highly supersaturated solutions. Our model of densely packed "brick-in-a-wall" surface fractal aggregates may well be the key precursor step in the formation of several types of mosaic- and meso-crystals.

Micro and MACRO Fractals Generated by Multi-Valued Dynamical Systems

NASA Astrophysics Data System (ADS)

Banakh, T.; Novosad, N.

2014-08-01

Given a multi-valued function Φ : X \\mumap X on a topological space X we study the properties of its fixed fractal \\malteseΦ, which is defined as the closure of the orbit Φω(*Φ) = ⋃n∈ωΦn(*Φ) of the set *Φ = {x ∈ X : x ∈ Φ(x)} of fixed points of Φ. A special attention is paid to the duality between micro-fractals and macro-fractals, which are fixed fractals \\maltese Φ and \\maltese {Φ -1} for a contracting compact-valued function Φ : X \\mumap X on a complete metric space X. With help of algorithms (described in this paper) we generate various images of macro-fractals which are dual to some well-known micro-fractals like the fractal cross, the Sierpiński triangle, Sierpiński carpet, the Koch curve, or the fractal snowflakes. The obtained images show that macro-fractals have a large-scale fractal structure, which becomes clearly visible after a suitable zooming.

Concept for Classifying Facade Elements Based on Material, Geometry and Thermal Radiation Using Multimodal Uav Remote Sensing

NASA Astrophysics Data System (ADS)

Ilehag, R.; Schenk, A.; Hinz, S.

2017-08-01

This paper presents a concept for classification of facade elements, based on the material and the geometry of the elements in addition to the thermal radiation of the facade with the usage of a multimodal Unmanned Aerial Vehicle (UAV) system. Once the concept is finalized and functional, the workflow can be used for energy demand estimations for buildings by exploiting existing methods for estimation of heat transfer coefficient and the transmitted heat loss. The multimodal system consists of a thermal, a hyperspectral and an optical sensor, which can be operational with a UAV. While dealing with sensors that operate in different spectra and have different technical specifications, such as the radiometric and the geometric resolution, the challenges that are faced are presented. Addressed are the different approaches of data fusion, such as image registration, generation of 3D models by performing image matching and the means for classification based on either the geometry of the object or the pixel values. As a first step towards realizing the concept, the result from a geometric calibration with a designed multimodal calibration pattern is presented.

Application to recognition of ferrography image with fractal neural network

NASA Astrophysics Data System (ADS)

Tian, Xianzhong; Hu, Tongsen; Zhang, Jian

2005-10-01

Because wear particles have fractal characteristics, it is necessary that adding fractal parameters to studying wear particles and diagnosing machine troubles. This paper discusses fractal parameters of wear particles, presents arithmetic calculating fractal dimension, and constructs a fractal neural network which can recognize wear particles image. It is proved by experiments that this fractal neural network can recognize some characteristics of wear particles image, and can also classify wear types.

Texture segmentation of non-cooperative spacecrafts images based on wavelet and fractal dimension

NASA Astrophysics Data System (ADS)

Wu, Kanzhi; Yue, Xiaokui

2011-06-01

With the increase of on-orbit manipulations and space conflictions, missions such as tracking and capturing the target spacecrafts are aroused. Unlike cooperative spacecrafts, fixing beacons or any other marks on the targets is impossible. Due to the unknown shape and geometry features of non-cooperative spacecraft, in order to localize the target and obtain the latitude, we need to segment the target image and recognize the target from the background. The data and errors during the following procedures such as feature extraction and matching can also be reduced. Multi-resolution analysis of wavelet theory reflects human beings' recognition towards images from low resolution to high resolution. In addition, spacecraft is the only man-made object in the image compared to the natural background and the differences will be certainly observed between the fractal dimensions of target and background. Combined wavelet transform and fractal dimension, in this paper, we proposed a new segmentation algorithm for the images which contains complicated background such as the universe and planet surfaces. At first, Daubechies wavelet basis is applied to decompose the image in both x axis and y axis, thus obtain four sub-images. Then, calculate the fractal dimensions in four sub-images using different methods; after analyzed the results of fractal dimensions in sub-images, we choose Differential Box Counting in low resolution image as the principle to segment the texture which has the greatest divergences between different sub-images. This paper also presents the results of experiments by using the algorithm above. It is demonstrated that an accurate texture segmentation result can be obtained using the proposed technique.

Fractal Theory and Field Cover Experiments: Implications for the Fractal Characteristics and Radon Diffusion Behavior of Soils and Rocks.

PubMed

Tan, Wanyu; Li, Yongmei; Tan, Kaixuan; Duan, Xianzhe; Liu, Dong; Liu, Zehua

2016-12-01

Radon diffusion and transport through different media is a complex process affected by many factors. In this study, the fractal theories and field covering experiments were used to study the fractal characteristics of particle size distribution (PSD) of six kinds of geotechnical materials (e.g., waste rock, sand, laterite, kaolin, mixture of sand and laterite, and mixture of waste rock and laterite) and their effects on radon diffusion. In addition, the radon diffusion coefficient and diffusion length were calculated. Moreover, new formulas for estimating diffusion coefficient and diffusion length functional of fractal dimension d of PSD were proposed. These results demonstrate the following points: (1) the fractal dimension d of the PSD can be used to characterize the property of soils and rocks in the studies of radon diffusion behavior; (2) the diffusion coefficient and diffusion length decrease with increasing fractal dimension of PSD; and (3) the effectiveness of final covers in reducing radon exhalation of uranium tailings impoundments can be evaluated on the basis of the fractal dimension of PSD of materials.

Variability of fractal dimension of solar radio flux

NASA Astrophysics Data System (ADS)

Bhatt, Hitaishi; Sharma, Som Kumar; Trivedi, Rupal; Vats, Hari Om

2018-04-01

In the present communication, the variation of the fractal dimension of solar radio flux is reported. Solar radio flux observations on a day to day basis at 410, 1415, 2695, 4995, and 8800 MHz are used in this study. The data were recorded at Learmonth Solar Observatory, Australia from 1988 to 2009 covering an epoch of two solar activity cycles (22 yr). The fractal dimension is calculated for the listed frequencies for this period. The fractal dimension, being a measure of randomness, represents variability of solar radio flux at shorter time-scales. The contour plot of fractal dimension on a grid of years versus radio frequency suggests high correlation with solar activity. Fractal dimension increases with increasing frequency suggests randomness increases towards the inner corona. This study also shows that the low frequency is more affected by solar activity (at low frequency fractal dimension difference between solar maximum and solar minimum is 0.42) whereas, the higher frequency is less affected by solar activity (here fractal dimension difference between solar maximum and solar minimum is 0.07). A good positive correlation is found between fractal dimension averaged over all frequencies and yearly averaged sunspot number (Pearson's coefficient is 0.87).

Diagnosis of cervical cells based on fractal and Euclidian geometrical measurements: Intrinsic Geometric Cellular Organization

PubMed Central

2014-01-01

Background Fractal geometry has been the basis for the development of a diagnosis of preneoplastic and neoplastic cells that clears up the undetermination of the atypical squamous cells of undetermined significance (ASCUS). Methods Pictures of 40 cervix cytology samples diagnosed with conventional parameters were taken. A blind study was developed in which the clinic diagnosis of 10 normal cells, 10 ASCUS, 10 L-SIL and 10 H-SIL was masked. Cellular nucleus and cytoplasm were evaluated in the generalized Box-Counting space, calculating the fractal dimension and number of spaces occupied by the frontier of each object. Further, number of pixels occupied by surface of each object was calculated. Later, the mathematical features of the measures were studied to establish differences or equalities useful for diagnostic application. Finally, the sensibility, specificity, negative likelihood ratio and diagnostic concordance with Kappa coefficient were calculated. Results Simultaneous measures of the nuclear surface and the subtraction between the boundaries of cytoplasm and nucleus, lead to differentiate normality, L-SIL and H-SIL. Normality shows values less than or equal to 735 in nucleus surface and values greater or equal to 161 in cytoplasm-nucleus subtraction. L-SIL cells exhibit a nucleus surface with values greater than or equal to 972 and a subtraction between nucleus-cytoplasm higher to 130. L-SIL cells show cytoplasm-nucleus values less than 120. The rank between 120–130 in cytoplasm-nucleus subtraction corresponds to evolution between L-SIL and H-SIL. Sensibility and specificity values were 100%, the negative likelihood ratio was zero and Kappa coefficient was equal to 1. Conclusions A new diagnostic methodology of clinic applicability was developed based on fractal and euclidean geometry, which is useful for evaluation of cervix cytology. PMID:24742118

Fractal fluctuations in gaze speed visual search.

PubMed

Stephen, Damian G; Anastas, Jason

2011-04-01

Visual search involves a subtle coordination of visual memory and lower-order perceptual mechanisms. Specifically, the fluctuations in gaze may provide support for visual search above and beyond what may be attributed to memory. Prior research indicates that gaze during search exhibits fractal fluctuations, which allow for a wide sampling of the field of view. Fractal fluctuations constitute a case of fast diffusion that may provide an advantage in exploration. We present reanalyses of eye-tracking data collected by Stephen and Mirman (Cognition, 115, 154-165, 2010) for single-feature and conjunction search tasks. Fluctuations in gaze during these search tasks were indeed fractal. Furthermore, the degree of fractality predicted decreases in reaction time on a trial-by-trial basis. We propose that fractality may play a key role in explaining the efficacy of perceptual exploration.

The concept of a hierarchical cosmos

NASA Astrophysics Data System (ADS)

Grujić, P. V.

2003-10-01

The idea of a hierachically structured cosmos can be traced back to the Presocratic Hellada. In the fifth century BC Anaxagoras from Clazomenae developed an idea of a sort of fractal material world, by introducing the concept of seeds (spermata), or homoeomeries as Aristotle dubbed it later (Grujić 2001). Anaxagoras ideas have been grossly neglected during the Middle Ages, to be invoked by a number of post-Renaissance thinkers, like Leibniz, Kant, etc, though neither of them referred to their Greek predecessor. But the real resurrections of the hierarchical paradigm started at the beginning of the last century, with Fournier and Charlier (Grujić 2002). Second half of the 20th century witnessed an intensive development of the theoretical models based on the (multi)fractal paradigm, as well as a considerable body of the observational evidence in favour of the hierarchical cosmos (Saar 1988). We overview the state of the art of the cosmological fractal concept, both within the astrophysical (Sylos Labini et al 1998), methodological (Ribeiro 2001) and epistemological (Ribeiro and Videira 1998) context.

Verifying the Dependence of Fractal Coefficients on Different Spatial Distributions

NASA Astrophysics Data System (ADS)

Gospodinov, Dragomir; Marekova, Elisaveta; Marinov, Alexander

2010-01-01

A fractal distribution requires that the number of objects larger than a specific size r has a power-law dependence on the size N(r) = C/rD∝r-D where D is the fractal dimension. Usually the correlation integral is calculated to estimate the correlation fractal dimension of epicentres. A `box-counting' procedure could also be applied giving the `capacity' fractal dimension. The fractal dimension can be an integer and then it is equivalent to a Euclidean dimension (it is zero of a point, one of a segment, of a square is two and of a cube is three). In general the fractal dimension is not an integer but a fractional dimension and there comes the origin of the term `fractal'. The use of a power-law to statistically describe a set of events or phenomena reveals the lack of a characteristic length scale, that is fractal objects are scale invariant. Scaling invariance and chaotic behavior constitute the base of a lot of natural hazards phenomena. Many studies of earthquakes reveal that their occurrence exhibits scale-invariant properties, so the fractal dimension can characterize them. It has first been confirmed that both aftershock rate decay in time and earthquake size distribution follow a power law. Recently many other earthquake distributions have been found to be scale-invariant. The spatial distribution of both regional seismicity and aftershocks show some fractal features. Earthquake spatial distributions are considered fractal, but indirectly. There are two possible models, which result in fractal earthquake distributions. The first model considers that a fractal distribution of faults leads to a fractal distribution of earthquakes, because each earthquake is characteristic of the fault on which it occurs. The second assumes that each fault has a fractal distribution of earthquakes. Observations strongly favour the first hypothesis. The fractal coefficients analysis provides some important advantages in examining earthquake spatial distribution, which are

a Fractal Network Model for Fractured Porous Media

NASA Astrophysics Data System (ADS)

Xu, Peng; Li, Cuihong; Qiu, Shuxia; Sasmito, Agus Pulung

2016-04-01

The transport properties and mechanisms of fractured porous media are very important for oil and gas reservoir engineering, hydraulics, environmental science, chemical engineering, etc. In this paper, a fractal dual-porosity model is developed to estimate the equivalent hydraulic properties of fractured porous media, where a fractal tree-like network model is used to characterize the fracture system according to its fractal scaling laws and topological structures. The analytical expressions for the effective permeability of fracture system and fractured porous media, tortuosity, fracture density and fraction are derived. The proposed fractal model has been validated by comparisons with available experimental data and numerical simulation. It has been shown that fractal dimensions for fracture length and aperture have significant effect on the equivalent hydraulic properties of fractured porous media. The effective permeability of fracture system can be increased with the increase of fractal dimensions for fracture length and aperture, while it can be remarkably lowered by introducing tortuosity at large branching angle. Also, a scaling law between the fracture density and fractal dimension for fracture length has been found, where the scaling exponent depends on the fracture number. The present fractal dual-porosity model may shed light on the transport physics of fractured porous media and provide theoretical basis for oil and gas exploitation, underground water, nuclear waste disposal and geothermal energy extraction as well as chemical engineering, etc.

The fractal nature of vacuum arc cathode spots

DOE Office of Scientific and Technical Information (OSTI.GOV)

Anders, Andre

2005-05-27

Cathode spot phenomena show many features of fractals, for example self-similar patterns in the emitted light and arc erosion traces. Although there have been hints on the fractal nature of cathode spots in the literature, the fractal approach to spot interpretation is underutilized. In this work, a brief review of spot properties is given, touching the differences between spot type 1 (on cathodes surfaces with dielectric layers) and spot type 2 (on metallic, clean surfaces) as well as the known spot fragment or cell structure. The basic properties of self-similarity, power laws, random colored noise, and fractals are introduced. Severalmore » points of evidence for the fractal nature of spots are provided. Specifically power laws are identified as signature of fractal properties, such as spectral power of noisy arc parameters (ion current, arc voltage, etc) obtained by fast Fourier transform. It is shown that fractal properties can be observed down to the cutoff by measurement resolution or occurrence of elementary steps in physical processes. Random walk models of cathode spot motion are well established: they go asymptotically to Brownian motion for infinitesimal step width. The power spectrum of the arc voltage noise falls as 1/f {sup 2}, where f is frequency, supporting a fractal spot model associated with Brownian motion.« less

Fractality à la carte: a general particle aggregation model.

PubMed

Nicolás-Carlock, J R; Carrillo-Estrada, J L; Dossetti, V

2016-01-19

In nature, fractal structures emerge in a wide variety of systems as a local optimization of entropic and energetic distributions. The fractality of these systems determines many of their physical, chemical and/or biological properties. Thus, to comprehend the mechanisms that originate and control the fractality is highly relevant in many areas of science and technology. In studying clusters grown by aggregation phenomena, simple models have contributed to unveil some of the basic elements that give origin to fractality, however, the specific contribution from each of these elements to fractality has remained hidden in the complex dynamics. Here, we propose a simple and versatile model of particle aggregation that is, on the one hand, able to reveal the specific entropic and energetic contributions to the clusters' fractality and morphology, and, on the other, capable to generate an ample assortment of rich natural-looking aggregates with any prescribed fractal dimension.

GENERATING FRACTAL PATTERNS BY USING p-CIRCLE INVERSION

NASA Astrophysics Data System (ADS)

Ramírez, José L.; Rubiano, Gustavo N.; Zlobec, Borut Jurčič

2015-10-01

In this paper, we introduce the p-circle inversion which generalizes the classical inversion with respect to a circle (p = 2) and the taxicab inversion (p = 1). We study some basic properties and we also show the inversive images of some basic curves. We apply this new transformation to well-known fractals such as Sierpinski triangle, Koch curve, dragon curve, Fibonacci fractal, among others. Then we obtain new fractal patterns. Moreover, we generalize the method called circle inversion fractal be means of the p-circle inversion.

FAST TRACK COMMUNICATION: Weyl law for fat fractals

NASA Astrophysics Data System (ADS)

Spina, María E.; García-Mata, Ignacio; Saraceno, Marcos

2010-10-01

It has been conjectured that for a class of piecewise linear maps the closure of the set of images of the discontinuity has the structure of a fat fractal, that is, a fractal with positive measure. An example of such maps is the sawtooth map in the elliptic regime. In this work we analyze this problem quantum mechanically in the semiclassical regime. We find that the fraction of states localized on the unstable set satisfies a modified fractal Weyl law, where the exponent is given by the exterior dimension of the fat fractal.

Fractal analysis of urban catchments and their representation in semi-distributed models: imperviousness and sewer system

NASA Astrophysics Data System (ADS)

Gires, Auguste; Tchiguirinskaia, Ioulia; Schertzer, Daniel; Ochoa-Rodriguez, Susana; Willems, Patrick; Ichiba, Abdellah; Wang, Lipen; Pina, Rui; Van Assel, Johan; Bruni, Guendalina; Murla Tuyls, Damian; ten Veldhuis, Marie-Claire

2017-04-01

Land use distribution and sewer system geometry exhibit complex scale dependent patterns in urban environment. This scale dependency is even more visible in a rasterized representation where only a unique class is affected to each pixel. Such features are well grasped with fractal tools, which are based scale invariance and intrinsically designed to characterise and quantify the space filled by a geometrical set exhibiting complex and tortuous patterns. Fractal tools have been widely used in hydrology but seldom in the specific context of urban hydrology. In this paper, they are used to analyse surface and sewer data from 10 urban or peri-urban catchments located in 5 European countries in the framework of the NWE Interreg RainGain project (www.raingain.eu). The aim was to characterise urban catchment properties accounting for the complexity and inhomogeneity typical of urban water systems. Sewer system density and imperviousness (roads or buildings), represented in rasterized maps of 2 m x 2 m pixels, were analysed to quantify their fractal dimension, characteristic of scaling invariance. It appears that both sewer density and imperviousness exhibit scale invariant features that can be characterized with the help of fractal dimensions ranging from 1.6 to 2, depending on the catchment. In a given area, consistent results were found for the two geometrical features, yielding a robust and innovative way of quantifying the level of urbanization. The representation of imperviousness in operational semi-distributed hydrological models for these catchments was also investigated by computing fractal dimensions of the geometrical sets made up of the sub-catchments with coefficients of imperviousness greater than a range of thresholds. It enables to quantify how well spatial structures of imperviousness are represented in the urban hydrological models.

Effective degrees of freedom of a random walk on a fractal.

PubMed

Balankin, Alexander S

2015-12-01

We argue that a non-Markovian random walk on a fractal can be treated as a Markovian process in a fractional dimensional space with a suitable metric. This allows us to define the fractional dimensional space allied to the fractal as the ν-dimensional space F(ν) equipped with the metric induced by the fractal topology. The relation between the number of effective spatial degrees of freedom of walkers on the fractal (ν) and fractal dimensionalities is deduced. The intrinsic time of random walk in F(ν) is inferred. The Laplacian operator in F(ν) is constructed. This allows us to map physical problems on fractals into the corresponding problems in F(ν). In this way, essential features of physics on fractals are revealed. Particularly, subdiffusion on path-connected fractals is elucidated. The Coulomb potential of a point charge on a fractal embedded in the Euclidean space is derived. Intriguing attributes of some types of fractals are highlighted.

Effective degrees of freedom of a random walk on a fractal

NASA Astrophysics Data System (ADS)

Balankin, Alexander S.

2015-12-01

We argue that a non-Markovian random walk on a fractal can be treated as a Markovian process in a fractional dimensional space with a suitable metric. This allows us to define the fractional dimensional space allied to the fractal as the ν -dimensional space Fν equipped with the metric induced by the fractal topology. The relation between the number of effective spatial degrees of freedom of walkers on the fractal (ν ) and fractal dimensionalities is deduced. The intrinsic time of random walk in Fν is inferred. The Laplacian operator in Fν is constructed. This allows us to map physical problems on fractals into the corresponding problems in Fν. In this way, essential features of physics on fractals are revealed. Particularly, subdiffusion on path-connected fractals is elucidated. The Coulomb potential of a point charge on a fractal embedded in the Euclidean space is derived. Intriguing attributes of some types of fractals are highlighted.

Fractal dust constrains the collisional history of comets

NASA Astrophysics Data System (ADS)

Fulle, M.; Blum, J.

2017-07-01

The fractal dust particles observed by Rosetta cannot form in the physical conditions observed today in comet 67P/Churyumov-Gerasimenko (67P hereinafter), being instead consistent with models of the pristine dust aggregates coagulated in the solar nebula. Since bouncing collisions in the protoplanetary disc restructure fractals into compact aggregates (pebbles), the only way to preserve fractals in a comet is the gentle gravitational collapse of a mixture of pebbles and fractals, which must occur before their mutual collision speeds overcome ≈1 m s-1. This condition fixes the pebble radius to ≲1 cm, as confirmed by Comet Nucleus Infrared and Visible Analyser onboard Philae. Here, we show that the flux of fractal particles measured by Rosetta constrains the 67P nucleus in a random packing of cm-sized pebbles, with all the voids among them filled by fractal particles. This structure is inconsistent with any catastrophic collision, which would have compacted or dispersed most fractals, thus leaving empty most voids in the reassembled nucleus. Comets are less numerous than current estimates, as confirmed by lacking small craters on Pluto and Charon. Bilobate comets accreted at speeds <1 m s-1 from cometesimals born in the same disc stream.

Perceptual and Physiological Responses to Jackson Pollock's Fractals

PubMed Central

Taylor, Richard P.; Spehar, Branka; Van Donkelaar, Paul; Hagerhall, Caroline M.

2011-01-01

Fractals have been very successful in quantifying the visual complexity exhibited by many natural patterns, and have captured the imagination of scientists and artists alike. Our research has shown that the poured patterns of the American abstract painter Jackson Pollock are also fractal. This discovery raises an intriguing possibility – are the visual characteristics of fractals responsible for the long-term appeal of Pollock's work? To address this question, we have conducted 10 years of scientific investigation of human response to fractals and here we present, for the first time, a review of this research that examines the inter-relationship between the various results. The investigations include eye tracking, visual preference, skin conductance, and EEG measurement techniques. We discuss the artistic implications of the positive perceptual and physiological responses to fractal patterns. PMID:21734876

A Fractal Dimension Survey of Active Region Complexity

NASA Technical Reports Server (NTRS)

McAteer, R. T. James; Gallagher, Peter; Ireland, Jack

2005-01-01

A new approach to quantifying the magnetic complexity of active regions using a fractal dimension measure is presented. This fully-automated approach uses full disc MDI magnetograms of active regions from a large data set (2742 days of the SoHO mission; 9342 active regions) to compare the calculated fractal dimension to both Mount Wilson classification and flare rate. The main Mount Wilson classes exhibit no distinct fractal dimension distribution, suggesting a self-similar nature of all active regions. Solar flare productivity exhibits an increase in both the frequency and GOES X-ray magnitude of flares from regions with higher fractal dimensions. Specifically a lower threshold fractal dimension of 1.2 and 1.25 exists as a necessary, but not sufficient, requirement for an active region to produce M- and X-class flares respectively .

Effect of Dust Coagulation Dynamics on the Geometry of Aggregates

NASA Technical Reports Server (NTRS)

Nakamura, R.

1996-01-01

Master equation gives a more fundamental description of stochastic coagulation processes rather than popular Smoluchowski's equation. In order to examine the effect of the dynamics on the geometry of resulting aggregates, we study Master equation with a rigorous Monte Carlo algorithm. It is found that Cluster-Cluster aggregation model is a good approximation of orderly growth and the aggregates have fluffy structures with a fractal dimension approx. 2. A scaling analysis of Smoluchowski's equation also supports this conclusion.

Geometry-Related Children's Literature Improves the Geometry Achievement and Attitudes of Second-Grade Students

ERIC Educational Resources Information Center

McAndrew, Erica M.; Morris, Wendy L.; Fennell, Francis

2017-01-01

Use of mathematics-related literature can engage students' interest and increase their understanding of mathematical concepts. A quasi-experimental study of two second-grade classrooms assessed whether daily inclusion of geometry-related literature in the classroom improved attitudes toward geometry and achievement in geometry. Consistent with the…

Wavelet-based 3D reconstruction of microcalcification clusters from two mammographic views: new evidence that fractal tumors are malignant and Euclidean tumors are benign.

PubMed

Batchelder, Kendra A; Tanenbaum, Aaron B; Albert, Seth; Guimond, Lyne; Kestener, Pierre; Arneodo, Alain; Khalil, Andre

2014-01-01

The 2D Wavelet-Transform Modulus Maxima (WTMM) method was used to detect microcalcifications (MC) in human breast tissue seen in mammograms and to characterize the fractal geometry of benign and malignant MC clusters. This was done in the context of a preliminary analysis of a small dataset, via a novel way to partition the wavelet-transform space-scale skeleton. For the first time, the estimated 3D fractal structure of a breast lesion was inferred by pairing the information from two separate 2D projected mammographic views of the same breast, i.e. the cranial-caudal (CC) and mediolateral-oblique (MLO) views. As a novelty, we define the "CC-MLO fractal dimension plot", where a "fractal zone" and "Euclidean zones" (non-fractal) are defined. 118 images (59 cases, 25 malignant and 34 benign) obtained from a digital databank of mammograms with known radiologist diagnostics were analyzed to determine which cases would be plotted in the fractal zone and which cases would fall in the Euclidean zones. 92% of malignant breast lesions studied (23 out of 25 cases) were in the fractal zone while 88% of the benign lesions were in the Euclidean zones (30 out of 34 cases). Furthermore, a Bayesian statistical analysis shows that, with 95% credibility, the probability that fractal breast lesions are malignant is between 74% and 98%. Alternatively, with 95% credibility, the probability that Euclidean breast lesions are benign is between 76% and 96%. These results support the notion that the fractal structure of malignant tumors is more likely to be associated with an invasive behavior into the surrounding tissue compared to the less invasive, Euclidean structure of benign tumors. Finally, based on indirect 3D reconstructions from the 2D views, we conjecture that all breast tumors considered in this study, benign and malignant, fractal or Euclidean, restrict their growth to 2-dimensional manifolds within the breast tissue.

Transport properties of electrons in fractal magnetic-barrier structures

NASA Astrophysics Data System (ADS)

Sun, Lifeng; Fang, Chao; Guo, Yong

2010-09-01

Quantum transport properties in fractal magnetically modulated structures are studied by the transfer-matrix method. It is found that the transmission spectra depend sensitively not only on the incident energy and the direction of the wave vector but also on the stage of the fractal structures. Resonance splitting, enhancement, and position shift of the resonance peaks under different magnetic modulation are observed at four different fractal stages, and the relationship between the conductance in the fractal structure and magnetic modulation is also revealed. The results indicate the spectra of the transmission can be considered as fingerprints for the fractal structures, which show the subtle correspondence between magnetic structures and transport behaviors.

Characterization of branch complexity by fractal analyses

USGS Publications Warehouse

Alados, C.L.; Escos, J.; Emlen, J.M.; Freeman, D.C.

1999-01-01

The comparison between complexity in the sense of space occupancy (box-counting fractal dimension D(c) and information dimension D1) and heterogeneity in the sense of space distribution (average evenness index f and evenness variation coefficient J(cv)) were investigated in mathematical fractal objects and natural branch structures. In general, increased fractal dimension was paired with low heterogeneity. Comparisons between branch architecture in Anthyllis cytisoides under different slope exposure and grazing impact revealed that branches were more complex and more homogeneously distributed for plants on northern exposures than southern, while grazing had no impact during a wet year. Developmental instability was also investigated by the statistical noise of the allometric relation between internode length and node order. In conclusion, our study demonstrated that fractal dimension of branch structure can be used to analyze the structural organization of plants, especially if we consider not only fractal dimension but also shoot distribution within the canopy (lacunarity). These indexes together with developmental instability analyses are good indicators of growth responses to the environment.

Undergraduate Experiment with Fractal Diffraction Gratings

ERIC Educational Resources Information Center

Monsoriu, Juan A.; Furlan, Walter D.; Pons, Amparo; Barreiro, Juan C.; Gimenez, Marcos H.

2011-01-01

We present a simple diffraction experiment with fractal gratings based on the triadic Cantor set. Diffraction by fractals is proposed as a motivating strategy for students of optics in the potential applications of optical processing. Fraunhofer diffraction patterns are obtained using standard equipment present in most undergraduate physics…

The analysis of the influence of fractal structure of stimuli on fractal dynamics in fixational eye movements and EEG signal

NASA Astrophysics Data System (ADS)

Namazi, Hamidreza; Kulish, Vladimir V.; Akrami, Amin

2016-05-01

One of the major challenges in vision research is to analyze the effect of visual stimuli on human vision. However, no relationship has been yet discovered between the structure of the visual stimulus, and the structure of fixational eye movements. This study reveals the plasticity of human fixational eye movements in relation to the ‘complex’ visual stimulus. We demonstrated that the fractal temporal structure of visual dynamics shifts towards the fractal dynamics of the visual stimulus (image). The results showed that images with higher complexity (higher fractality) cause fixational eye movements with lower fractality. Considering the brain, as the main part of nervous system that is engaged in eye movements, we analyzed the governed Electroencephalogram (EEG) signal during fixation. We have found out that there is a coupling between fractality of image, EEG and fixational eye movements. The capability observed in this research can be further investigated and applied for treatment of different vision disorders.

Quantitative assessment of early diabetic retinopathy using fractal analysis.

PubMed

Cheung, Ning; Donaghue, Kim C; Liew, Gerald; Rogers, Sophie L; Wang, Jie Jin; Lim, Shueh-Wen; Jenkins, Alicia J; Hsu, Wynne; Li Lee, Mong; Wong, Tien Y

2009-01-01

Fractal analysis can quantify the geometric complexity of the retinal vascular branching pattern and may therefore offer a new method to quantify early diabetic microvascular damage. In this study, we examined the relationship between retinal fractal dimension and retinopathy in young individuals with type 1 diabetes. We conducted a cross-sectional study of 729 patients with type 1 diabetes (aged 12-20 years) who had seven-field stereoscopic retinal photographs taken of both eyes. From these photographs, retinopathy was graded according to the modified Airlie House classification, and fractal dimension was quantified using a computer-based program following a standardized protocol. In this study, 137 patients (18.8%) had diabetic retinopathy signs; of these, 105 had mild retinopathy. Median (interquartile range) retinal fractal dimension was 1.46214 (1.45023-1.47217). After adjustment for age, sex, diabetes duration, A1C, blood pressure, and total cholesterol, increasing retinal vascular fractal dimension was significantly associated with increasing odds of retinopathy (odds ratio 3.92 [95% CI 2.02-7.61] for fourth versus first quartile of fractal dimension). In multivariate analysis, each 0.01 increase in retinal vascular fractal dimension was associated with a nearly 40% increased odds of retinopathy (1.37 [1.21-1.56]). This association remained after additional adjustment for retinal vascular caliber. Greater retinal fractal dimension, representing increased geometric complexity of the retinal vasculature, is independently associated with early diabetic retinopathy signs in type 1 diabetes. Fractal analysis of fundus photographs may allow quantitative measurement of early diabetic microvascular damage.

Relationship between mathematical abstraction in learning parallel coordinates concept and performance in learning analytic geometry of pre-service mathematics teachers: an investigation

NASA Astrophysics Data System (ADS)

Nurhasanah, F.; Kusumah, Y. S.; Sabandar, J.; Suryadi, D.

2018-05-01

As one of the non-conventional mathematics concepts, Parallel Coordinates is potential to be learned by pre-service mathematics teachers in order to give them experiences in constructing richer schemes and doing abstraction process. Unfortunately, the study related to this issue is still limited. This study wants to answer a research question “to what extent the abstraction process of pre-service mathematics teachers in learning concept of Parallel Coordinates could indicate their performance in learning Analytic Geometry”. This is a case study that part of a larger study in examining mathematical abstraction of pre-service mathematics teachers in learning non-conventional mathematics concept. Descriptive statistics method is used in this study to analyze the scores from three different tests: Cartesian Coordinate, Parallel Coordinates, and Analytic Geometry. The participants in this study consist of 45 pre-service mathematics teachers. The result shows that there is a linear association between the score on Cartesian Coordinate and Parallel Coordinates. There also found that the higher levels of the abstraction process in learning Parallel Coordinates are linearly associated with higher student achievement in Analytic Geometry. The result of this study shows that the concept of Parallel Coordinates has a significant role for pre-service mathematics teachers in learning Analytic Geometry.

Pore-wall roughness as a fractal surface and theoretical simulation of mercury intrusion/retraction in porous media

DOE Office of Scientific and Technical Information (OSTI.GOV)

Tsakiroglou, C.D.; Payatakes, A.C.

The mercury intrusion/retraction curves of many types of porous materials (e.g., sandstones) have sections of finite slope in the region of high and very high pressure. This feature is attributed to the existence of microroughness on the pore walls. In the present work pore-wall roughness features are added to a three-dimensional primary network of chambers-and-throats using ideas of fractal geometry. The roughness of the throats is modeled with a finite number of self-similar triangular prisms of progressively smaller sizes. The roughness of the chambers is modeled in a similar way using right circular cones instead of prisms. Three parameters sufficemore » for the complete characterization of the model of fractal roughness, namely, the number of features per unit length, the common angle of sharpness, and the number of layers (which is taken to be the same for throats and chambers). Analytical relations that give the surface area, pore volume, and mercury saturation of the pore network as functions of the fractal roughness parameters are developed for monolayer and multilayer arrangements. The chamber-and-throat network with fractal pore-wall roughness is used to develop an extended version of the computer-aided simulator of mercury porosimetry that has been reported in previous publications. This new simulator is used to investigate the effects of the roughness features on the form of mercury intrusion/retraction curves. It turns out that the fractal model of the porewall roughness gives an adequate representation of real porous media, and capillary pressure curves which are similar to the experimental ones for many typical porous materials such as sandstones. The method is demonstrated with the analysis of a Greek sandstone.« less

Using Peano Curves to Construct Laplacians on Fractals

NASA Astrophysics Data System (ADS)

Molitor, Denali; Ott, Nadia; Strichartz, Robert

2015-12-01

We describe a new method to construct Laplacians on fractals using a Peano curve from the circle onto the fractal, extending an idea that has been used in the case of certain Julia sets. The Peano curve allows us to visualize eigenfunctions of the Laplacian by graphing the pullback to the circle. We study in detail three fractals: the pentagasket, the octagasket and the magic carpet. We also use the method for two nonfractal self-similar sets, the torus and the equilateral triangle, obtaining appealing new visualizations of eigenfunctions on the triangle. In contrast to the many familiar pictures of approximations to standard Peano curves, that do no show self-intersections, our descriptions of approximations to the Peano curves have self-intersections that play a vital role in constructing graph approximations to the fractal with explicit graph Laplacians that give the fractal Laplacian in the limit.

Three-Dimensional Surface Parameters and Multi-Fractal Spectrum of Corroded Steel

PubMed Central

Shanhua, Xu; Songbo, Ren; Youde, Wang

2015-01-01

To study multi-fractal behavior of corroded steel surface, a range of fractal surfaces of corroded surfaces of Q235 steel were constructed by using the Weierstrass-Mandelbrot method under a high total accuracy. The multi-fractal spectrum of fractal surface of corroded steel was calculated to study the multi-fractal characteristics of the W-M corroded surface. Based on the shape feature of the multi-fractal spectrum of corroded steel surface, the least squares method was applied to the quadratic fitting of the multi-fractal spectrum of corroded surface. The fitting function was quantitatively analyzed to simplify the calculation of multi-fractal characteristics of corroded surface. The results showed that the multi-fractal spectrum of corroded surface was fitted well with the method using quadratic curve fitting, and the evolution rules and trends were forecasted accurately. The findings can be applied to research on the mechanisms of corroded surface formation of steel and provide a new approach for the establishment of corrosion damage constitutive models of steel. PMID:26121468

Three-Dimensional Surface Parameters and Multi-Fractal Spectrum of Corroded Steel.

PubMed

Shanhua, Xu; Songbo, Ren; Youde, Wang

2015-01-01

To study multi-fractal behavior of corroded steel surface, a range of fractal surfaces of corroded surfaces of Q235 steel were constructed by using the Weierstrass-Mandelbrot method under a high total accuracy. The multi-fractal spectrum of fractal surface of corroded steel was calculated to study the multi-fractal characteristics of the W-M corroded surface. Based on the shape feature of the multi-fractal spectrum of corroded steel surface, the least squares method was applied to the quadratic fitting of the multi-fractal spectrum of corroded surface. The fitting function was quantitatively analyzed to simplify the calculation of multi-fractal characteristics of corroded surface. The results showed that the multi-fractal spectrum of corroded surface was fitted well with the method using quadratic curve fitting, and the evolution rules and trends were forecasted accurately. The findings can be applied to research on the mechanisms of corroded surface formation of steel and provide a new approach for the establishment of corrosion damage constitutive models of steel.

Exploring fractal behaviour of blood oxygen saturation in preterm babies

NASA Astrophysics Data System (ADS)

Zahari, Marina; Hui, Tan Xin; Zainuri, Nuryazmin Ahmat; Darlow, Brian A.

2017-04-01

Recent evidence has been emerging that oxygenation instability in preterm babies could lead to an increased risk of retinal injury such as retinopathy of prematurity. There is a potential that disease severity could be better understood using nonlinear methods for time series data such as fractal theories [1]. Theories on fractal behaviours have been employed by researchers in various disciplines who were motivated to look into the behaviour or structure of irregular fluctuations in temporal data. In this study, an investigation was carried out to examine whether fractal behaviour could be detected in blood oxygen time series. Detection for the presence of fractals in oxygen data of preterm infants was performed using the methods of power spectrum, empirical probability distribution function and autocorrelation function. The results from these fractal identification methods indicate the possibility that these data exhibit fractal nature. Subsequently, a fractal framework for future research was suggested for oxygen time series.

Hierarchical socioeconomic fractality: The rich, the poor, and the middle-class

NASA Astrophysics Data System (ADS)

Eliazar, Iddo; Cohen, Morrel H.

2014-05-01

Since the seminal work of the Italian economist Vilfredo Pareto, the study of wealth and income has been a topic of active scientific exploration engaging researches ranging from economics and political science to econophysics and complex systems. This paper investigates the intrinsic fractality of wealth and income. To that end we introduce and characterize three forms of socioeconomic scale-invariance-poor fractality, rich fractality, and middle-class fractality-and construct hierarchical fractal approximations of general wealth and income distributions, based on the stitching of these three forms of fractality. Intertwining the theoretical results with real-world empirical data we then establish that the three forms of socioeconomic fractality-amalgamated into a composite hierarchical structure-underlie the distributions of wealth and income in human societies. We further establish that the hierarchical socioeconomic fractality of wealth and income is also displayed by empirical rank distributions observed across the sciences.

A Tutorial Review on Fractal Spacetime and Fractional Calculus

NASA Astrophysics Data System (ADS)

He, Ji-Huan

2014-11-01

This tutorial review of fractal-Cantorian spacetime and fractional calculus begins with Leibniz's notation for derivative without limits which can be generalized to discontinuous media like fractal derivative and q-derivative of quantum calculus. Fractal spacetime is used to elucidate some basic properties of fractal which is the foundation of fractional calculus, and El Naschie's mass-energy equation for the dark energy. The variational iteration method is used to introduce the definition of fractional derivatives. Fractal derivative is explained geometrically and q-derivative is motivated by quantum mechanics. Some effective analytical approaches to fractional differential equations, e.g., the variational iteration method, the homotopy perturbation method, the exp-function method, the fractional complex transform, and Yang-Laplace transform, are outlined and the main solution processes are given.

Fractal analysis of scatter imaging signatures to distinguish breast pathologies

NASA Astrophysics Data System (ADS)

Eguizabal, Alma; Laughney, Ashley M.; Krishnaswamy, Venkataramanan; Wells, Wendy A.; Paulsen, Keith D.; Pogue, Brian W.; López-Higuera, José M.; Conde, Olga M.

2013-02-01

Fractal analysis combined with a label-free scattering technique is proposed for describing the pathological architecture of tumors. Clinicians and pathologists are conventionally trained to classify abnormal features such as structural irregularities or high indices of mitosis. The potential of fractal analysis lies in the fact of being a morphometric measure of the irregular structures providing a measure of the object's complexity and self-similarity. As cancer is characterized by disorder and irregularity in tissues, this measure could be related to tumor growth. Fractal analysis has been probed in the understanding of the tumor vasculature network. This work addresses the feasibility of applying fractal analysis to the scattering power map (as a physical modeling) and principal components (as a statistical modeling) provided by a localized reflectance spectroscopic system. Disorder, irregularity and cell size variation in tissue samples is translated into the scattering power and principal components magnitude and its fractal dimension is correlated with the pathologist assessment of the samples. The fractal dimension is computed applying the box-counting technique. Results show that fractal analysis of ex-vivo fresh tissue samples exhibits separated ranges of fractal dimension that could help classifier combining the fractal results with other morphological features. This contrast trend would help in the discrimination of tissues in the intraoperative context and may serve as a useful adjunct to surgeons.

Multi-Scale Fractal Analysis of Image Texture and Pattern

NASA Technical Reports Server (NTRS)

Emerson, Charles W.

1998-01-01

Fractals embody important ideas of self-similarity, in which the spatial behavior or appearance of a system is largely independent of scale. Self-similarity is defined as a property of curves or surfaces where each part is indistinguishable from the whole, or where the form of the curve or surface is invariant with respect to scale. An ideal fractal (or monofractal) curve or surface has a constant dimension over all scales, although it may not be an integer value. This is in contrast to Euclidean or topological dimensions, where discrete one, two, and three dimensions describe curves, planes, and volumes. Theoretically, if the digital numbers of a remotely sensed image resemble an ideal fractal surface, then due to the self-similarity property, the fractal dimension of the image will not vary with scale and resolution. However, most geographical phenomena are not strictly self-similar at all scales, but they can often be modeled by a stochastic fractal in which the scaling and self-similarity properties of the fractal have inexact patterns that can be described by statistics. Stochastic fractal sets relax the monofractal self-similarity assumption and measure many scales and resolutions in order to represent the varying form of a phenomenon as a function of local variables across space. In image interpretation, pattern is defined as the overall spatial form of related features, and the repetition of certain forms is a characteristic pattern found in many cultural objects and some natural features. Texture is the visual impression of coarseness or smoothness caused by the variability or uniformity of image tone or color. A potential use of fractals concerns the analysis of image texture. In these situations it is commonly observed that the degree of roughness or inexactness in an image or surface is a function of scale and not of experimental technique. The fractal dimension of remote sensing data could yield quantitative insight on the spatial complexity and

A Complex Story: Universal Preference vs. Individual Differences Shaping Aesthetic Response to Fractals Patterns

PubMed Central

Street, Nichola; Forsythe, Alexandra M.; Reilly, Ronan; Taylor, Richard; Helmy, Mai S.

2016-01-01

Fractal patterns offer one way to represent the rough complexity of the natural world. Whilst they dominate many of our visual experiences in nature, little large-scale perceptual research has been done to explore how we respond aesthetically to these patterns. Previous research (Taylor et al., 2011) suggests that the fractal patterns with mid-range fractal dimensions (FDs) have universal aesthetic appeal. Perceptual and aesthetic responses to visual complexity have been more varied with findings suggesting both linear (Forsythe et al., 2011) and curvilinear (Berlyne, 1970) relationships. Individual differences have been found to account for many of the differences we see in aesthetic responses but some, such as culture, have received little attention within the fractal and complexity research fields. This two-study article aims to test preference responses to FD and visual complexity, using a large cohort (N = 443) of participants from around the world to allow universality claims to be tested. It explores the extent to which age, culture and gender can predict our preferences for fractally complex patterns. Following exploratory analysis that found strong correlations between FD and visual complexity, a series of linear mixed-effect models were implemented to explore if each of the individual variables could predict preference. The first tested a linear complexity model (likelihood of selecting the more complex image from the pair of images) and the second a mid-range FD model (likelihood of selecting an image within mid-range). Results show that individual differences can reliably predict preferences for complexity across culture, gender and age. However, in fitting with current findings the mid-range models show greater consistency in preference not mediated by gender, age or culture. This article supports the established theory that the mid-range fractal patterns appear to be a universal construct underlying preference but also highlights the fragility of

A Complex Story: Universal Preference vs. Individual Differences Shaping Aesthetic Response to Fractals Patterns.

PubMed

Street, Nichola; Forsythe, Alexandra M; Reilly, Ronan; Taylor, Richard; Helmy, Mai S

2016-01-01

Fractal patterns offer one way to represent the rough complexity of the natural world. Whilst they dominate many of our visual experiences in nature, little large-scale perceptual research has been done to explore how we respond aesthetically to these patterns. Previous research (Taylor et al., 2011) suggests that the fractal patterns with mid-range fractal dimensions (FDs) have universal aesthetic appeal. Perceptual and aesthetic responses to visual complexity have been more varied with findings suggesting both linear (Forsythe et al., 2011) and curvilinear (Berlyne, 1970) relationships. Individual differences have been found to account for many of the differences we see in aesthetic responses but some, such as culture, have received little attention within the fractal and complexity research fields. This two-study article aims to test preference responses to FD and visual complexity, using a large cohort (N = 443) of participants from around the world to allow universality claims to be tested. It explores the extent to which age, culture and gender can predict our preferences for fractally complex patterns. Following exploratory analysis that found strong correlations between FD and visual complexity, a series of linear mixed-effect models were implemented to explore if each of the individual variables could predict preference. The first tested a linear complexity model (likelihood of selecting the more complex image from the pair of images) and the second a mid-range FD model (likelihood of selecting an image within mid-range). Results show that individual differences can reliably predict preferences for complexity across culture, gender and age. However, in fitting with current findings the mid-range models show greater consistency in preference not mediated by gender, age or culture. This article supports the established theory that the mid-range fractal patterns appear to be a universal construct underlying preference but also highlights the fragility of

Optimization and experimental validation of electrostatic adhesive geometry

NASA Astrophysics Data System (ADS)

Ruffatto, D.; Shah, J.; Spenko, M.

This paper introduces a method to optimize the electrode geometry of electrostatic adhesives for robotic gripping, attachment, and manipulation applications. Electrostatic adhesion is achieved by applying a high voltage potential, on the order of kV, to a set of electrodes, which generates an electric field. The electric field polarizes the substrate material and creates an adhesion force. Previous attempts at creating electro-static adhesives have shown them to be effective, but researchers have made no effort to optimize the electrode configuration and geometry. We have shown that by optimizing the geometry of the electrode configuration, the electric field strength, and therefore the adhesion force, is enhanced. To accomplish this, Comsol Multiphysics was utilized to evaluate the average electric field generated by a given electrode geometry. Several electrode patterns were evaluated, including parallel conductors, concentric circles, Hilbert curves (a fractal geometry) and spirals. The arrangement of the electrodes in concentric circles with varying electrode widths proved to be the most effective. The most effective sizing was to use the smallest gap spacing allowable coupled with a variable electrode width. These results were experimentally validated on several different surfaces including drywall, wood, tile, glass, and steel. A new manufacturing process allowing for the fabrication of thin, conformal electro-static adhesive pads was utilized. By combining the optimized electrode geometry with the new fabrication process we are able to demonstrate a marked improvement of up to 500% in shear pressure when compared to previously published values.

Anomalous relaxation in fractal structures

DOE Office of Scientific and Technical Information (OSTI.GOV)

Fujiwara, S.; Yonezawa, F.

1995-03-01

For the purpose of studying some interesting properties of anomalous relaxation in fractal structures, we carry out Monte Carlo simulations of random walks on two-dimensional fractal structures (Sierpinski carpets with different cutouts and site-percolation clusters in a square lattice at the critical concentration). We find that the relaxation is of the Cole-Cole type [J. Chem. Phys. 9, 341 (1941)], which is one of the empirical laws of anomalous relaxation. Scaling properties are found in the relaxation function as well as in the particle density. We also find that, in strucures with almost the same fractal dimension, relaxation in structures withmore » dead ends is slower than that in structures without them. This paper ascertains that the essential aspects of the anomalous relaxation due to many-body effects can be explained in the framework of the one-body model.« less

On uses, misuses and potential abuses of fractal analysis in zooplankton behavioral studies: A review, a critique and a few recommendations

NASA Astrophysics Data System (ADS)

Seuront, Laurent

2015-08-01

Fractal analysis is increasingly used to describe, and provide further understanding to, zooplankton swimming behavior. This may be related to the fact that fractal analysis and the related fractal dimension D have the desirable properties to be independent of measurement scale and to be very sensitive to even subtle behavioral changes that may be undetectable to other behavioral variables. As early claimed by Coughlin et al. (1992), this creates "the need for fractal analysis" in behavioral studies, which has hence the potential to become a valuable tool in zooplankton behavioral ecology. However, this paper stresses that fractal analysis, as well as the more elaborated multifractal analysis, is also a risky business that may lead to irrelevant results, without paying extreme attention to a series of both conceptual and practical steps that are all likely to bias the results of any analysis. These biases are reviewed and exemplified on the basis of the published literature, and remedial procedures are provided not only for geometric and stochastic fractal analyses, but also for the more complicated multifractal analysis. The concept of multifractals is finally introduced as a direct, objective and quantitative tool to identify models of motion behavior, such as Brownian motion, fractional Brownian motion, ballistic motion, Lévy flight/walk and multifractal random walk. I finally briefly review the state of this emerging field in zooplankton behavioral research.

Pulse regime in formation of fractal fibers

DOE Office of Scientific and Technical Information (OSTI.GOV)

Smirnov, B. M., E-mail: bmsmirnov@gmail.com

The pulse regime of vaporization of a bulk metal located in a buffer gas is analyzed as a method of generation of metal atoms under the action of a plasma torch or a laser beam. Subsequently these atoms are transformed into solid nanoclusters, fractal aggregates and then into fractal fibers if the growth process proceeds in an external electric field. We are guided by metals in which transitions between s and d-electrons of their atoms are possible, since these metals are used as catalysts and filters in interaction with gas flows. The resistance of metal fractal structures to a gasmore » flow is evaluated that allows one to find optimal parameters of a fractal structure for gas flow propagation through it. The thermal regime of interaction between a plasma pulse or a laser beam and a metal surface is analyzed. It is shown that the basic energy from an external source is consumed on a bulk metal heating, and the efficiency of atom evaporation from the metal surface, that is the ratio of energy fluxes for vaporization and heating, is 10{sup –3}–10{sup –4} for transient metals under consideration. A typical energy flux (~10{sup 6} W/cm{sup 2}), a typical surface temperature (~3000 K), and a typical pulse duration (~1 μs) provide a sufficient amount of evaporated atoms to generate fractal fibers such that each molecule of a gas flow collides with the skeleton of fractal fibers many times.« less

Fractal tomography and its application in 3D vision

NASA Astrophysics Data System (ADS)

Trubochkina, N.

2018-01-01

A three-dimensional artistic fractal tomography method that implements a non-glasses 3D visualization of fractal worlds in layered media is proposed. It is designed for the glasses-free 3D vision of digital art objects and films containing fractal content. Prospects for the development of this method in art galleries and the film industry are considered.

Wetting characteristics of 3-dimensional nanostructured fractal surfaces

NASA Astrophysics Data System (ADS)

Davis, Ethan; Liu, Ying; Jiang, Lijia; Lu, Yongfeng; Ndao, Sidy

2017-01-01

This article reports the fabrication and wetting characteristics of 3-dimensional nanostructured fractal surfaces (3DNFS). Three distinct 3DNFS surfaces, namely cubic, Romanesco broccoli, and sphereflake were fabricated using two-photon direct laser writing. Contact angle measurements were performed on the multiscale fractal surfaces to characterize their wetting properties. Average contact angles ranged from 66.8° for the smooth control surface to 0° for one of the fractal surfaces. The change in wetting behavior was attributed to modification of the interfacial surface properties due to the inclusion of 3-dimensional hierarchical fractal nanostructures. However, this behavior does not exactly obey existing surface wetting models in the literature. Potential applications for these types of surfaces in physical and biological sciences are also discussed.

Teaching about Fractals.

ERIC Educational Resources Information Center

Willson, Stephen J.

1991-01-01

Described is a course designed to teach students about fractals using various teaching methods including the computer. Discussed are why the course drew students, prerequisites, clientele, textbook, grading, computer usage, and the syllabus. (KR)

Two-Dimensional Animal-Like Fractals in Thin Films

NASA Astrophysics Data System (ADS)

Gao, Hong-jun; Xue, Zeng-quan; Wu, Quan-de; Pang, Shi-jin

1996-02-01

We present a few unique animal-like fractal patterns in ionized-cluster-beam deposited fullerene-tetracyanoquinodimethane thin films. The fractal patterns consisting of animal-like aggregates such as "fishes" and "quasi-seahorses" have been characterized by transmission electron microscopy. The results indicate that the small aggregates of the animal-like body are composed of many single crystals whose crystalline directions are generally different. The formation of the fractal patterns can be attributed to the cluster-diffusion-limited aggregation.

Fractal-Inspired Subwavelength Geometric Inclusions for Improvement of High-Frequency Electromagnetic Devices

NASA Astrophysics Data System (ADS)

Smith, Kathryn Leigh

This dissertation presents research results demonstrating the efficacy of fractal-inspired subwavelength geometric inclusions for improvement of high-frequency electromagnetic devices. It begins with a review of the open literature in the area of fractal applications in antennas and metamaterials. This is followed by a detailed discussion of three high-frequency electromagnetic devices that demonstrate performance improvement through incorporation of subwavelength geometric design elements. The first of these devices is a spherical spiral metamaterial unit cell that was developed as a three-dimensional fractal expansion of the traditional split ring resonator, and is shown to be capable of producing broadband negative permeability, negative permittivity, or both, depending solely on the orientation of the unit cells with respect to the incident electric field. The second device is a ringed rectangular patch antenna that has four resonant frequencies. All four of these operative frequencies are shown to produce similar radiation patterns, which also closely match the pattern of a traditional patch antenna. Several minor geometric modifications of the basic shape of the device are also presented, and are shown to enable modification of the number of resonances, as well as tuning of frequencies of resonance. The third and final topic is a modified horn antenna that incorporates a spiral metamaterial as a phase-shifting device in order to achieve circularly polarized radiation. The handedness of the radiated wave is shown to be tunable through simple reorientation of the loading unit cells. In each of these cases, electrically-small geometric modification of existing device geometries is shown to greatly affect performance, either by increasing bandwidth, by inducing multiband behavior, or by enabling exotic radiation characteristics.

Elasticity of fractal materials using the continuum model with non-integer dimensional space

NASA Astrophysics Data System (ADS)

Tarasov, Vasily E.

2015-01-01

Using a generalization of vector calculus for space with non-integer dimension, we consider elastic properties of fractal materials. Fractal materials are described by continuum models with non-integer dimensional space. A generalization of elasticity equations for non-integer dimensional space, and its solutions for the equilibrium case of fractal materials are suggested. Elasticity problems for fractal hollow ball and cylindrical fractal elastic pipe with inside and outside pressures, for rotating cylindrical fractal pipe, for gradient elasticity and thermoelasticity of fractal materials are solved.

Multisource inverse-geometry CT. Part I. System concept and development

DOE Office of Scientific and Technical Information (OSTI.GOV)

De Man, Bruno, E-mail: deman@ge.com; Harrison, Dan

Purpose: This paper presents an overview of multisource inverse-geometry computed tomography (IGCT) as well as the development of a gantry-based research prototype system. The development of the distributed x-ray source is covered in a companion paper [V. B. Neculaes et al., “Multisource inverse-geometry CT. Part II. X-ray source design and prototype,” Med. Phys. 43, 4617–4627 (2016)]. While progress updates of this development have been presented at conferences and in journal papers, this paper is the first comprehensive overview of the multisource inverse-geometry CT concept and prototype. The authors also provide a review of all previous IGCT related publications. Methods: Themore » authors designed and implemented a gantry-based 32-source IGCT scanner with 22 cm field-of-view, 16 cm z-coverage, 1 s rotation time, 1.09 × 1.024 mm detector cell size, as low as 0.4 × 0.8 mm focal spot size and 80–140 kVp x-ray source voltage. The system is built using commercially available CT components and a custom made distributed x-ray source. The authors developed dedicated controls, calibrations, and reconstruction algorithms and evaluated the system performance using phantoms and small animals. Results: The authors performed IGCT system experiments and demonstrated tube current up to 125 mA with up to 32 focal spots. The authors measured a spatial resolution of 13 lp/cm at 5% cutoff. The scatter-to-primary ratio is estimated 62% for a 32 cm water phantom at 140 kVp. The authors scanned several phantoms and small animals. The initial images have relatively high noise due to the low x-ray flux levels but minimal artifacts. Conclusions: IGCT has unique benefits in terms of dose-efficiency and cone-beam artifacts, but comes with challenges in terms of scattered radiation and x-ray flux limits. To the authors’ knowledge, their prototype is the first gantry-based IGCT scanner. The authors summarized the design and implementation of the scanner and the authors

Multisource inverse-geometry CT. Part I. System concept and development

PubMed Central

De Man, Bruno; Uribe, Jorge; Baek, Jongduk; Harrison, Dan; Yin, Zhye; Longtin, Randy; Roy, Jaydeep; Waters, Bill; Wilson, Colin; Short, Jonathan; Inzinna, Lou; Reynolds, Joseph; Neculaes, V. Bogdan; Frutschy, Kristopher; Senzig, Bob; Pelc, Norbert

2016-01-01

Purpose: This paper presents an overview of multisource inverse-geometry computed tomography (IGCT) as well as the development of a gantry-based research prototype system. The development of the distributed x-ray source is covered in a companion paper [V. B. Neculaes et al., “Multisource inverse-geometry CT. Part II. X-ray source design and prototype,” Med. Phys. 43, 4617–4627 (2016)]. While progress updates of this development have been presented at conferences and in journal papers, this paper is the first comprehensive overview of the multisource inverse-geometry CT concept and prototype. The authors also provide a review of all previous IGCT related publications. Methods: The authors designed and implemented a gantry-based 32-source IGCT scanner with 22 cm field-of-view, 16 cm z-coverage, 1 s rotation time, 1.09 × 1.024 mm detector cell size, as low as 0.4 × 0.8 mm focal spot size and 80–140 kVp x-ray source voltage. The system is built using commercially available CT components and a custom made distributed x-ray source. The authors developed dedicated controls, calibrations, and reconstruction algorithms and evaluated the system performance using phantoms and small animals. Results: The authors performed IGCT system experiments and demonstrated tube current up to 125 mA with up to 32 focal spots. The authors measured a spatial resolution of 13 lp/cm at 5% cutoff. The scatter-to-primary ratio is estimated 62% for a 32 cm water phantom at 140 kVp. The authors scanned several phantoms and small animals. The initial images have relatively high noise due to the low x-ray flux levels but minimal artifacts. Conclusions: IGCT has unique benefits in terms of dose-efficiency and cone-beam artifacts, but comes with challenges in terms of scattered radiation and x-ray flux limits. To the authors’ knowledge, their prototype is the first gantry-based IGCT scanner. The authors summarized the design and implementation of the scanner and the authors presented

Universal Inverse Power-Law Distribution for Fractal Fluctuations in Dynamical Systems: Applications for Predictability of Inter-Annual Variability of Indian and USA Region Rainfall

NASA Astrophysics Data System (ADS)

Selvam, A. M.

2017-01-01

Dynamical systems in nature exhibit self-similar fractal space-time fluctuations on all scales indicating long-range correlations and, therefore, the statistical normal distribution with implicit assumption of independence, fixed mean and standard deviation cannot be used for description and quantification of fractal data sets. The author has developed a general systems theory based on classical statistical physics for fractal fluctuations which predicts the following. (1) The fractal fluctuations signify an underlying eddy continuum, the larger eddies being the integrated mean of enclosed smaller-scale fluctuations. (2) The probability distribution of eddy amplitudes and the variance (square of eddy amplitude) spectrum of fractal fluctuations follow the universal Boltzmann inverse power law expressed as a function of the golden mean. (3) Fractal fluctuations are signatures of quantum-like chaos since the additive amplitudes of eddies when squared represent probability densities analogous to the sub-atomic dynamics of quantum systems such as the photon or electron. (4) The model predicted distribution is very close to statistical normal distribution for moderate events within two standard deviations from the mean but exhibits a fat long tail that are associated with hazardous extreme events. Continuous periodogram power spectral analyses of available GHCN annual total rainfall time series for the period 1900-2008 for Indian and USA stations show that the power spectra and the corresponding probability distributions follow model predicted universal inverse power law form signifying an eddy continuum structure underlying the observed inter-annual variability of rainfall. On a global scale, man-made greenhouse gas related atmospheric warming would result in intensification of natural climate variability, seen immediately in high frequency fluctuations such as QBO and ENSO and even shorter timescales. Model concepts and results of analyses are discussed with reference

The Beauty of Geometry

ERIC Educational Resources Information Center

Morris, Barbara H.

2004-01-01

This article describes a geometry project that used the beauty of stained-glass-window designs to teach middle school students about geometric figures and concepts. Three honors prealgebra teachers and a middle school mathematics gifted intervention specialist created a geometry project that covered the curriculum and also assessed students'…

Surface and mass fractals in vapor-phase aggregates

NASA Astrophysics Data System (ADS)

Hurd, Alan J.; Schaefer, Dale W.; Martin, James E.

1987-03-01

Several types of fumed-silica aggregates with differing surface areas were studied over a wide range of spatial resolution by employing both light and neutron scattering. At intermediate length scales, between 100 and 1000 Å, the aggregates are mass fractals with Dm~=1.7-2.0, in basic agreement with simulations of aggregating clusters. At short length scales below 100 Å where the nature of the surfaces of the primary particles dominates the scattering, some of the samples appear to be fractally rough. In particular, a higher surface area seems to be correlated not with smaller primary particles in the aggregates, as previously assumed, but with fractally rough surfaces having Ds as high as 2.5. These may be the first materials discovered to have both mass and surface fractal structure.

Fractal analysis of bone structure with applications to osteoporosis and microgravity effects

NASA Astrophysics Data System (ADS)

Acharya, Raj S.; LeBlanc, Adrian; Shackelford, Linda; Swarnakar, Vivek; Krishnamurthy, Ram; Hausman, E.; Lin, Chin-Shoou

1995-05-01

We characterize the trabecular structure with the aid of fractal dimension. We use alternating sequential filters (ASF) to generate a nonlinear pyramid for fractal dimension computations. We do not make any assumptions of the statistical distributions of the underlying fractal bone structure. The only assumption of our scheme is the rudimentary definition of self-similarity. This allows us the freedom of not being constrained by statistical estimation schemes. With mathematical simulations, we have shown that the ASF methods outperform other existing methods for fractal dimension estimation. We have shown that the fractal dimension remains the same when computed with both the x-ray images and the MRI images of the patella. We have shown that the fractal dimension of osteoporotic subjects is lower than that of the normal subjects. In animal models, we have shown that the fractal dimension of osteoporotic rats was lower than that of the normal rats. In a 17 week bedrest study, we have shown that the subject's prebedrest fractal dimension is higher than that of the postbedrest fractal dimension.

Fractal Dimension Analysis of Subcortical Gray Matter Structures in Schizophrenia

PubMed Central

Sehatpour, Pejman; Long, Jun; Gui, Weihua; Qiao, Jianping; Javitt, Daniel C.; Wang, Zhishun

2016-01-01

A failure of adaptive inference—misinterpreting available sensory information for appropriate perception and action—is at the heart of clinical manifestations of schizophrenia, implicating key subcortical structures in the brain including the hippocampus. We used high-resolution, three-dimensional (3D) fractal geometry analysis to study subtle and potentially biologically relevant structural alterations (in the geometry of protrusions, gyri and indentations, sulci) in subcortical gray matter (GM) in patients with schizophrenia relative to healthy individuals. In particular, we focus on utilizing Fractal Dimension (FD), a compact shape descriptor that can be computed using inputs with irregular (i.e., not necessarily smooth) surfaces in order to quantify complexity (of geometrical properties and configurations of structures across spatial scales) of subcortical GM in this disorder. Probabilistic (entropy-based) information FD was computed based on the box-counting approach for each of the seven subcortical structures, bilaterally, as well as the brainstem from high-resolution magnetic resonance (MR) images in chronic patients with schizophrenia (n = 19) and age-matched healthy controls (n = 19) (age ranges: patients, 22.7–54.3 and healthy controls, 24.9–51.6 years old). We found a significant reduction of FD in the left hippocampus (median: 2.1460, range: 2.07–2.18 vs. median: 2.1730, range: 2.15–2.23, p<0.001; Cohen’s effect size, U3 = 0.8158 (95% Confidence Intervals, CIs: 0.6316, 1.0)), the right hippocampus (median: 2.1430, range: 2.05–2.19 vs. median: 2.1760, range: 2.12–2.21, p = 0.004; U3 = 0.8421 (CIs: 0.5263, 1)), as well as left thalamus (median: 2.4230, range: 2.40–2.44, p = 0.005; U3 = 0.7895 (CIs: 0.5789, 0.9473)) in schizophrenia patients, relative to healthy individuals. Our findings provide in-vivo quantitative evidence for reduced surface complexity of hippocampus, with reduced FD indicating a less complex, less regular GM

Fractals and the irreducibility of consciousness in plants and animals.

PubMed

Gardiner, John

2013-08-01

In both plants and animals consciousness is fractal. Since fractals can only pass information in one direction it is impossible to extrapolate backward to find the rule that governs the fractal. Thus, similarly, it will be impossible to completely determine the rule or rules that govern consciousness.

3D printing of wearable fractal-based sensor systems for neurocardiology and healthcare

NASA Astrophysics Data System (ADS)

Ramasamy, Mouli; Varadan, Vijay K.

2017-04-01

Neurocardiology is the pathophysiological interplay of nervous and cardiovascular systems. The communication between the heart and brain has revealed various methodologies in healthcare that could be investigated to study the heart-brain interactions and other cardiovascular and neurological diseases. A textile based wearable nanosensor system in the form of e-bra, e-shirt, e-headband, e-brief, underwear etc, was presented in this SPIE conferences earlier for noninvasive recording of EEG and EKG, and showing the correlation between the brain and heart signals. In this paper, the technology is expanded further using fractal based geometries using 3D printing system for low cost and flexible wearable sensor system for healthcare.

Fractality of eroded coastlines of correlated landscapes.

PubMed

Morais, P A; Oliveira, E A; Araújo, N A M; Herrmann, H J; Andrade, J S

2011-07-01

Using numerical simulations of a simple sea-coast mechanical erosion model, we investigate the effect of spatial long-range correlations in the lithology of coastal landscapes on the fractal behavior of the corresponding coastlines. In the model, the resistance of a coast section to erosion depends on the local lithology configuration as well as on the number of neighboring sea sides. For weak sea forces, the sea is trapped by the coastline and the eroding process stops after some time. For strong sea forces erosion is perpetual. The transition between these two regimes takes place at a critical sea force, characterized by a fractal coastline front. For uncorrelated landscapes, we obtain, at the critical value, a fractal dimension D=1.33, which is consistent with the dimension of the accessible external perimeter of the spanning cluster in two-dimensional percolation. For sea forces above the critical value, our results indicate that the coastline is self-affine and belongs to the Kardar-Parisi-Zhang universality class. In the case of landscapes generated with power-law spatial long-range correlations, the coastline fractal dimension changes continuously with the Hurst exponent H, decreasing from D=1.34 to 1.04, for H=0 and 1, respectively. This nonuniversal behavior is compatible with the multitude of fractal dimensions found for real coastlines.

Fractals and the irreducibility of consciousness in plants and animals

PubMed Central

Gardiner, John

2013-01-01

In both plants and animals consciousness is fractal. Since fractals can only pass information in one direction it is impossible to extrapolate backward to find the rule that governs the fractal. Thus, similarly, it will be impossible to completely determine the rule or rules that govern consciousness. PMID:23759545

Persistent fluctuations in stride intervals under fractal auditory stimulation.

PubMed

Marmelat, Vivien; Torre, Kjerstin; Beek, Peter J; Daffertshofer, Andreas

2014-01-01

Stride sequences of healthy gait are characterized by persistent long-range correlations, which become anti-persistent in the presence of an isochronous metronome. The latter phenomenon is of particular interest because auditory cueing is generally considered to reduce stride variability and may hence be beneficial for stabilizing gait. Complex systems tend to match their correlation structure when synchronizing. In gait training, can one capitalize on this tendency by using a fractal metronome rather than an isochronous one? We examined whether auditory cues with fractal variations in inter-beat intervals yield similar fractal inter-stride interval variability as isochronous auditory cueing in two complementary experiments. In Experiment 1, participants walked on a treadmill while being paced by either an isochronous or a fractal metronome with different variation strengths between beats in order to test whether participants managed to synchronize with a fractal metronome and to determine the necessary amount of variability for participants to switch from anti-persistent to persistent inter-stride intervals. Participants did synchronize with the metronome despite its fractal randomness. The corresponding coefficient of variation of inter-beat intervals was fixed in Experiment 2, in which participants walked on a treadmill while being paced by non-isochronous metronomes with different scaling exponents. As expected, inter-stride intervals showed persistent correlations similar to self-paced walking only when cueing contained persistent correlations. Our results open up a new window to optimize rhythmic auditory cueing for gait stabilization by integrating fractal fluctuations in the inter-beat intervals.

Methods of nanoassembly of a fractal polymer and materials formed thereby

DOE Office of Scientific and Technical Information (OSTI.GOV)

Newkome, George R; Moorefield, Charles N

2012-07-24

The invention relates to the formation of synthesized fractal constructs and the methods of chemical self-assembly for the preparation of a non-dendritic, nano-scale, fractal constructs or molecules. More particularly, the invention relates to fractal constructs formed by molecular self-assembly, to create synthetic, nanometer-scale fractal shapes. In an embodiment, a nanoscale Sierpinski hexagonal gasket is formed. This non-dendritic, perfectly self-similar fractal macromolecule is comprised of bisterpyridine building blocks that are bound together by coordination to 36 Ru and 6 Fe ions to form a nearly planar array of increasingly larger hexagons around a hollow center.

Methods of nanoassembly of a fractal polymer and materials formed thereby

DOE Office of Scientific and Technical Information (OSTI.GOV)

Newkome, George R; Moorefield, Charles N

2014-09-23

The invention relates to the formation of synthesized fractal constructs and the methods of chemical self-assembly for the preparation of a non-dendritic, nano-scale, fractal constructs or molecules. More particularly, the invention relates to fractal constructs formed by molecular self-assembly, to create synthetic, nanometer-scale fractal shapes. In an embodiment, a nanoscale Sierpinski hexagonal gasket is formed. This non-dendritic, perfectly self-similar fractal macromolecule is comprised of bisterpyridine building blocks that are bound together by coordination to (36) Ru and (6) Fe ions to form a nearly planar array of increasingly larger hexagons around a hollow center.

Multi-Scale Fractal Analysis of Image Texture and Pattern

NASA Technical Reports Server (NTRS)

Emerson, Charles W.; Lam, Nina Siu-Ngan; Quattrochi, Dale A.

1999-01-01

Analyses of the fractal dimension of Normalized Difference Vegetation Index (NDVI) images of homogeneous land covers near Huntsville, Alabama revealed that the fractal dimension of an image of an agricultural land cover indicates greater complexity as pixel size increases, a forested land cover gradually grows smoother, and an urban image remains roughly self-similar over the range of pixel sizes analyzed (10 to 80 meters). A similar analysis of Landsat Thematic Mapper images of the East Humboldt Range in Nevada taken four months apart show a more complex relation between pixel size and fractal dimension. The major visible difference between the spring and late summer NDVI images is the absence of high elevation snow cover in the summer image. This change significantly alters the relation between fractal dimension and pixel size. The slope of the fractal dimension-resolution relation provides indications of how image classification or feature identification will be affected by changes in sensor spatial resolution.

Multi-Scale Fractal Analysis of Image Texture and Pattern

NASA Technical Reports Server (NTRS)

Emerson, Charles W.; Lam, Nina Siu-Ngan; Quattrochi, Dale A.

1999-01-01

Analyses of the fractal dimension of Normalized Difference Vegetation Index (NDVI) images of homogeneous land covers near Huntsville, Alabama revealed that the fractal dimension of an image of an agricultural land cover indicates greater complexity as pixel size increases, a forested land cover gradually grows smoother, and an urban image remains roughly self-similar over the range of pixel sizes analyzed (10 to 80 meters). A similar analysis of Landsat Thematic Mapper images of the East Humboldt Range in Nevada taken four months apart show a more complex relation between pixel size and fractal dimension. The major visible difference between the spring and late summer NDVI images of the absence of high elevation snow cover in the summer image. This change significantly alters the relation between fractal dimension and pixel size. The slope of the fractal dimensional-resolution relation provides indications of how image classification or feature identification will be affected by changes in sensor spatial resolution.

Fractal rigidity in migraine

NASA Astrophysics Data System (ADS)

Latka, Miroslaw; Glaubic-Latka, Marta; Latka, Dariusz; West, Bruce J.

2004-04-01

We study the middle cerebral artery blood flow velocity (MCAfv) in humans using transcranial Doppler ultrasonography (TCD). Scaling properties of time series of the axial flow velocity averaged over a cardiac beat interval may be characterized by two exponents. The short time scaling exponent (STSE) determines the statistical properties of fluctuations of blood flow velocities in short-time intervals while the Hurst exponent describes the long-term fractal properties. In many migraineurs the value of the STSE is significantly reduced and may approach that of the Hurst exponent. This change in dynamical properties reflects the significant loss of short-term adaptability and the overall hyperexcitability of the underlying cerebral blood flow control system. We call this effect fractal rigidity.

Fractal analysis of bone structure with applications to osteoporosis and microgravity effects

DOE Office of Scientific and Technical Information (OSTI.GOV)

Acharya, R.S.; Swarnarkar, V.; Krishnamurthy, R.

1995-12-31

The authors characterize the trabecular structure with the aid of fractal dimension. The authors use Alternating Sequential filters to generate a nonlinear pyramid for fractal dimension computations. The authors do not make any assumptions of the statistical distributions of the underlying fractal bone structure. The only assumption of the scheme is the rudimentary definition of self similarity. This allows them the freedom of not being constrained by statistical estimation schemes. With mathematical simulations, the authors have shown that the ASF methods outperform other existing methods for fractal dimension estimation. They have shown that the fractal dimension remains the same whenmore » computed with both the X-Ray images and the MRI images of the patella. They have shown that the fractal dimension of osteoporotic subjects is lower than that of the normal subjects. In animal models, the authors have shown that the fractal dimension of osteoporotic rats was lower than that of the normal rats. In a 17 week bedrest study, they have shown that the subject`s prebedrest fractal dimension is higher than that of the postbedrest fractal dimension.« less

Is volcanic phenomena of fractal nature?

NASA Astrophysics Data System (ADS)

Quevedo, R.; Lopez, D. A. L.; Alparone, S.; Hernandez Perez, P. A.; Sagiya, T.; Barrancos, J.; Rodriguez-Santana, A. A.; Ramos, A.; Calvari, S.; Perez, N. M.

2016-12-01

A particular resonance waveform pattern has been detected beneath different physical volcano manifestations from recent 2011-2012 period of volcanic unrest at El Hierro Island, Canary Islands, and also from other worldwide volcanoes with different volcanic typology. This mentioned pattern appears to be a fractal time dependent waveform repeated in different time scales (periods of time). This time dependent feature suggests this resonance as a new approach to volcano phenomena for predicting such interesting matters as earthquakes, gas emission, deformation etc. as this fractal signal has been discovered hidden in a wide typical volcanic parameters measurements. It is known that the resonance phenomenon occurring in nature usually denote a structure, symmetry or a subjacent law (Fermi et al., 1952; and later -about enhanced cross-sections symmetry in protons collisions), which, in this particular case, may be indicative of some physical interactions showing a sequence not completely chaotic but cyclic provided with symmetries. The resonance and fractal model mentioned allowed the authors to make predictions in cycles from a few weeks to months. In this work an equation for this waveform has been described and also correlations with volcanic parameters and fractal behavior demonstration have been performed, including also some suggestive possible explanations of this signal origin.

Want to Play Geometry?

ERIC Educational Resources Information Center

Kaufmann, Matthew L.; Bomer, Megan A.; Powell, Nancy Norem

2009-01-01

Students enter the geometry classroom with a strong concept of fairness and a sense of what it means to "play by the rules," yet many students have difficulty understanding the postulates, or rules, of geometry and their implications. Although they may never have articulated the properties of an axiomatic system, they have gained a practical…

Fractal Music: The Mathematics Behind "Techno" Music

ERIC Educational Resources Information Center

Padula, Janice

2005-01-01

This article describes sound waves, their basis in the sine curve, Fourier's theorem of infinite series, the fractal equation and its application to the composition of music, together with algorithms (such as those employed by meteorologist Edward Lorenz in his discovery of chaos theory) that are now being used to compose fractal music on…

A Fractal Permeability Model for Shale Oil Reservoir

NASA Astrophysics Data System (ADS)

Zhang, Tao; Dong, Mingzhe; Li, Yajun

2018-01-01

In this work, a fractal analytical model is proposed to predict the permeability of shale reservoir. The proposed model explicitly relates the permeability to the micro-structural parameters (tortuosity, pore area fractal dimensions, porosity and slip velocity coefficient) of shale.

Dynamic fractals in spatial evolutionary games

NASA Astrophysics Data System (ADS)

Kolotev, Sergei; Malyutin, Aleksandr; Burovski, Evgeni; Krashakov, Sergei; Shchur, Lev

2018-06-01

We investigate critical properties of a spatial evolutionary game based on the Prisoner's Dilemma. Simulations demonstrate a jump in the component densities accompanied by drastic changes in average sizes of the component clusters. We argue that the cluster boundary is a random fractal. Our simulations are consistent with the fractal dimension of the boundary being equal to 2, and the cluster boundaries are hence asymptotically space filling as the system size increases.

Fractal continuum model for tracer transport in a porous medium.

PubMed

Herrera-Hernández, E C; Coronado, M; Hernández-Coronado, H

2013-12-01

A model based on the fractal continuum approach is proposed to describe tracer transport in fractal porous media. The original approach has been extended to treat tracer transport and to include systems with radial and uniform flow, which are cases of interest in geoscience. The models involve advection due to the fluid motion in the fractal continuum and dispersion whose mathematical expression is taken from percolation theory. The resulting advective-dispersive equations are numerically solved for continuous and for pulse tracer injection. The tracer profile and the tracer breakthrough curve are evaluated and analyzed in terms of the fractal parameters. It has been found in this work that anomalous transport frequently appears, and a condition on the fractal parameter values to predict when sub- or superdiffusion might be expected has been obtained. The fingerprints of fractality on the tracer breakthrough curve in the explored parameter window consist of an early tracer breakthrough and long tail curves for the spherical and uniform flow cases, and symmetric short tailed curves for the radial flow case.

Persistent Fluctuations in Stride Intervals under Fractal Auditory Stimulation

PubMed Central

Marmelat, Vivien; Torre, Kjerstin; Beek, Peter J.; Daffertshofer, Andreas

2014-01-01

Stride sequences of healthy gait are characterized by persistent long-range correlations, which become anti-persistent in the presence of an isochronous metronome. The latter phenomenon is of particular interest because auditory cueing is generally considered to reduce stride variability and may hence be beneficial for stabilizing gait. Complex systems tend to match their correlation structure when synchronizing. In gait training, can one capitalize on this tendency by using a fractal metronome rather than an isochronous one? We examined whether auditory cues with fractal variations in inter-beat intervals yield similar fractal inter-stride interval variability as isochronous auditory cueing in two complementary experiments. In Experiment 1, participants walked on a treadmill while being paced by either an isochronous or a fractal metronome with different variation strengths between beats in order to test whether participants managed to synchronize with a fractal metronome and to determine the necessary amount of variability for participants to switch from anti-persistent to persistent inter-stride intervals. Participants did synchronize with the metronome despite its fractal randomness. The corresponding coefficient of variation of inter-beat intervals was fixed in Experiment 2, in which participants walked on a treadmill while being paced by non-isochronous metronomes with different scaling exponents. As expected, inter-stride intervals showed persistent correlations similar to self-paced walking only when cueing contained persistent correlations. Our results open up a new window to optimize rhythmic auditory cueing for gait stabilization by integrating fractal fluctuations in the inter-beat intervals. PMID:24651455

Fractal and multifractal analyses of bipartite networks

NASA Astrophysics Data System (ADS)

Liu, Jin-Long; Wang, Jian; Yu, Zu-Guo; Xie, Xian-Hua

2017-03-01

Bipartite networks have attracted considerable interest in various fields. Fractality and multifractality of unipartite (classical) networks have been studied in recent years, but there is no work to study these properties of bipartite networks. In this paper, we try to unfold the self-similarity structure of bipartite networks by performing the fractal and multifractal analyses for a variety of real-world bipartite network data sets and models. First, we find the fractality in some bipartite networks, including the CiteULike, Netflix, MovieLens (ml-20m), Delicious data sets and (u, v)-flower model. Meanwhile, we observe the shifted power-law or exponential behavior in other several networks. We then focus on the multifractal properties of bipartite networks. Our results indicate that the multifractality exists in those bipartite networks possessing fractality. To capture the inherent attribute of bipartite network with two types different nodes, we give the different weights for the nodes of different classes, and show the existence of multifractality in these node-weighted bipartite networks. In addition, for the data sets with ratings, we modify the two existing algorithms for fractal and multifractal analyses of edge-weighted unipartite networks to study the self-similarity of the corresponding edge-weighted bipartite networks. The results show that our modified algorithms are feasible and can effectively uncover the self-similarity structure of these edge-weighted bipartite networks and their corresponding node-weighted versions.

Fractal and multifractal analyses of bipartite networks.

PubMed

Liu, Jin-Long; Wang, Jian; Yu, Zu-Guo; Xie, Xian-Hua

2017-03-31

Bipartite networks have attracted considerable interest in various fields. Fractality and multifractality of unipartite (classical) networks have been studied in recent years, but there is no work to study these properties of bipartite networks. In this paper, we try to unfold the self-similarity structure of bipartite networks by performing the fractal and multifractal analyses for a variety of real-world bipartite network data sets and models. First, we find the fractality in some bipartite networks, including the CiteULike, Netflix, MovieLens (ml-20m), Delicious data sets and (u, v)-flower model. Meanwhile, we observe the shifted power-law or exponential behavior in other several networks. We then focus on the multifractal properties of bipartite networks. Our results indicate that the multifractality exists in those bipartite networks possessing fractality. To capture the inherent attribute of bipartite network with two types different nodes, we give the different weights for the nodes of different classes, and show the existence of multifractality in these node-weighted bipartite networks. In addition, for the data sets with ratings, we modify the two existing algorithms for fractal and multifractal analyses of edge-weighted unipartite networks to study the self-similarity of the corresponding edge-weighted bipartite networks. The results show that our modified algorithms are feasible and can effectively uncover the self-similarity structure of these edge-weighted bipartite networks and their corresponding node-weighted versions.

Fractal and multifractal analyses of bipartite networks

PubMed Central

Liu, Jin-Long; Wang, Jian; Yu, Zu-Guo; Xie, Xian-Hua

2017-01-01

Bipartite networks have attracted considerable interest in various fields. Fractality and multifractality of unipartite (classical) networks have been studied in recent years, but there is no work to study these properties of bipartite networks. In this paper, we try to unfold the self-similarity structure of bipartite networks by performing the fractal and multifractal analyses for a variety of real-world bipartite network data sets and models. First, we find the fractality in some bipartite networks, including the CiteULike, Netflix, MovieLens (ml-20m), Delicious data sets and (u, v)-flower model. Meanwhile, we observe the shifted power-law or exponential behavior in other several networks. We then focus on the multifractal properties of bipartite networks. Our results indicate that the multifractality exists in those bipartite networks possessing fractality. To capture the inherent attribute of bipartite network with two types different nodes, we give the different weights for the nodes of different classes, and show the existence of multifractality in these node-weighted bipartite networks. In addition, for the data sets with ratings, we modify the two existing algorithms for fractal and multifractal analyses of edge-weighted unipartite networks to study the self-similarity of the corresponding edge-weighted bipartite networks. The results show that our modified algorithms are feasible and can effectively uncover the self-similarity structure of these edge-weighted bipartite networks and their corresponding node-weighted versions. PMID:28361962

A contact angle hysteresis model based on the fractal structure of contact line.

PubMed

Wu, Shuai; Ma, Ming

2017-11-01

Contact angle is one of the most popular concept used in fields such as wetting, transport and microfludics. In practice, different contact angles such as equilibrium, receding and advancing contact angles are observed due to hysteresis. The connection among these contact angles is important in revealing the chemical and physical properties of surfaces related to wetting. Inspired by the fractal structure of contact line, we propose a single parameter model depicting the connection of the three angles. This parameter is decided by the fractal structure of the contact line. The results of this model agree with experimental observations. In certain cases, it can be reduced to other existing models. It also provides a new point of view in understanding the physical nature of the contact angle hysteresis. Interestingly, some counter-intuitive phenomena, such as the binary receding angles, are indicated in this model, which are waited to be validated by experiments. Copyright © 2017 Elsevier Inc. All rights reserved.

Hyper-Fractal Analysis: A visual tool for estimating the fractal dimension of 4D objects

NASA Astrophysics Data System (ADS)

Grossu, I. V.; Grossu, I.; Felea, D.; Besliu, C.; Jipa, Al.; Esanu, T.; Bordeianu, C. C.; Stan, E.

2013-04-01

This work presents a new version of a Visual Basic 6.0 application for estimating the fractal dimension of images and 3D objects (Grossu et al. (2010) [1]). The program was extended for working with four-dimensional objects stored in comma separated values files. This might be of interest in biomedicine, for analyzing the evolution in time of three-dimensional images. New version program summaryProgram title: Hyper-Fractal Analysis (Fractal Analysis v03) Catalogue identifier: AEEG_v3_0 Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AEEG_v3_0.html Program obtainable from: CPC Program Library, Queen’s University, Belfast, N. Ireland Licensing provisions: Standard CPC license, http://cpc.cs.qub.ac.uk/licence/licence.html No. of lines in distributed program, including test data, etc.: 745761 No. of bytes in distributed program, including test data, etc.: 12544491 Distribution format: tar.gz Programming language: MS Visual Basic 6.0 Computer: PC Operating system: MS Windows 98 or later RAM: 100M Classification: 14 Catalogue identifier of previous version: AEEG_v2_0 Journal reference of previous version: Comput. Phys. Comm. 181 (2010) 831-832 Does the new version supersede the previous version? Yes Nature of problem: Estimating the fractal dimension of 4D images. Solution method: Optimized implementation of the 4D box-counting algorithm. Reasons for new version: Inspired by existing applications of 3D fractals in biomedicine [3], we extended the optimized version of the box-counting algorithm [1, 2] to the four-dimensional case. This might be of interest in analyzing the evolution in time of 3D images. The box-counting algorithm was extended in order to support 4D objects, stored in comma separated values files. A new form was added for generating 2D, 3D, and 4D test data. The application was tested on 4D objects with known dimension, e.g. the Sierpinski hypertetrahedron gasket, Df=ln(5)/ln(2) (Fig. 1). The algorithm could be extended, with minimum effort, to

Passenger flow analysis of Beijing urban rail transit network using fractal approach

NASA Astrophysics Data System (ADS)

Li, Xiaohong; Chen, Peiwen; Chen, Feng; Wang, Zijia

2018-04-01

To quantify the spatiotemporal distribution of passenger flow and the characteristics of an urban rail transit network, we introduce four radius fractal dimensions and two branch fractal dimensions by combining a fractal approach with passenger flow assignment model. These fractal dimensions can numerically describe the complexity of passenger flow in the urban rail transit network and its change characteristics. Based on it, we establish a fractal quantification method to measure the fractal characteristics of passenger follow in the rail transit network. Finally, we validate the reasonability of our proposed method by using the actual data of Beijing subway network. It has been shown that our proposed method can effectively measure the scale-free range of the urban rail transit network, network development and the fractal characteristics of time-varying passenger flow, which further provides a reference for network planning and analysis of passenger flow.

Ulam method and fractal Weyl law for Perron-Frobenius operators

NASA Astrophysics Data System (ADS)

Ermann, L.; Shepelyansky, D. L.

2010-06-01

We use the Ulam method to study spectral properties of the Perron-Frobenius operators of dynamical maps in a chaotic regime. For maps with absorption we show numerically that the spectrum is characterized by the fractal Weyl law recently established for nonunitary operators describing poles of quantum chaotic scattering with the Weyl exponent ν = d-1, where d is the fractal dimension of corresponding strange set of trajectories nonescaping in future times. In contrast, for dissipative maps we numerically find the Weyl exponent ν = d/2 where d is the fractal dimension of strange attractor. The Weyl exponent can be also expressed via the relation ν = d0/2 where d0 is the fractal dimension of the invariant sets. We also discuss the properties of eigenvalues and eigenvectors of such operators characterized by the fractal Weyl law.

Resource Letter FR-1: Fractals

NASA Astrophysics Data System (ADS)

Hurd, Alan J.

1988-11-01

This Resource Letter provides a guide to the literature on fractals. Although ``fractal'' is a relatively new term in science, unifying many new ideas with established ones, its wide application and general popularity have made it one of the fastest growing fields in statistical physics. The letter E after an item indicates elementary level or material of general interest to persons becoming informed in the field; the letter I, for intermediate level, indicates material of somewhat more specialized nature; and the letter A indicates rather specialized or advanced material. An asterisk (*) indicates those articles to be included in an accompanying Reprint Book.

Fractal universe and quantum gravity.

PubMed

Calcagni, Gianluca

2010-06-25

We propose a field theory which lives in fractal spacetime and is argued to be Lorentz invariant, power-counting renormalizable, ultraviolet finite, and causal. The system flows from an ultraviolet fixed point, where spacetime has Hausdorff dimension 2, to an infrared limit coinciding with a standard four-dimensional field theory. Classically, the fractal world where fields live exchanges energy momentum with the bulk with integer topological dimension. However, the total energy momentum is conserved. We consider the dynamics and the propagator of a scalar field. Implications for quantum gravity, cosmology, and the cosmological constant are discussed.

Surface areas of fractally rough particles studied by scattering

NASA Astrophysics Data System (ADS)

Hurd, Alan J.; Schaefer, Dale W.; Smith, Douglas M.; Ross, Steven B.; Le Méhauté, Alain; Spooner, Steven

1989-05-01

The small-angle scattering from fractally rough surfaces has the potential to give information on the surface area at a given resolution. By use of quantitative neutron and x-ray scattering, a direct comparison of surface areas of fractally rough powders was made between scattering and adsorption techniques. This study supports a recently proposed correction to the theory for scattering from fractal surfaces. In addition, the scattering data provide an independent calibration of molecular adsorbate areas.

a New Method for Calculating Fractal Dimensions of Porous Media Based on Pore Size Distribution

NASA Astrophysics Data System (ADS)

Xia, Yuxuan; Cai, Jianchao; Wei, Wei; Hu, Xiangyun; Wang, Xin; Ge, Xinmin

Fractal theory has been widely used in petrophysical properties of porous rocks over several decades and determination of fractal dimensions is always the focus of researches and applications by means of fractal-based methods. In this work, a new method for calculating pore space fractal dimension and tortuosity fractal dimension of porous media is derived based on fractal capillary model assumption. The presented work establishes relationship between fractal dimensions and pore size distribution, which can be directly used to calculate the fractal dimensions. The published pore size distribution data for eight sandstone samples are used to calculate the fractal dimensions and simultaneously compared with prediction results from analytical expression. In addition, the proposed fractal dimension method is also tested through Micro-CT images of three sandstone cores, and are compared with fractal dimensions by box-counting algorithm. The test results also prove a self-similar fractal range in sandstone when excluding smaller pores.

Process for applying control variables having fractal structures

DOEpatents

Bullock, IV, Jonathan S.; Lawson, Roger L.

1996-01-01

A process and apparatus for the application of a control variable having a fractal structure to a body or process. The process of the present invention comprises the steps of generating a control variable having a fractal structure and applying the control variable to a body or process reacting in accordance with the control variable. The process is applicable to electroforming where first, second and successive pulsed-currents are applied to cause the deposition of material onto a substrate, such that the first pulsed-current, the second pulsed-current, and successive pulsed currents form a fractal pulsed-current waveform.

Process for applying control variables having fractal structures

DOEpatents

Bullock, J.S. IV; Lawson, R.L.

1996-01-23

A process and apparatus are disclosed for the application of a control variable having a fractal structure to a body or process. The process of the present invention comprises the steps of generating a control variable having a fractal structure and applying the control variable to a body or process reacting in accordance with the control variable. The process is applicable to electroforming where first, second and successive pulsed-currents are applied to cause the deposition of material onto a substrate, such that the first pulsed-current, the second pulsed-current, and successive pulsed currents form a fractal pulsed-current waveform. 3 figs.

Comprehensive Fractal Description of Porosity of Coal of Different Ranks

PubMed Central

Ren, Jiangang; Zhang, Guocheng; Song, Zhimin; Liu, Gaofeng; Li, Bing

2014-01-01

We selected, as the objects of our research, lignite from the Beizao Mine, gas coal from the Caiyuan Mine, coking coal from the Xiqu Mine, and anthracite from the Guhanshan Mine. We used the mercury intrusion method and the low-temperature liquid nitrogen adsorption method to analyze the structure and shape of the coal pores and calculated the fractal dimensions of different aperture segments in the coal. The experimental results show that the fractal dimension of the aperture segment of lignite, gas coal, and coking coal with an aperture of greater than or equal to 10 nm, as well as the fractal dimension of the aperture segment of anthracite with an aperture of greater than or equal to 100 nm, can be calculated using the mercury intrusion method; the fractal dimension of the coal pore, with an aperture range between 2.03 nm and 361.14 nm, can be calculated using the liquid nitrogen adsorption method, of which the fractal dimensions bounded by apertures of 10 nm and 100 nm are different. Based on these findings, we defined and calculated the comprehensive fractal dimensions of the coal pores and achieved the unity of fractal dimensions for full apertures of coal pores, thereby facilitating, overall characterization for the heterogeneity of the coal pore structure. PMID:24955407

H-fractal seismic metamaterial with broadband low-frequency bandgaps

NASA Astrophysics Data System (ADS)

Du, Qiujiao; Zeng, Yi; Xu, Yang; Yang, Hongwu; Zeng, Zuoxun

2018-03-01

The application of metamaterial in civil engineering to achieve isolation of a building by controlling the propagation of seismic waves is a substantial challenge because seismic waves, a superposition of longitudinal and shear waves, are more complex than electromagnetic and acoustic waves. In this paper, we design a broadband seismic metamaterial based on H-shaped fractal pillars and report numerical simulation of band structures for seismic surface waves propagating. Comparative study on the band structures of H-fractal seismic metamaterials with different levels shows that a new level of fractal structure creates new band gap, widens the total band gaps and shifts the same band gap towards lower frequencies. Moreover, the vibration modes for H-fractal seismic metamaterials are computed and analyzed to clarify the mechanism of widening band gaps. A numerical investigation of seismic surface waves propagation on a 2D array of fractal unit cells on the surface of semi-infinite substrate is proposed to show the efficiency of earthquake shielding in multiple complete band gaps.

Fractal Dimensions of Umbral and Penumbral Regions of Sunspots

NASA Astrophysics Data System (ADS)

Rajkumar, B.; Haque, S.; Hrudey, W.

2017-11-01

The images of sunspots in 16 active regions taken at the University College of the Cayman Islands (UCCI) Observatory on Grand Cayman during June-November 2015 were used to determine their fractal dimensions using the perimeter-area method for the umbral and the penumbral region. Scale-free fractal dimensions of 2.09 ±0.42 and 1.72 ±0.4 were found, respectively. This value was higher than the value determined by Chumak and Chumak ( Astron. Astrophys. Trans. 10, 329, 1996), who used a similar method, but only for the penumbral region of their sample set. The umbral and penumbral fractal dimensions for the specific sunspots are positively correlated with r = 0.58. Furthermore, a similar time-series analysis was performed on eight images of AR 12403, from 21 August 2015 to 28 August 2015 taken from the Debrecen Photoheliographic Data (DPD). The correlation is r = 0.623 between the umbral and penumbral fractal dimensions in the time series, indicating that the complexity in morphology indicated by the fractal dimension between the umbra and penumbra followed each other in time as well.

A fractal approach to dynamic inference and distribution analysis

PubMed Central

van Rooij, Marieke M. J. W.; Nash, Bertha A.; Rajaraman, Srinivasan; Holden, John G.

2013-01-01

Event-distributions inform scientists about the variability and dispersion of repeated measurements. This dispersion can be understood from a complex systems perspective, and quantified in terms of fractal geometry. The key premise is that a distribution's shape reveals information about the governing dynamics of the system that gave rise to the distribution. Two categories of characteristic dynamics are distinguished: additive systems governed by component-dominant dynamics and multiplicative or interdependent systems governed by interaction-dominant dynamics. A logic by which systems governed by interaction-dominant dynamics are expected to yield mixtures of lognormal and inverse power-law samples is discussed. These mixtures are described by a so-called cocktail model of response times derived from human cognitive performances. The overarching goals of this article are twofold: First, to offer readers an introduction to this theoretical perspective and second, to offer an overview of the related statistical methods. PMID:23372552

Anisotropic fractal media by vector calculus in non-integer dimensional space

DOE Office of Scientific and Technical Information (OSTI.GOV)

Tarasov, Vasily E., E-mail: tarasov@theory.sinp.msu.ru

2014-08-15

A review of different approaches to describe anisotropic fractal media is proposed. In this paper, differentiation and integration non-integer dimensional and multi-fractional spaces are considered as tools to describe anisotropic fractal materials and media. We suggest a generalization of vector calculus for non-integer dimensional space by using a product measure method. The product of fractional and non-integer dimensional spaces allows us to take into account the anisotropy of the fractal media in the framework of continuum models. The integration over non-integer-dimensional spaces is considered. In this paper differential operators of first and second orders for fractional space and non-integer dimensionalmore » space are suggested. The differential operators are defined as inverse operations to integration in spaces with non-integer dimensions. Non-integer dimensional space that is product of spaces with different dimensions allows us to give continuum models for anisotropic type of the media. The Poisson's equation for fractal medium, the Euler-Bernoulli fractal beam, and the Timoshenko beam equations for fractal material are considered as examples of application of suggested generalization of vector calculus for anisotropic fractal materials and media.« less

Anisotropic fractal media by vector calculus in non-integer dimensional space

NASA Astrophysics Data System (ADS)

Tarasov, Vasily E.

2014-08-01

A review of different approaches to describe anisotropic fractal media is proposed. In this paper, differentiation and integration non-integer dimensional and multi-fractional spaces are considered as tools to describe anisotropic fractal materials and media. We suggest a generalization of vector calculus for non-integer dimensional space by using a product measure method. The product of fractional and non-integer dimensional spaces allows us to take into account the anisotropy of the fractal media in the framework of continuum models. The integration over non-integer-dimensional spaces is considered. In this paper differential operators of first and second orders for fractional space and non-integer dimensional space are suggested. The differential operators are defined as inverse operations to integration in spaces with non-integer dimensions. Non-integer dimensional space that is product of spaces with different dimensions allows us to give continuum models for anisotropic type of the media. The Poisson's equation for fractal medium, the Euler-Bernoulli fractal beam, and the Timoshenko beam equations for fractal material are considered as examples of application of suggested generalization of vector calculus for anisotropic fractal materials and media.

Turbulent premixed flames on fractal-grid-generated turbulence

NASA Astrophysics Data System (ADS)

Soulopoulos, N.; Kerl, J.; Sponfeldner, T.; Beyrau, F.; Hardalupas, Y.; Taylor, A. M. K. P.; Vassilicos, J. C.

2013-12-01

A space-filling, low blockage fractal grid is used as a novel turbulence generator in a premixed turbulent flame stabilized by a rod. The study compares the flame behaviour with a fractal grid to the behaviour when a standard square mesh grid with the same effective mesh size and solidity as the fractal grid is used. The isothermal gas flow turbulence characteristics, including mean flow velocity and rms of velocity fluctuations and Taylor length, were evaluated from hot-wire measurements. The behaviour of the flames was assessed with direct chemiluminescence emission from the flame and high-speed OH-laser-induced fluorescence. The characteristics of the two flames are considered in terms of turbulent flame thickness, local flame curvature and turbulent flame speed. It is found that, for the same flow rate and stoichiometry and at the same distance downstream of the location of the grid, fractal-grid-generated turbulence leads to a more turbulent flame with enhanced burning rate and increased flame surface area.

Fractal cometary dust - a window into the early Solar system

NASA Astrophysics Data System (ADS)

Mannel, T.; Bentley, M. S.; Schmied, R.; Jeszenszky, H.; Levasseur-Regourd, A. C.; Romstedt, J.; Torkar, K.

2016-11-01

The properties of dust in the protoplanetary disc are key to understanding the formation of planets in our Solar system. Many models of dust growth predict the development of fractal structures which evolve into non-fractal, porous dust pebbles representing the main component for planetesimal accretion. In order to understand comets and their origins, the Rosetta orbiter followed comet 67P/Churyumov-Gerasimenko for over two years and carried a dedicated instrument suite for dust analysis. One of these instruments, the MIDAS (Micro-Imaging Dust Analysis System) atomic force microscope, recorded the 3D topography of micro- to nanometre-sized dust. All particles analysed to date have been found to be hierarchical agglomerates. Most show compact packing; however, one is extremely porous. This paper contains a structural description of a compact aggregate and the outstanding porous one. Both particles are tens of micrometres in size and show rather narrow subunit size distributions with noticeably similar mean values of 1.48^{+0.13}_{-0.59} μm for the porous particle and 1.36^{+0.15}_{-0.59} μm for the compact. The porous particle allows a fractal analysis, where a density-density correlation function yields a fractal dimension of Df = 1.70 ± 0.1. GIADA, another dust analysis instrument on board Rosetta, confirms the existence of a dust population with a similar fractal dimension. The fractal particles are interpreted as pristine agglomerates built in the protoplanetary disc and preserved in the comet. The similar subunits of both fractal and compact dust indicate a common origin which is, given the properties of the fractal, dominated by slow agglomeration of equally sized aggregates known as cluster-cluster agglomeration.

A Whirlwind Tour of Computational Geometry.

ERIC Educational Resources Information Center

Graham, Ron; Yao, Frances

1990-01-01

Described is computational geometry which used concepts and results from classical geometry, topology, combinatorics, as well as standard algorithmic techniques such as sorting and searching, graph manipulations, and linear programing. Also included are special techniques and paradigms. (KR)

Influence of Turbulent Flow and Fractal Scaling on Effective Permeability of Fracture Network

NASA Astrophysics Data System (ADS)

Zhu, J.

2017-12-01

A new approach is developed to calculate hydraulic gradient dependent effective permeability of a fractal fracture network where both laminar and turbulent flows may occur in individual fractures. A critical fracture length is used to distinguish flow characteristics in individual fractures. The developed new solutions can be used for the case of a general scaling relationship, an extension to the linear scaling. We examine the impact on the effective permeability of the network of fractal fracture network characteristics, which include the fractal scaling coefficient and exponent, fractal dimension, ratio of minimum over maximum fracture lengths. Results demonstrate that the developed solution can explain more variations of the effective permeability in relation to the fractal dimensions estimated from the field observations. At high hydraulic gradient the effective permeability decreases with the fractal scaling exponent, but increases with the fractal scaling exponent at low gradient. The effective permeability increases with the scaling coefficient, fractal dimension, fracture length ratio and maximum fracture length.

Fractal based curves in musical creativity: A critical annotation

NASA Astrophysics Data System (ADS)

Georgaki, Anastasia; Tsolakis, Christos

In this article we examine fractal curves and synthesis algorithms in musical composition and research. First we trace the evolution of different approaches for the use of fractals in music since the 80's by a literature review. Furthermore, we review representative fractal algorithms and platforms that implement them. Properties such as self-similarity (pink noise), correlation, memory (related to the notion of Brownian motion) or non correlation at multiple levels (white noise), can be used to develop hierarchy of criteria for analyzing different layers of musical structure. L-systems can be applied in the modelling of melody in different musical cultures as well as in the investigation of musical perception principles. Finally, we propose a critical investigation approach for the use of artificial or natural fractal curves in systematic musicology.

Variable geometry Darrieus wind machine

NASA Astrophysics Data System (ADS)

Pytlinski, J. T.; Serrano, D.

1983-08-01

A variable geometry Darrieus wind machine is proposed. The lower attachment of the blades to the rotor can move freely up and down the axle allowing the blades of change shape during rotation. Experimental data for a 17 m. diameter Darrieus rotor and a theoretical model for multiple streamtube performance prediction were used to develop a computer simulation program for studying parameters that affect the machine's performance. This new variable geometry concept is described and interrelated with multiple streamtube theory through aerodynamic parameters. The computer simulation study shows that governor behavior of a Darrieus turbine can not be attained by a standard turbine operating within normally occurring rotational velocity limits. A second generation variable geometry Darrieus wind turbine which uses a telescopic blade is proposed as a potential improvement on the studied concept.

Fractal 1/f Dynamics Suggest Entanglement of Measurement and Human Performance

ERIC Educational Resources Information Center

Holden, John G.; Choi, Inhyun; Amazeen, Polemnia G.; Van Orden, Guy

2011-01-01

Variability of repeated measurements in human performances exhibits fractal 1/f noise. Yet the relative strength of this fractal pattern varies widely across conditions, tasks, and individuals. Four experiments illustrate how subtle details of the conditions of measurement change the fractal patterns observed across task conditions. The results…

Study on Conversion Between Momentum and Contrarian Based on Fractal Game

NASA Astrophysics Data System (ADS)

Wu, Xu; Song, Guanghui; Deng, Yan; Xu, Lin

2015-06-01

Based on the fractal game which is performed by the majority and the minority, the fractal market theory (FMT) is employed to describe the features of investors' decision-making. Accordingly, the process of fractal games is formed in order to analyze the statistical features of conversion between momentum and contrarian. The result shows that among three fractal game mechanisms, the statistical feature of simulated return rate series is much more similar to log returns on actual series. In addition, the conversion between momentum and contrarian is also extremely similar to real situation, which can reflect the effectiveness of using fractal game in analyzing the conversion between momentum and contrarian. Moreover, it also provides decision-making reference which helps investors develop effective investment strategy.

Is the co-seismic slip distribution fractal?

NASA Astrophysics Data System (ADS)

Milliner, Christopher; Sammis, Charles; Allam, Amir; Dolan, James

2015-04-01

Co-seismic along-strike slip heterogeneity is widely observed for many surface-rupturing earthquakes as revealed by field and high-resolution geodetic methods. However, this co-seismic slip variability is currently a poorly understood phenomenon. Key unanswered questions include: What are the characteristics and underlying causes of along-strike slip variability? Do the properties of slip variability change from fault-to-fault, along-strike or at different scales? We cross-correlate optical, pre- and post-event air photos using the program COSI-Corr to measure the near-field, surface deformation pattern of the 1992 Mw 7.3 Landers and 1999 Mw 7.1 Hector Mine earthquakes in high-resolution. We produce the co-seismic slip profiles of both events from over 1,000 displacement measurements and observe consistent along-strike slip variability. Although the observed slip heterogeneity seems apparently complex and disordered, a spectral analysis reveals that the slip distributions are indeed self-affine fractal i.e., slip exhibits a consistent degree of irregularity at all observable length scales, with a 'short-memory' and is not random. We find a fractal dimension of 1.58 and 1.75 for the Landers and Hector Mine earthquakes, respectively, indicating that slip is more heterogeneous for the Hector Mine event. Fractal slip is consistent with both dynamic and quasi-static numerical simulations that use non-planar faults, which in turn causes heterogeneous along-strike stress, and we attribute the observed fractal slip to fault surfaces of fractal roughness. As fault surfaces are known to smooth over geologic time due to abrasional wear and fracturing, we also test whether the fractal properties of slip distributions alters between earthquakes from immature to mature fault systems. We will present results that test this hypothesis by using the optical image correlation technique to measure historic, co-seismic slip distributions of earthquakes from structurally mature, large

Curriculum Forms: On the Assumed Shapes of Knowing and Knowledge.

ERIC Educational Resources Information Center

Davis, Brent; Sumara, Dennis J.

2000-01-01

Draws on the new field of mathematical study called fractal geometry. Illustrates the pervasiveness and constraining tendencies of classical geometries. Suggests that fractal geometry is a mathematical analogue to fields such as post-modernism, post-structuralism, and ecological theory. Examines how fractal geometry can complement other emergent…

Fractal Analyses of High-Resolution Cloud Droplet Measurements.

NASA Astrophysics Data System (ADS)

Malinowski, Szymon P.; Leclerc, Monique Y.; Baumgardner, Darrel G.

1994-02-01

Fractal analyses of individual cloud droplet distributions using aircraft measurements along one-dimensional horizontal cross sections through clouds are performed. Box counting and cluster analyses are used to determine spatial scales of inhomogeneity of cloud droplet spacing. These analyses reveal that droplet spatial distributions do not exhibit a fractal behavior. A high variability in local droplet concentration in cloud volumes undergoing mixing was found. In these regions, thin filaments of cloudy air with droplet concentration close to those observed in cloud cores were found. Results suggest that these filaments may be anisotropic. Additional box counting analyses performed for various classes of cloud droplet diameters indicate that large and small droplets are similarly distributed, except for the larger characteristic spacing of large droplets.A cloud-clear air interface defined by a certain threshold of total droplet count (TDC) was investigated. There are indications that this interface is a convoluted surface of a fractal nature, at least in actively developing cumuliform clouds. In contrast, TDC in the cloud interior does not have fractal or multifractal properties. Finally a random Cantor set (RCS) was introduced as a model of a fractal process with an ill-defined internal scale. A uniform measure associated with the RCS after several generations was introduced to simulate the TDC records. Comparison of the model with real TDC records indicates similar properties of both types of data series.

On the fractal morphology of combustion-generated soot aggregates

DOE Office of Scientific and Technical Information (OSTI.GOV)

Koylu, U.O.

1995-12-31

The fractal properties of soot aggregates were investigated using ex-situ and in-situ experimental methods as well as computer simulations. Ex-situ experiments involved thermophoretic sampling and analysis by transmission electron microscopy (TEM), while in-situ measurements employed angular static light scattering and data inversion based on Rayleigh-Debye-Gans (RDG) approximation. Computer simulations used a sequential algorithm which mimics mass fractal-like structures. So from a variety of hydrocarbon-fueled laminar and turbulent nonpremixed flame environments were considered in the present study. The TEM analysis of projected soot images sampled from fuel-rich conditions of buoyant and weakly-buoyant laminar flames indicated that the fractal dimension of sootmore » was relatively independent of position in flames, fuel type and flame condition. These measurements yielded an average fractal dimension of 1.8, although other structure parameters such as the primary particle diameters and number of primary particles in aggregates had wide range of values. Fractal prefactor (lacunarity) was also measured for soot sampled from the fuel-lean conditions of turbulent flames, considering the actual morphology by tilting the samples during TEM analysis. These measurements yielded a fractal dimension of 1.65 and a lacunarity of 8.5, with experimental uncertainties (95% confidence) of 0.08 and 0.5, respectively. Relationships between the actual and projected structure properties of soot were also developed by combining TEM observations with numerical simulations. Practical approximate formulae were suggested to find radius of gyration of an aggregate from its maximum dimension, and number of primary particles in an aggregate from projected area. Finally, the fractal dimension and lacunarity of soot were obtained using light scattering for the same conditions of the above TEM measurements.« less

Fractal markets: Liquidity and investors on different time horizons

NASA Astrophysics Data System (ADS)

Li, Da-Ye; Nishimura, Yusaku; Men, Ming

2014-08-01

In this paper, we propose a new agent-based model to study the source of liquidity and the “emergent” phenomenon in financial market with fractal structure. The model rests on fractal market hypothesis and agents with different time horizons of investments. What is interesting is that though the agent-based model reveals that the interaction between these heterogeneous agents affects the stability and liquidity of the financial market the real world market lacks detailed data to bring it to light since it is difficult to identify and distinguish the investors with different time horizons in the empirical approach. results show that in a relatively short period of time fractal market provides liquidity from investors with different horizons and the market gains stability when the market structure changes from uniformity to diversification. In the real world the fractal structure with the finite of horizons can only stabilize the market within limits. With the finite maximum horizons, the greater diversity of the investors and the fractal structure will not necessarily bring more stability to the market which might come with greater fluctuation in large time scale.

A comparison of the fractal and JPEG algorithms

NASA Technical Reports Server (NTRS)

Cheung, K.-M.; Shahshahani, M.

1991-01-01

A proprietary fractal image compression algorithm and the Joint Photographic Experts Group (JPEG) industry standard algorithm for image compression are compared. In every case, the JPEG algorithm was superior to the fractal method at a given compression ratio according to a root mean square criterion and a peak signal to noise criterion.

a Fractal Permeability Model Coupling Boundary-Layer Effect for Tight Oil Reservoirs

NASA Astrophysics Data System (ADS)

Wang, Fuyong; Liu, Zhichao; Jiao, Liang; Wang, Congle; Guo, Hu

A fractal permeability model coupling non-flowing boundary-layer effect for tight oil reservoirs was proposed. Firstly, pore structures of tight formations were characterized with fractal theory. Then, with the empirical equation of boundary-layer thickness, Hagen-Poiseuille equation and fractal theory, a fractal torturous capillary tube model coupled with boundary-layer effect was developed, and verified with experimental data. Finally, the parameters influencing effective liquid permeability were quantitatively investigated. The research results show that effective liquid permeability of tight formations is not only decided by pore structures, but also affected by boundary-layer distributions, and effective liquid permeability is the function of fluid type, fluid viscosity, pressure gradient, fractal dimension, tortuosity fractal dimension, minimum pore radius and maximum pore radius. For the tight formations dominated with nanoscale pores, boundary-layer effect can significantly reduce effective liquid permeability, especially under low pressure gradient.

A Fractal Excursion.

ERIC Educational Resources Information Center

Camp, Dane R.

1991-01-01

After introducing the two-dimensional Koch curve, which is generated by simple recursions on an equilateral triangle, the process is extended to three dimensions with simple recursions on a regular tetrahedron. Included, for both fractal sequences, are iterative formulae, illustrations of the first several iterations, and a sample PASCAL program.…

Geometry of modified release formulations during dissolution--influence on performance of dosage forms with diclofenac sodium.

PubMed

Dorożyński, Przemysław; Kulinowski, Piotr; Jamróz, Witold; Juszczyk, Ewelina

2014-12-30

The objectives of the work included: presentation of magnetic resonance imaging (MRI) and fractal analysis based approach to comparison of dosage forms of different composition, structure, and assessment of the influence of the compositional factors i.e., matrix type, excipients etc., on properties and performance of the dosage form during drug dissolution. The work presents the first attempt to compare MRI data obtained for tablet formulations of different composition and characterized by distinct differences in hydration and drug dissolution mechanisms. The main difficulty, in such a case stems from differences in hydration behavior and tablet's geometry i.e., swelling, cracking, capping etc. A novel approach to characterization of matrix systems i.e., quantification of changes of geometrical complexity of the matrix shape during drug dissolution has been developed. Using three chosen commercial modified release tablet formulations with diclofenac sodium we present the method of parameterization of their geometrical complexity on the base of fractal analysis. The main result of the study is the correlation between the hydrating tablet behavior and drug dissolution - the increase of geometrical complexity expressed as fractal dimension relates to the increased variability of drug dissolution results. Copyright © 2014 Elsevier B.V. All rights reserved.

A fractal growth model: Exploring the connection pattern of hubs in complex networks

NASA Astrophysics Data System (ADS)

Li, Dongyan; Wang, Xingyuan; Huang, Penghe

2017-04-01

Fractal is ubiquitous in many real-world networks. Previous researches showed that the strong disassortativity between the hub-nodes on all length scales was the key principle that gave rise to the fractal architecture of networks. Although fractal property emerged in some models, there were few researches about the fractal growth model and quantitative analyses about the strength of the disassortativity for fractal model. In this paper, we proposed a novel inverse renormalization method, named Box-based Preferential Attachment (BPA), to build the fractal growth models in which the Preferential Attachment was performed at box level. The proposed models provided a new framework that demonstrated small-world-fractal transition. Also, we firstly demonstrated the statistical characteristic of connection patterns of the hubs in fractal networks. The experimental results showed that, given proper growing scale and added edges, the proposed models could clearly show pure small-world or pure fractal or both of them. It also showed that the hub connection ratio showed normal distribution in many real-world networks. At last, the comparisons of connection pattern between the proposed models and the biological and technical networks were performed. The results gave useful reference for exploring the growth principle and for modeling the connection patterns for real-world networks.

A fractal analysis of quaternary, Cenozoic-Mesozoic, and Late Pennsylvanian sea level changes

NASA Technical Reports Server (NTRS)

Hsui, Albert T.; Rust, Kelly A.; Klein, George D.

1993-01-01

Sea level changes are related to both climatic variations and tectonic movements. The fractal dimensions of several sea level curves were compared to a modern climatic fractal dimension of 1.26 established for annual precipitation records. A similar fractal dimension (1.22) based on delta(O-18/O-16) in deep-sea sediments has been suggested to characterize climatic change during the past 2 m.y. Our analysis indicates that sea level changes over the past 150,000 to 250,000 years also exhibit comparable fractal dimensions. Sea level changes for periods longer than about 30 m.y. are found to produce fractal dimensions closer to unity and Missourian (Late Pennsylvanian) sea level changes yield a fractal dimension of 1.41. The fact that these sea level curves all possess fractal dimensions less than 1.5 indicates that sea level changes exhibit nonperiodic, long-run persistence. The different fractal dimensions calculated for the various time periods could be the result of a characteristic overprinting of the sediment recored by prevailing processes during deposition. For example, during the Quaternary, glacio-eustatic sea level changes correlate well with the present climatic signature. During the Missourian, however, mechanisms such as plate reorganization may have dominated, resulting in a significantly different fractal dimension.

Hydrophobicity of silver surfaces with microparticle geometry

NASA Astrophysics Data System (ADS)

Macko, Ján; Oriňaková, Renáta; Oriňak, Andrej; Kovaľ, Karol; Kupková, Miriam; Erdélyi, Branislav; Kostecká, Zuzana; Smith, Roger M.

2016-11-01

The effect of the duration of the current deposition cycle and the number of current pulses on the geometry of silver microstructured surfaces and on the free surface energy, polarizability, hydrophobicity and thus adhesion force of the silver surfaces has been investigated. The changes in surface hydrophobicity were entirely dependent on the size and density of the microparticles on the surface. The results showed that formation of the silver microparticles was related to number of current pulses, while the duration of one current pulse played only a minor effect on the final surface microparticle geometry and thus on the surface tension and hydrophobicity. The conventional geometry of the silver particles has been transformed to the fractal dimension D. The surface hydrophobicity depended predominantly on the length of the dendrites not on their width. The highest silver surface hydrophobicity was observed on a surface prepared by 30 current pulses with a pulse duration of 1 s, the lowest one when deposition was performed by 10 current pulses with a duration of 0.1 s. The partial surface tension coefficients γDS and polarizability kS of the silver surfaces were calculated. Both parameters can be applied in future applications in living cells adhesion prediction and spectral method selection. Silver films with microparticle geometry showed a lower variability in final surface hydrophobicity when compared to nanostructured surfaces. The comparisons could be used to modify surfaces and to modulate human cells and bacterial adhesion on body implants, surgery instruments and clean surfaces.

Numerical study of electromagnetic scattering from one-dimensional nonlinear fractal sea surface

NASA Astrophysics Data System (ADS)

Xie, Tao; He, Chao; William, Perrie; Kuang, Hai-Lan; Zou, Guang-Hui; Chen, Wei

2010-02-01

In recent years, linear fractal sea surface models have been developed for the sea surface in order to establish an electromagnetic backscattering model. Unfortunately, the sea surface is always nonlinear, particularly at high sea states. We present a nonlinear fractal sea surface model and derive an electromagnetic backscattering model. Using this model, we numerically calculate the normalized radar cross section (NRCS) of a nonlinear sea surface. Comparing the averaged NRCS between linear and nonlinear fractal models, we show that the NRCS of a linear fractal sea surface underestimates the NRCS of the real sea surface, especially for sea states with high fractal dimensions, and for dominant ocean surface gravity waves that are either very short or extremely long.

Analysis of fractal dimensions of rat bones from film and digital images

NASA Technical Reports Server (NTRS)

Pornprasertsuk, S.; Ludlow, J. B.; Webber, R. L.; Tyndall, D. A.; Yamauchi, M.

2001-01-01

OBJECTIVES: (1) To compare the effect of two different intra-oral image receptors on estimates of fractal dimension; and (2) to determine the variations in fractal dimensions between the femur, tibia and humerus of the rat and between their proximal, middle and distal regions. METHODS: The left femur, tibia and humerus from 24 4-6-month-old Sprague-Dawley rats were radiographed using intra-oral film and a charge-coupled device (CCD). Films were digitized at a pixel density comparable to the CCD using a flat-bed scanner. Square regions of interest were selected from proximal, middle, and distal regions of each bone. Fractal dimensions were estimated from the slope of regression lines fitted to plots of log power against log spatial frequency. RESULTS: The fractal dimensions estimates from digitized films were significantly greater than those produced from the CCD (P=0.0008). Estimated fractal dimensions of three types of bone were not significantly different (P=0.0544); however, the three regions of bones were significantly different (P=0.0239). The fractal dimensions estimated from radiographs of the proximal and distal regions of the bones were lower than comparable estimates obtained from the middle region. CONCLUSIONS: Different types of image receptors significantly affect estimates of fractal dimension. There was no difference in the fractal dimensions of the different bones but the three regions differed significantly.

Surface layer protein characterization by small angle x-ray scattering and a fractal mean force concept: from protein structure to nanodisk assemblies.

PubMed

Horejs, Christine; Pum, Dietmar; Sleytr, Uwe B; Peterlik, Herwig; Jungbauer, Alois; Tscheliessnig, Rupert

2010-11-07

Surface layers (S-layers) are the most commonly observed cell surface structure of prokaryotic organisms. They are made up of proteins that spontaneously self-assemble into functional crystalline lattices in solution, on various solid surfaces, and interfaces. While classical experimental techniques failed to recover a complete structural model of an unmodified S-layer protein, small angle x-ray scattering (SAXS) provides an opportunity to study the structure of S-layer monomers in solution and of self-assembled two-dimensional sheets. For the protein under investigation we recently suggested an atomistic structural model by the use of molecular dynamics simulations. This structural model is now refined on the basis of SAXS data together with a fractal assembly approach. Here we show that a nondiluted critical system of proteins, which crystallize into monomolecular structures, might be analyzed by SAXS if protein-protein interactions are taken into account by relating a fractal local density distribution to a fractal local mean potential, which has to fulfill the Poisson equation. The present work demonstrates an important step into the elucidation of the structure of S-layers and offers a tool to analyze the structure of self-assembling systems in solution by means of SAXS and computer simulations.

Surface layer protein characterization by small angle x-ray scattering and a fractal mean force concept: From protein structure to nanodisk assemblies

NASA Astrophysics Data System (ADS)

Horejs, Christine; Pum, Dietmar; Sleytr, Uwe B.; Peterlik, Herwig; Jungbauer, Alois; Tscheliessnig, Rupert

2010-11-01

Surface layers (S-layers) are the most commonly observed cell surface structure of prokaryotic organisms. They are made up of proteins that spontaneously self-assemble into functional crystalline lattices in solution, on various solid surfaces, and interfaces. While classical experimental techniques failed to recover a complete structural model of an unmodified S-layer protein, small angle x-ray scattering (SAXS) provides an opportunity to study the structure of S-layer monomers in solution and of self-assembled two-dimensional sheets. For the protein under investigation we recently suggested an atomistic structural model by the use of molecular dynamics simulations. This structural model is now refined on the basis of SAXS data together with a fractal assembly approach. Here we show that a nondiluted critical system of proteins, which crystallize into monomolecular structures, might be analyzed by SAXS if protein-protein interactions are taken into account by relating a fractal local density distribution to a fractal local mean potential, which has to fulfill the Poisson equation. The present work demonstrates an important step into the elucidation of the structure of S-layers and offers a tool to analyze the structure of self-assembling systems in solution by means of SAXS and computer simulations.

Surface layer protein characterization by small angle x-ray scattering and a fractal mean force concept: From protein structure to nanodisk assemblies

DOE Office of Scientific and Technical Information (OSTI.GOV)

Horejs, Christine; Pum, Dietmar; Sleytr, Uwe B.

2010-11-07

Surface layers (S-layers) are the most commonly observed cell surface structure of prokaryotic organisms. They are made up of proteins that spontaneously self-assemble into functional crystalline lattices in solution, on various solid surfaces, and interfaces. While classical experimental techniques failed to recover a complete structural model of an unmodified S-layer protein, small angle x-ray scattering (SAXS) provides an opportunity to study the structure of S-layer monomers in solution and of self-assembled two-dimensional sheets. For the protein under investigation we recently suggested an atomistic structural model by the use of molecular dynamics simulations. This structural model is now refined on themore » basis of SAXS data together with a fractal assembly approach. Here we show that a nondiluted critical system of proteins, which crystallize into monomolecular structures, might be analyzed by SAXS if protein-protein interactions are taken into account by relating a fractal local density distribution to a fractal local mean potential, which has to fulfill the Poisson equation. The present work demonstrates an important step into the elucidation of the structure of S-layers and offers a tool to analyze the structure of self-assembling systems in solution by means of SAXS and computer simulations.« less

Universal characteristics of fractal fluctuations in prime number distribution

NASA Astrophysics Data System (ADS)

Selvam, A. M.

2014-11-01

The frequency of occurrence of prime numbers at unit number spacing intervals exhibits self-similar fractal fluctuations concomitant with inverse power law form for power spectrum generic to dynamical systems in nature such as fluid flows, stock market fluctuations and population dynamics. The physics of long-range correlations exhibited by fractals is not yet identified. A recently developed general systems theory visualizes the eddy continuum underlying fractals to result from the growth of large eddies as the integrated mean of enclosed small scale eddies, thereby generating a hierarchy of eddy circulations or an inter-connected network with associated long-range correlations. The model predictions are as follows: (1) The probability distribution and power spectrum of fractals follow the same inverse power law which is a function of the golden mean. The predicted inverse power law distribution is very close to the statistical normal distribution for fluctuations within two standard deviations from the mean of the distribution. (2) Fractals signify quantum-like chaos since variance spectrum represents probability density distribution, a characteristic of quantum systems such as electron or photon. (3) Fractal fluctuations of frequency distribution of prime numbers signify spontaneous organization of underlying continuum number field into the ordered pattern of the quasiperiodic Penrose tiling pattern. The model predictions are in agreement with the probability distributions and power spectra for different sets of frequency of occurrence of prime numbers at unit number interval for successive 1000 numbers. Prime numbers in the first 10 million numbers were used for the study.

A tale of two fractals: The Hofstadter butterfly and the integral Apollonian gaskets

NASA Astrophysics Data System (ADS)

Satija, Indubala I.

2016-11-01

This paper unveils a mapping between a quantum fractal that describes a physical phenomena, and an abstract geometrical fractal. The quantum fractal is the Hofstadter butterfly discovered in 1976 in an iconic condensed matter problem of electrons moving in a two-dimensional lattice in a transverse magnetic field. The geometric fractal is the integer Apollonian gasket characterized in terms of a 300 BC problem of mutually tangent circles. Both of these fractals are made up of integers. In the Hofstadter butterfly, these integers encode the topological quantum numbers of quantum Hall conductivity. In the Apollonian gaskets an infinite number of mutually tangent circles are nested inside each other, where each circle has integer curvature. The mapping between these two fractals reveals a hidden D3 symmetry embedded in the kaleidoscopic images that describe the asymptotic scaling properties of the butterfly. This paper also serves as a mini review of these fractals, emphasizing their hierarchical aspects in terms of Farey fractions.

Applications of Fractal Analytical Techniques in the Estimation of Operational Scale

NASA Technical Reports Server (NTRS)

Emerson, Charles W.; Quattrochi, Dale A.

2000-01-01

The observational scale and the resolution of remotely sensed imagery are essential considerations in the interpretation process. Many atmospheric, hydrologic, and other natural and human-influenced spatial phenomena are inherently scale dependent and are governed by different physical processes at different spatial domains. This spatial and operational heterogeneity constrains the ability to compare interpretations of phenomena and processes observed in higher spatial resolution imagery to similar interpretations obtained from lower resolution imagery. This is a particularly acute problem, since longterm global change investigations will require high spatial resolution Earth Observing System (EOS), Landsat 7, or commercial satellite data to be combined with lower resolution imagery from older sensors such as Landsat TM and MSS. Fractal analysis is a useful technique for identifying the effects of scale changes on remotely sensed imagery. The fractal dimension of an image is a non-integer value between two and three which indicates the degree of complexity in the texture and shapes depicted in the image. A true fractal surface exhibits self-similarity, a property of curves or surfaces where each part is indistinguishable from the whole, or where the form of the curve or surface is invariant with respect to scale. Theoretically, if the digital numbers of a remotely sensed image resemble an ideal fractal surface, then due to the self-similarity property, the fractal dimension of the image will not vary with scale and resolution, and the slope of the fractal dimension-resolution relationship would be zero. Most geographical phenomena, however, are not self-similar at all scales, but they can be modeled by a stochastic fractal in which the scaling properties of the image exhibit patterns that can be described by statistics such as area-perimeter ratios and autocovariances. Stochastic fractal sets relax the self-similarity assumption and measure many scales and

BOOK REVIEW: Critical Phenomena in Natural Sciences: Chaos, Fractals, Selforganization and Disorder: Concepts and Tools

NASA Astrophysics Data System (ADS)

Franz, S.

2004-10-01

variables with power law distribution, random walks, fractals and multifractal formalisms, etc, are discussed in an immediate and direct way so as to provide ready-to-use tools for analysing and representing power law behaviour in natural phenomena. The exposition then continues discussing the main developments, allowing the reader to understand theoretically and model strongly correlated behaviour. After a concise, but useful, introduction to the fundamentals of statistical physics a discussion of equilibrium critical phenomena and the renormalization group is proposed to the reader. With the centrality of the problem of non-equilibrium behaviour in mind, a discussion is devoted to tentative applications of the concept of temperature in the off-equilibrium context. Particular emphasis is given to the development of long range correlation and of precursors of phase transitions, and their role in the prediction of catastrophic events. Then, basic models such as percolation and rupture models are described. A central position in the book is occupied by a chapter on mechanisms for power laws and a subsequent one on self-organized criticality as a general paradigm for critical behaviour as proposed by P Bak and collaborators. The book concludes with a chapter on the prediction of fields generated by a random distribution of sources. The book maintains the promise of the title of providing concepts and tools to tackle criticality and self-organization. The second edition, while retaining the structure of the first edition, considerably extends the scope with new examples and applications of a research field which is constantly growing. Any scientific book has to solve the dichotomy between the depth of discussion, the pedagogical character of exposition and the quantity of material discussed. In general the book, which evolved from a graduate student course, favours these last two aspects at the expense of the first one. This makes the book very readable and means that, while

Temporal fractals in seabird foraging behaviour: diving through the scales of time

PubMed Central

MacIntosh, Andrew J. J.; Pelletier, Laure; Chiaradia, Andre; Kato, Akiko; Ropert-Coudert, Yan

2013-01-01

Animal behaviour exhibits fractal structure in space and time. Fractal properties in animal space-use have been explored extensively under the Lévy flight foraging hypothesis, but studies of behaviour change itself through time are rarer, have typically used shorter sequences generated in the laboratory, and generally lack critical assessment of their results. We thus performed an in-depth analysis of fractal time in binary dive sequences collected via bio-logging from free-ranging little penguins (Eudyptula minor) across full-day foraging trips (216 data points; 4 orders of temporal magnitude). Results from 4 fractal methods show that dive sequences are long-range dependent and persistent across ca. 2 orders of magnitude. This fractal structure correlated with trip length and time spent underwater, but individual traits had little effect. Fractal time is a fundamental characteristic of penguin foraging behaviour, and its investigation is thus a promising avenue for research on interactions between animals and their environments. PMID:23703258

High-Directivity Emissions with Flexible Beam Numbers and Beam Directions Using Gradient-Refractive-Index Fractal Metamaterial

PubMed Central

Xu, He-Xiu; Wang, Guang-Ming; Tao, Zui; Cui, Tie Jun

2014-01-01

A three-dimensional (3D) highly-directive emission system is proposed to enable beam shaping and beam steering capabilities in wideband frequencies. It is composed of an omnidirectional source antenna and several 3D gradient-refractive-index (GRIN) lenses. To engineer a broadband impedance match, the design method for these 3D lenses is established under the scenario of free-space excitation by using a planar printed monopole. For realizations and demonstrations, a kind of GRIN metamaterial is proposed, which is constructed by non-uniform fractal geometries. Due to the non-resonant and deep-subwavelength features of the fractal elements, the resulting 3D GRIN metamaterial lenses have extra wide bandwidth (3 to 7.5 GHz), and are capable of manipulating electromagnetic wavefronts accurately, advancing the state of the art of available GRIN lenses. The proposal for the versatile highly-directive emissions has been confirmed by simulations and measurements, showing that not only the number of beams can be arbitrarily tailored but also the beam directions can be steerable. The proposal opens a new way to control broadband highly-directive emissions with pre-designed directions, promising great potentials in modern wireless communication systems. PMID:25034268

Noninvasive fractal biomarker of clock neurotransmitter disturbance in humans with dementia

PubMed Central

Hu, Kun; Harper, David G.; Shea, Steven A.; Stopa, Edward G.; Scheer, Frank A. J. L.

2013-01-01

Human motor activity has a robust, intrinsic fractal structure with similar patterns from minutes to hours. The fractal activity patterns appear to be physiologically important because the patterns persist under different environmental conditions but are significantly altered/reduced with aging and Alzheimer's disease (AD). Here, we report that dementia patients, known to have disrupted circadian rhythmicity, also have disrupted fractal activity patterns and that the disruption is more pronounced in patients with more amyloid plaques (a marker of AD severity). Moreover, the degree of fractal activity disruption is strongly associated with vasopressinergic and neurotensinergic neurons (two major circadian neurotransmitters) in postmortem suprachiasmatic nucleus (SCN), and can better predict changes of the two neurotransmitters than traditional circadian measures. These findings suggest that the SCN impacts human activity regulation at multiple time scales and that disrupted fractal activity may serve as a non-invasive biomarker of SCN neurodegeneration in dementia. PMID:23863985

Analysis of Fractional Flow for Transient Two-Phase Flow in Fractal Porous Medium

NASA Astrophysics Data System (ADS)

Lu, Ting; Duan, Yonggang; Fang, Quantang; Dai, Xiaolu; Wu, Jinsui

2016-03-01

Prediction of fractional flow in fractal porous medium is important for reservoir engineering and chemical engineering as well as hydrology. A physical conceptual fractional flow model of transient two-phase flow is developed in fractal porous medium based on the fractal characteristics of pore-size distribution and on the approximation that porous medium consist of a bundle of tortuous capillaries. The analytical expression for fractional flow for wetting phase is presented, and the proposed expression is the function of structural parameters (such as tortuosity fractal dimension, pore fractal dimension, maximum and minimum diameters of capillaries) and fluid properties (such as contact angle, viscosity and interfacial tension) in fractal porous medium. The sensitive parameters that influence fractional flow and its derivative are formulated, and their impacts on fractional flow are discussed.

On fractal properties of arterial trees.

PubMed

Zamir, M

1999-04-21

The question of fractal properties of arterial trees is considered in light of data from the extensive tree structure of the right coronary artery of a human heart. Because of the highly non-uniform structure of this tree, the study focuses on the purely geometrical rather than statistical aspects of fractal properties. The large number of arterial bifurcations comprising the tree were found to have a mixed degree of asymmetry at all levels of the tree, including the depth of the tree where it has been generally supposed that they would be symmetrical. Cross-sectional area ratios of daughter to parent vessels were also found to be highly mixed at all levels, having values both above and below 1.0, rather than consistently above as has been generally supposed in the past. Calculated values of the power law index which describes the theoretical relation between the diameters of the three vessel segments at an arterial bifurcation were found to range far beyond the two values associated with the cube and square laws, and not clearly favoring one or the other. On the whole the tree structure was found to have what we have termed "pseudo-fractal" properties, in the sense that vessels of different calibers displayed the same branching pattern but with a range of values of the branching parameters. The results suggest that a higher degree of fractal character, one in which the branching parameters are constant throughout the tree structure, is unlikely to be attained in non-uniform vascular structures. Copyright 1999 Academic Press.

Fractal and Multifractal Analysis of Human Gait

NASA Astrophysics Data System (ADS)

Muñoz-Diosdado, A.; del Río Correa, J. L.; Angulo-Brown, F.

2003-09-01

We carried out a fractal and multifractal analysis of human gait time series of young and old individuals, and adults with three illnesses that affect the march: The Parkinson's and Huntington's diseases and the amyotrophic lateral sclerosis (ALS). We obtained cumulative plots of events, the correlation function, the Hurst exponent and the Higuchi's fractal dimension of these time series and found that these fractal markers could be a factor to characterize the march, since we obtained different values of these quantities for youths and adults and they are different also for healthy and ill persons and the most anomalous values belong to ill persons. In other physiological signals there is complexity lost related with the age and the illness, in the case of the march the opposite occurs. The multifractal analysis could be also a useful tool to understand the dynamics of these and other complex systems.

Fractal analysis of narwhal space use patterns.

PubMed

Laidre, Kristin L; Heide-Jørgensen, Mads P; Logsdon, Miles L; Hobbs, Roderick C; Dietz, Rune; VanBlaricom, Glenn R

2004-01-01

Quantifying animal movement in response to a spatially and temporally heterogeneous environment is critical to understanding the structural and functional landscape influences on population viability. Generalities of landscape structure can easily be extended to the marine environment, as marine predators inhabit a patchy, dynamic system, which influences animal choice and behavior. An innovative use of the fractal measure of complexity, indexing the linearity of movement paths over replicate temporal scales, was applied to satellite tracking data collected from narwhals (Monodon monoceros) (n = 20) in West Greenland and the eastern Canadian high Arctic. Daily movements of individuals were obtained using polar orbiting satellites via the ARGOS data location and collection system. Geographic positions were filtered to obtain a daily good quality position for each whale. The length of total pathway was measured over seven different temporal length scales (step lengths), ranging from one day to one week, and a seasonal mean was calculated. Fractal dimension (D) was significantly different between seasons, highest during summer (D = 1.61, SE 0.04) and winter (D = 1.69, SE 0.06) when whales made convoluted movements in focal areas. Fractal dimension was lowest during fall (D = 1.34, SE 0.03) when whales were migrating south ahead of the forming sea ice. There were no significant effects of size category or sex on fractal dimension by season. The greater linearity of movement during the migration period suggests individuals do not intensively forage on patchy resources until they arrive at summer or winter sites. The highly convoluted movements observed during summer and winter suggest foraging or searching efforts in localized areas. Significant differences between the fractal dimensions on two separate wintering grounds in Baffin Bay suggest differential movement patterns in response to the dynamics of sea ice.

Fractal characterization and wettability of ion treated silicon surfaces

NASA Astrophysics Data System (ADS)

Yadav, R. P.; Kumar, Tanuj; Baranwal, V.; Vandana, Kumar, Manvendra; Priya, P. K.; Pandey, S. N.; Mittal, A. K.

2017-02-01

Fractal characterization of surface morphology can be useful as a tool for tailoring the wetting properties of solid surfaces. In this work, rippled surfaces of Si (100) are grown using 200 keV Ar+ ion beam irradiation at different ion doses. Relationship between fractal and wetting properties of these surfaces are explored. The height-height correlation function extracted from atomic force microscopic images, demonstrates an increase in roughness exponent with an increase in ion doses. A steep variation in contact angle values is found for low fractal dimensions. Roughness exponent and fractal dimensions are found correlated with the static water contact angle measurement. It is observed that after a crossover of the roughness exponent, the surface morphology has a rippled structure. Larger values of interface width indicate the larger ripples on the surface. The contact angle of water drops on such surfaces is observed to be lowest. Autocorrelation function is used for the measurement of ripple wavelength.

Two Dimensional Drug Diffusion Between Nanoparticles and Fractal Tumors

NASA Astrophysics Data System (ADS)

Samioti, S. E.; Karamanos, K.; Tsiantis, A.; Papathanasiou, A.; Sarris, I.

2017-11-01

Drug delivery methods based on nanoparticles are some of the most promising medical applications in nanotechnology to treat cancer. It is observed that drug released by nanoparticles to the cancer tumors may be driven by diffusion. A fractal tumor boundary of triangular Von Koch shape is considered here and the diffusion mechanism is studied for different drug concentrations and increased fractality. A high order Finite Elements method based on the Fenics library is incorporated in fine meshes to fully resolve these irregular boundaries. Drug concentration, its transfer rates and entropy production are calculated in an up to forth order fractal iteration boundaries. We observed that diffusion rate diminishes for successive prefractal generations. Also, the entropy production around the system changes greatly as the order of the fractal curve increases. Results indicate with precision where the active sites are, in which most of the diffusion takes place and thus drug arrives to the tumor.

Integral geometry and holography

DOE PAGES

Czech, Bartlomiej; Lamprou, Lampros; McCandlish, Samuel; ...

2015-10-27

We present a mathematical framework which underlies the connection between information theory and the bulk spacetime in the AdS 3/CFT 2 correspondence. A key concept is kinematic space: an auxiliary Lorentzian geometry whose metric is defined in terms of conditional mutual informations and which organizes the entanglement pattern of a CFT state. When the field theory has a holographic dual obeying the Ryu-Takayanagi proposal, kinematic space has a direct geometric meaning: it is the space of bulk geodesics studied in integral geometry. Lengths of bulk curves are computed by kinematic volumes, giving a precise entropic interpretation of the length ofmore » any bulk curve. We explain how basic geometric concepts -- points, distances and angles -- are reflected in kinematic space, allowing one to reconstruct a large class of spatial bulk geometries from boundary entanglement entropies. In this way, kinematic space translates between information theoretic and geometric descriptions of a CFT state. As an example, we discuss in detail the static slice of AdS 3 whose kinematic space is two-dimensional de Sitter space.« less

Small-angle scattering from the Cantor surface fractal on the plane and the Koch snowflake

NASA Astrophysics Data System (ADS)

Cherny, Alexander Yu.; Anitas, Eugen M.; Osipov, Vladimir A.; Kuklin, Alexander I.

The small-angle scattering (SAS) from the Cantor surface fractal on the plane and Koch snowflake is considered. We develop the construction algorithm for the Koch snowflake, which makes possible the recurrence relation for the scattering amplitude. The surface fractals can be decomposed into a sum of surface mass fractals for arbitrary fractal iteration, which enables various approximations for the scattering intensity. It is shown that for the Cantor fractal, one can neglect with a good accuracy the correlations between the mass fractal amplitudes, while for the Koch snowflake, these correlations are important. It is shown that nevertheless, the correlations can be build in the mass fractal amplitudes, which explains the decay of the scattering intensity $I(q)\\sim q^{D_{\\mathrm{s}}-4}$ with $1 < D_{\\mathrm{s}} < 2$ being the fractal dimension of the perimeter. The curve $I(q)q^{4-D_{\\mathrm{s}}}$ is found to be log-periodic in the fractal region with the period equal to the scaling factor of the fractal. The log-periodicity arises from the self-similarity of sizes of basic structural units rather than from correlations between their distances. A recurrence relation is obtained for the radius of gyration of Koch snowflake, which is solved in the limit of infinite iterations. The present analysis allows us to obtain additional information from SAS data, such as the edges of the fractal regions, the fractal iteration number and the scaling factor.

Fractality and the law of the wall

NASA Astrophysics Data System (ADS)

Xu, Haosen H. A.; Yang, X. I. A.

2018-05-01

Fluid motions in the inertial range of isotropic turbulence are fractal, with their space-filling capacity slightly below regular three-dimensional objects, which is a consequence of the energy cascade. Besides the energy cascade, the other often encountered cascading process is the momentum cascade in wall-bounded flows. Despite the long-existing analogy between the two processes, many of the thoroughly investigated aspects of the energy cascade have so far received little attention in studies of the momentum counterpart, e.g., the possibility of the momentum-transferring scales in the logarithmic region being fractal has not been considered. In this work, this possibility is pursued, and we discuss one of its implications. Following the same dimensional arguments that lead to the D =2.33 fractal dimension of wrinkled surfaces in isotropic turbulence, we show that the large-scale momentum-carrying eddies may also be fractal and non-space-filling, which then leads to the power-law scaling of the mean velocity profile. The logarithmic law of the wall, on the other hand, corresponds to space-filling eddies, as suggested by Townsend [The Structure of Turbulent Shear Flow (Cambridge University Press, Cambridge, 1980)]. Because the space-filling capacity is an integral geometric quantity, the analysis presented in this work provides us with a low-order quantity, with which, one would be able to distinguish between the logarithmic law and the power law.

Vector calculus in non-integer dimensional space and its applications to fractal media

NASA Astrophysics Data System (ADS)

Tarasov, Vasily E.

2015-02-01

We suggest a generalization of vector calculus for the case of non-integer dimensional space. The first and second orders operations such as gradient, divergence, the scalar and vector Laplace operators for non-integer dimensional space are defined. For simplification we consider scalar and vector fields that are independent of angles. We formulate a generalization of vector calculus for rotationally covariant scalar and vector functions. This generalization allows us to describe fractal media and materials in the framework of continuum models with non-integer dimensional space. As examples of application of the suggested calculus, we consider elasticity of fractal materials (fractal hollow ball and fractal cylindrical pipe with pressure inside and outside), steady distribution of heat in fractal media, electric field of fractal charged cylinder. We solve the correspondent equations for non-integer dimensional space models.

Complex Patterns in Financial Time Series Through HIGUCHI’S Fractal Dimension

NASA Astrophysics Data System (ADS)

Grace Elizabeth Rani, T. G.; Jayalalitha, G.

2016-11-01

This paper analyzes the complexity of stock exchanges through fractal theory. Closing price indices of four stock exchanges with different industry sectors are selected. Degree of complexity is assessed through Higuchi’s fractal dimension. Various window sizes are considered in evaluating the fractal dimension. It is inferred that the data considered as a whole represents random walk for all the four indices. Analysis of financial data through windowing procedure exhibits multi-fractality. Attempts to apply moving averages to reduce noise in the data revealed lower estimates of fractal dimension, which was verified using fractional Brownian motion. A change in the normalization factor in Higuchi’s algorithm did improve the results. It is quintessential to focus on rural development to realize a standard and steady growth of economy. Tools must be devised to settle the issues in this regard. Micro level institutions are necessary for the economic growth of a country like India, which would induce a sporadic development in the present global economical scenario.

Fractal scaling in bottlenose dolphin (Tursiops truncatus) echolocation: A case study

NASA Astrophysics Data System (ADS)

Perisho, Shaun T.; Kelty-Stephen, Damian G.; Hajnal, Alen; Houser, Dorian; Kuczaj, Stan A., II

2016-02-01

Fractal scaling patterns, which entail a power-law relationship between magnitude of fluctuations in a variable and the scale at which the variable is measured, have been found in many aspects of human behavior. These findings have led to advances in behavioral models (e.g. providing empirical support for cascade-driven theories of cognition) and have had practical medical applications (e.g. providing new methods for early diagnosis of medical conditions). In the present paper, fractal analysis is used to investigate whether similar fractal scaling patterns exist in inter-click interval and peak-peak amplitude measurements of bottlenose dolphin click trains. Several echolocation recordings taken from two male bottlenose dolphins were analyzed using Detrended Fluctuation Analysis and Higuchi's (1988) method for determination of fractal dimension. Both animals were found to exhibit fractal scaling patterns near what is consistent with persistent long range correlations. These findings suggest that recent advances in human cognition and medicine may have important parallel applications to echolocation as well.

Fuzzy fractals, chaos, and noise

DOE Office of Scientific and Technical Information (OSTI.GOV)

Zardecki, A.

1997-05-01

To distinguish between chaotic and noisy processes, the authors analyze one- and two-dimensional chaotic mappings, supplemented by the additive noise terms. The predictive power of a fuzzy rule-based system allows one to distinguish ergodic and chaotic time series: in an ergodic series the likelihood of finding large numbers is small compared to the likelihood of finding them in a chaotic series. In the case of two dimensions, they consider the fractal fuzzy sets whose {alpha}-cuts are fractals, arising in the context of a quadratic mapping in the extended complex plane. In an example provided by the Julia set, the conceptmore » of Hausdorff dimension enables one to decide in favor of chaotic or noisy evolution.« less

New 5-adic Cantor sets and fractal string.

PubMed

Kumar, Ashish; Rani, Mamta; Chugh, Renu

2013-01-01

In the year (1879-1884), George Cantor coined few problems and consequences in the field of set theory. One of them was the Cantor ternary set as a classical example of fractals. In this paper, 5-adic Cantor one-fifth set as an example of fractal string have been introduced. Moreover, the applications of 5-adic Cantor one-fifth set in string theory have also been studied.

Dimensional flow in discrete quantum geometries

NASA Astrophysics Data System (ADS)

Calcagni, Gianluca; Oriti, Daniele; Thürigen, Johannes

2015-04-01

In various theories of quantum gravity, one observes a change in the spectral dimension from the topological spatial dimension d at large length scales to some smaller value at small, Planckian scales. While the origin of such a flow is well understood in continuum approaches, in theories built on discrete structures a firm control of the underlying mechanism is still missing. We shed some light on the issue by presenting a particular class of quantum geometries with a flow in the spectral dimension, given by superpositions of states defined on regular complexes. For particular superposition coefficients parametrized by a real number 0 <α geometries may be considered as fractal only when α =1 , where the "magic number" DS≃2 for the spectral dimension of spacetime, appearing so often in quantum gravity, is reproduced as well. These results apply, in particular, to special superpositions of spin-network states in loop quantum gravity, and they provide more solid indications of dimensional flow in this approach.

An organic jelly made fractal logic gate with an infinite truth table

PubMed Central

Ghosh, Subrata; Fujita, Daisuke; Bandyopadhyay, Anirban

2015-01-01

Widely varying logic gates invented over a century are all finite. As data deluge problem looms large on the information processing and communication industry, the thrust to explore radical concepts is increasing rapidly. Here, we design and synthesis a molecule, wherein, the input energy transmits in a cycle inside the molecular system, just like an oscillator, then, we use the molecule to make a jelly that acts as chain of oscillators with a fractal like resonance band. Hence, with the increasing detection resolution, in the vacant space between two energy levels of a given resonance band, a new band appears, due to fractal nature, generation of newer energy levels never stops. This is natural property of a linear chain oscillator. As we correlate each energy level of the resonance band of organic jelly, as a function of pH and density of the jelly, we realize a logic gate, whose truth table is finite, but if we zoom any small part, a new truth table appears. In principle, zooming of truth table would continue forever. Thus, we invent a new class of infinite logic gate for the first time. PMID:26086417

A simple method for estimating the size of nuclei on fractal surfaces

NASA Astrophysics Data System (ADS)

Zeng, Qiang

2017-10-01

Determining the size of nuclei on complex surfaces remains a big challenge in aspects of biological, material and chemical engineering. Here the author reported a simple method to estimate the size of the nuclei in contact with complex (fractal) surfaces. The established approach was based on the assumptions of contact area proportionality for determining nucleation density and the scaling congruence between nuclei and surfaces for identifying contact regimes. It showed three different regimes governing the equations for estimating the nucleation site density. Nuclei in the size large enough could eliminate the effect of fractal structure. Nuclei in the size small enough could lead to the independence of nucleation site density on fractal parameters. Only when nuclei match the fractal scales, the nucleation site density is associated with the fractal parameters and the size of the nuclei in a coupling pattern. The method was validated by the experimental data reported in the literature. The method may provide an effective way to estimate the size of nuclei on fractal surfaces, through which a number of promising applications in relative fields can be envisioned.

Aqueous synthesis of LiFePO4 with Fractal Granularity.

PubMed

Cabán-Huertas, Zahilia; Ayyad, Omar; Dubal, Deepak P; Gómez-Romero, Pedro

2016-06-03

Lithium iron phosphate (LiFePO4) electrodes with fractal granularity are reported. They were made from a starting material prepared in water by a low cost, easy and environmentally friendly hydrothermal method, thus avoiding the use of organic solvents. Our method leads to pure olivine phase, free of the impurities commonly found after other water-based syntheses. The fractal structures consisted of nanoparticles grown into larger micro-sized formations which in turn agglomerate leading to high tap density electrodes, which is beneficial for energy density. These intricate structures could be easily and effectively coated with a thin and uniform carbon layer for increased conductivity, as it is well established for simpler microstructures. Materials and electrodes were studied by means of XRD, SEM, TEM, SAED, XPS, Raman and TGA. Last but not least, lithium transport through fractal LiFePO4 electrodes was investigated based upon fractal theory. These water-made fractal electrodes lead to high-performance lithium cells (even at high rates) tested by CV and galvanostatic charge-discharge, their performance is comparable to state of the art (but less environmentally friendly) electrodes.

Aqueous synthesis of LiFePO4 with Fractal Granularity

PubMed Central

Cabán-Huertas, Zahilia; Ayyad, Omar; Dubal, Deepak P.; Gómez-Romero, Pedro

2016-01-01

Lithium iron phosphate (LiFePO4) electrodes with fractal granularity are reported. They were made from a starting material prepared in water by a low cost, easy and environmentally friendly hydrothermal method, thus avoiding the use of organic solvents. Our method leads to pure olivine phase, free of the impurities commonly found after other water-based syntheses. The fractal structures consisted of nanoparticles grown into larger micro-sized formations which in turn agglomerate leading to high tap density electrodes, which is beneficial for energy density. These intricate structures could be easily and effectively coated with a thin and uniform carbon layer for increased conductivity, as it is well established for simpler microstructures. Materials and electrodes were studied by means of XRD, SEM, TEM, SAED, XPS, Raman and TGA. Last but not least, lithium transport through fractal LiFePO4 electrodes was investigated based upon fractal theory. These water-made fractal electrodes lead to high-performance lithium cells (even at high rates) tested by CV and galvanostatic charge-discharge, their performance is comparable to state of the art (but less environmentally friendly) electrodes. PMID:27256504

Aqueous synthesis of LiFePO4 with Fractal Granularity

NASA Astrophysics Data System (ADS)

Cabán-Huertas, Zahilia; Ayyad, Omar; Dubal, Deepak P.; Gómez-Romero, Pedro

2016-06-01

Lithium iron phosphate (LiFePO4) electrodes with fractal granularity are reported. They were made from a starting material prepared in water by a low cost, easy and environmentally friendly hydrothermal method, thus avoiding the use of organic solvents. Our method leads to pure olivine phase, free of the impurities commonly found after other water-based syntheses. The fractal structures consisted of nanoparticles grown into larger micro-sized formations which in turn agglomerate leading to high tap density electrodes, which is beneficial for energy density. These intricate structures could be easily and effectively coated with a thin and uniform carbon layer for increased conductivity, as it is well established for simpler microstructures. Materials and electrodes were studied by means of XRD, SEM, TEM, SAED, XPS, Raman and TGA. Last but not least, lithium transport through fractal LiFePO4 electrodes was investigated based upon fractal theory. These water-made fractal electrodes lead to high-performance lithium cells (even at high rates) tested by CV and galvanostatic charge-discharge, their performance is comparable to state of the art (but less environmentally friendly) electrodes.

Electrical conductivity modeling in fractal non-saturated porous media

NASA Astrophysics Data System (ADS)

Wei, W.; Cai, J.; Hu, X.; Han, Q.

2016-12-01

The variety of electrical conductivity in non-saturated conditions is important to study electric conduction in natural sedimentary rocks. The electrical conductivity in completely saturated porous media is a porosity-function representing the complex connected behavior of single conducting phases (pore fluid). For partially saturated conditions, the electrical conductivity becomes even more complicated since the connectedness of pore. Archie's second law is an empirical electrical conductivity-porosity and -saturation model that has been used to predict the formation factor of non-saturated porous rock. However, the physical interpretation of its parameters, e.g., the cementation exponent m and the saturation exponent n, remains questionable. On basis of our previous work, we combine the pore-solid fractal (PSF) model to build an electrical conductivity model in non-saturated porous media. Our theoretical porosity- and saturation-dependent models contain endmember properties, such as fluid electrical conductivities, pore fractal dimension and tortuosity fractal dimension (representing the complex degree of electrical flowing path). We find the presented model with non-saturation-dependent electrical conductivity datasets indicate excellent match between theory and experiments. This means the value of pore fractal dimension and tortuosity fractal dimension change from medium to medium and depends not only on geometrical properties of pore structure but also characteristics of electrical current flowing in the non-saturated porous media.

Fractal structure of the interplanetary magnetic field

NASA Technical Reports Server (NTRS)

Burlaga, L. F.; Klein, L. W.

1985-01-01

Under some conditions, time series of the interplanetary magnetic field strength and components have the properties of fractal curves. Magnetic field measurements made near 8.5 AU by Voyager 2 from June 5 to August 24, 1981 were self-similar over time scales from approximately 20 sec to approximately 3 x 100,000 sec, and the fractal dimension of the time series of the strength and components of the magnetic field was D = 5/3, corresponding to a power spectrum P(f) approximately f sup -5/3. Since the Kolmogorov spectrum for homogeneous, isotropic, stationary turbulence is also f sup -5/3, the Voyager 2 measurements are consistent with the observation of an inertial range of turbulence extending over approximately four decades in frequency. Interaction regions probably contributed most of the power in this interval. As an example, one interaction region is discussed in which the magnetic field had a fractal dimension D = 5/3.

Assessment of disintegrant efficacy with fractal dimensions from real-time MRI.

PubMed

Quodbach, Julian; Moussavi, Amir; Tammer, Roland; Frahm, Jens; Kleinebudde, Peter

2014-11-20

An efficient disintegrant is capable of breaking up a tablet in the smallest possible particles in the shortest time. Until now, comparative data on the efficacy of different disintegrants is based on dissolution studies or the disintegration time. Extending these approaches, this study introduces a method, which defines the evolution of fractal dimensions of tablets as surrogate parameter for the available surface area. Fractal dimensions are a measure for the tortuosity of a line, in this case the upper surface of a disintegrating tablet. High-resolution real-time MRI was used to record videos of disintegrating tablets. The acquired video images were processed to depict the upper surface of the tablets and a box-counting algorithm was used to estimate the fractal dimensions. The influence of six different disintegrants, of different relative tablet density, and increasing disintegrant concentration was investigated to evaluate the performance of the novel method. Changing relative densities hardly affect the progression of fractal dimensions, whereas an increase in disintegrant concentration causes increasing fractal dimensions during disintegration, which are also reached quicker. Different disintegrants display only minor differences in the maximal fractal dimension, yet the kinetic in which the maximum is reached allows a differentiation and classification of disintegrants. Copyright © 2014 Elsevier B.V. All rights reserved.

A novel fractal approach for predicting G-protein-coupled receptors and their subfamilies with support vector machines.

PubMed

Nie, Guoping; Li, Yong; Wang, Feichi; Wang, Siwen; Hu, Xuehai

2015-01-01

G-protein-coupled receptors (GPCRs) are seven membrane-spanning proteins and regulate many important physiological processes, such as vision, neurotransmission, immune response and so on. GPCRs-related pathways are the targets of a large number of marketed drugs. Therefore, the design of a reliable computational model for predicting GPCRs from amino acid sequence has long been a significant biomedical problem. Chaos game representation (CGR) reveals the fractal patterns hidden in protein sequences, and then fractal dimension (FD) is an important feature of these highly irregular geometries with concise mathematical expression. Here, in order to extract important features from GPCR protein sequences, CGR algorithm, fractal dimension and amino acid composition (AAC) are employed to formulate the numerical features of protein samples. Four groups of features are considered, and each group is evaluated by support vector machine (SVM) and 10-fold cross-validation test. To test the performance of the present method, a new non-redundant dataset was built based on latest GPCRDB database. Comparing the results of numerical experiments, the group of combined features with AAC and FD gets the best result, the accuracy is 99.22% and Matthew's correlation coefficient (MCC) is 0.9845 for identifying GPCRs from non-GPCRs. Moreover, if it is classified as a GPCR, it will be further put into the second level, which will classify a GPCR into one of the five main subfamilies. At this level, the group of combined features with AAC and FD also gets best accuracy 85.73%. Finally, the proposed predictor is also compared with existing methods and shows better performances.

Fractal fluctuations in spatiotemporal variables when walking on a self-paced treadmill.

PubMed

Choi, Jin-Seung; Kang, Dong-Won; Seo, Jeong-Woo; Tack, Gye-Rae

2017-12-08

This study investigated the fractal dynamic properties of stride time (ST), stride length (SL) and stride speed (SS) during walking on a self-paced treadmill (STM) in which the belt speed is automatically controlled by the walking speed. Twelve healthy young subjects participated in the study. The subjects walked at their preferred walking speed under four conditions: STM, STM with a metronome (STM+met), fixed-speed (conventional) treadmill (FTM), and FTM with a metronome (FTM+met). To compare the fractal dynamics between conditions, the mean, variability, and fractal dynamics of ST, SL, and SS were compared. Moreover, the relationship among the variables was examined under each walking condition using three types of surrogates. The mean values of all variables did not differ between the two treadmills, and the variability of all variables was generally larger for STM than for FTM. The use of a metronome resulted in a decrease in variability in ST and SS for all conditions. The fractal dynamic characteristics of SS were maintained with STM, in contrast to FTM, and only the fractal dynamic characteristics of ST disappeared when using a metronome. In addition, the fractal dynamic patterns of the cross-correlated surrogate results were identical to those of all variables for the two treadmills. In terms of the fractal dynamic properties, STM walking was generally closer to overground walking than FTM walking. Although further research is needed, the present results will be useful in research on gait fractal dynamics and rehabilitation. Copyright © 2017 Elsevier Ltd. All rights reserved.

A fractal process of hydrogen diffusion in a-Si:H with exponential energy distribution

NASA Astrophysics Data System (ADS)

Hikita, Harumi; Ishikawa, Hirohisa; Morigaki, Kazuo

2017-04-01

Hydrogen diffusion in a-Si:H with exponential distribution of the states in energy exhibits the fractal structure. It is shown that a probability P(t) of the pausing time t has a form of tα (α: fractal dimension). It is shown that the fractal dimension α = Tr/T0 (Tr: hydrogen temperature, T0: a temperature corresponding to the width of exponential distribution of the states in energy) is in agreement with the Hausdorff dimension. A fractal graph for the case of α ≤ 1 is like the Cantor set. A fractal graph for the case of α > 1 is like the Koch curves. At α = ∞, hydrogen migration exhibits Brownian motion. Hydrogen diffusion in a-Si:H should be the fractal process.

Fractal-like thickness and topography of the salt layer in a pillows province of the southern North Sea

NASA Astrophysics Data System (ADS)

Hernandez Maya, K.; Mitchell, N. C.; Huuse, M.

2017-12-01

Salt topography and thickness variations are important for testing theories of how halokinetic deformation proceeds. The ability to predict thickness variations of salt at small scale is also important for reservoir evaluations, as breach of the salt layer can lead to loss of petroleum fluids and can be difficult to evaluate from seismic reflection data. Relevant to these issues, we here report analysis of data on salt layer topography and thickness from the southern North Sea, where the salt is organized into pillows. These data were derived by the Geological Survey of the Netherlands (TNO) from industry 3D seismic reflection data combined with a dense network of well information. Highs and lows in the topography of the upper salt interface occur spaced over a variety of lengthscales. Power spectral analysis of the interface topography reveals a simple inverse power law relationship between power spectral density and spatial wave number. The relationship suggests that the interface is a self-affine fractal with a fractal dimension of 2.85. A similar analysis of the salt layer thickness also suggests a fractal-like power law. Whereas the layer thickness power law is unsurprising as the underlying basement topography dominates the thickness and it also has a fractal-like power spectrum, the salt topography is not so easily explained as not all the basement faults are overlaid by salt pillows, instead some areas of the dataset salt thinning overlies faults. We consider instead whether a spatially varied loading of the salt layer may have caused this fractal-like geometry. Varied density and thickness of overburdening layers seem unlikely causes, as thicknesses of layers and their reflectivities do not vary sympathetically with the topography of the interface. The composition of the salt layer varies with the relative proportions of halite and denser anhydrite and other minerals. Although limited in scope and representing the mobilized salt layer, the information from

Prediction of pork quality parameters by applying fractals and data mining on MRI.

PubMed

Caballero, Daniel; Pérez-Palacios, Trinidad; Caro, Andrés; Amigo, José Manuel; Dahl, Anders B; ErsbØll, Bjarne K; Antequera, Teresa

2017-09-01

This work firstly investigates the use of MRI, fractal algorithms and data mining techniques to determine pork quality parameters non-destructively. The main objective was to evaluate the capability of fractal algorithms (Classical Fractal algorithm, CFA; Fractal Texture Algorithm, FTA and One Point Fractal Texture Algorithm, OPFTA) to analyse MRI in order to predict quality parameters of loin. In addition, the effect of the sequence acquisition of MRI (Gradient echo, GE; Spin echo, SE and Turbo 3D, T3D) and the predictive technique of data mining (Isotonic regression, IR and Multiple linear regression, MLR) were analysed. Both fractal algorithm, FTA and OPFTA are appropriate to analyse MRI of loins. The sequence acquisition, the fractal algorithm and the data mining technique seems to influence on the prediction results. For most physico-chemical parameters, prediction equations with moderate to excellent correlation coefficients were achieved by using the following combinations of acquisition sequences of MRI, fractal algorithms and data mining techniques: SE-FTA-MLR, SE-OPFTA-IR, GE-OPFTA-MLR, SE-OPFTA-MLR, with the last one offering the best prediction results. Thus, SE-OPFTA-MLR could be proposed as an alternative technique to determine physico-chemical traits of fresh and dry-cured loins in a non-destructive way with high accuracy. Copyright © 2017. Published by Elsevier Ltd.

Fractal Theory for Permeability Prediction, Venezuelan and USA Wells

NASA Astrophysics Data System (ADS)

Aldana, Milagrosa; Altamiranda, Dignorah; Cabrera, Ana

2014-05-01

Inferring petrophysical parameters such as permeability, porosity, water saturation, capillary pressure, etc, from the analysis of well logs or other available core data has always been of critical importance in the oil industry. Permeability in particular, which is considered to be a complex parameter, has been inferred using both empirical and theoretical techniques. The main goal of this work is to predict permeability values on different wells using Fractal Theory, based on a method proposed by Pape et al. (1999). This approach uses the relationship between permeability and the geometric form of the pore space of the rock. This method is based on the modified equation of Kozeny-Carman and a fractal pattern, which allows determining permeability as a function of the cementation exponent, porosity and the fractal dimension. Data from wells located in Venezuela and the United States of America are analyzed. Employing data of porosity and permeability obtained from core samples, and applying the Fractal Theory method, we calculated the prediction equations for each well. At the beginning, this was achieved by training with 50% of the data available for each well. Afterwards, these equations were tested inferring over 100% of the data to analyze possible trends in their distribution. This procedure gave excellent results in all the wells in spite of their geographic distance, generating permeability models with the potential to accurately predict permeability logs in the remaining parts of the well for which there are no core samples, using even porority logs. Additionally, empirical models were used to determine permeability and the results were compared with those obtained by applying the fractal method. The results indicated that, although there are empirical equations that give a proper adjustment, the prediction results obtained using fractal theory give a better fit to the core reference data.

Optimal spinneret layout in Von Koch curves of fractal theory based needleless electrospinning process

NASA Astrophysics Data System (ADS)

Yang, Wenxiu; Liu, Yanbo; Zhang, Ligai; Cao, Hong; Wang, Yang; Yao, Jinbo

2016-06-01

Needleless electrospinning technology is considered as a better avenue to produce nanofibrous materials at large scale, and electric field intensity and its distribution play an important role in controlling nanofiber diameter and quality of the nanofibrous web during electrospinning. In the current study, a novel needleless electrospinning method was proposed based on Von Koch curves of Fractal configuration, simulation and analysis on electric field intensity and distribution in the new electrospinning process were performed with Finite element analysis software, Comsol Multiphysics 4.4, based on linear and nonlinear Von Koch fractal curves (hereafter called fractal models). The result of simulation and analysis indicated that Second level fractal structure is the optimal linear electrospinning spinneret in terms of field intensity and uniformity. Further simulation and analysis showed that the circular type of Fractal spinneret has better field intensity and distribution compared to spiral type of Fractal spinneret in the nonlinear Fractal electrospinning technology. The electrospinning apparatus with the optimal Von Koch fractal spinneret was set up to verify the theoretical analysis results from Comsol simulation, achieving more uniform electric field distribution and lower energy cost, compared to the current needle and needleless electrospinning technologies.

The Fractal Patterns of Words in a Text: A Method for Automatic Keyword Extraction.

PubMed

Najafi, Elham; Darooneh, Amir H

2015-01-01

A text can be considered as a one dimensional array of words. The locations of each word type in this array form a fractal pattern with certain fractal dimension. We observe that important words responsible for conveying the meaning of a text have dimensions considerably different from one, while the fractal dimensions of unimportant words are close to one. We introduce an index quantifying the importance of the words in a given text using their fractal dimensions and then ranking them according to their importance. This index measures the difference between the fractal pattern of a word in the original text relative to a shuffled version. Because the shuffled text is meaningless (i.e., words have no importance), the difference between the original and shuffled text can be used to ascertain degree of fractality. The degree of fractality may be used for automatic keyword detection. Words with the degree of fractality higher than a threshold value are assumed to be the retrieved keywords of the text. We measure the efficiency of our method for keywords extraction, making a comparison between our proposed method and two other well-known methods of automatic keyword extraction.

The Fractal Patterns of Words in a Text: A Method for Automatic Keyword Extraction

PubMed Central

Najafi, Elham; Darooneh, Amir H.

2015-01-01

A text can be considered as a one dimensional array of words. The locations of each word type in this array form a fractal pattern with certain fractal dimension. We observe that important words responsible for conveying the meaning of a text have dimensions considerably different from one, while the fractal dimensions of unimportant words are close to one. We introduce an index quantifying the importance of the words in a given text using their fractal dimensions and then ranking them according to their importance. This index measures the difference between the fractal pattern of a word in the original text relative to a shuffled version. Because the shuffled text is meaningless (i.e., words have no importance), the difference between the original and shuffled text can be used to ascertain degree of fractality. The degree of fractality may be used for automatic keyword detection. Words with the degree of fractality higher than a threshold value are assumed to be the retrieved keywords of the text. We measure the efficiency of our method for keywords extraction, making a comparison between our proposed method and two other well-known methods of automatic keyword extraction. PMID:26091207

Communities and classes in symmetric fractals

NASA Astrophysics Data System (ADS)

Krawczyk, Małgorzata J.

2015-07-01

Two aspects of fractal networks are considered: the community structure and the class structure, where classes of nodes appear as a consequence of a local symmetry of nodes. The analyzed systems are the networks constructed for two selected symmetric fractals: the Sierpinski triangle and the Koch curve. Communities are searched for by means of a set of differential equations. Overlapping nodes which belong to two different communities are identified by adding some noise to the initial connectivity matrix. Then, a node can be characterized by a spectrum of probabilities of belonging to different communities. Our main goal is that the overlapping nodes with the same spectra belong to the same class.

[Modeling continuous scaling of NDVI based on fractal theory].

PubMed

Luan, Hai-Jun; Tian, Qing-Jiu; Yu, Tao; Hu, Xin-Li; Huang, Yan; Du, Ling-Tong; Zhao, Li-Min; Wei, Xi; Han, Jie; Zhang, Zhou-Wei; Li, Shao-Peng

2013-07-01

Scale effect was one of the very important scientific problems of remote sensing. The scale effect of quantitative remote sensing can be used to study retrievals' relationship between different-resolution images, and its research became an effective way to confront the challenges, such as validation of quantitative remote sensing products et al. Traditional up-scaling methods cannot describe scale changing features of retrievals on entire series of scales; meanwhile, they are faced with serious parameters correction issues because of imaging parameters' variation of different sensors, such as geometrical correction, spectral correction, etc. Utilizing single sensor image, fractal methodology was utilized to solve these problems. Taking NDVI (computed by land surface radiance) as example and based on Enhanced Thematic Mapper Plus (ETM+) image, a scheme was proposed to model continuous scaling of retrievals. Then the experimental results indicated that: (a) For NDVI, scale effect existed, and it could be described by fractal model of continuous scaling; (2) The fractal method was suitable for validation of NDVI. All of these proved that fractal was an effective methodology of studying scaling of quantitative remote sensing.

Evolution of fractality in space plasmas of interest to geomagnetic activity

NASA Astrophysics Data System (ADS)

Muñoz, Víctor; Domínguez, Macarena; Alejandro Valdivia, Juan; Good, Simon; Nigro, Giuseppina; Carbone, Vincenzo

2018-03-01

We studied the temporal evolution of fractality for geomagnetic activity, by calculating fractal dimensions from the Dst data and from a magnetohydrodynamic shell model for turbulent magnetized plasma, which may be a useful model to study geomagnetic activity under solar wind forcing. We show that the shell model is able to reproduce the relationship between the fractal dimension and the occurrence of dissipative events, but only in a certain region of viscosity and resistivity values. We also present preliminary results of the application of these ideas to the study of the magnetic field time series in the solar wind during magnetic clouds, which suggest that it is possible, by means of the fractal dimension, to characterize the complexity of the magnetic cloud structure.

Self-organized network of fractal-shaped components coupled through statistical interaction.

PubMed

Ugajin, R

2001-09-01

A dissipative dynamics is introduced to generate self-organized networks of interacting objects, which we call coupled-fractal networks. The growth model is constructed based on a growth hypothesis in which the growth rate of each object is a product of the probability of receiving source materials from faraway and the probability of receiving adhesives from other grown objects, where each object grows to be a random fractal if isolated, but connects with others if glued. The network is governed by the statistical interaction between fractal-shaped components, which can only be identified in a statistical manner over ensembles. This interaction is investigated using the degree of correlation between fractal-shaped components, enabling us to determine whether it is attractive or repulsive.

Jupiter Fractal Art

NASA Image and Video Library

2017-08-10

See Jupiter's Great Red Spot as you've never seen it before in this new Jovian work of art. Artist Mik Petter created this unique, digital artwork using data from the JunoCam imager on NASA's Juno spacecraft. The art form, known as fractals, uses mathematical formulas to create art with an infinite variety of form, detail, color and light. The tumultuous atmospheric zones in and around the Great Red Spot are highlighted by the author's use of colorful fractals. Vibrant colors of various tints and hues, combined with the almost organic-seeming shapes, make this image seem to be a colorized and crowded petri dish of microorganisms, or a close-up view of microscopic and wildly-painted seashells. The original JunoCam image was taken on July 10, 2017 at 7:10 p.m. PDT (10:10 p.m. EDT), as the Juno spacecraft performed its seventh close flyby of Jupiter. The spacecraft captured the image from about 8,648 miles (13,917 kilometers) above the tops of the clouds of the planet at a latitude of -32.6 degrees. https://photojournal.jpl.nasa.gov/catalog/PIA21777

Geometry in the Adult Education Classroom.

ERIC Educational Resources Information Center

Markus, Nancy

2001-01-01

For many adults, geometry is a mathematics topic that immediately makes sense to them and gives them confidence in their ability to learn, while other adult learners identify geometry with failure. Most adults, however, do recognize the need for measurement, and many have a basic understanding of measurement concepts, although they may need to…

A user-friendly modified pore-solid fractal model

PubMed Central

Ding, Dian-yuan; Zhao, Ying; Feng, Hao; Si, Bing-cheng; Hill, Robert Lee

2016-01-01

The primary objective of this study was to evaluate a range of calculation points on water retention curves (WRC) instead of the singularity point at air-entry suction in the pore-solid fractal (PSF) model, which additionally considered the hysteresis effect based on the PSF theory. The modified pore-solid fractal (M-PSF) model was tested using 26 soil samples from Yangling on the Loess Plateau in China and 54 soil samples from the Unsaturated Soil Hydraulic Database. The derivation results showed that the M-PSF model is user-friendly and flexible for a wide range of calculation point options. This model theoretically describes the primary differences between the soil moisture desorption and the adsorption processes by the fractal dimensions. The M-PSF model demonstrated good performance particularly at the calculation points corresponding to the suctions from 100 cm to 1000 cm. Furthermore, the M-PSF model, used the fractal dimension of the particle size distribution, exhibited an accepted performance of WRC predictions for different textured soils when the suction values were ≥100 cm. To fully understand the function of hysteresis in the PSF theory, the role of allowable and accessible pores must be examined. PMID:27996013

Fractal-Based Analysis of the Influence of Music on Human Respiration

NASA Astrophysics Data System (ADS)

Reza Namazi, H.

An important challenge in respiration related studies is to investigate the influence of external stimuli on human respiration. Auditory stimulus is an important type of stimuli that influences human respiration. However, no one discovered any trend, which relates the characteristics of the auditory stimuli to the characteristics of the respiratory signal. In this paper, we investigate the correlation between auditory stimuli and respiratory signal from fractal point of view. We found out that the fractal structure of respiratory signal is correlated with the fractal structure of the applied music. Based on the obtained results, the music with greater fractal dimension will result in respiratory signal with smaller fractal dimension. In order to verify this result, we benefit from approximate entropy. The results show the respiratory signal will have smaller approximate entropy by choosing the music with smaller approximate entropy. The method of analysis could be further investigated to analyze the variations of different physiological time series due to the various types of stimuli when the complexity is the main concern.

A fractal analysis of pathogen detection by biosensors

NASA Astrophysics Data System (ADS)

Doke, Atul M.; Sadana, Ajit

2006-05-01

A fractal analysis is presented for the detection of pathogens such as Franscisela tularensis, and Yersinia pestis (the bacterium that causes plague) using a CANARY (cellular analysis and notification of antigens risks and yields) biosensor (Rider et al., 2003). In general, the binding and dissociation rate coefficients may be adequately described by either a single- or a dual-fractal analysis. An attempt is made to relate the binding rate coefficient to the degree of heterogeneity (fractal dimension value) present on the biosensor surface. Binding and dissociation rate coefficient values obtained are presented. The kinetics aspects along with the affinity values presented are of interest, and should along with the rate coefficients presented for the binding and the dissociation phase be of significant interest in help designing better biosensors for an application area that is bound to gain increasing importance in the future.

Design of inside cut von koch fractal UWB MIMO antenna

NASA Astrophysics Data System (ADS)

Tharani, V.; Shanmuga Priya, N.; Rajesh, A.

2017-11-01

An Inside Cut Hexagonal Von Koch fractal MIMO antenna is designed for UWB applications and its characteristics behaviour are studied. Self-comparative and space filling properties of Koch fractal structure are utilized in the antenna design which leads to the desired miniaturization and wideband characteristics. The hexagonal shaped Von Koch Fractal antenna with Defected Ground Structure (DGS) is designed on FR4 substrate with a compact size of 30mm x 20mm x 1.6mm. The antenna achieves a maximum of -44dB and -51dB at 7.1GHz for 1-element and 2-element case respectively.

Fractal properties and denoising of lidar signals from cirrus clouds

NASA Astrophysics Data System (ADS)

van den Heuvel, J. C.; Driesenaar, M. L.; Lerou, R. J. L.

2000-02-01

Airborne lidar signals of cirrus clouds are analyzed to determine the cloud structure. Climate modeling and numerical weather prediction benefit from accurate modeling of cirrus clouds. Airborne lidar measurements of the European Lidar in Space Technology Experiment (ELITE) campaign were analyzed by combining shots to obtain the backscatter at constant altitude. The signal at high altitude was analyzed for horizontal structure of cirrus clouds. The power spectrum and the structure function show straight lines on a double logarithmic plot. This behavior is characteristic for a Brownian fractal. Wavelet analysis using the Haar wavelet confirms the fractal aspects. It is shown that the horizontal structure of cirrus can be described by a fractal with a dimension of 1.8 over length scales that vary 4 orders of magnitude. We use the fractal properties in a new denoising method. Denoising is required for future lidar measurements from space that have a low signal to noise ratio. Our wavelet denoising is based on the Haar wavelet and uses the statistical fractal properties of cirrus clouds in a method based on the maximum a posteriori (MAP) probability. This denoising based on wavelets is tested on airborne lidar signals from ELITE using added Gaussian noise. Superior results with respect to averaging are obtained.

Fractal Dimension Analysis of Transient Visual Evoked Potentials: Optimisation and Applications.

PubMed

Boon, Mei Ying; Henry, Bruce Ian; Chu, Byoung Sun; Basahi, Nour; Suttle, Catherine May; Luu, Chi; Leung, Harry; Hing, Stephen

2016-01-01

The visual evoked potential (VEP) provides a time series signal response to an external visual stimulus at the location of the visual cortex. The major VEP signal components, peak latency and amplitude, may be affected by disease processes. Additionally, the VEP contains fine detailed and non-periodic structure, of presently unclear relevance to normal function, which may be quantified using the fractal dimension. The purpose of this study is to provide a systematic investigation of the key parameters in the measurement of the fractal dimension of VEPs, to develop an optimal analysis protocol for application. VEP time series were mathematically transformed using delay time, τ, and embedding dimension, m, parameters. The fractal dimension of the transformed data was obtained from a scaling analysis based on straight line fits to the numbers of pairs of points with separation less than r versus log(r) in the transformed space. Optimal τ, m, and scaling analysis were obtained by comparing the consistency of results using different sampling frequencies. The optimised method was then piloted on samples of normal and abnormal VEPs. Consistent fractal dimension estimates were obtained using τ = 4 ms, designating the fractal dimension = D2 of the time series based on embedding dimension m = 7 (for 3606 Hz and 5000 Hz), m = 6 (for 1803 Hz) and m = 5 (for 1000Hz), and estimating D2 for each embedding dimension as the steepest slope of the linear scaling region in the plot of log(C(r)) vs log(r) provided the scaling region occurred within the middle third of the plot. Piloting revealed that fractal dimensions were higher from the sampled abnormal than normal achromatic VEPs in adults (p = 0.02). Variances of fractal dimension were higher from the abnormal than normal chromatic VEPs in children (p = 0.01). A useful analysis protocol to assess the fractal dimension of transformed VEPs has been developed.

Improving the signal subtle feature extraction performance based on dual improved fractal box dimension eigenvectors

NASA Astrophysics Data System (ADS)

Chen, Xiang; Li, Jingchao; Han, Hui; Ying, Yulong

2018-05-01

Because of the limitations of the traditional fractal box-counting dimension algorithm in subtle feature extraction of radiation source signals, a dual improved generalized fractal box-counting dimension eigenvector algorithm is proposed. First, the radiation source signal was preprocessed, and a Hilbert transform was performed to obtain the instantaneous amplitude of the signal. Then, the improved fractal box-counting dimension of the signal instantaneous amplitude was extracted as the first eigenvector. At the same time, the improved fractal box-counting dimension of the signal without the Hilbert transform was extracted as the second eigenvector. Finally, the dual improved fractal box-counting dimension eigenvectors formed the multi-dimensional eigenvectors as signal subtle features, which were used for radiation source signal recognition by the grey relation algorithm. The experimental results show that, compared with the traditional fractal box-counting dimension algorithm and the single improved fractal box-counting dimension algorithm, the proposed dual improved fractal box-counting dimension algorithm can better extract the signal subtle distribution characteristics under different reconstruction phase space, and has a better recognition effect with good real-time performance.

A fractal model for nuclear organization: current evidence and biological implications

PubMed Central

Bancaud, Aurélien; Lavelle, Christophe; Huet, Sébastien; Ellenberg, Jan

2012-01-01

Chromatin is a multiscale structure on which transcription, replication, recombination and repair of the genome occur. To fully understand any of these processes at the molecular level under physiological conditions, a clear picture of the polymorphic and dynamic organization of chromatin in the eukaryotic nucleus is required. Recent studies indicate that a fractal model of chromatin architecture is consistent with both the reaction-diffusion properties of chromatin interacting proteins and with structural data on chromatin interminglement. In this study, we provide a critical overview of the experimental evidence that support a fractal organization of chromatin. On this basis, we discuss the functional implications of a fractal chromatin model for biological processes and propose future experiments to probe chromatin organization further that should allow to strongly support or invalidate the fractal hypothesis. PMID:22790985

Terahertz response of fractal meta-atoms based on concentric rectangular square resonators

DOE Office of Scientific and Technical Information (OSTI.GOV)

Song, Zhiqiang; Zhao, Zhenyu, E-mail: zyzhao@shnu.edu.cn; Shi, Wangzhou

We investigate the terahertz electromagnetic responses of fractal meta-atoms (MAs) induced by different mode coupling mechanisms. Two types of MAs based on concentric rectangular square (CRS) resonators are presented: independent CRS (I-CRS) and junctional-CRS (J-CRS). In I-CRS, each resonator works as an independent dipole so as to result in the multiple resonance modes when the fractal level is above 1. In J-CRS, however, the generated layer is rotated by π/2 radius to the adjacent CRS in one MA. The multiple resonance modes are coupled into a single mode resonance. The fractal level increasing induces resonance modes redshift in I-CRS whilemore » blueshift in J-CRS. When the fractal level is below 4, the mode Q factor of J-CRS is in between the two modes of I-CRS; when the fractal level is 4 or above, the mode Q factor of J-CRS exceeds the two modes of I-CRS. Furthermore, the modulation depth (MD) decreases in I-CRS while it increases in J-CRS with the increase in fractal levels. The surface currents analysis reveals that the capacitive coupling of modes in I-CRS results in the modes redshift, while the conductive coupling of modes in J-CRS induces the mode blueshift. A high Q mode with large MD can be achieved via conductive coupling between the resonators of different scales in a fractal MA.« less

On the Gompertzian growth in the fractal space-time.

PubMed

Molski, Marcin; Konarski, Jerzy

2008-06-01

An analytical approach to determination of time-dependent temporal fractal dimension b(t)(t) and scaling factor a(t)(t) for the Gompertzian growth in the fractal space-time is presented. The derived formulae take into account the proper boundary conditions and permit a calculation of the mean values b(t)(t) and a(t)(t) at any period of time. The formulae derived have been tested on experimental data obtained by Schrek for the Brown-Pearce rabbit's tumor growth. The results obtained confirm a possibility of successful mapping of the experimental Gompertz curve onto the fractal power-law scaling function y(t)=a(t)tb(t) and support a thesis that Gompertzian growth is a self-similar and allometric process of a holistic nature.

[Lithology feature extraction of CASI hyperspectral data based on fractal signal algorithm].

PubMed

Tang, Chao; Chen, Jian-Ping; Cui, Jing; Wen, Bo-Tao

2014-05-01

Hyperspectral data is characterized by combination of image and spectrum and large data volume dimension reduction is the main research direction. Band selection and feature extraction is the primary method used for this objective. In the present article, the authors tested methods applied for the lithology feature extraction from hyperspectral data. Based on the self-similarity of hyperspectral data, the authors explored the application of fractal algorithm to lithology feature extraction from CASI hyperspectral data. The "carpet method" was corrected and then applied to calculate the fractal value of every pixel in the hyperspectral data. The results show that fractal information highlights the exposed bedrock lithology better than the original hyperspectral data The fractal signal and characterized scale are influenced by the spectral curve shape, the initial scale selection and iteration step. At present, research on the fractal signal of spectral curve is rare, implying the necessity of further quantitative analysis and investigation of its physical implications.

Fluvial drainage networks: the fractal approach as an improvement of quantitative geomorphic analyses

NASA Astrophysics Data System (ADS)

Melelli, Laura; Liucci, Luisa; Vergari, Francesca; Ciccacci, Sirio; Del Monte, Maurizio

2014-05-01

Drainage basins are primary landscape units for geomorphological investigations. Both hillslopes and river drainage system are fundamental components in drainage basins analysis. As other geomorphological systems, also the drainage basins aim to an equilibrium condition where the sequence of erosion, transport and sedimentation approach to a condition of minimum energy effort. This state is revealed by a typical geometry of landforms and of drainage net. Several morphometric indexes can measure how much a drainage basin is far from the theoretical equilibrium configuration, revealing possible external disarray. In active tectonic areas, the drainage basins have a primary importance in order to highlight style, amount and rate of tectonic impulses, and morphometric indexes allow to estimate the tectonic activity classes of different sectors in a study area. Moreover, drainage rivers are characterized by a self-similarity structure; this promotes the use of fractals theory to investigate the system. In this study, fractals techniques are employed together with quantitative geomorphological analysis to study the Upper Tiber Valley (UTV), a tectonic intermontane basin located in northern Apennines (Umbria, central Italy). The area is the result of different tectonic phases. From Late Pliocene until present time the UTV is strongly controlled by a regional uplift and by an extensional phase with different sets of normal faults playing a fundamental role in basin morphology. Thirty-four basins are taken into account for the quantitative analysis, twenty on the left side of the basin, the others on the right side. Using fractals dimension of drainage networks, Horton's laws results, concavity and steepness indexes, and hypsometric curves, this study aims to obtain an evolutionary model of the UTV, where the uplift is compared to local subsidence induced by normal fault activity. The results highlight a well defined difference between western and eastern tributary basins

Microvascular fractal dimension predicts prognosis and response to chemotherapy in glioblastoma: an automatic image analysis study.

PubMed

Chen, Cong; He, Zhi-Cheng; Shi, Yu; Zhou, Wenchao; Zhang, Xia; Xiao, Hua-Liang; Wu, Hai-Bo; Yao, Xiao-Hong; Luo, Wan-Chun; Cui, You-Hong; Bao, Shideng; Kung, Hsiang-Fu; Bian, Xiu-Wu; Ping, Yi-Fang

2018-05-15

The microvascular profile has been included in the WHO glioma grading criteria. Nevertheless, microvessels in gliomas of the same WHO grade, e.g., WHO IV glioblastoma (GBM), exhibit heterogeneous and polymorphic morphology, whose possible clinical significance remains to be determined. In this study, we employed a fractal geometry-derived parameter, microvascular fractal dimension (mvFD), to quantify microvessel complexity and developed a home-made macro in Image J software to automatically determine mvFD from the microvessel-stained immunohistochemical images of GBM. We found that mvFD effectively quantified the morphological complexity of GBM microvasculature. Furthermore, high mvFD favored the survival of GBM patients as an independent prognostic indicator and predicted a better response to chemotherapy of GBM patients. When investigating the underlying relations between mvFD and tumor growth by deploying Ki67/mvFD as an index for microvasculature-normalized tumor proliferation, we discovered an inverse correlation between mvFD and Ki67/mvFD. Furthermore, mvFD inversely correlated with the expressions of a glycolytic marker, LDHA, which indicated poor prognosis of GBM patients. Conclusively, we developed an automatic approach for mvFD measurement, and demonstrated that mvFD could predict the prognosis and response to chemotherapy of GBM patients.

A novel small-angle neutron scattering detector geometry

PubMed Central

Kanaki, Kalliopi; Jackson, Andrew; Hall-Wilton, Richard; Piscitelli, Francesco; Kirstein, Oliver; Andersen, Ken H.

2013-01-01

A novel 2π detector geometry for small-angle neutron scattering (SANS) applications is presented and its theoretical performance evaluated. Such a novel geometry is ideally suited for a SANS instrument at the European Spallation Source (ESS). Motivated by the low availability and high price of 3He, the new concept utilizes gaseous detectors with 10B as the neutron converter. The shape of the detector is inspired by an optimization process based on the properties of the conversion material. Advantages over the detector geometry traditionally used on SANS instruments are discussed. The angular and time resolutions of the proposed detector concept are shown to satisfy the requirements of the particular SANS instrument. PMID:24046504

Fractal planetary rings: Energy inequalities and random field model

NASA Astrophysics Data System (ADS)

Malyarenko, Anatoliy; Ostoja-Starzewski, Martin

2017-12-01

This study is motivated by a recent observation, based on photographs from the Cassini mission, that Saturn’s rings have a fractal structure in radial direction. Accordingly, two questions are considered: (1) What Newtonian mechanics argument in support of such a fractal structure of planetary rings is possible? (2) What kinematics model of such fractal rings can be formulated? Both challenges are based on taking planetary rings’ spatial structure as being statistically stationary in time and statistically isotropic in space, but statistically nonstationary in space. An answer to the first challenge is given through an energy analysis of circular rings having a self-generated, noninteger-dimensional mass distribution [V. E. Tarasov, Int. J. Mod Phys. B 19, 4103 (2005)]. The second issue is approached by taking the random field of angular velocity vector of a rotating particle of the ring as a random section of a special vector bundle. Using the theory of group representations, we prove that such a field is completely determined by a sequence of continuous positive-definite matrix-valued functions defined on the Cartesian square F2 of the radial cross-section F of the rings, where F is a fat fractal.

a Fractal Analysis for Net Present Value of Multi-Stage Hydraulic Fractured Horizontal Well

NASA Astrophysics Data System (ADS)

Lu, Hong-Lin; Zhang, Ji-Jun; Tan, Xiao-Hua; Li, Xiao-Ping; Zhao, Jia-Hui

Because of the low permeability, multi-stage hydraulic fractured horizontal wells (MHFHWs) occupy a dominant position among production wells in tight gas reservoir. However, net present value (NPV) estimation method for MHFHW in tight gas reservoirs often ignores the effect of heterogeneity in microscopic pore structure. Apart from that, a new fractal model is presented for NPV of MHFHW, based on the fractal expressions of formation parameters. First, with the aid of apparent permeability model, a pseudo pressure expression considering both reservoir fractal features and slippage effect is derived, contributing to establish the productivity model. Secondly, economic assessment method is built based on the fractal productivity model, in order to obtain the NPV of MHFHW. Thirdly, the type curves are illustrated and the influences of different fractal parameters are discussed. The pore fractal dimensions Df and the capillary tortuosity fractal dimensions DT have significant effects on the NPV of an MHFHW. Finally, the proposed model in this paper provides a new methodology for analyzing and predicting the NPV of an MHFHW and may be conducive to a better understanding of the optimal design of MHFHW.

Characterizing Tityus discrepans scorpion venom from a fractal perspective: Venom complexity, effects of captivity, sexual dimorphism, differences among species.

PubMed

D'Suze, Gina; Sandoval, Moisés; Sevcik, Carlos

2015-12-15

A characteristic of venom elution patterns, shared with many other complex systems, is that many their features cannot be properly described with statistical or euclidean concepts. The understanding of such systems became possible with Mandelbrot's fractal analysis. Venom elution patterns were produced using the reversed phase high performance liquid chromatography (HPLC) with 1 mg of venom. One reason for the lack of quantitative analyses of the sources of venom variability is parametrizing the venom chromatograms' complexity. We quantize this complexity by means of an algorithm which estimates the contortedness (Q) of a waveform. Fractal analysis was used to compare venoms and to measure inter- and intra-specific venom variability. We studied variations in venom complexity derived from gender, seasonal and environmental factors, duration of captivity in the laboratory, technique used to milk venom. Copyright © 2015 Elsevier Ltd. All rights reserved.

The Use of Geometry Learning Media Based on Augmented Reality for Junior High School Students

NASA Astrophysics Data System (ADS)

Rohendi, D.; Septian, S.; Sutarno, H.

2018-02-01

Understanding the geometry especially of three-dimensional space is still considered difficult by some students. Therefore, a learning innovation is required to overcome students’ difficulties in learning geometry. In this research, we developed geometry learning media based on augmented reality in android flatform’s then it was implemented in teaching three-dimensional objects for some junior high school students to find out: how is the students response in using this new media in geometry and is this media can solve the student’s difficulties in understanding geometry concept. The results showed that the use of geometry learning media based on augmented reality in android flatform is able to get positive responses from the students in learning geometry concepts especially three-dimensional objects and students more easy to understand concept of diagonal in geometry than before using this media.

Fractal Simulations of African Design in Pre-College Computing Education

ERIC Educational Resources Information Center

Eglash, Ron; Krishnamoorthy, Mukkai; Sanchez, Jason; Woodbridge, Andrew

2011-01-01

This article describes the use of fractal simulations of African design in a high school computing class. Fractal patterns--repetitions of shape at multiple scales--are a common feature in many aspects of African design. In African architecture we often see circular houses grouped in circular complexes, or rectangular houses in rectangular…

Fractality and growth of He bubbles in metals

NASA Astrophysics Data System (ADS)

Kajita, Shin; Ito, Atsushi M.; Ohno, Noriyasu

2017-08-01

Pinholes are formed on surfaces of metals by the exposure to helium plasmas, and they are regarded as the initial process of the growth of fuzzy nanostructures. In this study, number density of the pinholes is investigated in detail from the scanning electron microscope (SEM) micrographs of tungsten and tantalum exposed to the helium plasmas. A power law relation was identified between the number density and the size of pinholes. From the slope and the region where the power law was satisfied, the fractal dimension D and smin, which characterize the SEM images, are deduced. Parametric dependences and material dependence of D and smin are revealed. To explain the fractality, simple Monte-Carlo simulations including random walks of He atoms and absorption on bubble was introduced. It is shown that the initial position of the random walk is one of the key factors to deduce the fractality. The results indicated that new nucleations of bubbles are necessary to reproduce the number-density distribution of bubbles.

Fractal analysis on human dynamics of library loans

NASA Astrophysics Data System (ADS)

Fan, Chao; Guo, Jin-Li; Zha, Yi-Long

2012-12-01

In this paper, the fractal characteristic of human behaviors is investigated from the perspective of time series constructed with the amount of library loans. The values of the Hurst exponent and length of non-periodic cycle calculated through rescaled range analysis indicate that the time series of human behaviors and their sub-series are fractal with self-similarity and long-range dependence. Then the time series are converted into complex networks by the visibility algorithm. The topological properties of the networks such as scale-free property and small-world effect imply that there is a close relationship among the numbers of repetitious behaviors performed by people during certain periods of time. Our work implies that there is intrinsic regularity in the human collective repetitious behaviors. The conclusions may be helpful to develop some new approaches to investigate the fractal feature and mechanism of human dynamics, and provide some references for the management and forecast of human collective behaviors.

Monte Carlo Sampling in Fractal Landscapes

NASA Astrophysics Data System (ADS)

Leitão, Jorge C.; Lopes, J. M. Viana Parente; Altmann, Eduardo G.

2013-05-01

We design a random walk to explore fractal landscapes such as those describing chaotic transients in dynamical systems. We show that the random walk moves efficiently only when its step length depends on the height of the landscape via the largest Lyapunov exponent of the chaotic system. We propose a generalization of the Wang-Landau algorithm which constructs not only the density of states (transient time distribution) but also the correct step length. As a result, we obtain a flat-histogram Monte Carlo method which samples fractal landscapes in polynomial time, a dramatic improvement over the exponential scaling of traditional uniform-sampling methods. Our results are not limited by the dimensionality of the landscape and are confirmed numerically in chaotic systems with up to 30 dimensions.

Fractal dimension of spatially extended systems

NASA Astrophysics Data System (ADS)

Torcini, A.; Politi, A.; Puccioni, G. P.; D'Alessandro, G.

1991-10-01

Properties of the invariant measure are numerically investigated in 1D chains of diffusively coupled maps. The coarse-grained fractal dimension is carefully computed in various embedding spaces, observing an extremely slow convergence towards the asymptotic value. This is in contrast with previous simulations, where the analysis of an insufficient number of points led the authors to underestimate the increase of fractal dimension with increasing the dimension of the embedding space. Orthogonal decomposition is also performed confirming that the slow convergence is intrinsically related to local nonlinear properties of the invariant measure. Finally, the Kaplan-Yorke conjecture is tested for short chains, showing that, despite the noninvertibility of the dynamical system, a good agreement is found between Lyapunov dimension and information dimension.

Fractal and topological sustainable methods of overcoming expected uncertainty in the radiolocation of low-contrast targets and in the processing of weak multi-dimensional signals on the background of high-intensity noise: A new direction in the statistical decision theory

NASA Astrophysics Data System (ADS)

Potapov, A. A.

2017-11-01

The main purpose of this work is to interpret the main directions of radio physics, radio engineering and radio location in “fractal” language that makes new ways and generalizations on future promising radio systems. We introduce a new kind and approach of up-to-date radiolocation: fractal-scaling or scale-invariant radiolocation. The new topologic signs and methods of detecting the low-contrast objects against the high-intensity noise background are presented. It leads to basic changes in the theoretical radiolocation structure itself and also in its mathematical apparatus. The fractal radio systems conception, sampling topology, global fractal-scaling approach and the fractal paradigm underlie the scientific direction established by the author in Russia and all over the world for the first time ever.

About Schrödinger Equation on Fractals Curves Imbedding in R 3

NASA Astrophysics Data System (ADS)

Golmankhaneh, Alireza Khalili; Golmankhaneh, Ali Khalili; Baleanu, Dumitru

2015-04-01

In this paper we introduced the quantum mechanics on fractal time-space. In a suggested formalism the time and space vary on Cantor-set and Von-Koch curve, respectively. Using Feynman path method in quantum mechanics and F α -calculus we find Schrëdinger equation on on fractal time-space. The Hamiltonian and momentum fractal operator has been indicated. More, the continuity equation and the probability density is given in view of F α -calculus.

Studies in Mathematics, Volume V. Concepts of Informal Geometry. Preliminary Edition.

ERIC Educational Resources Information Center

Anderson, Richard D.

The main purpose of this book is to provide background material in geometry for teachers or prospective teachers who know little or no geometry. It should be suitable as a text for a one-semester course for teachers of junior high school or upper elementary school students. Chapters contain developmental material and exercises. Chapters include:…

Space-coiling fractal metamaterial with multi-bandgaps on subwavelength scale

NASA Astrophysics Data System (ADS)

Man, Xianfeng; Liu, Tingting; Xia, Baizhan; Luo, Zhen; Xie, Longxiang; Liu, Jian

2018-06-01

Acoustic metamaterials are remarkably different from conventional materials, as they can flexibly manipulate and control the propagation of sound waves. Unlike the locally resonant metamaterials introduced in earlier studies, we designed an ultraslow artificial structure with a sound speed much lower than that in air. In this paper, the space-coiling approach is proposed for achieving artificial metamaterial for extremely low-frequency airborne sound. In addition, the self-similar fractal technique is utilized for designing space-coiling Mie-resonance-based metamaterials (MRMMs) to obtain a band-dispersive spectrum. The band structures of two-dimensional (2D) acoustic metamaterials with different fractal levels are illustrated using the finite element method. The low-frequency bandgap can easily be formed, and multi-bandgap properties are observed in high-level fractals. Furthermore, the designed MRMMs with higher order fractal space coiling shows a good robustness against irregular arrangement. Besides, the proposed artificial structure was found to modify and control the radiation field arbitrarily. Thus, this work provides useful guidelines for the design of acoustic filtering devices and acoustic wavefront shaping applications on the subwavelength scale.

Characterization of branch complexity by fractal analyses and detect plant functional adaptations

USGS Publications Warehouse

Alados, C.L.; Escos, J.; Emlen, J.M.; Freeman, D.C.

1999-01-01

The comparison between complexity in the sense of space occupancy (box-counting fractal dimension Dc and information dimension DI ) and heterogeneity in the sense of space distribution (average evenness index and evenness variation coefficient JCV) were investigated in mathematical fractal objects and natural branch ¯ J structures. In general, increased fractal dimension was paired with low heterogeneity. Comparisons between branch architecture in Anthyllis cytisoides under different slope exposure and grazing impact revealed that branches were more complex and more homogeneously distributed for plants on northern exposures than southern, while grazing had no impact during a wet year. Developmental instability was also investigated by the statistical noise of the allometric relation between internode length and node order. In conclusion, our study demonstrated that fractal dimension of branch structure can be used to analyze the structural organization of plants, especially if we consider not only fractal dimension but also shoot distribution within the canopy (lacunarity). These indexes together with developmental instability analyses are good indicators of growth responses to the environment.

Fractal structure of sequential behaviour patterns: an indicator of stress

USGS Publications Warehouse

Alados, C.L.; Escos, J.M; Emlen, J.M.

1996-01-01

The detection of stress arising from parasitic infection bySarcoptes scabieisand from pregnancy is explored, using a fractal analysis of head lifting behaviour and feeding–non-feeding activity sequences in female Spanish ibex,Capra pyrenaica, under natural conditions. Because organisms under stress increase their metabolic rate and, in consequence, energy consumption, it follows that stress will, generally, lead to a reduction in complexity (fractal dimension) of exploratory behaviour. In the present study the fractal dimension of the three measures of complexity used declined with stress, both from pregnancy and from parasitic infection. This observation provides a new and effective way to assess the general state of animals’ health in the field, without the need for capture and handling.

Archaeon and archaeal virus diversity classification via sequence entropy and fractal dimension

NASA Astrophysics Data System (ADS)

Tremberger, George, Jr.; Gallardo, Victor; Espinoza, Carola; Holden, Todd; Gadura, N.; Cheung, E.; Schneider, P.; Lieberman, D.; Cheung, T.

2010-09-01

Archaea are important potential candidates in astrobiology as their metabolism includes solar, inorganic and organic energy sources. Archaeal viruses would also be expected to be present in a sustainable archaeal exobiological community. Genetic sequence Shannon entropy and fractal dimension can be used to establish a two-dimensional measure for classification and phylogenetic study of these organisms. A sequence fractal dimension can be calculated from a numerical series consisting of the atomic numbers of each nucleotide. Archaeal 16S and 23S ribosomal RNA sequences were studied. Outliers in the 16S rRNA fractal dimension and entropy plot were found to be halophilic archaea. Positive correlation (R-square ~ 0.75, N = 18) was observed between fractal dimension and entropy across the studied species. The 16S ribosomal RNA sequence entropy correlates with the 23S ribosomal RNA sequence entropy across species with R-square 0.93, N = 18. Entropy values correspond positively with branch lengths of a published phylogeny. The studied archaeal virus sequences have high fractal dimensions of 2.02 or more. A comparison of selected extremophile sequences with archaeal sequences from the Humboldt Marine Ecosystem database (Wood-Hull Oceanography Institute, MIT) suggests the presence of continuous sequence expression as inferred from distributions of entropy and fractal dimension, consistent with the diversity expected in an exobiological archaeal community.

Comparison of Pore Fractal Characteristics Between Marine and Continental Shales

NASA Astrophysics Data System (ADS)

Liu, Jun; Yao, Yanbin; Liu, Dameng; Cai, Yidong; Cai, Jianchao

Fractal characterization offers a quantitative evaluation on the heterogeneity of pore structure which greatly affects gas adsorption and transportation in shales. To compare the fractal characteristics between marine and continental shales, nine samples from the Lower Silurian Longmaxi formation in the Sichuan basin and nine from the Middle Jurassic Dameigou formation in the Qaidam basin were collected. Reservoir properties and fractal dimensions were characterized for all the collected samples. In this study, fractal dimensions were originated from the Frenkel-Halsey-Hill (FHH) model with N2 adsorption data. Compared to continental shale, marine shale has greater values of quartz content, porosity, specific surface area and total pore volume but lower level of clay minerals content, permeability, average pore diameter and methane adsorption capacity. The quartz in marine shale is mostly associated with biogenic origin, while that in continental shale is mainly due to terrigenous debris. The N2 adsorption-desorption isotherms exhibit that marine shale has fewer inkbottle-shaped pores but more plate-like and slit-shaped pores than continental shale. Two fractal dimensions (D1 and D2) were obtained at P/Po of 0-0.5 and 0.5-1. The dimension D2 is commonly greater than D1, suggesting that larger pores (diameter >˜ 4nm) have more complex structures than small pores (diameter <˜ 4nm). The fractal dimensions (both D1 and D2) positively correlate to clay minerals content, specific surface area and methane adsorption capacity, but have negative relationships with porosity, permeability and average pore diameter. The fractal dimensions increase proportionally with the increasing quartz content in marine shale but have no obvious correlation with that in continental shale. The dimension D1 is correlative to the TOC content and permeability of marine shale at a similar degree with dimension D2, while the dimension D1 is more sensitive to those of continental shale than

Experimental criteria for the determination of fractal parameters of premixed turbulent flames

NASA Astrophysics Data System (ADS)

Shepherd, I. G.; Cheng, Robert K.; Talbot, L.

1992-10-01

The influence of spatial resolution, digitization noise, the number of records used for averaging, and the method of analysis on the determination of the fractal parameters of a high Damköhler number, methane/air, premixed, turbulent stagnation-point flame are investigated in this paper. The flow exit velocity was 5 m/s and the turbulent Reynolds number was 70 based on a integral scale of 3 mm and a turbulent intensity of 7%. The light source was a copper vapor laser which delivered 20 nsecs, 5 mJ pulses at 4 kHz and the tomographic cross-sections of the flame were recorded by a high speed movie camera. The spatial resolution of the images is 155 × 121 μm/pixel with a field of view of 50 × 65 mm. The stepping caliper technique for obtaining the fractal parameters is found to give the clearest indication of the cutoffs and the effects of noise. It is necessary to ensemble average the results from more than 25 statistically independent images to reduce sufficiently the scatter in the fractal parameters. The effects of reduced spatial resolution on fractal plots are estimated by artificial degradation of the resolution of the digitized flame boundaries. The effect of pixel resolution, an apparent increase in flame length below the inner scale rolloff, appears in the fractal plots when the measurent scale is less than approximately twice the pixel resolution. Although a clearer determination of fractal parameters is obtained by local averaging of the flame boundaries which removes digitization noise, at low spatial resolution this technique can reduce the fractal dimension. The degree of fractal isotropy of the flame surface can have a significant effect on the estimation of the flame surface area and hence burning rate from two-dimensional images. To estimate this isotropy a determination of the outer cutoff is required and three-dimensional measurements are probably also necessary.

Fractal analysis of the hydraulic conductivity on a sandy porous media reproduced in a laboratory facility.

NASA Astrophysics Data System (ADS)

de Bartolo, S.; Fallico, C.; Straface, S.; Troisi, S.; Veltri, M.

2009-04-01

The complexity characterization of the porous media structure, in terms of the "pore" phase and the "solid" phase, can be carried out by means of the fractal geometry which is able to put in relationship the soil structural properties and the water content. It is particularly complicated to describe analytically the hydraulic conductivity for the irregularity of the porous media structure. However these can be described by many fractal models considering the soil structure as the distribution of particles dimensions, the distribution of the solid aggregates, the surface of the pore-solid interface and the fractal mass of the "pore" and "solid" phases. In this paper the fractal model of Yu and Cheng (2002) and Yu and Liu (2004), for a saturated bidispersed porous media, was considered. This model, using the Sierpinsky-type gasket scheme, doesn't contain empiric constants and furnishes a well accord with the experimental data. For this study an unconfined aquifer was reproduced by means of a tank with a volume of 10 Ã- 7 Ã- 3 m3, filled with a homogeneous sand (95% of SiO2), with a high percentage (86.4%) of grains between 0.063mm and 0.125mm and a medium-high permeability. From the hydraulic point of view, 17 boreholes, a pumping well and a drainage ring around its edge were placed. The permeability was measured utilizing three different methods, consisting respectively in pumping test, slug test and laboratory analysis of an undisturbed soil cores, each of that involving in the measurement a different support volume. The temporal series of the drawdown obtained by the pumping test were analyzed by the Neuman-type Curve method (1972), because the saturated part above the bottom of the facility represents an unconfined aquifer. The data analysis of the slug test were performed by the Bouwer & Rice (1976) method and the laboratory analysis were performed on undisturbed saturated soil samples utilizing a falling head permeameter. The obtained values either of the

The generalized 20/80 law using probabilistic fractals applied to petroleum field size

USGS Publications Warehouse

Crovelli, R.A.

1995-01-01

Fractal properties of the Pareto probability distribution are used to generalize "the 20/80 law." The 20/80 law is a heuristic law that has evolved over the years into the following rule of thumb for many populations: 20 percent of the population accounts for 80 percent of the total value. The general p100/q100 law in probabilistic form is defined with q as a function of p, where p is the population proportion and q is the proportion of total value. Using the Pareto distribution, the p100/q100 law in fractal form is derived with the parameter q being a fractal, where q unexpectedly possesses the scale invariance property. The 20/80 law is a special case of the p100/q100 law in fractal form. The p100/q100 law in fractal form is applied to petroleum fieldsize data to obtain p and q such that p100% of the oil fields greater than any specified scale or size in a geologic play account for q100% of the total oil of the fields. The theoretical percentages of total resources of oil using the fractal q are extremely close to the empirical percentages from the data using the statistic q. Also, the empirical scale invariance property of the statistic q for the petroleum fieldsize data is in excellent agreement with the theoretical scale invariance property of the fractal q. ?? 1995 Oxford University Press.

Improved visibility graph fractality with application for the diagnosis of Autism Spectrum Disorder

NASA Astrophysics Data System (ADS)

Ahmadlou, Mehran; Adeli, Hojjat; Adeli, Amir

2012-10-01

Recently, the visibility graph (VG) algorithm was proposed for mapping a time series to a graph to study complexity and fractality of the time series through investigation of the complexity of its graph. The visibility graph algorithm converts a fractal time series to a scale-free graph. VG has been used for the investigation of fractality in the dynamic behavior of both artificial and natural complex systems. However, robustness and performance of the power of scale-freeness of VG (PSVG) as an effective method for measuring fractality has not been investigated. Since noise is unavoidable in real life time series, the robustness of a fractality measure is of paramount importance. To improve the accuracy and robustness of PSVG to noise for measurement of fractality of time series in biological time-series, an improved PSVG is presented in this paper. The proposed method is evaluated using two examples: a synthetic benchmark time series and a complicated real life Electroencephalograms (EEG)-based diagnostic problem, that is distinguishing autistic children from non-autistic children. It is shown that the proposed improved PSVG is less sensitive to noise and therefore more robust compared with PSVG. Further, it is shown that using improved PSVG in the wavelet-chaos neural network model of Adeli and c-workers in place of the Katz fractality dimension results in a more accurate diagnosis of autism, a complicated neurological and psychiatric disorder.

Fractal modeling of fluidic leakage through metal sealing surfaces

NASA Astrophysics Data System (ADS)

Zhang, Qiang; Chen, Xiaoqian; Huang, Yiyong; Chen, Yong

2018-04-01

This paper investigates the fluidic leak rate through metal sealing surfaces by developing fractal models for the contact process and leakage process. An improved model is established to describe the seal-contact interface of two metal rough surface. The contact model divides the deformed regions by classifying the asperities of different characteristic lengths into the elastic, elastic-plastic and plastic regimes. Using the improved contact model, the leakage channel under the contact surface is mathematically modeled based on the fractal theory. The leakage model obtains the leak rate using the fluid transport theory in porous media, considering that the pores-forming percolation channels can be treated as a combination of filled tortuous capillaries. The effects of fractal structure, surface material and gasket size on the contact process and leakage process are analyzed through numerical simulations for sealed ring gaskets.

Fractal Stiffening

NASA Technical Reports Server (NTRS)

Harper, David William (Inventor)

2017-01-01

A structural support having fractal-stiffening and method of fabricating the support is presented where an optimized location of at least three nodes is predetermined prior to fabricating the structural support where a first set of webs is formed on one side of the support and joined to the nodes to form a first pocket region. A second set of webs is formed within the first pocket region forming a second pocket region where the height of the first set of webs extending orthogonally from the side of the support is greater than the second set of webs extending orthogonally from the support.

Creating Dynamic Learning Environment to Enhance Students’ Engagement in Learning Geometry

NASA Astrophysics Data System (ADS)

Sariyasa

2017-04-01

Learning geometry gives many benefits to students. It strengthens the development of deductive thinking and reasoning; it also provides an opportunity to improve visualisation and spatial ability. Some studies, however, have pointed out the difficulties that students encountered when learning geometry. A preliminary study by the author in Bali revealed that one of the main problems was teachers’ difficulties in delivering geometry instruction. It was partly due to the lack of appropriate instructional media. Coupling with dynamic geometry software, dynamic learning environments is a promising solution to this problem. Employing GeoGebra software supported by the well-designed instructional process may result in more meaningful learning, and consequently, students are motivated to engage in the learning process more deeply and actively. In this paper, we provide some examples of GeoGebra-aided learning activities that allow students to interactively explore and investigate geometry concepts and the properties of geometry objects. Thus, it is expected that such learning environment will enhance students’ internalisation process of geometry concepts.

Long-range (fractal) correlations in the LEDA database.

NASA Astrophysics Data System (ADS)

di Nella, H.; Montuori, M.; Paturel, G.; Pietronero, L.; Sylos Labini, F.

1996-04-01

All the recent redshift surveys show highly irregular patterns of galaxies on scales of hundreds of megaparsecs such as chains, walls and cells. One of the most powerful catalog of galaxies is represented by the LEDA database that contains more than 36,000 galaxies with redshift. We study the correlation properties of such a sample finding that galaxy distribution shows well defined fractal nature up to R_S_~150h^-1^Mpc with fractal dimension D~2. We test the consistency of these results versus the incompleteness in the sample.

Time Series Analysis OF SAR Image Fractal Maps: The Somma-Vesuvio Volcanic Complex Case Study

NASA Astrophysics Data System (ADS)

Pepe, Antonio; De Luca, Claudio; Di Martino, Gerardo; Iodice, Antonio; Manzo, Mariarosaria; Pepe, Susi; Riccio, Daniele; Ruello, Giuseppe; Sansosti, Eugenio; Zinno, Ivana

2016-04-01

The fractal dimension is a significant geophysical parameter describing natural surfaces representing the distribution of the roughness over different spatial scale; in case of volcanic structures, it has been related to the specific nature of materials and to the effects of active geodynamic processes. In this work, we present the analysis of the temporal behavior of the fractal dimension estimates generated from multi-pass SAR images relevant to the Somma-Vesuvio volcanic complex (South Italy). To this aim, we consider a Cosmo-SkyMed data-set of 42 stripmap images acquired from ascending orbits between October 2009 and December 2012. Starting from these images, we generate a three-dimensional stack composed by the corresponding fractal maps (ordered according to the acquisition dates), after a proper co-registration. The time-series of the pixel-by-pixel estimated fractal dimension values show that, over invariant natural areas, the fractal dimension values do not reveal significant changes; on the contrary, over urban areas, it correctly assumes values outside the natural surfaces fractality range and show strong fluctuations. As a final result of our analysis, we generate a fractal map that includes only the areas where the fractal dimension is considered reliable and stable (i.e., whose standard deviation computed over the time series is reasonably small). The so-obtained fractal dimension map is then used to identify areas that are homogeneous from a fractal viewpoint. Indeed, the analysis of this map reveals the presence of two distinctive landscape units corresponding to the Mt. Vesuvio and Gran Cono. The comparison with the (simplified) geological map clearly shows the presence in these two areas of volcanic products of different age. The presented fractal dimension map analysis demonstrates the ability to get a figure about the evolution degree of the monitored volcanic edifice and can be profitably extended in the future to other volcanic systems with

Diagnostics of multi-fractality of magnetized plasma inside coronal holes and quiet sun areas

NASA Astrophysics Data System (ADS)

Abramenko, Valentyna

Turbulent and multi-fractal properties of magnetized plasma in solar Coronal Holes (CHs) and Quiet Sun (QS) photosphere were explored using high-resolution magnetograms measured with the New Solar Telescope (NST) at the Big Bear Solar Observatory (BBSO, USA), Hinode/SOT and SDO/HMI instruments. Distribution functions of size and magnetic flux measured for small-scale magnetic elements follow the log-normal law, which implies multi-fractal organization of the magnetic field and the absence of a unique power law for all scales. The magnetograms show multi-fractality in CHs on scales 400 - 10000 km, which becomes better pronounced as the spatial resolution of data improves. Photospheric granulation measured with NST exhibits multi-fractal properties on very small scales of 50 - 600 km. While multi-fractal nature of solar active regions is well known, newly established multi-fractality of weakest magnetic fields on the solar surface, i.e., in CHs and QS, leads us to a conclusion that the entire variety of solar magnetic fields is generated by a unique nonlinear dynamical process.

Fractal Tempo Fluctuation and Pulse Prediction

PubMed Central

Rankin, Summer K.; Large, Edward W.; Fink, Philip W.

2010-01-01

WE INVESTIGATED PEOPLES’ ABILITY TO ADAPT TO THE fluctuating tempi of music performance. In Experiment 1, four pieces from different musical styles were chosen, and performances were recorded from a skilled pianist who was instructed to play with natural expression. Spectral and rescaled range analyses on interbeat interval time-series revealed long-range (1/f type) serial correlations and fractal scaling in each piece. Stimuli for Experiment 2 included two of the performances from Experiment 1, with mechanical versions serving as controls. Participants tapped the beat at ¼- and ⅛-note metrical levels, successfully adapting to large tempo fluctuations in both performances. Participants predicted the structured tempo fluctuations, with superior performance at the ¼-note level. Thus, listeners may exploit long-range correlations and fractal scaling to predict tempo changes in music. PMID:25190901

Fractal Geometry in Physics

DTIC Science & Technology

1993-09-30

AD-A273 271 OFFICE OF NAVAL RESEARCH 3-0 1993 CONTRACT NUMBER: N00014-90-J- 1 30 MODIFICATION NO: P00005 19 R&T PROJECT: 4123041-10 & 11 A S.O. CODE...investigated. Indirectly related to these topics is a papaer by D. Stauffer, A. Aharony and B. Man- delbrot in Phy•sica A 196, 1993, 1-5. It evaluates

Performance assessment of methods for estimation of fractal dimension from scanning electron microscope images.

PubMed

Risović, Dubravko; Pavlović, Zivko

2013-01-01

Processing of gray scale images in order to determine the corresponding fractal dimension is very important due to widespread use of imaging technologies and application of fractal analysis in many areas of science, technology, and medicine. To this end, many methods for estimation of fractal dimension from gray scale images have been developed and routinely used. Unfortunately different methods (dimension estimators) often yield significantly different results in a manner that makes interpretation difficult. Here, we report results of comparative assessment of performance of several most frequently used algorithms/methods for estimation of fractal dimension. To that purpose, we have used scanning electron microscope images of aluminum oxide surfaces with different fractal dimensions. The performance of algorithms/methods was evaluated using the statistical Z-score approach. The differences between performances of six various methods are discussed and further compared with results obtained by electrochemical impedance spectroscopy on the same samples. The analysis of results shows that the performance of investigated algorithms varies considerably and that systematically erroneous fractal dimensions could be estimated using certain methods. The differential cube counting, triangulation, and box counting algorithms showed satisfactory performance in the whole investigated range of fractal dimensions. Difference statistic is proved to be less reliable generating 4% of unsatisfactory results. The performances of the Power spectrum, Partitioning and EIS were unsatisfactory in 29%, 38%, and 75% of estimations, respectively. The results of this study should be useful and provide guidelines to researchers using/attempting fractal analysis of images obtained by scanning microscopy or atomic force microscopy. © Wiley Periodicals, Inc.

Proliferative diabetic retinopathy characterization based on fractal features: Evaluation on a publicly available dataset.

PubMed

Orlando, José Ignacio; van Keer, Karel; Barbosa Breda, João; Manterola, Hugo Luis; Blaschko, Matthew B; Clausse, Alejandro

2017-12-01

Diabetic retinopathy (DR) is one of the most widespread causes of preventable blindness in the world. The most dangerous stage of this condition is proliferative DR (PDR), in which the risk of vision loss is high and treatments are less effective. Fractal features of the retinal vasculature have been previously explored as potential biomarkers of DR, yet the current literature is inconclusive with respect to their correlation with PDR. In this study, we experimentally assess their discrimination ability to recognize PDR cases. A statistical analysis of the viability of using three reference fractal characterization schemes - namely box, information, and correlation dimensions - to identify patients with PDR is presented. These descriptors are also evaluated as input features for training ℓ1 and ℓ2 regularized logistic regression classifiers, to estimate their performance. Our results on MESSIDOR, a public dataset of 1200 fundus photographs, indicate that patients with PDR are more likely to exhibit a higher fractal dimension than healthy subjects or patients with mild levels of DR (P≤1.3×10-2). Moreover, a supervised classifier trained with both fractal measurements and red lesion-based features reports an area under the ROC curve of 0.93 for PDR screening and 0.96 for detecting patients with optic disc neovascularizations. The fractal dimension of the vasculature increases with the level of DR. Furthermore, PDR screening using multiscale fractal measurements is more feasible than using their derived fractal dimensions. Code and further resources are provided at https://github.com/ignaciorlando/fundus-fractal-analysis. © 2017 American Association of Physicists in Medicine.

A Lossless hybrid wavelet-fractal compression for welding radiographic images.

PubMed

Mekhalfa, Faiza; Avanaki, Mohammad R N; Berkani, Daoud

2016-01-01

In this work a lossless wavelet-fractal image coder is proposed. The process starts by compressing and decompressing the original image using wavelet transformation and fractal coding algorithm. The decompressed image is removed from the original one to obtain a residual image which is coded by using Huffman algorithm. Simulation results show that with the proposed scheme, we achieve an infinite peak signal to noise ratio (PSNR) with higher compression ratio compared to typical lossless method. Moreover, the use of wavelet transform speeds up the fractal compression algorithm by reducing the size of the domain pool. The compression results of several welding radiographic images using the proposed scheme are evaluated quantitatively and compared with the results of Huffman coding algorithm.

Association between stride time fractality and gait adaptability during unperturbed and asymmetric walking.

PubMed

Ducharme, Scott W; Liddy, Joshua J; Haddad, Jeffrey M; Busa, Michael A; Claxton, Laura J; van Emmerik, Richard E A

2018-04-01

Human locomotion is an inherently complex activity that requires the coordination and control of neurophysiological and biomechanical degrees of freedom across various spatiotemporal scales. Locomotor patterns must constantly be altered in the face of changing environmental or task demands, such as heterogeneous terrains or obstacles. Variability in stride times occurring at short time scales (e.g., 5-10 strides) is statistically correlated to larger fluctuations occurring over longer time scales (e.g., 50-100 strides). This relationship, known as fractal dynamics, is thought to represent the adaptive capacity of the locomotor system. However, this has not been tested empirically. Thus, the purpose of this study was to determine if stride time fractality during steady state walking associated with the ability of individuals to adapt their gait patterns when locomotor speed and symmetry are altered. Fifteen healthy adults walked on a split-belt treadmill at preferred speed, half of preferred speed, and with one leg at preferred speed and the other at half speed (2:1 ratio asymmetric walking). The asymmetric belt speed condition induced gait asymmetries that required adaptation of locomotor patterns. The slow speed manipulation was chosen in order to determine the impact of gait speed on stride time fractal dynamics. Detrended fluctuation analysis was used to quantify the correlation structure, i.e., fractality, of stride times. Cross-correlation analysis was used to measure the deviation from intended anti-phasing between legs as a measure of gait adaptation. Results revealed no association between unperturbed walking fractal dynamics and gait adaptability performance. However, there was a quadratic relationship between perturbed, asymmetric walking fractal dynamics and adaptive performance during split-belt walking, whereby individuals who exhibited fractal scaling exponents that deviated from 1/f performed the poorest. Compared to steady state preferred walking

Fractal dimension analysis of weight-bearing bones of rats during skeletal unloading

NASA Technical Reports Server (NTRS)

Pornprasertsuk, S.; Ludlow, J. B.; Webber, R. L.; Tyndall, D. A.; Sanhueza, A. I.; Yamauchi, M.

2001-01-01

Fractal analysis was used to quantify changes in trabecular bone induced through the use of a rat tail-suspension model to simulate microgravity-induced osteopenia. Fractal dimensions were estimated from digitized radiographs obtained from tail-suspended and ambulatory rats. Fifty 4-month-old male Sprague-Dawley rats were divided into groups of 24 ambulatory (control) and 26 suspended (test) animals. Rats of both groups were killed after periods of 1, 4, and 8 weeks. Femurs and tibiae were removed and radiographed with standard intraoral films and digitized using a flatbed scanner. Square regions of interest were cropped at proximal, middle, and distal areas of each bone. Fractal dimensions were estimated from slopes of regression lines fitted to circularly averaged plots of log power vs. log spatial frequency. The results showed that the computed fractal dimensions were significantly greater for images of trabecular bones from tail-suspended groups than for ambulatory groups (p < 0.01) at 1 week. Periods between 1 and 4 weeks likewise yielded significantly different estimates (p < 0.05), consistent with an increase in bone loss. In the tibiae, the proximal regions of the suspended group produced significantly greater fractal dimensions than other regions (p < 0.05), which suggests they were more susceptible to unloading. The data are consistent with other studies demonstrating osteopenia in microgravity environments and the regional response to skeletal unloading. Thus, fractal analysis could be a useful technique to evaluate the structural changes of bone.

The Twinkling Fractal Theory of the Glass Transition: Applications to Soft Matter

NASA Astrophysics Data System (ADS)

Wool, Richard

2012-02-01

The Twinkling Fractal Theory (TFT) of the glass transition has recently been demonstrated experimentally [J.F. Stanzione et al., J. Non Cryst. Sol., (2011, 357,311]. The hard to-soft matter transition is characterized by the presence of solid fractal clusters with liquid-like pools that are dynamically interchanging via their anharmonic intermolecular potentials with Boltzmann energy populations with a characteristic temperature dependent vibrational density of states g(φ) ˜ φ^df . The twinkling fractal frequencies φ cover a range of 10^12 Hz to 10-10Hz and the fractal solid clusters of size R have a lifetime τ ˜ R^Df/df, where the fractal dimension Df 2.4 and the fracton dimension df = 4/3. Here we explore its application to a number of soft matter issues. These include (a) Confinement effects on Tg reduction in thin films of thickness h, where by virtue of large cluster exclusion, δTg ˜ 1/h^Df/df; (b) Tg gradients near bulk surfaces, where the smaller clusters on the surface have a faster relaxation time; (c) Effect of twinkling surfaces on cell growth, where at T Tg + 20 C, there exists a twinkling fractal range that leads to bell-shaped enhancement of cell growth and chemical up-regulation via the twinkling surfaces ``communicating `` with the cells through their vibrations; and (d) adhesion above and below Tg where topological fluctuations associated with g(φ) promotes the development of nano-nails at the interface.

Fractal dimension of microbead assemblies used for protein detection.

PubMed

Hecht, Ariel; Commiskey, Patrick; Lazaridis, Filippos; Argyrakis, Panos; Kopelman, Raoul

2014-11-10

We use fractal analysis to calculate the protein concentration in a rotating magnetic assembly of microbeads of size 1 μm, which has optimized parameters of sedimentation, binding sites and magnetic volume. We utilize the original Forrest-Witten method, but due to the relatively small number of bead particles, which is of the order of 500, we use a large number of origins and also a large number of algorithm iterations. We find a value of the fractal dimension in the range 1.70-1.90, as a function of the thrombin concentration, which plays the role of binding the microbeads together. This is in good agreement with previous results from magnetorotation studies. The calculation of the fractal dimension using multiple points of reference can be used for any assembly with a relatively small number of particles. © 2014 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim.

Fractal serpentine-shaped design for stretchable wireless strain sensors

NASA Astrophysics Data System (ADS)

Dong, Wentao; Cheng, Xiao; Wang, Xiaoming; Zhang, Hailiang

2018-07-01

Stretchable sensors have been widely applied to biological fields due to their unique capacity to integrate with soft materials and curvilinear surfaces. The article presents the fractal serpentine-shaped design for stretchable wireless strain sensor which is operating around 1.6 GHz. The wireless passive LC sensor is formed by a fractal serpentine-shaped inductor coil and a concentric coplanar capacitor. The inductance of the fractal serpentine-shaped coil varies with the deformation of the wireless sensor, and the resonance frequency also varies with the applied strain of the wireless sensor embedded in soft substrate. The 40% stretchability of wireless sensor is verified by finite element analysis (FEA). Strain response of the stretchable wireless sensor has been characterized by experiments and demonstrates high strain responsivity about 6.74 MHz/1%. The stretchable wireless sensor has the potential to be used in biological and wearable applications.

Excitation of Terahertz Charge Transfer Plasmons in Metallic Fractal Structures

NASA Astrophysics Data System (ADS)

Ahmadivand, Arash; Gerislioglu, Burak; Sinha, Raju; Vabbina, Phani Kiran; Karabiyik, Mustafa; Pala, Nezih

2017-08-01

There have been extensive researches on terahertz (THz) plasmonic structures supporting resonant modes to demonstrate nano and microscale devices with high efficiency and responsivity as well as frequency selectivity. Here, using antisymmetric plasmonic fractal Y-shaped (FYS) structures as building blocks, we introduce a highly tunable four-member fractal assembly to support charge transfer plasmons (CTPs) and classical dipolar resonant modes with significant absorption cross section in the THz domain. We first present that the unique geometrical nature of the FYS system and corresponding spectral response allow for supporting intensified dipolar plasmonic modes under polarised light exposure in a standalone structure. In addition to classical dipolar mode, for the very first time, we demonstrated CTPs in the THz domain due to the direct shuttling of the charges across the metallic fractal microantenna which led to sharp resonant absorption peaks. Using both numerical and experimental studies, we have investigated and confirmed the excitation of the CTP modes and highly tunable spectral response of the proposed plasmonic fractal structure. This understanding opens new and promising horizons for tightly integrated THz devices with high efficiency and functionality.

Down syndrome's brain dynamics: analysis of fractality in resting state.

PubMed

Hemmati, Sahel; Ahmadlou, Mehran; Gharib, Masoud; Vameghi, Roshanak; Sajedi, Firoozeh

2013-08-01

To the best knowledge of the authors there is no study on nonlinear brain dynamics of down syndrome (DS) patients, whereas brain is a highly complex and nonlinear system. In this study, fractal dimension of EEG, as a key characteristic of brain dynamics, showing irregularity and complexity of brain dynamics, was used for evaluation of the dynamical changes in the DS brain. The results showed higher fractality of the DS brain in almost all regions compared to the normal brain, which indicates less centrality and higher irregular or random functioning of the DS brain regions. Also, laterality analysis of the frontal lobe showed that the normal brain had a right frontal laterality of complexity whereas the DS brain had an inverse pattern (left frontal laterality). Furthermore, the high accuracy of 95.8 % obtained by enhanced probabilistic neural network classifier showed the potential of nonlinear dynamic analysis of the brain for diagnosis of DS patients. Moreover, the results showed that the higher EEG fractality in DS is associated with the higher fractality in the low frequencies (delta and theta), in broad regions of the brain, and the high frequencies (beta and gamma), majorly in the frontal regions.

Design and analysis microstrip dipole using fractal Koch for 433 MHz applications

NASA Astrophysics Data System (ADS)

Zulfin, M.; Rambe, A. H.; Budi, B.

2018-02-01

This paper discussed the dipole microstrip antenna design using fractal Koch for working on frequency of 433 MHz. The fractal Koch was used to reduce the size of the microstrip antenna. The smaller the antenna size, the lighter the equipment. AWR simulator was employed to evaluate antenna parameters such as return loss, gain and radiation pattern. The antenna was designed on a FR4 substrate with relative permittivity of 4.4 and thickness 1.6 mm. The result shows that the fractal Koch reduce antenna size about 41.2% and decrease return loss about 30%.

Loss of 'complexity' and aging. Potential applications of fractals and chaos theory to senescence

NASA Technical Reports Server (NTRS)

Lipsitz, L. A.; Goldberger, A. L.

1992-01-01

The concept of "complexity," derived from the field of nonlinear dynamics, can be adapted to measure the output of physiologic processes that generate highly variable fluctuations resembling "chaos." We review data suggesting that physiologic aging is associated with a generalized loss of such complexity in the dynamics of healthy organ system function and hypothesize that such loss of complexity leads to an impaired ability to adapt to physiologic stress. This hypothesis is supported by observations showing an age-related loss of complex variability in multiple physiologic processes including cardiovascular control, pulsatile hormone release, and electroencephalographic potentials. If further research supports this hypothesis, measures of complexity based on chaos theory and the related geometric concept of fractals may provide new ways to monitor senescence and test the efficacy of specific interventions to modify the age-related decline in adaptive capacity.

A spectrum fractal feature classification algorithm for agriculture crops with hyper spectrum image

NASA Astrophysics Data System (ADS)

Su, Junying

2011-11-01

A fractal dimension feature analysis method in spectrum domain for hyper spectrum image is proposed for agriculture crops classification. Firstly, a fractal dimension calculation algorithm in spectrum domain is presented together with the fast fractal dimension value calculation algorithm using the step measurement method. Secondly, the hyper spectrum image classification algorithm and flowchart is presented based on fractal dimension feature analysis in spectrum domain. Finally, the experiment result of the agricultural crops classification with FCL1 hyper spectrum image set with the proposed method and SAM (spectral angle mapper). The experiment results show it can obtain better classification result than the traditional SAM feature analysis which can fulfill use the spectrum information of hyper spectrum image to realize precision agricultural crops classification.

The fractal architecture of cytoplasmic organization: scaling, kinetics and emergence in metabolic networks.

PubMed

Aon, Miguel Antonio; O'Rourke, Brian; Cortassa, Sonia

2004-01-01

In this work, we highlight the links between fractals and scaling in cells and explore the kinetic consequences for biochemical reactions operating in fractal media. Based on the proposal that the cytoskeletal architecture is organized as a percolation lattice, with clusters emerging as fractal forms, the analysis of kinetics in percolation clusters is especially emphasized. A key consequence of this spatiotemporal cytoplasmic organization is that enzyme reactions following Michaelis-Menten or allosteric type kinetics exhibit higher rates in fractal media (for short times and at lower substrate concentrations) at the percolation threshold than in Euclidean media. As a result, considerably faster and higher amplification of enzymatic activity is obtained. Finally, we describe some of the properties bestowed by cytoskeletal organization and dynamics on metabolic networks.

Fractality of pulsatile flow in speckle images

NASA Astrophysics Data System (ADS)

Nemati, M.; Kenjeres, S.; Urbach, H. P.; Bhattacharya, N.

2016-05-01

The scattering of coherent light from a system with underlying flow can be used to yield essential information about dynamics of the process. In the case of pulsatile flow, there is a rapid change in the properties of the speckle images. This can be studied using the standard laser speckle contrast and also the fractality of images. In this paper, we report the results of experiments performed to study pulsatile flow with speckle images, under different experimental configurations to verify the robustness of the techniques for applications. In order to study flow under various levels of complexity, the measurements were done for three in-vitro phantoms and two in-vivo situations. The pumping mechanisms were varied ranging from mechanical pumps to the human heart for the in vivo case. The speckle images were analyzed using the techniques of fractal dimension and speckle contrast analysis. The results of these techniques for the various experimental scenarios were compared. The fractal dimension is a more sensitive measure to capture the complexity of the signal though it was observed that it is also extremely sensitive to the properties of the scattering medium and cannot recover the signal for thicker diffusers in comparison to speckle contrast.

Interfacial contact stiffness of fractal rough surfaces.

PubMed

Zhang, Dayi; Xia, Ying; Scarpa, Fabrizio; Hong, Jie; Ma, Yanhong

2017-10-09

In this work we describe a theoretical model that predicts the interfacial contact stiffness of fractal rough surfaces by considering the effects of elastic and plastic deformations of the fractal asperities. We also develop an original test rig that simulates dovetail joints for turbo machinery blades, which can fine tune the normal contact load existing between the contacting surfaces of the blade root. The interfacial contact stiffness is obtained through an inverse identification method in which finite element simulations are fitted to the experimental results. Excellent agreement is observed between the contact stiffness predicted by the theoretical model and by the analogous experimental results. We demonstrate that the contact stiffness is a power law function of the normal contact load with an exponent α within the whole range of fractal dimension D(1 < D < 2). We also show that for 1 < D < 1.5 the Pohrt-Popov behavior (α = 1/(3 - D)) is valid, however for 1.5 < D < 2, the exponent α is different and equal to 2(D - 1)/D. The diversity between the model developed in the work and the Pohrt-Popov one is explained in detail.

Fractal Patterns of Neural Activity Exist within the Suprachiasmatic Nucleus and Require Extrinsic Network Interactions

PubMed Central

Hu, Kun; Meijer, Johanna H.; Shea, Steven A.; vanderLeest, Henk Tjebbe; Pittman-Polletta, Benjamin; Houben, Thijs; van Oosterhout, Floor; Deboer, Tom; Scheer, Frank A. J. L.

2012-01-01

The mammalian central circadian pacemaker (the suprachiasmatic nucleus, SCN) contains thousands of neurons that are coupled through a complex network of interactions. In addition to the established role of the SCN in generating rhythms of ∼24 hours in many physiological functions, the SCN was recently shown to be necessary for normal self-similar/fractal organization of motor activity and heart rate over a wide range of time scales—from minutes to 24 hours. To test whether the neural network within the SCN is sufficient to generate such fractal patterns, we studied multi-unit neural activity of in vivo and in vitro SCNs in rodents. In vivo SCN-neural activity exhibited fractal patterns that are virtually identical in mice and rats and are similar to those in motor activity at time scales from minutes up to 10 hours. In addition, these patterns remained unchanged when the main afferent signal to the SCN, namely light, was removed. However, the fractal patterns of SCN-neural activity are not autonomous within the SCN as these patterns completely broke down in the isolated in vitro SCN despite persistence of circadian rhythmicity. Thus, SCN-neural activity is fractal in the intact organism and these fractal patterns require network interactions between the SCN and extra-SCN nodes. Such a fractal control network could underlie the fractal regulation observed in many physiological functions that involve the SCN, including motor control and heart rate regulation. PMID:23185285

Evidence of diffusive fractal aggregation of TiO2 nanoparticles by femtosecond laser ablation at ambient conditions

NASA Astrophysics Data System (ADS)

Celardo, G. L.; Archetti, D.; Ferrini, G.; Gavioli, L.; Pingue, P.; Cavaliere, E.

2017-01-01

The specific mechanisms which lead to the formation of fractal nanostructures by pulsed laser deposition remain elusive despite intense research efforts, motivated mainly by the technological interest in obtaining tailored nanostructures with simple and scalable production methods. Here we focus on fractal nanostructures of titanium dioxide, TiO2, a strategic material for many applications, obtained by femtosecond laser ablation at ambient conditions. We compare a theoretical model of fractal formation with experimental data. The comparison of theory and experiment confirms that fractal aggregates are formed after landing of the ablated material on the substrate surface by a simple diffusive mechanism. We model the fractal formation through extensive Monte Carlo simulations based on a set of minimal assumptions: TiO2 nanoparticles arrive already formed on the substrate, then they diffuse in a size/mass independent way and stick irreversibly upon touching, thus forming fractal clusters. Despite its simplicity, our model explains the main features of the fractal structures arising from the complex interaction of large TiO2 nanoparticles with different substrates. Indeed our model is able to reproduce both the fractal dimensions and the area distributions of the nanostructures for different densities of the ablated material. Finally we discuss the role of the thermal conductivity of the substrate and the laser fluence on the properties of the fractal nanostructures. Our results represent an advancement towards controlling the production of fractal nanostructures by pulsed laser deposition.

[Features of fractal dynamics EEG of alpha-rhythm in patients with neurotic and neurosis-like disorders].

PubMed

Shul'ts, E V; Baburin, I N; Karavaeva, T A; Karvasarskiĭ, B D; Slezin, V B

2011-01-01

Fifty-five patients with neurotic and neurosis-like disorders and 20 healthy controls, aged 17-64 years, have been examined. The basic research method was electroencephalography (EEG) with the fractal analysis of alpha power fluctuations. In patients, the changes in the fractal structure were of the same direction: the decrease of fractal indexes of low-frequency fluctuations and the increase of fractal indexes of mid-frequency fluctuations. Patients with neurosis-like disorders, in comparison to those with neurotic disorders, were characterized by more expressed (quantitative) changes in fractal structures of more extended character. It suggests the presence of deeper pathological changes in patients with neurosis-like disorders.

[Fractal research of neurite growth in immunofluorescent images].

PubMed

Tang, Min; Wang, Huinan

2008-12-01

Fractal dimension has been widely used in medical images processing and analysis. The neurite growth of cultured dorsal root ganglion (DRG) was detected by fluorescent immunocytochemistry treated with nerve regeneration factor (0.1, 0.5, 2.0 mg/L). A novel method based on triangular prism surface area (TPSA) was introduced and adopted to calculate the fractal dimension of the two-dimensional immunofluorescent images. Experimental results demonstrate that this method is easy to understand and convenient to operate, and the quantititve results are concordant with the observational findings under microscope. This method can be guidelines for analyzing and deciding experimental results.

Nonlinear dynamics, fractals, cardiac physiology and sudden death

NASA Technical Reports Server (NTRS)

Goldberger, Ary L.

1987-01-01

The authors propose a diametrically opposite viewpoint to the generally accepted tendency of equating healthy function with order and disease with chaos. With regard to the question of sudden cardiac death and chaos, it is suggested that certain features of dynamical chaos related to fractal structure and fractal dynamics may be important organizing principles in normal physiology and that certain pathologies, including ventricular fibrillation, represent a class of 'pathological periodicities'. Some laboratory work bearing on the relation of nonlinear analysis to physiological and pathophysiological data is briefly reviewed, with tentative theories and models described in reference to the mechanism of ventricular fibrillation.

Super Water-Repellent Fractal Surfaces of a Photochromic Diarylethene Induced by UV Light

NASA Astrophysics Data System (ADS)

Izumi, Norikazu; Minami, Takayuki; Mayama, Hiroyuki; Takata, Atsushi; Nakamura, Shinichiro; Yokojima, Satoshi; Tsujii, Kaoru; Uchida, Kingo

2008-09-01

Photochromic diarylethene forms super water-repellent surfaces upon irradiation with UV light. Microfibril-like crystals grow on the solid diarylethene surface after UV irradiation, and the contact angle of water on the surface becomes larger with increasing surface roughness with time. The fractal analysis was made by the box-counting method for the rough surfaces. There are three regions in the roughness size having the fractal dimension of ca. 2.4 (size of roughness smaller than 5 µm), of ca. 2.2 (size of roughness between 5-40 µm), and of ca. 2.0 (size of roughness larger than 40 µm). The fractal dimension of ca. 2.4 was due to the fibril-like structures generated gradually by UV irradiation on diarylethene surfaces accompanied with an increase in the contact angle. The surface structure with larger fractal dimension mainly contributes to realizing the super water-repellency of the diarylethene surfaces. This mechanism of spontaneous formation of fractal surfaces is similar to that for triglyceride and alkylketene dimer waxes.

Static friction between rigid fractal surfaces

NASA Astrophysics Data System (ADS)

Alonso-Marroquin, Fernando; Huang, Pengyu; Hanaor, Dorian A. H.; Flores-Johnson, E. A.; Proust, Gwénaëlle; Gan, Yixiang; Shen, Luming

2015-09-01

Using spheropolygon-based simulations and contact slope analysis, we investigate the effects of surface topography and atomic scale friction on the macroscopically observed friction between rigid blocks with fractal surface structures. From our mathematical derivation, the angle of macroscopic friction is the result of the sum of the angle of atomic friction and the slope angle between the contact surfaces. The latter is obtained from the determination of all possible contact slopes between the two surface profiles through an alternative signature function. Our theory is validated through numerical simulations of spheropolygons with fractal Koch surfaces and is applied to the description of frictional properties of Weierstrass-Mandelbrot surfaces. The agreement between simulations and theory suggests that for interpreting macroscopic frictional behavior, the descriptors of surface morphology should be defined from the signature function rather than from the slopes of the contacting surfaces.

Fractal Properties of Some Machined Surfaces

NASA Astrophysics Data System (ADS)

Thomas, T. R.; Rosén, B.-G.

Many surface profiles are self-affine fractals defined by fractal dimension D and topothesy Λ. Traditionally these parameters are derived laboriously from the slope and intercept of the profile's structure function. Recently a quicker and more convenient derivation from standard roughness parameters has been suggested. Based on this derivation, it is shown that D and Λ depend on two dimensionless numbers: the ratio of the mean peak spacing to the rms roughness, and the ratio of the mean local peak spacing to the sampling interval. Using this approach, values of D and Λ are calculated for 125 profiles produced by polishing, plateau honing and various single-point machining processes. Different processes are shown to occupy different regions in D-Λ space, and polished surfaces show a relationship between D and Λ which is independent of the surface material.

Theoretical Study on Stress Sensitivity of Fractal Porous Media with Irreducible Water

NASA Astrophysics Data System (ADS)

Lei, Gang; Dong, Zhenzhen; Li, Weirong; Wen, Qingzhi; Wang, Cai

The couple flow deformation behavior in porous media has drawn tremendous attention in various scientific and engineering fields. However, though the coupled flow deformation mechanism has been intensively investigated in the last decades, the essential controls on stress sensitivity are not determined. It is of practical significance to use analytic methods to study stress sensitivity of porous media. Unfortunately, because of the disordered and extremely complicated microstructures of porous media, the theoretical model for stress sensitivity is scarce. The goal of this work is to establish a novel and reasonable quantitative model to determine the essential controls on stress sensitivity. The predictions of the theoretical model, derived from the Hertzian contact theory and fractal geometry, agree well with the available experimental data. Compared with the previous models, our model takes into account more factors, including the influence of the water saturation and the microstructural parameters of the pore space. The proposed models can reveal more mechanisms that affect the coupled flow deformation behavior in fractal porous media. The results show that the irreducible water saturation increases with the increase of effective stress, and decreases with the increased rock elastic modulus (or increased power law index) at a given effective stress. The effect of stress variation on porosity is smaller than that on permeability. Under a given effective stress, the normalized permeability (or the normalized porosity) becomes smaller with the decrease of rock elastic modulus (or the decrease of power law index). And a lower capillary pressure will correspond to an increased rock elastic modulus (or an increased power law index) under a given water saturation.

Multi-year encoding of daily rainfall and streamflow via the fractal-multifractal method

NASA Astrophysics Data System (ADS)

Puente, C. E.; Maskey, M.; Sivakumar, B.

2017-12-01

A deterministic geometric approach, the fractal-multifractal (FM) method, which has been proven to be faithful in encoding daily geophysical sets over a year, is used to describe records over multiple years at a time. Looking for FM parameter trends over longer periods, the present study shows FM descriptions of daily rainfall and streamflow gathered over five consecutive years optimizing deviations on accumulated sets. The results for 100 and 60 sets of five years for rainfall streamflow, respectively, near Sacramento, California illustrate that: (a) encoding of both types of data sets may be accomplished with relatively small errors; and (b) predicting the geometry of both variables appears to be possible, even five years ahead, training neural networks on the respective FM parameters. It is emphasized that the FM approach not only captures the accumulated sets over successive pentades but also preserves other statistical attributes including the overall "texture" of the records.

Automatic localization of cerebral cortical malformations using fractal analysis.

PubMed

De Luca, A; Arrigoni, F; Romaniello, R; Triulzi, F M; Peruzzo, D; Bertoldo, A

2016-08-21

Malformations of cortical development (MCDs) encompass a variety of brain disorders affecting the normal development and organization of the brain cortex. The relatively low incidence and the extreme heterogeneity of these disorders hamper the application of classical group level approaches for the detection of lesions. Here, we present a geometrical descriptor for a voxel level analysis based on fractal geometry, then define two similarity measures to detect the lesions at single subject level. The pipeline was applied to 15 normal children and nine pediatric patients affected by MCDs following two criteria, maximum accuracy (WACC) and minimization of false positives (FPR), and proved that our lesion detection algorithm is able to detect and locate abnormalities of the brain cortex with high specificity (WACC  =  85%, FPR  =  96%), sensitivity (WACC  =  83%, FPR  =  63%) and accuracy (WACC  =  85%, FPR  =  90%). The combination of global and local features proves to be effective, making the algorithm suitable for the detection of both focal and diffused malformations. Compared to other existing algorithms, this method shows higher accuracy and sensitivity.

Automatic localization of cerebral cortical malformations using fractal analysis

NASA Astrophysics Data System (ADS)

De Luca, A.; Arrigoni, F.; Romaniello, R.; Triulzi, F. M.; Peruzzo, D.; Bertoldo, A.

2016-08-01

Malformations of cortical development (MCDs) encompass a variety of brain disorders affecting the normal development and organization of the brain cortex. The relatively low incidence and the extreme heterogeneity of these disorders hamper the application of classical group level approaches for the detection of lesions. Here, we present a geometrical descriptor for a voxel level analysis based on fractal geometry, then define two similarity measures to detect the lesions at single subject level. The pipeline was applied to 15 normal children and nine pediatric patients affected by MCDs following two criteria, maximum accuracy (WACC) and minimization of false positives (FPR), and proved that our lesion detection algorithm is able to detect and locate abnormalities of the brain cortex with high specificity (WACC  =  85%, FPR  =  96%), sensitivity (WACC  =  83%, FPR  =  63%) and accuracy (WACC  =  85%, FPR  =  90%). The combination of global and local features proves to be effective, making the algorithm suitable for the detection of both focal and diffused malformations. Compared to other existing algorithms, this method shows higher accuracy and sensitivity.

An Evaluation of Fractal Surface Measurement Methods for Characterizing Landscape Complexity from Remote-Sensing Imagery

NASA Technical Reports Server (NTRS)

Lam, Nina Siu-Ngan; Qiu, Hong-Lie; Quattrochi, Dale A.; Emerson, Charles W.; Arnold, James E. (Technical Monitor)

2001-01-01

The rapid increase in digital data volumes from new and existing sensors necessitates the need for efficient analytical tools for extracting information. We developed an integrated software package called ICAMS (Image Characterization and Modeling System) to provide specialized spatial analytical functions for interpreting remote sensing data. This paper evaluates the three fractal dimension measurement methods: isarithm, variogram, and triangular prism, along with the spatial autocorrelation measurement methods Moran's I and Geary's C, that have been implemented in ICAMS. A modified triangular prism method was proposed and implemented. Results from analyzing 25 simulated surfaces having known fractal dimensions show that both the isarithm and triangular prism methods can accurately measure a range of fractal surfaces. The triangular prism method is most accurate at estimating the fractal dimension of higher spatial complexity, but it is sensitive to contrast stretching. The variogram method is a comparatively poor estimator for all of the surfaces, particularly those with higher fractal dimensions. Similar to the fractal techniques, the spatial autocorrelation techniques are found to be useful to measure complex images but not images with low dimensionality. These fractal measurement methods can be applied directly to unclassified images and could serve as a tool for change detection and data mining.

Simultaneous estimation of deterministic and fractal stochastic components in non-stationary time series

NASA Astrophysics Data System (ADS)

García, Constantino A.; Otero, Abraham; Félix, Paulo; Presedo, Jesús; Márquez, David G.

2018-07-01

In the past few decades, it has been recognized that 1 / f fluctuations are ubiquitous in nature. The most widely used mathematical models to capture the long-term memory properties of 1 / f fluctuations have been stochastic fractal models. However, physical systems do not usually consist of just stochastic fractal dynamics, but they often also show some degree of deterministic behavior. The present paper proposes a model based on fractal stochastic and deterministic components that can provide a valuable basis for the study of complex systems with long-term correlations. The fractal stochastic component is assumed to be a fractional Brownian motion process and the deterministic component is assumed to be a band-limited signal. We also provide a method that, under the assumptions of this model, is able to characterize the fractal stochastic component and to provide an estimate of the deterministic components present in a given time series. The method is based on a Bayesian wavelet shrinkage procedure that exploits the self-similar properties of the fractal processes in the wavelet domain. This method has been validated over simulated signals and over real signals with economical and biological origin. Real examples illustrate how our model may be useful for exploring the deterministic-stochastic duality of complex systems, and uncovering interesting patterns present in time series.

[A method to estimate the short-term fractal dimension of heart rate variability based on wavelet transform].

PubMed

Zhonggang, Liang; Hong, Yan

2006-10-01

A new method of calculating fractal dimension of short-term heart rate variability signals is presented. The method is based on wavelet transform and filter banks. The implementation of the method is: First of all we pick-up the fractal component from HRV signals using wavelet transform. Next, we estimate the power spectrum distribution of fractal component using auto-regressive model, and we estimate parameter 7 using the least square method. Finally according to formula D = 2- (gamma-1)/2 estimate fractal dimension of HRV signal. To validate the stability and reliability of the proposed method, using fractional brown movement simulate 24 fractal signals that fractal value is 1.6 to validate, the result shows that the method has stability and reliability.

Fractal spectral triples on Kellendonk's C∗-algebra of a substitution tiling

NASA Astrophysics Data System (ADS)

Mampusti, Michael; Whittaker, Michael F.

2017-02-01

We introduce a new class of noncommutative spectral triples on Kellendonk's C∗-algebra associated with a nonperiodic substitution tiling. These spectral triples are constructed from fractal trees on tilings, which define a geodesic distance between any two tiles in the tiling. Since fractals typically have infinite Euclidean length, the geodesic distance is defined using Perron-Frobenius theory, and is self-similar with scaling factor given by the Perron-Frobenius eigenvalue. We show that each spectral triple is θ-summable, and respects the hierarchy of the substitution system. To elucidate our results, we construct a fractal tree on the Penrose tiling, and explicitly show how it gives rise to a collection of spectral triples.

Fractal dynamics of heartbeat time series of young persons with metabolic syndrome

NASA Astrophysics Data System (ADS)

Muñoz-Diosdado, A.; Alonso-Martínez, A.; Ramírez-Hernández, L.; Martínez-Hernández, G.

2012-10-01

Many physiological systems have been in recent years quantitatively characterized using fractal analysis. We applied it to study heart variability of young subjects with metabolic syndrome (MS); we examined the RR time series (time between two R waves in ECG) with the detrended fluctuation analysis (DFA) method, the Higuchi's fractal dimension method and the multifractal analysis to detect the possible presence of heart problems. The results show that although the young persons have MS, the majority do not present alterations in the heart dynamics. However, there were cases where the fractal parameter values differed significantly from the healthy people values.